Mathematical Structures in Physics

Winter term 2009/10

Christoph Schweigert

Hamburg University

Department of Mathematics

Section Algebra and Number Theory

and

Center for Mathematical Physics(as of 02.02.2010)

Contents

1 Newtonian mechanics 11.1 Galilei space, equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Lagrangian mechanics 102.1 Variational calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Systems with constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Lagrangian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Symmetries and Noether identities . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Natural geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Classical field theories 423.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Electrodynamics as a gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Hamiltonian mechanics 554.1 (Pre-) symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Hamiltonian dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Quantum mechanics 685.1 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 C∗-algebras and states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 Strong quantizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.4 Schrodinger picture and examples . . . . . . . . . . . . . . . . . . . . . . . . . . 86

i

6 A glimpse to quantum field theory 91

A Differentiable Manifolds 94A.1 Definition of differentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . 94A.2 Tangent vectors and differentiation . . . . . . . . . . . . . . . . . . . . . . . . . 97A.3 Fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100A.4 Vector fields and Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A.5 Differential forms and the de Rham complex . . . . . . . . . . . . . . . . . . . . 106A.6 Riemannian manifolds and the Hodge dual . . . . . . . . . . . . . . . . . . . . . 113

The current version of these notes can be found underhttp://www.math.uni-hamburg.de/home/schweigert/ws09/pskript.pdf

as a pdf file.Please send comments and corrections to schweigert@math.uni-hamburg.de!

These notes are based on lectures delivered at the University of Hamburg in the fall term2007/2008 and in the fall term 2009/2010 as part of the master programs in physics andmathematical physics. I am grateful to Linda Sass for notes taken in 2007/08.

References:References per Chapter:1. Classical Mechanics

• Arnold, Vladimir I.: Mathematical methods of classical mechanics, Springer GraduateText in Mathematics 60, Springer, New York, 1978.

• Goldstein, Herbert; Poole, Charles and Safko, John: Classical mechanics, Pear-son/Addison Wesley, Upper Saddle River, 2002

• etc.

2. Variational bicomplex and Lagrangian mechanics

• Anderson, Ian M.: Introduction to the variational bicomplex, in M. Gotay, J. Marsdenand V. Moncrief (eds), Mathematical Aspects of Classical Field Theory, ContemporaryMathematics 132, Amer. Math. Soc., Providence, 1992, pp. 5173. http://www.math.usu.edu/~fg_mp/Publications/ComtempMath/IntroVariationalBicomplex.ps

• Olver, Peter: Applications of Lie Groups to Differential Equations. Springer GraduateTexts in Mathematics 107, 1986

• Reyes, Enrique G.: On covariant phase space and the variational bicomplex. Internat. J.Theoret. Phys. 43 (2004), no. 5, 1267–1286. http://www.springerlink.com/content/q151731w71l71q23/

3. Hamiltonian mechanics and symplectic geometry

• Berndt, Rolf: An introduction to symplectic geometry, American Mathematical Society,Providence, 2001

ii

• Souriau, Jean-Marie: Structure of dynamical systems. Birkhauser, Boston, 1997. http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm

4. Quantum mechanics

• Takhtajan, Leon A.: Quantum Mechanics for Mathematicians. Graduate Studies in Math-ematics, American Mathematical Society, 2008.

• Messiah, Albert: Quantum mechanics. Dover Publications, 1999.

• Strocchi, F.: An introduction to the mathematical structure of quantum mechanics. Ad-vanced Series in mathematical physics 27, World Scientific, Singapur, 2005.

• Giulini, Domenico: That Strange Procedure Called Quantisation. Lecture Notes in Physics631, Springer, Berlin, 2003. http://de.arxiv.org/abs/quant-ph/0304202

5. Electrodynamics

• Romer, Hartmann, M. Forger Elementare Feldtheorie. VCH, Weinheim 1993.http://www.freidok.uni-freiburg.de/volltexte/405/pdf/efeld1.pdf

Appendix A: Manifolds

• Bredon, Glen E.: Topology and geometry. Springer Graduate Texts in Mathematics 139,1993.

• Madsen, Ib and Tornehave, Jørgen: From Calculus to cohomology: de Rham cohomologyand characteristic classes. Cambridge University Press 1997.

• Bott, Raoul and Tu, Loring W.: Differential forms in algebraic topology. Springer Grad-uate Texts in Mathematics 82, New York, 1982.

iii

1 Newtonian mechanics

1.1 Galilei space, equations of motion

A basic postulate of classical physics requires (empty) space and time to be homogenous. Inclassical physics, the idea of empty space should be accepted as central. This is afforded by thenotion of an affine space over a k-vector space V . Here k is a field, in our application the fieldR of real numbers.

Definition 1.1.1

(i) An affine space is a pair (A, V ), consisting of a set A and a k-vector space V togetherwith an action of the abelian group (V,+) underlying V on the set A that is transitiveand free.

In more detail, an action is a map

ρ : V × A→ A

such thatρ(v + w, a) = ρ(v, ρ(w, a)) for all v, w ∈ V, a ∈ A .

An action is called transitive, if for all p, q ∈ A exists v ∈ V such that ρ(v, p) = q; anaction is called free, if this v ∈ V is unique.

We call dimk V the dimension of A and write dim A = dimk V .

We also say that the affine space (A, V ) is modelled over the vector space V .

(ii) We introduce the notation ρ(v, p) = p + v and, if v is the unique vector in V suchthat q = p + v, we write q − p = v. We also say that A is a (V,+)-Torsor or aprincipal homogenous space for the group (V,+). The group (V,+) is also called thedifference space of the affine space.

(iii) As morphisms between affine spaces (A1, V1)→ (A2, V2) modelled over vector spaces V1, V2

over the same field, we consider affine maps, i.e. maps

ϕ : A1 → A2

for which there exists a linear map Aϕ : V1 → V2 such that

ϕ(p)− ϕ(q) = Aϕ(p− q) for all p, q ∈ A .

Remarks 1.1.2.

1. Note that the (V,+)-equivariant morphisms are those morphisms for which Aϕ = idV .

2. While a vector space has the zero vector as a distinguished element, there is no distin-guished element in an affine space.

3. Any two affine spaces over the same vector space are isomorphic, but not canonicallyisomorphic.

1

4. Recall the definition of a semi-direct product of two groups H,N . Given a group homo-morphism ϕ : H → Aut(N) the set N ×H can be endowed with the structure of a groupN oϕ H by

(n, h) · (n′, h′) := (nϕh(n), hh′) .

The automorphism group of an affine space is isomorphic to the semi-direct product

Aut(A) = V o GL(V ) ,

where V acts on A by translations.

5. The choice of any point p ∈ A induces a bijection ρ(·, p) : V → A of sets. As a finite-dimensional R-vector space, V has a unique topology as a normed vector space. By con-sidering preimages of open sets in V as open in A, we get a topology on A that does notdepend on the choice of base point. We endow A with this topology.

6. We will see in the appendix that affine space An is an n-dimensional manifold. The choiceof a point p ∈ A provides a natural global coordinate chart.

Suppose that we wish to model space; since space is three-dimensional, we have n = 3.To be able to talk about lengths, we have to model measurements with rods. Our idealizationimplies that there are infinitely long rods and rulers. In a first step, the edge of the ruler has adistinguished direction and a marked point on it.

Definition 1.1.3

1. A ray L in a real vector space V is a subset of the form L = R≥0v with v ∈ V \ 0.

2. A halfplane in a real vector space V is a subset H ⊂ V such that there are linearlyindependent vectors v, w ∈ V with H = Rv + R≥0w. The boundary of a halfplane is theonly line through the origin contained in it. (For the halfplane just given, this is the lineRv.)

3. A rotation group for a real three-dimensional vector space V is a subgroup D ⊂ GL(V )which acts transitively and freely on the set of pairs consisting of a halfplane and a rayon its boundary.

Proposition 1.1.4.For any three-dimensional vector space, the map

Scalar products on V /R>0 → Rotation groups

which maps the scalar product b to its special orthogonal group SO(V, b) is a bijection.

Proof:We have to construct an inverse of the map. If one assumes that the rotation group is compact,an invariant scalar product can be obtained by integration. For an elementary (but lengthy)proof without this assumption, we refer to W. Soergel, Herleitung von Skalarprodukten ausSymmetrieprinzipien. Mathematische Semesterberichte 68 (2008) 197-202.

2

Next we wish to talk about length scales. Such a length scale is e.g. the prototype meter.It consists of a ruler with two marked points. Fixing one point, the other point moves on theorbit of the rotation group.

Definition 1.1.5

1. Given a rotation group D, we call an orbit l ⊂ V \ 0 a unit length.

2. Given a unit length, we define a norm on V as follows. We first remark that any rayintersects a given unit length l in precisely one point. If the ray through w ∈ V intersectsl in v ∈ l and w = λv with λ ≥ 0, we define the norm on W by |w| := λ.

We introduce the measurements of lengths with the help of a ruler with two marked points;we find that the three-dimensional space of our intuition should have the following mathematicalstructure:

Definition 1.1.6

1. An n-dimensional Euclidean space En is an n-dimensional affine space An together withthe structure of a Euclidean vector space on the difference space.

2. As morphisms of Euclidean spaces, we only admit those affine maps ϕ for which the linearmap Aϕ is an isometry, i.e. an orthogonal map.

3. The group of automorphisms of a Euclidean space is called a Euclidean group.

Proposition 1.1.7.The Euclidean group of a Euclidean vector space En is a semi-direct product of the subgroup Vof translations given by the action of V and the rotation group SO(V, b):

Aut(E) = V o SO(V, b)

It is a non-compact Lie group of dimension n+ n(n−1)2

.

Including also time, we obtain the model of empty space in classical physics:

Definition 1.1.8

(i) A Galilei space (A, V, t, 〈·, ·〉) consists of

• An affine space A over a real four-dimensional vector space V . The elements of Aare called events or space time points.

• A non-zero linear functionalt : V → R

called absolute time difference function.

• The structure of a Euclidean vector space with positive definite scalar product 〈·, ·〉on ker t.

(ii) t(a− b) ∈ R is called the time difference between the events a, b ∈ A.

3

(iii) Two events a, b ∈ A with t(a− b) = 0 are called simultaneous. This gives an equivalencerelation on Galilei space. The equivalence class

Cont(a) = b ∈ A | t(b− a) = 0

is the subset of events simultaneous to a ∈ A.

Remarks 1.1.9.

(a) Cont(a) for any a ∈ A is a three-dimensional Euclidean space.

(b) While it makes sense to talk about simultaneous events a, b at different places, the phrase:“The two events a, b are at different time, but at the same place in three-dimensional space.”does not make sense.

Definition 1.1.10As the morphisms of two Galilei spaces (A1, V1, t1, 〈·, ·〉1) and (A2, V2, t2, 〈·, ·〉2) we consider thoseaffine maps

ϕ : A1 → A2

which respect time differences

t1(b− a) = t2(ϕ(b)− ϕ(a)) for all a, b ∈ A1,

and the Euclidean structure on space in the sense that the restriction Aϕ : ker t1 → ker t2 is anorthogonal linear map.

Remarks 1.1.11.

(a) An example for a Galilei space is given by the Galileian coordinate space G = (R1 ×R3,R4, pr1, 〈·, ·〉). This is R×R3 seen as an affine space over the vector space R4 ∼= R×R3

with the projection on the first component

t = pr1 : R4 ∼= R1 × R3 → R

as the time difference functional and the standard Euclidean scalar product 〈·, ·〉 on R3.

(b) Two Galilei spaces are isomorphic, but not canonically isomorphic. The automorphismgroup Aut(G) of the Galileian coordinate space G is called the Galilei group.

(c) We consider three classes of automorphisms in Aut(G):

• Uniform motions with velocity v ∈ R3:

g1(t, x) = (t, x+ vt)

• Spacial translations by x0 ∈ R3 combined with time translations by t0 ∈ R:

g2(t, x) = (t+ t0, x+ x0)

4

• Rotations and reflections in space with R ∈ O(3):

g3(t, x) = (t, Rx)

One can show that Aut(G) is generated by these elements. It is a ten-dimensional non-compact Lie group.

Definition 1.1.12

1. Let X be any set andφ : X → G

be a bijection of sets. (On the right hand side, it would be formally more correct to writethe set underlying G.) We say that φ provides a global Galilean coordinate system on X.

2. We say that two Galilean coordinate systems φ1, φ2 : X → G are inrelative uniform motion, if

φ1 φ−12 ∈ Aut(G) .

Any global Galilean coordinate system φ : X → G endows the set X with the structure ofa Galilean space over the vector space R4:we define on X the structure of an affine space over R4 by requiring φ to be an affine map:

x+ v := φ−1(φ(x) + v) for all x ∈ X, v ∈ R4 .

Then we use the standard time difference functional pr1 : R4 ∼= R× R3 → R and the standardEuclidean structure on R3 to get a Galilei space (X,R4)φ that depends on φ. For this Galileispace, we have an isomorphism

(X,R4)φφ→ G

of Galilei spaces such that the linear map Aφ associated to φ is the identity.

Lemma 1.1.13.

1. Being in relative uniform motion is an equivalence relation on the set of global Galileancoordinate systems on X.

2. Two Galilean coordinate systems φ1, φ2 in relative uniform motion endow the set X withthe structures of Galilei spaces for which the identity is an isomorphism of Galilei spaces.

This leads us to the following

Definition 1.1.14

1. A Galilean structure on a set X is an equivalence class of Galilean coordinate systems.

2. Given a Galilean structure on a set X, any coordinate system of the defining equivalenceclass is called an inertial system or inertial frame for this Galilean structure.

Remarks 1.1.15.

5

1. By definition, two different inertial systems for the same Galilean structure are in uniformrelative motion.

2. There are no distinguished inertial systems.

Principle 1.1.16. (“Galilean principle of relativity”)All laws of nature are of the same form in all inertial systems.

To make this principle more precise for mechanical system, we have to describe the motionsof mass points.

Definition 1.1.17

(i) A trajectory of a mass point in R3 is an (at least twice) differentiable map

x : I → R3

with I ⊂ R an interval.

(ii) A trajectory of N mass points in R3 is an N -tuple of (at least twice) differentiable maps

x(i) : I → R3

with I ⊂ R an interval. Equivalently, we can consider an (at least twice) differentiablemap

~x : I → (R3)N .

(iii) The velocity in t0 ∈ I is defined as the derivative:

x(t0) =dx

dt(t0) .

(iv) The acceleration in t0 ∈ I is defined as the second derivative:

x(t0) =d2x

dt2(t0) .

(v) The graph(t, x(t) | t ∈ I ⊂ R× R3

of a trajectory is called the world line of the mass point. We consider a world line as asubset of Galilean coordinate space G.

(vi) If we parameterize a world line by a different function I → G with I an interval, we callthe parameter τ ∈ I the eigentime of the mass point.

The following principle is the basic axiom of Newtonian mechanics. It cannot be derivedmathematically but should rather seen as description of observations in nature.

Definition 1.1.18 [Newtonian determinism]

6

(a) A Newtonian trajectory of a point particle

σ : I → R3, with I = (t0, t1) ⊂ R

is completely determined by the initial position σ(τ0) and the inital velocity

d

dtϕ σ

∣∣∣t=t0

.

(b) In particular, the acceleration at t0 is determined by the initial position and the initialvelocity. As a consequence, there exists a function

F : R3N × R3N × R→ R3N ,

where the first factor are spacial coordinates, the second are velocities and the third is time,such that for all Newtonian trajectories the Newtonian equation

d2

dt2xi = xi = F i(x, x, t), i = 1, ..., N.

holds. This is a second order ordinary differential equation that completely determines thesystem.

Remarks 1.1.19.

(a) The function F has to be measured experimentally and describes the acting forces. It de-termines the physical system. For the particular case F = 0, no forces act and x = 0. Asa consequence, the trajectory is a uniform motion. (Aristotle had a different idea about thesituation!)

(b) Standard mathematical propositions assure that a sufficiently smooth function F deter-mines, together with the initial conditions on position and velocity, the trajectory. Implic-itly, we will frequently assume the existence of global solutions or of solutions for sufficientlylarge times.

(c) We will see that not all graphs (t, x(t)) ⊂ G describe physical motions.

So far we have described trajectories using maps σ : I → R3 giving rise to a graph (t, x(t) ∈G. Suppose we are given a Galilei space A with internal frames ψ : A→ G. We then get a map

Definition 1.1.20Let A be a Galilei space, I an interval of eigentime and σ : I → A a smooth function. Then σis called a physical motion if for all inertial frames ψ : A → G the function ψ σ : I → G isthe graph of a Newtonian trajectory.

This is can be seen as a more precise version of the Newtonian principle of relativity.

Remarks 1.1.21.

1. We immediately have the following consequences of the Newtonian principle of relativity.

• Invariance under time translations: the force F does not depend on time t.

7

• Invariance under spacial translations: F depends only on the relative coordinatesxi − x1.

• Invariance under relative uniform motion: F only depends on the relative velocities.

• Invariance under rotations: the force F is a vector, i.e. transforms in the samerepresentation of the rotation group as x and x.

2. Altogether, this implies that the motion of n points is described by a function

x(k)(t) = f(x(j) − x(l), x(s) − x(r))

with x(i) : I → R3.

3. We deduce Newton’s first law: a system consisting of a single point is described in aninertial system by a uniform motion (t, x0 + tv0). In particular, the acceleration vanishes,x = 0.

1.2 Examples

It should be emphasized that the discussion of Newtonian relativity was for a system in emptyspace. In general, it is interesting to consider systems where F is a function violating theseprinciples. In this way, we can conveniently summarize the effect of other mass points in thesystem. For example, one might switch the perspective from considering the two-body systemconsisting of the sun and the earth by considering just the motion of earth in the backgroundof the force exerced by the mass of the sun.

We investigate some examples in a fixed inertial system with the help of a function x : R→R3. We call R3 the configuration space of the system.

Examples 1.2.1.

(a) Freely falling particle:x = −gx3 with g ≈ 9, 81 ms−2.

One should notice that a direction is distinguished by the unit vector x3 in the directionof the x3-axis. Hence the isotropy of space is violated. We introduce a function, calledpotential energy

U(x1, x2, x3) : R3 → R such that −grad U = F ;

in our case U(x1, x2, x3) = gx3. The function U is not unique, but can be added by anyfunction whose derivative vanishes. The equations of motion then read x = −grad U .

(b) To write down Newton’s law of gravity, we consider a potential energy depending only onthe distance r from the center of gravity:

U(x1, x2, x3) = − k

rwith r :=

√x2

1 + x22 + x2

3

x = − grad U = − k

r3x

8

(c) One-dimensional oscillator. To investigate the equation of motion x = −α2x, we introducethe potential U = α2x2

2. This can be realized e.g. using a mass connected to a spring.

Experimentally, one finds that for equal springs with equal initial conditions, but differentmasses, the ratio

x1

x2

= const1,2

only depends on the balls, but not on the initial conditions. We put

x1

x2

=m2

m1

with a quantity mi called inertial mass which is a property of the i-the ball. Again, weintroduce a physical unit by comparing to a standard, e.g. the prototype kilogram in Paris.

Observation 1.2.2.Consider a mechanical system given by a force F (x, x, t) which we suppose right away to begiven by a potential of the form

U : R3 → R

and thus independent of t and x. We have to study the coupled system of ordinary differentialequations

x = −grad U(x) (∗)

of second order. For any solution x : I → R3 of (∗) we consider the real-valued function

ε : I → R

ε(t) =1

2‖x(t)‖2 + U(x(t)) .

For its derivative, we find

d

dtε(t) = 〈x, x〉 + 〈gradU , x〉 = 0.

This suggests the following point of view:

Lemma 1.2.3.The system (∗) is equivalent to the following system of ordinary differential equation of firstorder for the function y : I → R6

y1 = y2, y2 = −grady1U(y1, y3, y5)y3 = y4, y4 = −grady3U(y1, y3, y5)y5 = y6, y6 = −grady5U(y1, y3, y5)

The solutions of this system are called phase curves.

We thus enlarged the space by considering the first derivatives as independent geometricquantities.

9

Observation 1.2.4.This suggests to introduce R6 with coordinates x, y, z, ux, uy, uz. We call this the phase space Pof the system. We equip the phase space P with the energy function

E(x1, x2, x3, ux1 , ux2 , ux3) :=1

2(ux1

2, ux2

2, ux3

2) + U(x1, x2, x3) .

It should be appreciated that the first term in E is a quadratic form.On phase space, we consider the ordinary differential equations

dxi

dt= uxi

,duxi

dt= −gradiU(x1, x2, x3) (∗∗) .

The solutions of (∗∗) are called phase curves. They are contained in subspaces of P of constantvalue of E.

Observation 1.2.5.1. Assume that for any point M ∈ P a global solution xM(t) of (∗∗) with initial conditions

M exists. This allows us to define a mapping

gt : P→ P by gt(M) = xM(t).

2. Standard theorems about ordinary differential equations imply that gt is a diffeomorphism,i.e. a differential map with differentiable inverse mapping.

3. We find gt1 gt2 = gt1+t2 and thus a smooth action

g : R× P→ P(t,M) 7−→ gt(M)

called the phase flow.

2 Lagrangian mechanics

2.1 Variational calculus

We start by considering the following mathematical problem:

Observation 2.1.1.1. We wish to associate to trajectories ϕ : I → RN , I = [t1, t2] scalar values. Examples of

such functionals are the length of a trajectory:

L1(ϕ) =

∫I

‖ϕ‖dt,

or its energy

L2(ϕ) =

∫I

‖ϕ‖2dt .

One can ask for which trajectories one obtains extremal values for a given functional.This leads to an ordinary differential equation for the trajectory. In fact, this is a con-venient way to describe (certain) ordinary differential equations in terms of scalar-valuedfunctionals which are sometimes much easier to handle, in particular as far as covarianceproperties and coordinate transformations are concerned.

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2. The functional we are interested in is allowed to depend on the trajectory ϕ and its deriva-tives. To this end, we need apart from position space

RN with coordinates (x1, ..., xN)

also recipients for higher derivatives of ϕ. We therefore introduce

J1(RN) = RN ⊕ RN

with coordinates (x1, ..., xN , x1t , ..., x

Nt ). This comes with a natural injection

RN → J1(RN)

(x1, ..., xN) 7→ (x1, ..., xN , 0, ..., 0).

Since the equations of motion are second order differential equations, we continue byadding a recipient for the second derivatives:

J2(RN) = RN ⊕ RN ⊕ RN with coordinates (xi, xit, xitt) .

We also use the abbreviation xi2 := xitt. If we continue this way, we get a system of vectorspaces for derivatives up to order α

Jα(RN) ∼= (RN)α

with embeddingsJα → Jα+1(RN).

3. Suppose we are given a smooth trajectory

ϕ : I → RN .

It comes with derivative functions of all orders which we can use to extend ϕ to a functionto the larger spaces we just constructed:

jαϕ : I → JαRN

where the components are given by

(jαϕ)in =dn

dtnϕi.

4. The functionals of length and energy can now be expressed in terms of integrals over thefollowing functions on J1RN

l1, l2 : J1(RN)→ R

l1(xi, xit) =

( N∑i=1

(xit)2) 1

2

l2(xi, xit) =

N∑i=1

(xit)2

as

Li(ϕ) =

∫I

dt li (j1ϕ).

11

5. The variational problem consists of finding trajectories

ϕ : I → RN

which for a given functionl : Jα(RN)→ R

are stationary points for the functional on trajectories

L(ϕ) =

∫I

dt l(jαϕ).

To make the problem well-defined, we fix boundary values ai, bi ∈ RN , i = 0, 1, ..., α − 1and restrict ourselves to trajectories with I = [t1, t2] in the subset

T(α)(ai)(bi)

=ϕ : I → RN :

diϕ

dti(t1) = ai,

diϕ

dti(t2) = bi

.

6. We wish to find extremal trajectories in T(α)(ai)(bi)

. To this end, we consider differentiableone-parameter families of trajectories

ϕ : I × (−ε0, ε0)→ RN .

We introduce for ε ∈ (−ε0, ε0) the notation

ϕε : I → RN

t 7→ ϕ(t, ε),

where we require that for all values of ε that ϕε ∈ T(α)(ai)(bi)

. An application of the chainrule yields

d

dε

∣∣∣ε=0

∫I

l jα(ϕε) =d

dε

∣∣∣ε=0

∫I

dt l(ϕε, ϕε, ϕε, . . .)

=

∫I

∂l

∂xi

∣∣∣xi

β=dβ ϕi

εdtβ

d

dεϕiε +

∂l

∂xit

d

dεϕiε + ...

We assume for simplicity that l restricts to a function on J1RN . We notice that

d

dεϕiε =

∂2ϕ(t, ε)

∂ε∂t=∂2ϕ(t, ε)

∂t∂ε

By partial integration with respect to the variable t, we find

=

∫I

( ∂l∂xi

∣∣∣xi

β=dβ ϕi

dtβ

− d

dt

∂l

∂xit

∣∣∣xi

β=dβ ϕi

εdtβ

) d

dεϕiε +

∫I

d

dt

( ∂l∂xi

dϕiεdε

)=

∫I

( ∂l

∂xi−∑j,β

∂2l

∂xjβ∂xit

xjβ+1︸ ︷︷ ︸important expression

) j1(ϕ) · d

dεϕiε +

∫I

dt( ∂l∂xi

dϕiεdε

).

12

Definition 2.1.2To any function

l : J1(RN)→ R

the Euler-Lagrange operator associates an RN -valued function

E(l) : J2(RN)→ RN

E(l)i =∂l

∂xi−∑j,β

∂2l

∂xjβ∂xit

xjβ+1

We rewrite it using a total derivative operator

D :=∑iβ

xiβ+1

∂

∂xiβ

on functions of x, xt, xtt, . . . by

E(l)i =∂l

∂xi−D ∂l

∂xit.

Given a trajectory in T(α)(ai)(bi)

, we obtain an RN -valued function on I ⊂ R:

ϕ 7→ E(l)i jα(ϕ) =: E(l) [ϕ]

In terms of this function, the derivative with respect to ε becomes

d

dεL(ϕ) =

∫I

⟨E(l) [ϕε=0] ,

d

dεϕ⟩.

The boundary conditions at t0 and t1 enforce

d

dε

∣∣∣t=t0,t1

ϕε = 0.

We are now ready to apply the next lemma which uses standard facts from real analysis:

Lemma 2.1.3.A continuous real-valued function

f : I = [t1, t2]→ R

which has the property that for any smooth function

h : I → R

vanishing at the end points of the interval, h(t1) = h(t2) = 0, the integral over the productvanishes, ∫ t2

t1

f · h dt = 0 ,

is identically zero, f(t) = 0 for all t ∈ [t1, t2].

13

Proof:Suppose that f does not vanish identically. Possibly after replacing f by −f , we find t0 <t∗ < t1 such that f(t∗) > 0. Since f is continuous, we find a neighbourhood U of t∗ such thatf(t) > c > 0 for all t ∈ U .

We can now find a smooth function h with support in U such that h|U = 1 for someneighborhood of t∗ contained in U . We thus find the inequalities∫ t1

t0

f · h =

∫U

f · h >∫U

f > |U |c > 0

contradicting our assumptions.

From this we deduce the following

Proposition 2.1.4.A trajectory ϕ : I → RN is a stationary point for a functional on trajectories given by a function

l : Jα(RN)→ R,

if and only if the N ordinary differential equations

E(l) [ϕ] = 0

hold. This set of ordinary differential equations is called Euler-Lagrange equations for the func-tion l on the trajectory ϕ.

Examples 2.1.5.(a) A so-called natural system of classical mechanics is given by a function of the form

l(x, xt) =1

2

N∑i=1

(xit)2 − V (xi)

We compute∂l

∂xi= −gradiV

∂l

∂xit= xit

and find for the Euler-Lagrange operator

E(l)i = −gradiV − xitt .

This gives us a system of N equations

xitt = −gradiV

which gives the following Euler-Lagrange equations on a trajectory

ϕ i = −gradiV(ϕ(τ)

),

i.e. Newton’s equations of motion in the presence of a force given by the gradient of thepotential V . If there is a potential for the forces of a system, all information about theequations of motion is thus contained in the function

l : J(RN)→ R

the Lagrangian.

14

(b) The length of the curve

γ =(t, x) : x = ϕ(t) mit t0 6 t 6 t1

⊂ R2

is given by

L(γ) =

∫ t1

t0

√1 + ϕ2dt .

We thus consider the functionl(x, xt) =

√1 + x2

t .

To find the Euler-Lagrange equations, we compute

∂l

∂x= 0 and

∂l

∂xt=

xt√1 + x2

t

and find∂

∂xt

∂l

∂xtxtt =

xtt

(1 + x2t )

32

= 0

which reduces to xtt = 0. Hence we get the differential equation ϕ = 0 on the trajectorieswhich have as solutions ϕ(t) = ct+ d, i.e. the stationary points are straight lines.

(c) To illustrate that a Lagrangian can be more easily re-expressed in other coordinates thanan equation of motion, we consider the two-dimensional case without potential given inCartesian coordinates by

l(x1, x2, x1t , x

2t ) =

1

2

((x1

t )2 + (x2

t )2).

In polar coordinates, we find by applying the chain rule

x1,t = (r cosϕ)t = rt cosϕ− r sinϕϕt

x2,t = (r sinϕ)t = rt sinϕ+ r cosϕϕt

(1)

and thus for the function

l(r, ϕ, rt, ϕt) =1

2

(rt cosϕ− r sinϕϕt)

2 +1

2

(r2t sinϕ+ r cosϕϕt)

2 =1

2(r2t + r2ϕ2

t ).

It has as solutions straight lines. In radial coordinates, we compute

∂l

∂ϕ= 0 and

∂l

∂ϕt= r2ϕt

which implies the equation for the trajectory

d

dt(r2ϕ) = 0

that expresses the conservation of angular momentum with respect to the origin. For theradial coordinate we find

∂l

∂r= rϕ2

t and∂l

∂rt= rt and thus r = rϕ2.

15

(d) To investigate the motion of earth around the sun, we consider Kepler’s problem that isdefined by the following function

lk : J1(R3)→ Rlk(x, xt) = 1

2‖xt‖2 + k

‖x‖ .

Here k is a constant that we keep for convenience. Instead of solving the correspondingequation of motion exactly, we show how to use a scaling argument to learn more aboutsolutions.

To this end, we consider for any λ > 0 the map

sλ : J1(R3)→ J1(R3)

sλ(x, xt) := (λ2x, λ−1xt).

It is chosen such that for the Lagrangian we have

l sλ = λ−2l

and therefore for the Euler-Lagrange operators

E [l sλ] = λ−1E [l] .

Suppose we have a trajectory ϕ : R → RN that is a global solution of the Euler-Lagrangeequations

E [l] j1ϕ = 0 .

Consider for any λ ∈ R>0 the trajectory

ϕλ : R→ R3

ϕλ(t) := λ2ϕ(λ−3t).

We then findj1ϕλ(t) = (λ2ϕ(λ−3t), λ2λ−3ϕ(λ−3t)) = sλ j1ϕ(λ−3t)).

We findE [l] j1ϕλ(t) = E [l] sλ j1ϕ(λ3t) = λ−1E [l] j1ϕ(λ3t) = 0.

and conclude that the trajectory ϕλ is a solution of the equations of motion as well.An explicit calculation reveals that there solutions for which the trajectory is an ellipsis

with focal point x = 0. The scaling affects the semimajor axis as

aλ = λ2a.

If T is the time for one period,ϕ(t+ T ) = ϕ(t),

thenϕλ(t+ λ3T ) = λ2ϕ(λ−3t+ T ) = λ2ϕ(λ−3t) = ϕλ(t),

and hence we find for the periodsTλ = λ−3T.

16

We have found from scaling considerations Kepler’s third law:

a3λ

T 2λ

=λ6a3

λ6T 2=a3

T 2

is independent of λ.We introduce some terminology:

Definition 2.1.6

1. Given a Lagrangian l(xi, xit, t), we call L(ϕ) =∫Idt l jϕ the corresponding action for the

trajectory ϕ.

2. A coordinate function xi is called a generalized coordinate, xit a generalized velocity. ∂l∂xi

tis

called the generalized momentum canonically conjugate to xi. ∂l∂xi is called the generalized

force.

3. A coordinate is called cyclic if the Lagrangian does not depend on it, i.e.

∂l

∂xi= 0.

Proposition 2.1.7.The momentum canonically conjugate to a cyclic coordinate is constant for any solution of theEuler-Lagrange equations.

Proof:For any solution

ϕ : I → RN

of the Euler-Lagrange equations, we have for a cyclic coordinate xi

d

d t

∂l

∂xit j1ϕ =

∂l

∂xi j1ϕ = 0.

2.2 Systems with constraints

Frequently, one observes that N particles can only move on a submanifold of R3N . Examplesinclude:

• A roller coaster is (hopefully) more or less obliged to move on a one-dimensional subman-ifold.

• The points of a rigid body are required to move in such a way that their distances remainconstant.

17

In each situation, this is achieved by forces that are not known explicitly, but whose effect isknown.

Hence we consider an embedded submanifold M ⊂ RN which we can at least locally describeas the zero locus of r functions

gα : RN → R, α = 1, ..., r ,

i.e. asM =

x ∈ RN |gα(x) = 0 for all α

.

For example, a rigid body consisting of N distinct particles is described by N(N−1)2

functions,one for each pair of distinct points,

‖xi − xj‖2 − cij = gij(x1, ..., xN)

with constants cij ∈ R>0. As a regularity assumption we require the rank of the Jacobian to bemaximal,

rang∂gα∂xi

∣∣∣x∈M

= r = maximal.

As a warm-up problem, we want restrict a smooth function f : RN → R to M and describethe stationary points of this restriction f |M .

Proposition 2.2.1.Consider the auxiliary function

f : RN × Rr → R(x, λ) 7→ f(x) +

∑rα=1 λαgα(x) .

The additional parameters λ ∈ Rr are called Lagrangian multipliers. Then the restriction f |Mhas a stationary point in x0 ∈M , if and only if the function f has a stationary point in (x0, λ0)for some λ0 ∈ Rr.

Proof:The function f has a stationary point, if and only if the two equations hold

0 =∂f

∂λα= gα(x0) and

∂f

∂xi= −

r∑α=1

λα∂

∂xigα

The first equation is equivalent to x0 ∈ M while the second equation requires the gradi-ent of f to be normal to M in x0, ensuring that x0 is a stationary point of the restriction f |M .

Observation 2.2.2.

1. We now wish to consider the natural system on RN given by a potential U : RN → R,

l(x, xt) =1

2

N∑i=1

(xit)2 − U(x) ,

18

that is constrained by unknown forces to a submanifold M ⊂ RN of dimension r. Weassume that M is given by N − r smooth functions gα = gα(x

i).

We thus consider trajectoriesϕ : I → RN

with im ϕ ⊂M . For any family of trajectories ϕε we are looking for stationary points of∫I

l j1ϕε

subject to the constraints gα (ϕε) = 0 for all α = 1, ..., r and all ε. We introduce Lagrangianmultipliers and minimize the functional

f(ε, λ) =

∫I

dt(l j1ϕε +

∑λαgα(ϕε)

).

The derivative with respect to ε yields the following additional terms∑α

λα∂gα∂xi

dϕiεdε

∣∣∣ε=0

so that the equation of motion becomes

E(l) j1(ϕ) = xtt + gradU = −∑

λα grad gα . (2)

The right hand side describes additional forces whose concrete form is unknown thatconstraint the motion to the submanifold M .

2. We need to describe the geometry of the situation: using the theorem of implicit functions,we find an open subset U ⊂ RN−r with coordinates qαα=1,...,r and a chart with localcoordinates (qα) on U ⊂M that locally parametrizes M . We identify TpM with a subspaceof TpRN and express the basis vectors as

∂

∂qα=∂xi

∂qα∂

∂xi, α = 1, ..., N − r.

In subsequent calculations, the notation is simplified by introducing the vector x ∈ RN

with coordinates xi and writing∂

∂qα=

∂x

∂qα. (3)

3. Next, we have to relate the jet spaces J2M and J2RN . Since they are designed as recipientsof trajectories and their derivatives, we consider a trajectory with values in U ⊂M

ϕ : I → U ⊂M

which we continue to a trajectory ϕ with values in RN : The chain rule yields

dϕi

dτ=∂xi

∂qα· dϕ

α

dτ.

19

This is, of course, just the usual transformation of tangent vectors under a smooth map,in the case the embedding M → RN . From this, we deduce the following transformationrules for the coordinates xt:

xit = xit(qα, qαt , t) =

∂xi

∂qαqαt (4)

between the coordinates qα, qαt on J1M and xit on J1RN . We find as an obvious consequence

∂xit∂qαt

=∂xi

∂qα.

Similarly, to find the transformation rules for the coordinates xitt describing the secondderivative, we compute the second derivative of a trajectory:

d2ϕ

dt2=

∂2xi

∂qβ∂qα· dϕ

β

dt

dϕα

dt+∂xi

∂qαd2ϕα

dt2.

This yields the following relation between the coordinates qαtt on J2M and xitt on J2RN :

xitt = xitt(qα, qαt , q

αtt, t) =

∂xi

∂qαqαtt +

∂2xi

∂qα∂qβqβt q

αt . (5)

4. Since we do not know the right hand side of the equation of motion (2), we take the scalarproduct in RN with the tangent vectors (3). This yields N − r equations⟨

xtt,∂x

∂qα

⟩= −

⟨gradU,

∂x

∂qα

⟩(∗)

Our goal is to rewrite this as the Euler-Lagrange equation for a function on J1M . Theright hand side is easily rewritten using the chain rule:⟨

gradU,∂x

∂qα

⟩=∂U

∂xi∂xi

∂qα=∂U

∂qα

This suggests to take the restriction of U to M as the potential for the Lagrangian systemon M .

5. To rewrite the left hand side, we use the total differential operator on smooth functionsof xi, xit, x

itt, . . . introduced in definition 2.1.2:

Dx =∑i,β

xiβ+1

∂

∂qiβ.

Note that the operator is linear and obeys the Leibniz rule. Recall that the Euler-Lagrangeequations read in terms of this operator

Dx∂l

∂xit=

∂l

∂xi.

Similarly, we introduce the differential operator

D =∑α,β

qαβ+1

∂

∂qαβ.

on smooth functions of qα, qαt , qαtt, . . .

20

6. We need the following result:

D

(⟨xt,

∂x

∂qα

⟩)=

⟨xtt,

∂x

∂qα

⟩+

⟨xt,

∂xt∂qα

⟩.

To see this, we first apply the Leibniz rule for D to find

D

⟨xt,

∂x

∂qα

⟩=

⟨xt,D

∂x

∂qα

⟩+

⟨Dxt,

∂x

∂qα

⟩and compute for the second term

Dxt =∂xt∂qα

qαt +∂xt∂qαt

qαtt =∂2x

∂qβ∂qαqαt q

βt +

∂x

∂qαqαtt = xtt

according to (3) and (5). For the first term, we compute

D

(∂xi

∂qα

)=

∂2xi

∂qβ∂qαqβt ,

which turns out to be the same as

∂xit∂qα

=∂

∂qα

(∂xi

∂qβqβt

)=

∂2xi

∂qα∂qβqβt .

7. We now introduce the following function on J1M depending on the coordinates qα, qαt :

T =1

2

N∑i=1

(xit)2 =

1

2

N∑i=1

xit(qα, qαt )2

To compute Euler-Lagrange equations for T in terms of q and qt, we first note that thechain rule implies

∂T

∂qα=

⟨xt,

∂xt∂qα

⟩. (6)

Next, we calculate, again using the chain rule,

∂T

∂qαt=

⟨xt,

∂xt∂qαt

⟩=

⟨xt,

∂x

∂qα

⟩where we have used the relation derived in the third step. The left hand side of the Euler-Lagrange equation for T then reads

D

(∂

∂qαtT

)= D

⟨xt,

∂x

∂qα

⟩=

⟨xtt,

∂x

∂qα

⟩+

⟨xt,

∂xt∂qα

⟩,

where we used in the last equalities identities derived in step 6. We rewrite the firstsummand using the equation of motion (∗) and the replace the second summand by identity(6):

D

(∂

∂qαtT

)= − ∂U

∂qα+∂T

∂qα.

Taking into account that U is independent of xit and thus of qαt , we realize that these arethe Euler-Lagrange equations for the Lagrangian

L(qα, qατ ) = T (xt(q, qt))− U(x(q)) , .

21

We have thus shown:

Proposition 2.2.3.Suppose a natural system on RN with potential U(x) is constrained by unknown forces to asubmanifold M ⊂ RN . Then the equations of motion on M are the Euler-Lagrange equationsfor a Lagrangian l(q, qt) on J1M obtained from the function on J1RN by restricting U to Mand transforming the coordinates qt as in (4).

This result clearly shows that, for the description of derivatives, we should associate bundlesto a Lagrangian system that can be pulled back consistently.

2.3 Lagrangian systems

We will now look at Lagrangian systems in a more abstract way. We start with the following

Observation 2.3.1.

1. A first datum in the description of a mechanical system is a surjective submersion π :E → I, where I ⊂ R is an interval of “eigentime” parametrizing the system and E is theextended configuration space.

2. Global trajectories are global sections of π. Local sections give local trajectories. They forma sheaf on I.

3. A system with constraints is described by a submanifold ι : E ′ → E such that π restrictsto a surjective submersion on E ′.

Remarks 2.3.2.

1. Upon choosing a space-time point p0 ∈ A in a Galilei space A, the motion of a pointparticle in A is described by taking as the surjective submersion the absolute time differencefunctional

A → Rp 7→ t(p− p0)

2. A mass point moving in a manifold M with motion parametrized by an interval I ⊂ R isdescribed by the surjective submersion p1 : I ×M → I given by the projection on the firstfactor.

3. We do not require E to be a bundle over I. Any smooth bundle over I is globally trivial,but E might carry additional structure that depends on I. Moreover, we will generalizethe situation.

Observation 2.3.3.

1. It turns out that we do not gain much by requiring the target of the surjection to be aninterval. We therefore consider the case of surjective submersions π : E →M with M anysmooth manifold. One application is classical field theory: then M has the interpretationof space-time that parametrizes a field like the electromagnetic field E(x) that dependson the point x ∈ M in space time. In this situation, the (local) sections of π : E → Mare called (local) field configurations. They generalize trajectories which arise in the casewhen dimM = 1.

22

2. The following crucial difference between field theory and mechanics should be noted: inmechanics, there is one parameter which can be chosen to be identical to time. Local spacialcoordinates are coordinate functions on the manifold E of the extended configuration space.In a field theory, on the other hand, space and time are locally defined parameters andare to be considered as local coordinate functions on M .

3. In certain situations, it might be necessary to endow either of the manifolds E or M withmore structure. Examples in field theory include metrics endowing E with the structureof a Riemannian manifold or, to be able to include fermions, a spin structure on M .

4. To be able to formulate a dynamical principle in terms of a partial differential equation,we need to work with derivatives of any order of local sections. This is achieved by thefollowing construction.

Definition 2.3.4

1. Let E,M be smooth manifolds, dimM = m and let π : E → M be a surjective submer-sion. We say that π defines a fibre bundle in the category of smooth manifolds.

2. For any open subset U ⊂ M , we denote by Γπ(U) the set of local smooth sections of π,i.e. the set of smooth functions sU : U → E such that π sU = idU . These sets form asheaf.

3. Consider a point p ∈ M and two open neighbourhoods U,U ′ of p. We identify two localsections ϕ ∈ Γπ(U) and ϕ′ ∈ Γπ(U

′) if they agree on a common open refinement U ⊂U ∩ U ′. The equivalence classes are called germs of local sections in the point p. The setof equivalence classes is denoted by Γπ(p).

4. For any multi-index I = (I(1), ..., I(m)), we introduce the derivative operators

|I| :=m∑i=1

I(i)

∂|I|

∂xI:=

m∏i=1

(∂

∂xi

)I(i)I = ∅ is admitted as a value, |∅| = 0 and the corresponding derivative operator is theidentity.

5. Let p ∈M and U,U ′ be open neighborhoods of M . Consider two local sections ϕ ∈ Γπ(U)and ϕ′ ∈ Γπ(U

′) whose value in p agrees. Possibly after a further restriction, we canassume that the two sections are defined on the same coordinate neighborhood of p thatwe call for simplicity U . We fix local coordinates xi on M around p and yα on E aroundϕ(p) = ϕ′(p).

We say that ϕ and ϕ′ have the same r-jet in p, if in local coordinates all partial derivativesup to order r coincide,

∂|I|ϕα

∂xI

∣∣∣p

=∂|I|(ϕ′)α

∂xI

∣∣∣p, 0 6 |I| 6 r .

23

Since differentiation is a local operation, local sections with the same germ in p ∈M giverise to the same r-jet. We have thus defined an equivalence relation on germs of sectionsthat is moreover independent of the choice of local coordinates.

6. An r-jet with representative ϕ will be denoted by jrpϕ; the natural number r is called theorder of the jet. The point p ∈M is called the source and ϕ(p) ∈ E the target of the jet.

7. We introduce the setJrπ :=

jrpϕ∣∣p ∈M,ϕ ∈ Γπ(p)

(The notation JrE is also in use, but we wish to stress that the set is determined by amorphism π and not only by the manifold E.)

We endow these sets with more structure

Observation 2.3.5.

1. We have the following obvious projections:

πr : Jrπ → M (source projection)jrp(ϕ) 7→ p

πr,0 : Jrπ → E (target projection)jrp(ϕ) 7→ ϕ(p)

and for r > k > 1πr,k : Jrπ → Jkπ (jet projections)

jrpϕ 7→ jkpϕ

Such that the following diagram commutes:

Jrππr,r−1−−−→ Jr−1π −−−→ ... −−−→ J1π

π1,0−−−→ Eyπr

yπr−1

y yπ1

yπM M ... M M

2. In a next step, we introduce local coordinates to endow Jrπ with the structure of a smoothmanifold and the surjection πr : Jrπ → M with the structure of a bundle map, thejet bundle of π with fibre Jrp π. We include the case J0π := E.

To this end, we choose a bundle chart (U, u) for E, where U ⊂M is an open subset of Mand u is a pair u = (xi, uα) consisting of a local coordinate chart x : U ⊆M → Rm of Mand u : π−1(U)→ Rn of E. A common terminology is to call the local coordinates x of Mthe independent coordinates and the local coordinates u of E the dependent coordinates.This language is motivated by the situation where we describe local sections s : M ⊃U → E leading to an expression of the local coordinates uα of E in terms of the localcoordinates xi of E.

3. The induced bundle chart of Jrπ is defined on

U r :=jrpϕ∣∣ϕ ∈ Γπ(U)

24

and consists of maps ur = (xi, uαI ) with |I| ≤ r with |i| ≤ r defined by the coordinates xi

of the base point p ∈Mxi(jrpϕ)

= xi(p)

and the partial derivatives of the jet in local coordinates (uα, xi),

uαI(jrpϕ)

=∂|I|ϕα

∂xI

∣∣∣p.

The jet bundle Jr is in particular finite-dimensional.

4. Let ϕ be a local section of π defined on an open subset U ⊂ M . Its r-th jet prolongationis the local section

jrϕ : U → Jrπ

given by(jrϕ) (p) = jrpϕ .

In local coordinates, it is given by (ϕα,

∂|I|

∂xIϕα).

Lemma 2.3.6.Let E be a smooth manifold and p1 : R × E → R a trivial bundle. This situation is relevantfor classical mechanics with time-independent configuration space E. Then there is a canonicalisomorphism

J1π → R× TE .

Given any section

ϕ : R→ R× Et 7→ (t, ϕ(t))

where ϕ is a smooth function R→ E, the isomorphism maps the one-jet of ϕ to the pair withsecond entry the tangent vector

j1t0ϕ 7→

(t0,

dϕ

dt

∣∣∣∣t=t0

).

This allows us to relate 1-jets to the tangent bundle. Certain authors, e.g. Arnold, therefore usethe tangent bundle as a framework for discussing Lagrangian mechanics. This has, however,the disadvantage that one cannot describe geometrically equations of motions. Moreover, thelanguage does not generalize to field theory, i.e. higher dimensional parameter spaces.

Proof:The isomorphy is immediate in coordinates:

(t0, x

i, xit) j1tϕ =

(t0, x

iϕ(t),d

dt

∣∣∣t=t0

xi ϕ(t)

).

25

Definition 2.3.7The infinite jet bundle π∞ : J∞π → M is defined as the projective limit (in the category oftopological spaces) of the jet bundles

M ←−πE ←−−

π1,0

J1π ←−−π2,1

J2π ← ...

It comes with a family of natural projections

π∞,rJ∞π → Jrπ .

Proposition 2.3.8.The infinite jet bundle has the following universal property: for any bundle ν : F → M withbundle morphisms sn : F → Jnπ such that all diagrams of the form

Fsr

((RRRRRRRRRRRRRRRR

ss

111

1111

1111

1111

Jrπ

πs,0

Jsπ πs,0

// E

commute, there exists a unique bundle morphism µ : F → J∞π such that the following diagramcommutes:

Fsr

))RRRRRRRRRRRRRRRRR

ss

222

2222

2222

2222

µ""D

DD

D

J∞π π∞,r//

π∞,s

Jrπ

πs,0

Jsπ πs,0

// E

Locally, the infinite jet bundle J∞π is described by coordinates (xi, uαI ) and all multi-indices I,where we admit I = ∅. In particular, the jet bundle is infinite-dimensional.

Proposition 2.3.9.Let E → M be a fibre bundle and U ⊂M be an open subset. Any local section ϕ : U → E canbe extended to a section j∞(s) = j∞s of J∞π. In local coordinates, this extension is given bycollecting all partial derivatives: (

xi, sα(xi), ...,∂|I|

∂xIsα(xi)...

)In other words, the extension to j∞(s) amounts to keeping all partial derivatives, or put

differently, to keeping the Taylor series as a formal series.

Definition 2.3.10

26

1. A real-valued function f : J∞π → R is called smooth, if it factorizes through an finite jetbundle Jrπ, i.e. there exists r ∈ N and a smooth function fr : Jrπ → R such that

f = fr π∞r .

Note that this implies that the algebra of smooth functions on J∞π is defined as theinductive limit of the injections

π∗k+1,k : C∞ (Jkπ)→ C∞ (Jk+1π).

It has the structure of a filtered commutative algebra.

2. A (geometric) partial differential equation of order r is a closed submanifold ρ ⊂ Jrπ. Alocal solution on an open subset U ⊂ M is a section ϕ : U → E with jrpϕ ∈ ρ for allp ∈ W . Sometimes, this submanifold is also called the shell. This terminology comes fromfield theory jargon where one speaks of the “mass shell” and of “on shell conditions” forrelations holding only on the submanifold.

Example 2.3.11.Consider the bundle p1 : Rn×R→ Rn whose sections are real-valued functions on Rn. We haveglobal bundle coordinates (x1, ..., xn, u). Consider as a submanifold the zero locus of the smoothfunction

F : J2π → R

F (xi, u, ui, uij) =n∑i=1

uii

This submanifold describes Laplace’s equation.

Definition 2.3.12

(i) A vector field on the jet bundle J∞π is defined as a derivation on the ring C∞(J∞π). Inlocal coordinates (xi, uαI ) on J∞π, this is a formal series of the form

X = Ai∂

∂xi+∑α,I

BαI

∂

∂uαI

with smooth functions Ai = Ai(xj, uαI ) and Bα

I = BαI (xj, uαI ).

(ii) The prolongation of a vector field X ∈ Γ(U) on U ⊂M is the unique vector field pr∞X ∈Γ(J∞π) acting in the point j∞(s)(p) ∈ J∞π given by the prolongation of the sections ∈ Γπ(U) like as the following derivation on a smooth function f ∈ C∞(J∞π)

pr∞Xj∞s(p)f := Xp (f j∞s) .

Notice that f j∞s is a real-valued function on U ⊂M .

27

Proposition 2.3.13.Let (x, u) be local bundle coordinates for a bundle π : E →M . Then

Dj := pr∞(

∂

∂xj

)=

∂

∂xj+ uαI+(j)

∂

∂uαI.

In particular, when M = I ⊆ R is an interval, we obtain the differential operator D appearingin the Euler-Lagrange operator in proposition 2.1.2

Proof:We compute using f ∈ C∞(J∞π)(

pr∞∂

∂xj

)j∞s(p)

· f =∂

∂xj(f j∞s) =

∂f

∂xj+

∂f

∂uαI

∂|I|+1s

∂x|I|+(j)

Our next goal is to endow the jet bundle J∞π with a connection:

Definition 2.3.14Let π : E → M be a fibre bundle. Then V = V (π) := ker (dπ : TE → TM) ⊂ TE is vectorsubbundle of rank dimE − dimM over E, the vertical bundle of π. It comes with a naturalsurjective submersion to M .

(i) An Ehresmann connection on E is a smooth vector subbundle H ⊆ TE over E such that

TE = H ⊕ V

where the direct sum is to be understood fibrewise. H is then called the horizontal bundleof the Ehresmann connection. The rank as a vector bundle over E is dimM .

(ii) Equivalently, an Ehresmann connection can be defined as an TE-valued 1-form ν on E,i.e. an endomorphism of the vector bundle TE such that

• ν2 = ν (idempotent)

• im ν = V

Then H := ker ν is the horizontal subbundle.

Given an Ehresmann connection, we can decompose any vector field X ∈ vect(E) on E inits horizontal part XH and its vertical part XV :

X = XH +XV .

We need a few properties of Ehresmann connections:

Remarks 2.3.15.Let π : E →M be a smooth fibre bundle with an Ehresmann connection.

28

(a) Let γ : [0, 1] → M be a smooth curve in M . A lift of γ is a smooth curve γ : [0, 1] → Esuch that π γ = γ. As a diagram:

E

π

[0, 1]

γ //

γ<<yyyyyyyyy

M

A lift is called horizontal, if we have for the derivative vector

γ′(t) ∈ Hγ(t) for all t ∈ [0, 1].

By standard theorems about ordinary differential equations, horizontal lifts locally exist fora given initial condition γ(0) ∈ π−1(0). If global lifts exist for all curves, the connection iscalled complete.

(b) Given an Ehresmann connection, we define a 2-form R with on E with values in TE whichis called the curvature of the Ehresmann connection. The 2-form is defined by its value ontwo vector fields X, Y ∈ vect(E):

R(X, Y ) = [XH , YH ]V

(c) An Ehresmann connection is called flat, if the 2-form R vanishes. Flat connections areexactly Frobenius integrable connections.

We next wish to convince ourselves that the notion of Ehresmann connection comprisesother notions of connection if we combine it with suitable compatibility conditions.

Observation 2.3.16.Consider a real vector bundle over M , i.e. a bundle π : E →M where each fibre π−1(x) carriesthe structure of an R-vector space. (The case of complex vector bundles is completely analogous.)

1. For any x ∈M , the fibre Ex = π−1(x) has the structure of an R-vector space, i.e. we havefor each e ∈ Ex a translation operator Te : Ex → Ex. Similarly, we have for any scalars ∈ R a scalar multiplication Ss : Ex → Ex. For a linear connection, the subspaces Ee fordifferent e ∈ Ex in the same fibre are required to be linearly related:

He+e′ = (dTe′)eHe and Hse = dSs(He) .

Moreover, we require the vertical lift

vlE : E ×M E → V

vlE(ux, vx) =d

dt

∣∣∣t=0

(ux + tvx) with ux, vx ∈ Ex

to be an isomorphism of vector bundles over M .

2. This allows us to define the connector of a connection as the following homomorphism ofvector bundles over M :

K : TE −→ν

V∼−−→vl−1

E

E ×M E −−→pr2

E

29

3. Let now N be any smooth manifold. We can now define the covariant derivative for asmooth map f : N → E and a vector field X ∈ vectN :

OXf : N −→X

TN −→Tf

TE −→K

E.

In the special case N = M and the tangent bundle, E = TM , the smooth map f : M →TM is just a vector field on M and we have the usual notion of the covariant derivativeof a vector field.

In the next step, we will see that any jet bundle carries a natural Ehresmann connection.

Proposition 2.3.17.Let π : E → M be a fibre bundle and J∞π be the corresponding jet bundle. For any x ∈ J∞πconsider the subspace of TxJ

∞π spanned by all prolongations pr∞X of tangent vectors X ∈Tπ∞xM . The the following holds:

1. The subspaces endow J∞π with an Ehresmann connection, the so-calledCartan connection.

2. One haspr∞ [X1, X2] = [pr∞X1, pr∞X2] ;

hence the Cartan connection is flat.

In local coordinates, one can be more explicit:

Definition 2.3.18

1. Given a multi-index I = (i1, i2, . . . , ir), we define a differential operator DI := Di1 Di2 . . .Dir . Given finitely many local functions ZI , we call a differential operator of the formZIDI acting on local functions a total differential operator.

2. Local differential forms on the jet bundle J∞π are defined as the inductive limit of thesystem

π∗k+1,k : Ω• (Jkπ)→ Ω• (Jk+1π)

i.e.Ω• (J∞π) = lim ind Ω• (Jkπ) .

A local p-form on J∞π is thus a finite linear combination of expressions of the form

A dxi1 ∧ ... ∧ dxip ∧ duα1j1...jpq

∧ ... ∧ duαq

l1...lpq

with p+ q = k and with A a smooth function on J∞π.

3. A differential form ω ∈ Ω• (J∞π) is called a contact form, if

j∞(s)∗ω = 0 ∈ Ω•(M)

for all local sections s ∈ Γ(E).

Remarks 2.3.19.

30

(a) The subspace of contact forms is a differential ideal of Ω• (J∞π), the contact ideal I.

(b) One can check that a vector field on the jet bundle X ∈ Vect (J∞π) is horizontal, if andonly if its inner product with all contact forms vanishes,

ιXω = 0 for all ω ∈ I

(c) Locally, the contact ideal I is generated by the one-forms

ΘαI := duαI − uαI+(j)dx

j .

We convince ourselves that these differential forms are indeed contact forms: given a localsection s : U → E, we consider its prolongation

f := j∞s : U → J∞π

and pull back using the usual formula ωαdyα for one-forms:

f ∗ω = ωα∂fα

∂xjdxj .

This yields

f ∗ΘαI =

∂(j∞s)αI∂xj

dxj − uαI+(j)

∣∣∣j∞(s)

dxj = 0.

We are now in a position to introduce a natural splitting on the complex of local differentialforms on a jet space.

Definition 2.3.20

(i) A p-form ω ∈ Ωp (J∞π) is said to be of type (r, s), if at all points σ = j∞s ∈ J∞π one has

ω(X1, ..., Xp) = 0,

as soon as either more than s vectors are vertical or more than r vectors are horizontal.In local coordinates, ω is then a sum of terms

f dxi1 ∧ ... ∧ dxir ∧Θα1I1∧ ... ∧Θαs

Is

with a smooth function f . The subspace of local differential forms of type (r, s) is denotedby Ωr,s (J∞π).

(ii) We decompose

Ωp (J∞) =⊕r+s=p

Ωr,s (J∞π) .

Correspondingly, the exterior derivative

d : Ωp (J∞π)→ Ωp+1 (J∞π)

decomposes into horizontal and vertical derivatives

d = dH + dV

31

with

dH : Ωr,s (J∞π)→ Ωr+1,s (J∞π)

dV : Ωr,s (J∞π)→ Ωr,s+1 (J∞π)

The relation d2 = 0 implies the three identities

d2H = 0, d2

V = 0, dHdV = −dV dH .

Remarks 2.3.21.(a) On smooth functions, we have the following formula in terms of local coordinates:

dV f =∂f

∂uαIΘαI

dHf = (Dif) dxi =

[∂f

∂xi+ uαI+(i)

∂f

∂uαI

]dxi

Moreover, one checksdH(dxi) = 0 and dV (dxi) = 0

as well asdHΘα

I = dxj ∧ΘαI+(j) and dV Θα

I = 0.

(b) We wish to comment on the variational problem given by a smooth function on a jet bundleJ∞π. Again we consider a one-parameter family sε of local sections of E −→

πM and want

to determined

dε

∣∣∣ε=0f (j∞sε) .

Then the “infinitesimal version” of the variation

X :=

[dsαεdε

∣∣∣ε=0

]∂

∂uα

gives us a vertical vector field on E. We denote by pr∞EX the unique vector field on J∞πwhich acts on functions on E as X and preserves the contact ideal. Then the variationalformula expresses the derivative as the contraction of a vector field and a one-form on J∞π:

d

dε

∣∣∣ε=0f (j∞sε) = 〈 pr∞EX, dV f 〉.

(c) Given a total differential operator Z, we define a new total differential operator Z+ calledthe adjoint of Z by∫

M

(j∞φ)∗(FZ(G))dvolM =

∫M

(j∞φ)∗(Z+(F )G)dvolM

for all sections ϕ : M → E and all local functions F,G ∈ LocE. It follows that

FZ(G)dvolM = Z+(F )GdvolM + dHζ

for some ζ ∈ Ωn−1,0(E).

If Z = ZJDJ in local coordinates, then integration by parts yields Z+(F ) = (−D)J(ZJF ).

32

Definition 2.3.22

(i) The variational bicomplex of a fibre bundle π : E →M is the double complex

↑ ↑ ↑0 → Ω0,2 →

dH

Ω1,2 → ... →dH

Ωn,2

↑ dV ↑ dV ↑ dV0 → Ω0,1 →

dH

Ω1,1 → ... →dH

Ωn,1

↑ dV ↑ dV ↑ dV0 → R → Ω0,0 →

dH

Ω1,0 →dH

Ω2,0 → ... → Ωn−1,0 →dH

Ωn,0

(ii) A Lagrangian system consists of a smooth fibre bundle π : E → M together with a forml ∈ Ωn,0 (J∞π), called the Lagrange function or Lagrangian density of the system. Themanifold E is called the (extended) configuration space of the system.

(iii) A Lagrangian system is called mechanical, if M is an interval I ⊂ R.

(iv) A global section ϕ : M → E is a called a motion of the Lagrangian system, if it is astationary point for the functional, called the action

F (s) =

∫M

j∞(s)∗l .

(v) Given a Lagrangian density

l = l(xi, uαI

)dx1 ∧ ... ∧ dxn ∈ Ωn,0 (J∞π) ,

the Euler-Lagrange form E(l) ∈ Ωn,1 (J∞π) is defined as

E(l) = Eα(l)Θα ∧ dx1 ∧ ... ∧ dxn

with

Eα(l) =∂l

∂uα−Di

∂l

∂uαi+ Dij

∂l

∂uαij+− . . . .

Example 2.3.23.1. Consider a Riemannian manifold (E, g). The metric gives us a smooth function

T : TE → Rv 7→ 1

2gp(v, v) for v ∈ TpE

We wish to construct a classical Lagrangian system on p1 : E := I × E → I withI ⊂ R an open interval. Using the canonical identification J1π ∼= R × TE, we pullback T along the projection R × E → E to a function on J1π. This function is calledthe kinetic energy and the mechanical Lagrangian system described by it is called the

free system on the Riemannian manifold (E, g). The corresponding Lagrangian density is

T (xt)dt ∧ ωg ∈ Ωn,0(E) with n := dimE = dim E + 1.

The trajectories for this system can be seen to be geodesics for the Riemannian manifold.

33

2. A smooth functionU : E → R

is called a potential energy. (In interesting problems, one might be obliged to allow U

to have singularities and to impose more conditions on U .) Again, we pull back U to afunction U on J1π and obtain a n-form on E by combining it with the volume form.

3. A Lagrangian system with Lagrangian l := T − U is called a natural system.

We next formulate the algebraic Poincare lemma:

Proposition 2.3.24 (algebraic Poincare lemma).On good open subsets of J∞π, we have dHα = 0 for α ∈ Ωs,0 if and only if there are localfunctions jµ on J∞π such that

α = ∂µjµd1x ∧ . . . ∧ dxs .

Let M be a smooth manifold of dimension n and π : E →M a fibre bundle. Forms on J∞πof type (n, s) with s ≥ 1 are dH-closed, but not even locally exact.

To improve the situation, we introduce for s ≥ 1 coaugmentations, called theinner Euler operators

I : Ωn,s (J∞π)→ Ωn,s (J∞π)

by taking the inner product

I(ω) :=1

sΘα ∧

∑I

(−1)|I|ι ∂∂uα

I

ω.

Lemma 2.3.25.We have the following identities for the inner Euler operators:

(i) For η ∈ Ωn−1,s (J∞π), we haveI (dHη) = 0 .

(ii) For any ω ∈ Ωn,s (J∞π) there exists η ∈ Ωn−1 (J∞π) such that

ω = I(ω) + dHη.

(iii) I is an idempotent, I2 = I.

(iv) For any Lagrangian density Lagrange-Dichte λ ∈ Ωn,0 (J∞π), we have

E(λ) = I (dV λ)

This suggests to extend the bicomplex. We introduce the following subspace of Ωn,s((J∞π)):

F s (J∞π) := im I : Ωn,s → Ωn,s = ω ∈ Ωn,s (J∞π) : Iω = ω

The elements of F s (J∞π) are called source forms.Moreover, we introduce maps

δV : F (J∞π)→ F s+1 (J∞π)

δV (ω) = I(dV ω)

One can check that the following identity holds: δ2V = 0.

Definition 2.3.26

34

(i) The augmented variational bicomplex for a fibre bundle π : E →M is the double complex

0 → Ω0,3

↑ ↑ ↑ ↑0 → Ω0,2 → Ω1,2 → ... → Ωn,2 →

IF2 → 0

↑ ↑ ↑ ↑ δV0 → Ω0,1 → Ω1,1 → ... Ωn−1,1 → Ωn,1 →

IF1 → 0

↑dV ↑ ↑ ↑ E

0 → R → Ω0,0 →dH

Ω1,0 → ... → Ωn−1,0 → Ωn,0

(ii) The Euler-Lagrange complex is the edge complex

0 → R → Ω0,0 dH−→ Ω1,0 → ...dH−→ Ωn,0 E−→ F1 δV−→ F2 → ...

where δV is called the Helmholtz operator.

We recognize Ωn,0 as the recipient for the Lagrangian densities and F1 as the recipientfor the equations of motion. The size of the kernel of δV describes the obstruction to findingLagrangian densities for equations of motion.

2.4 Symmetries and Noether identities

Our goal in this subsection is to describe symmetries of a Lagrangian system (π : E →M, l).

Observation 2.4.1.Suppose a Lie group G acts by automorphisms of a vector bundle E −→ M (over the identityof M) in such a way that it leaves the action S of a Lagrangian l : J∞π −→ R invariant.

Then the group action induces a vertical vector field η on E, for each element η of the Liealgebra of G, such that the prolongation prE(η) of η to J∞π has the property that dl(prE(η))volMis dH exact. Here volM denotes both a volume on M and its pullback to J∞π via the projectionJ∞π −→M .

We only need infinitesimal aspects of the group action and hence formulate symmetries interms of vector fields. We introduce the following notions:

Definition 2.4.2

1. An evolutionary vector field is a mapping from J∞π into the vertical vector fields on E.

In local coordinates Q = Qα ∂∂uα where the functions Qa are local functions on J∞π.

It can be thought of as an “infinitesimal variation depending on the fields and theirderivatives”.

For every evolutionary vector field Q on E, there exists its prolongation, prE(Q), theunique vector field on J∞π preserving the contact ideal such that (dπ∞E )(prE(Q)) = Q.

2. An evolutionary vector field QE on E is called a variational symmetry of a Lagrangian l,if it has the property that dl(pr(QE))volM = pr(QE)(l)volM is dH exact.

35

Remarks 2.4.3.

1. In local coordinates, QE is a variational symmetry iff

prE(QE)(l) = DK(QaE)

∂l

∂uaK

is a divergence on jet space, i.e., iff it is equal to Dµjµ for some set jµ of local functions

defined on J∞π.

2. Integrating by parts shows that this condition is equivalent to requiring thatQaE(−D)K( ∂l

∂uaK

) be a divergence.

3. But the Euler Lagrange operator Ea acting on the Lagrangian l is defined by the equationEa(l) = (−D)K( ∂l

∂uaK

). Thus an evolutionary vector field QE is a variational symmetry of

a Lagrangian L iff QaEEa(l) is a divergence.

We rewrite these results and obtain Noether’s first theorem:

Theorem 2.4.4.Let QE be a variational symmetry, i.e. there are local functions jK on jet space such that

prE(QE)(l) = DK(QaE)

∂l

∂uaK= DKj

K .

Then we have on jet space

DK

(QaE

∂l

∂uaK− jK

)= 0

so that the function on jet space

QaE

∂l

∂uaK− jK

is conserved for all solutions s : M → E of the Lagrangian system:

∂

∂xK(Qa

E

∂l

∂uaK− jK) j∞s = DK(Qa

E

∂l

∂uaK− jK) ∂

Ks

∂xK= 0

Example 2.4.5.Consider a natural mechanical system on E = RN , i.e. l(x, xt) = 1

2|xt|2−U(x), and suppose that

the potential U is invariant under the family of translations given by a fixed vector T ∈ RN \0:

TT : R× RN → RN

x 7→ x+ tT ,

i.e. T i ∂U∂xi = 0. Then ( ∂

∂t, T ) is a variational vector field on E = R× E and

T i∂l

∂xit= T i(xt)i

is a conserved quantity. It is called momentum in the direction T .

Our next example is the rigid body:

36

Observation 2.4.6.

• A rigid body is a system of N ≥ 3 mass points moving in R3 with the standard Euclideanstructure, subject to the constraints

‖xi − xj‖ = rij = const.

To avoid degenerate cases, we require that not all points are on a line.

To determine the configuration space, we choose three mass points x[1], x[2], x[3] not onaline whose position determine the position of the body. As the first three coordinates, wetake x(1)

Next consider the differences v1 = x2 − x1, v2 = x3 − x1; there is exactly one rotationin SO(3) which maps these vectors to a fixed reference pair (w1, w2) of vectors having thesame lengths and including the same angle:

‖v1‖ = ‖w1‖, ‖v2‖ = ‖w2‖ and 〈v1, v2〉 = 〈w1, w2〉

i.e. g(v1, v2)wi = vi. As coordinates, we choose (x1, g(v1, v2)) and identify the configurationspace with R3 × SO(3).

• We consider the system in the absence of external forces, i.e. we assume that it is describedby the Lagrangian:

l =N∑α=1

1

2mα〈xα,t, xα,t〉 with mα > 0

Noether’s theorem gives three conserved quantities for the translations in R3: the vectorfield on E = R3N induced by a translation by T i is

Qi,αE = T i .

with i = 1, ..., 3 and for all α = 1, ..., N . For the derivative of the action, we find

∂l

∂xiαt= mαx

iαt

Hence the conserved quantity is

I =∑i,α

∂l

∂xiαt

(dhs

ds

)i,α=∑i,α

mαxi,αt T i .

Since this applies to all T ∈ R3, we have three conserved quantities

I i =N∑α=1

mαxi,αt i = 1, ..., 3

Denote by M :=∑N

α=1mα > 0 the total mass of the body and introduce the center of mass

xcm :=1

M

N∑α=1

mαxα

37

of the rigid body, then the conservation law implies

d2

dt2xcm = 0.

In other words, the center of mass is in uniform motion.

• This allows us to continue our discussion in a reference frame where the center of massis at rest.

The invariance under rotations gives, again by Noether’s theorem three further conservedquantities; one more conserved quantity is energy, as follows from the time independenceof the action. These four functions determine for any given solution of the equations ofmotion a submanifold C of the six-dimensional phase space SO(3). It is

– Two-dimensional, dimC = 2.

– Bounds on the tangent vectors from energy conservation imply that it is a compactsubmanifold of the compact manifold SO(3).

– Orientable, since T SO(3) ∼= SO(3)×R3 as a smooth manifold. Thus T SO(3)is par-allelizable and in particular orientable and any submanifold defined by global regularequations is orientable as well.

– The derivative of the motion gives a vector field on C that for non-vanishing energyvanishes nowhere.

A standard theorem of topology then implies that C is a torus. One can choose coordinatesϕ1, ϕ2 ∈ R/2πZ on C such that the equations of motion have the form

ϕ1 = ω1 ϕ2 = ω2 .

Obviously, the two angular frequencies ωi depend on the initial conditions.

Noether’s second theorem deals with more structure than just symmetries.

Definition 2.4.7A gauge symmetry of a Lagrangian system (π : E →M, l) consists of a family of local functions

RaI : J∞π −→ R

such that for any local functionε : J∞π −→ R

the vector field RaI(DIε)∂∂ua on E is a variational symmetry of l.

Remarks 2.4.8.

1. Loosely speaking, a gauge symmetry is a linear mapping from local functions on J∞π intothe variational vector fields on E.

2. Notice that the coefficients of the vector field depend linearly on ε and on its derivatives.

3. By the results just obtained, it follows that being a gauge symmetry is equivalent to re-quiring (RaIDIε)Ea(l) to be a divergence for each local function ε on J∞π.

38

4. This in turn is equivalent to saying that ε(RaIDI)+(Ea(l)) is a divergence for each local

function ε, where (RaIDI)+ is the formal adjoint of the differential operator RaIDI .

Since the adjoint of a total differential operator is again a total differential operator, thereexist local functions R+aI : J∞E −→ R such that (RaIDI)

+ = R+aIDI . These functionscan be found by working out the iterated total derivatives (−D)I(R

aIF ). In many cases itis easier to use an “integration by parts” procedure to obtain the coefficient functions onJ∞π, R+aI.

5. It follows that ε 7→ RaI(DIε)∂∂ua defines a gauge symmetry iff εR+aIDI(Ea(l)) is a diver-

gence for each ε. This condition is equivalent to saying that R+aIDI(Ea(l)) is identicallyzero on the jet bundle.

Such an identity is called a Noether identities One thus has a one-one correspondencebetween gauge symmetries of a Lagrangian and Noether identities.

We have thus found:

Theorem 2.4.9 (Noether).For a given Lagrangian system (π : E →M, l) and for local real-valued functions RaI definedon J∞π, the following statements are equivalent:

1. The functions RaI define a gauge symmetry of l, i.e., RaI(DIε)∂∂ua is a variational

symmetry of l for any local function ε : J∞π −→ R.

2. RaI(DIε)Ea(l) is a divergence for any local function ε,

3. The functions RaI define Noether identities of l, i.e. R+aIDI(Ea(l)) is identically zeroon the jet bundle.

One should be aware that gauge symmetries induce differential identities between the equa-tions of motions, i.e. dependencies between the equations of motions and their derivatives.

2.5 Natural geometry

A natural differential operator is a recipe that constructs from a geometric object another one,in a natural fashion, and which is locally a function of coordinates and their derivatives. Theyare thus intimately related to jet bundles and have to be compatible with smooth maps betweenmanifolds.

Examples 2.5.1.1. Let M be a n-dimensional smooth manifold. The classical Lie bracket X, Y 7→ [X, Y ] is a

natural operation that constructs from two vector fields on M a third one. Given a localcoordinate system (x1, . . . , xn) on M , the vector fields X and Y are locally expressionsX =

∑1≤i≤nX

i∂/∂xi, Y =∑

1≤i≤n Yi∂/∂xi, where X i, Y i are smooth functions on M .

If we define X ij := ∂X i/∂xj and Y i

j := ∂Y i/∂xj, 1 ≤ i, j ≤ n, then the Lie bracket is

locally given by the formula [X, Y ] =∑

1≤i,j≤n(XjY i

j − Y jX ij

)∂/∂xi.

2. The covariant derivative (Γ, X, Y ) 7→ ∇XY is a natural operator that constructs from alinear connection Γ and vector fields X and Y , a vector field ∇XY . In local coordinates,

∇XY =(ΓijkX

jY k +XjY ij

) ∂

∂xi, (7)

where Γijk are Christoffel symbols.

39

3. Natural operations can be composed into more complicated ones. Examples of ‘com-posed’ operations are the torsion T (X, Y ) := ∇XY − ∇YX − [X, Y ] and the curvatureR(X, Y )Z := ∇[X,Y ]Z − [∇X ,∇Y ]Z of the linear connection Γ.

4. Let X be a vector field and ω a 1-form on M . Denote by ω(X) ∈ C∞(M) the evaluationof the form ω on X. Then (X,ω) 7→ exp(ω(X)) defines a natural differential operatorwith values in smooth functions. Clearly, the exponential can be replaced by an arbitrarysmooth function φ : R→ R, giving rise to a natural operator Oφ(X,ω) := φ(ω(X)).

5. ‘Randomly’ generated local formulas need not lead to natural operators. As we will seelater, neither O1(X, Y ) = X1

3Y4∂/∂x2 nor O2(X, Y ) = XjY i

j ∂/∂xi behaves properly under

coordinate changes, so they do not give rise to vector-field valued natural operators.

In the examples, the natural differential operators are recipes given as a smooth function incoordinates and derivatives that are covariant under changes of local coordinates.

Definition 2.5.2

1. Denote by Mann the category of n-dimensional manifolds and open embeddings. Let Fibnbe the category of smooth fiber bundles over n-dimensional manifolds with morphismsdifferentiable maps covering morphisms of their bases in Mann.

2. A natural bundle is a functor B : Mann → Fibn such that for each M ∈ Mann, B(M) is abundle over M . Moreover, B(M ′) is the restriction of B(M) for each open submanifoldM ′ ⊂ M , the map B(M ′)→ B(M) induced by M ′ → M being the inclusion B(M ′) →B(M).

For each s ≥ 1 we denote by GL(s)n the group of s-jets of local diffeomorphisms Rn → Rn at 0,

so that GL(1)n is the ordinary general linear group GLn of linear invertible maps A : Rn → Rn.

Let Frs(M) be the bundle of s-jets of frames on M whose fiber over z ∈M consist of s-jets oflocal diffeomorphisms of neighborhoods of 0 ∈ Rn with neighborhoods of z ∈M . It is clear thatFrs(M) is a principal GL(s)

n -bundle and Fr1(M) the ordinary GLn-bundle of frames Fr(M).

Theorem 2.5.3 (Krupka, Palais, Terng).For each natural bundle B, there exists l ≥ 1 and a manifold B with a smooth GL(l)

n -actionsuch that there is a functorial isomorphism

B(M) ∼= Frl(M)×GL

(l)nB := (Frl(M)×B)/GL(l)

n . (8)

Conversely, each smooth GL(l)n -manifold B induces, a natural bundle B. We will call B the

fiber of the natural bundle B. If the action of GL(l)n on B does not reduce to an action of the

quotient GL(l−1)n we say that B has order l.

Examples 2.5.4.

1. Vector fields are sections of the tangent bundle T (M). The fiber of this bundle is Rn, withthe standard action of GLn. The description T (M) ∼= Fr(M)×GLn Rn is classical.

2. De Rham m-forms are sections of the bundle Ωm(M) whose fiber is the space of anti-symmetric m-linear maps Lin(m(Rn),R), with the obvious induced GLn-action. The pre-sentation

Ωm(M) ∼= Fr(M)×GLn Lin(Λm(Rn),R)

40

is also classical. A particular case is Ω0(M) ∼= Fr(M)×GLn R ∼= M ×R, the bundle whosesections are smooth functions. We will denote this natural bundle by R, believing therewill be no confusion with the symbol for the reals.

3. Linear connections are sections of the bundle of connections Con(M) which we recallbelow. Let us first describe the group GL(2)

n . Its elements are expressions of the formA = A1 +A2, where A1 : Rn → Rn is a linear invertible map and A2 is a linear map fromthe symmetric product Rn Rn to Rn. The multiplication in GL(2)

n is given by

(A1 + A2)(B1 +B2) := A1(B1) + A1(B2) + A2(B1, B1).

The unit of GL(2)n is idRn + 0 and the inverse is given by the formula

(A1 + A2)−1 = A−1

1 − A−11 (A2(A

−11 , A−1

1 )).

Let C be the space of linear maps Lin(Rn⊗Rn,Rn), with the left action of GL(2)n given as

(Af)(u⊗ v) := A1f(A−11 (u), A−1

1 (v))− A2(A−11 (u), A−1

1 (v)), (9)

for f ∈ Lin(Rn⊗Rn,Rn), A = A1 +A2 ∈ GL(2)n and u, v ∈ Rn. The bundle of connections

is then the order 2 natural bundle represented as

Con(M) := Fr2(M)×GL

(2)nC .

Observe that, while the action of GL(2)n on the vector space C is not linear, the restricted

action of GLn ⊂ GL(2)n on C is the standard action of the general linear group on the

space of bilinear maps.

For k ≥ 0 we denote by B(k) the bundle of k-jets of local sections of the natural bundle B

so that B(0) = B. If g is represented in this way, then

B(k)(M) ∼= Fr(k+l)(M)×GL

(k+l)n

B(k) ,

where B(k) is the space of k-jets of local diffeomorphisms Rn → B defined in a neighborhoodof 0 ∈ Rn.

Definition 2.5.5Let F and G be natural bundles. A (finite order) natural differential operator O : F → G isa natural transformation (denoted by the same symbol) O : F(k) → G, for some k ≥ 1. Wedenote the space of all natural differential operators F→ G by Nat(F,G).

If F and G are natural bundles of order ≤ l, with fibers F and G, respectively, then eachnatural operator in Definition is induced by an GL(k+l)

n -equivariant map O : F (k) → G, forsome k ≥ 0. Conversely, such an equivariant map induces an operator D : F→ G. This meansthat the study of natural operators is reduced to the study of equivariant maps. The proceduredescribed above is therefore called the IT reduction (from invariant-theoretic).

Examples 2.5.6.1. Given natural bundles B′ and B′′ with fibers B′ resp. B′′, there is an obviously defined

natural bundle B′×B′′ with fiber B′×B′′. With this notation, the Lie bracket is a naturaloperator [−,−] : T×T → T and the covariant derivative an operator ∇ : Con×T×T → T ,where T is the tangent space functor and Con the bundle of connections. The correspondingequivariant maps of fibers can be easily read off from local formulas given in Examples.

41

2. The operator Dφ : T × Ω1 → C∞ from the Example above is induced by the GLn-equivariant map Oφ : Rn × (Rn)∗ → R given by oφ(v, α) := φ(α(v)).

3 Classical field theories

3.1 Maxwell’s equations

We start considering Maxwell’s equation on a Galilei space A and will see only later that thisis not an appropriate conceptual setting, because the laws of electrodynamics are incompatiblewith the Galilean principle of relativity.

Observation 3.1.1.(a) Electrodynamics leads to a new conserved quantity: electric charge. We introduce a charge

density. It assigns to a space-time point (x, t) ∈ A the density of electric charges. Given anymeasurable subset U of a hypersurface Ht ⊂ A of simultaneous events, the electric chargepresent in the space region U should be given by an integral over U . It is therefore naturalto consider the charge density ρt as a three-form on Ht:

ρt := ρ dx1 ∧ dx2 ∧ dx3 ∈ Ω3(Ht)

We will endow the hypersurfaces Ht with an orientation. By the basic axioms of Galileispace, these hypersurfaces are endowed with a Euclidian metric. We can thus use three-dimensional Hodge duality to identify it with a function ρ on Ht.

(b) The charge density is not constant in time. If we take a volume V ⊂ Ht with smoothboundary ∂V , then electric charge can pass through its boundary ∂V . The amount of chargepassing per time should be described by the integral over a two form j ∈ Ω2(Ht). We thenhave the natural conservation law for any closed volume V ∈ Ht

d

dt

∫V

ρ = −∫∂V

j

which by Stokes’ theorem takes the form

d

dt

∫V

ρ = −∫∂V

j = −∫V

dj .

Since this holds for all volumes V ⊂ Ht, we have the infinitesimal form of the conservationlaw

d

dtρ+ dj = 0

which is an equality of three-forms on Ht for all t. We can write the conservation law morecompactly in terms of the three-form form

j := ρ− dt ∧ j ∈ Ω3(A)

defined on the four-dimensional space A as dj = 0.

Using the Hodge star on Ht, we can relate the three-form j on Ht to the 1-form ∗j whichin turn can be identified with a vector ~j. This allows us to recover the conservation law inthe classical form

∂ρ

∂t+ div~j = 0 .

42

(c) Moreover, observations tell us that distributions of electric charge experience a force. Forsimplicity, we consider a point particle carrying an electric charge q and moving with ve-locity

~v(t) =d~x(t)

dt∈ Tx(t)Ht

It turns out that the force is proportional to q and depends on two R3-valued functions on A,the electric field ~E(~x, t) and the magnetic field ~B(~x, t). The Lorentz force force experiencedis

~FLor(x(t)) = q ~E(x(t) + q~v(t) ∧ ~B(x(t)

Using the metric on A.

The electric and the magnetic field are to be thought of as sources of this force. Again usingthe Euclidean structure on the hypersurface Ht of simultaneous events in a Galilei space A,we identify the vector of force with a one-form on Ht and thus introduce a one-form for theelectric field

E = Ei dxi ∈ Ω1(Ht) .

To understand the second term, we introduce a 2-form to describe the magnetic field

B :=1

2εklmBk dxl ∧ dxm ∈ Ω2(Ht)

and compute its contraction with the vector ~v of velocity:

ι~vB = ι~v(Bxdy ∧ dz +Bydz ∧ dx+Bzdx ∧ dy= Bxvydz −Bxvzdy +Byvzdx−Byvxdz +Bzvxdy −Bzvydx= (Byvz −Bzvy)dx+ (Bzvx −Bxvz)dy + (Bxvy −Byvx)dz

This gives a one-form ι~v(t)B ∈ T ∗x(t)Ht on the world line of the charged point particle.

Again, we would like to rewrite the equation in a more uniform way on the four-dimensionalGalilei space A. To this end we introduce a vector on A by

j = −q ∂∂t

+ q~v ∈ vect(A)

that combines the electric charge and the electric current associated to the moving pointlinkesource. We have added a minus-sign for later convenience. We also combine electricand magnetic field to get a two-form on the four-dimensional Galilei space A, called theelectromagnetic field strength

F := −dt∧E+B = −3∑i=1

Eidt∧dxi+B1dx2∧dx3 +B2dx

3∧dx1 +B3dx1∧dx2) ∈ Ω2(A) .

We then find in terms of 1-forms the Lorentz force as a contraction:

ι(FLor) = ιjF .

43

(d) Maxwell’s equations conversely describe the dynamics of the field for given charges andsources.

To state them concisely, we endow Galilei space A with the structure of a pseudo-Euclideanmanifold by extending the metric on the spacial hypersurfaces Ht in such a way that theglobal vector field ∂

∂tis normal to all surfaces Ht and normalized by

g(∂

∂t,∂

∂t) = −1

The minus sign is here just an ad hoc convention.

This allows us to introduce a one-form on A

j := −ρdt+ j1dx+ j2dy + j3dz ∈ Ω1(A)

with the property that j = ∗j where the Hodge star is now the four-dimensional Hodgeoperator. We compute explicitly the Hodge dual

∗F = Bidt ∧ dxi − E1dx2 ∧ dx3 − E2dx

3 ∧ dx1 − E3dx1 ∧ dx2

This two-form is sometimes called the dual field strength.

Maxwell’s equations can be stated concisely as

dF = 0 δF = ∗d ∗ F = j

Here the first equality is in Ω3(A) and the second equality in Ω1(A).

(e) To make the comparison to the classical formulation, we compute

dF = (divB)dx ∧ dy ∧ dz + (∂tB1 + ∂2E3 − ∂3E2)dt ∧ x2 ∧ dx3

+(∂tB2 + ∂3E1 − ∂3E1)dt ∧ x3 ∧ dx1 + (∂tB3 + ∂1E2 − ∂2E1)dt ∧ x1 ∧ dx2

we finddivB = 0 and ∂tB + rot ~E = 0

The first equation expresses the absence of magnetic charges. The second equation is Fara-day’s law of induction. They are the homogeneous Maxwell equations.

(f) Similarly, we compute

∗d ∗ F = −div ~Edt+ (−∂tE1 + ∂2B3 − ∂3B2)dx+(−∂tE2 + ∂3B1 − ∂1B3)dy + (−∂tE3 + ∂1B2 − ∂2B1)dz

We thus finddivE = ρ and rotB − ∂tE = ~j

These equations are called inhomogeneous Maxwell equations. The first equation is calledGauß’ law and the second equation Ampere’s law with Maxwell’s term.

We spend a moment to use Stoke’s theorem discuss the interpretation of these equationswhich are the basis of all electrical engineering.

Interpretation 3.1.2.

44

(i) Gauß’ law reads in integral form

Q =

∫V

ρ d3x =

∫Σ

~E d~j.

The electric flux through a surface is thus proportional to the electric charge included bythe surface. It should be appreciated that this holds even for time dependent electric fields.

We use this law to determine the electric field of a point charge in the origin:

ρ(~x, t) = qδ0(~x).

Symmetry considerations lead to the ansatz:

~E(~x, t) = f(r)~er .

Integrating over the two-sphere with center 0 and radius r, we find

q =

∫S2

r

~E d~f = ε0 4π r2 f(r)

and thus Coulomb’s law:

~E(~x, t) =q

4π

1

r2~er .

In the case of static field, Coulombs law and the principle of superposition of chargesimplies the first Maxwell equation.

(ii) Faraday’s law of induction yields a relation between the induced voltage along a loop ∂Fbounding a surface F

Uind ≡∫∂F

dx ~E

and the magnetic flux

Φmag =

∫F

d~f ~B

through the surface which reads

Uind ≡∫∂F

dx ~E = − d

dt

(∫F

d~j ~B

)= − d

dtΦmagf ≡ − d

dt

(∫F

d~f ~B

)This law is the basis of the electric motor and the electrical generator.

Remarks 3.1.3.1. We consider the Maxwell equation on a star-shaped region U ⊂ A on which we can apply

Poincare’s lemma. From dF = 0, we conclude that there exists a one-form A ∈ Ω1(U)such that dA = F |U . The one-form A is called a gauge potential for the electromagneticfield strength F .

A is not unique: taking any function λ ∈ Ω0(U), we find that

A′ := A+ dλΩ1(U)

also obeys the equation dA′ = F |U . This arbitrariness in choosing A is called thegauge freedom and choosing one A is called a choice of gauge.

One can impose additional gauge conditions on A to restrict the choice. For example, fora Lorentz gauge, one requires δA = 0.

45

2. We also write these equations in the three-dimensional language of vector calculus

~B = rotA

where A is the 3-vector dual to the spacial part of the one-form A ∈ Ω1(U). The field ~Ais called a vector potential. For the electric field, we find

~E = −dΦ− ∂A

∂t.

with a scalar function Φ called scalar potential. It is given by the time component of theone-form A ∈ Ω1(U). Indeed, the homogeneous Maxwell equation yields

d( ~E +∂A

∂t) = 0,

and by Poincare’s lemma, we write the expression in parenthesis locally as the gradientof a smooth function.

3. In this language, the gauge freedom is expressed by the fact that vector potential and scalarpotential are not unique, but can be change for any smooth function χ(~x, t) to

~A′ = ~A+ gradχ and Φ′ = Φ− dχ

dt

which are again potentials for the same field strength.

4. Typical gauge conditions then read:

• the Coulomb gauge: div ~A = 0for which the remaining gauge transformations are those with 4χ = 0.

• the Lorentz gauge: div ~A+ 1κc2

∂χ∂t

for which the remaining gauge transformations are those with χ = 1c2

∂x∂t2−4χ = 0.

5. We pause a moment to consider the special case of so called static situations in whichfields, currents and charge densities are time independent, the Maxwell equations decouple:

div ~E = ρε0

div ~B = 0

rot ~E = 0 rot ~B = κµ0~j with div~j = 0

On a star-shaped region, we find E = −grad Φ with 4Φ = −ρε. Mathematically, electro-

statics is thus the theory of the (inhomogeneous) Poisson equation. Similarly, we find for

the magnetic field in terms of the vector potential ~B = rot ~A the equation

4A− grad divA = −κµ0~j,

which reduces in the Coulomb gauge to 4A = −κµ0~j.

A characteristic feature of electrodynamics is the existence of a quantity c having the di-mension of a velocity.

Observation 3.1.4.

46

1. The force between two point charges q1 and q2 at a distance r is∣∣~Fm∣∣ =

1

4πε0

q1q2r2

.

The magnetostatic force between two parallel conductors with currents I1 and I2 at adistance r can be seen to be ∣∣~Fm

∣∣ =κ2µ0

4π

2lI1I2r

.

The ratio of these two forces is dimensionless and independent of the system of units.Hence the ratio

κ2ε0µ0 =:1

c2

is a quantity with the dimension of a velocity that is characteristic for eletrodynamics.

2. Another consideration yielding this velocity concerns Maxwell’s equations with vanishingexternal sources ρ = 0, ~v = 0.

div ~E = 0 rot ~E = −∂ ~B∂t

div ~B = 0 rot ~B = ∂ ~E∂t

Taking the rotation of the second equation yields

4E = 4E − grad(divE) = −rot rotE =

= rot∂

∂t~B =

∂

∂trot ~B =

∂2E

∂t2,

A dimensional analysis shows that we should have a factor with the dimension of a velocityc2. Hence we get

~E :=

(1

c2∂2

∂t2−4

)~E = 0

with the so-called d’Alembert operator. Similarly, one finds

~B :=

(1

c2∂2

∂t2−4

)~B = 0.

To find solutions, we make the ansatz

~E(t, ~x) := ~E0e−i(ωt−~k~x)

~B(t, ~x) := ~B0e−i(ωt−~k~x)

of plane waves with direction of propagation~k

|~k|. These waves decribe the propagation of

light or other electromagnetic radiation (radio waves, gamma rays, . . . ) in empty space.Hence c can be identified with the velocity of light.

We also get a dispersion relation

ω2 = c2k2

between wave length λ = 2πk

and frequency ν = ω2π

.

47

Maxwell’s equations imply moreover that

~k ~E0 = 0 ~k ~B0 = 0~k ∧ ~E0 = κω ~B0

~k ∧ ~B0 = − ωκc2

~E0.

In other words, (~k, ~E, ~B) form an oriented basis of R3: electromagnetic waves are transver-sal. Depending on the initial conditions, they can be polarized, either linearly or elliptically,in particular in a circular way.

3.2 Special relativity

The existence of a distinguished velocity that is the same in all reference frames in relativeuniform motion has been experimentally verified to very high precision. But it is incompatiblewith the transformation of velocities under the Galilei group. It thus forces us to revise ourideas about space and time and gives rise to the theory of special relativity.

Our postulates are

(i) Space and time are homogeneous and spacially isotropic, i.e. our model for space time isstill still be the one of an affine space A over R4.

(ii) The laws of physics should be of the same form in all reference frames in relative uniformmotion.

(iii) In particular, the velocity of light is fixed by Maxwell’s equations and thus should be thesame for all reference frames, independently of the motion of the source of light. It shouldbe a universal limit velocity that cannot be reached by an observer at rest.

Homogeneity in the first postulate is taken into account by modelling space-time again byan affine space over the vector space R4. It has as a symmetry group R4 o GL(4,R). Spacialisotropy means that we distinguish a subgroup isomorphic to the rotation group SO(3). It isimportant to note that the notion of spacial depends a priori on an observer We are lead to asymmetry group R4 oG where G should contain a subgroup isomorphic top SO(3) which mightdepend on the observer.

We have to specify the set of reference frames. They correspond to “observers” moving aset of distinguished lines that is invariant under the action of R4 oG. It will be our postulatesthat these are certain affine lines.

Together with postulate (iii), one can show that this leads to the unique solution:

Definition 3.2.1(i) An affine space M over R4 together with a metric of signature (−1,+1,+1,+1) on its

difference space is called a Minkowski space.

(ii) Standard Minkowski space is R4 with the diagonal metric. It plays the role of standardGalilei space. Using the velocity of light, we endow it with coordinates (ct, x1, x2, x3) whosedimension is length.

(iii) A Lorentz system of M is an affine map

φ : M→ R4

which is an isometry on the difference space.

48

(iv) The light cone in TpM is the subset

LCp := x ∈ TpM∣∣ ηp(x, x) = 0.

(v) A Poincare transformation is a diffeomorphism of M whose differentials respect the lightcones in all tangent spaces, the so-called “causal structure”. A Lorentz transformation isthe linear map induced on the difference space.

Lemma 3.2.2.Let

Λ : M→Mbe a diffeomorphism that preserves light cones. Then Λ is an affine mapping and we have ontangent spaces

ηp(Λpx,Λpy) = ap(Λ)ηp(x, y) for all x, y ∈ TpM.

Neglecting translations, we have thus the direct product

R>0 × SO(3, 1),

with R>0 the subgroup of dilatations.

Remarks 3.2.3.

(a) The Lorentz group L = O(3, 1) is a Lie group with four connected components:

Lηε =Λ ∈ L

∣∣ detΛ = ε, Λ00 = η|Λ0

0|

with ε, η = ±1.

(b) The connected component of the neutral element, L+1+1, are called proper orthochronous

Lorentz transformations.

Using the parity transformation

P = diag(+1,−1,−1,−1)

and time reversalT = diag(−1,+1,+1,+1)

we can write every element of L uniquely in the form

Λ = P nTmΛ0 with n,m ∈ 0, 1, Λ0 ∈ L+1+1.

(c) The Lorentz group L+1+1 is not compact. A maximal compact subgroup is the group of rotations

of the Euclidean subspace spanned by span(e1, e2, e3) which is isomorphic to the group SO(3)of rotations. Since the Lorentz group is semi-simple, any other maximal compact subgroupis conjugate to this subgroup.

Consider the element

Λ1(θ) =

cosh θ − sinh θ 0 0− sinh θ cosh θ 0 0

0 0 1 00 0 0 1

∈ Λ+1+1.

49

It is called a boost in e1-direction with rapidity θ. For y = Λ(θ)x, we have y2 = x2, y3 = x3

and

y0 = x0 cosh θ − x1 sinh θ

y1 = x1 cosh θ − x0 sinh θ

In particular, y1 = 0 is equivalent to x1 = ct tanh θ = vt so that the origin of the ycoordinate system is in uniform relative motion, with velocity v = c tanh θ. In particular,we find the velocity c of light as a limit that cannot be reached: |v| < |c|. Similarly, onedefines boost Λn(θ) in any direction n and finds

Λn(θ1)Λn(θ2) = Λn(θ1 + θ2) = Λn(θ)

which reads in terms of coordinates

v =v1 + v2

1 + v1v2c2

.

(d) Every Lorentz transformation Λ can be written uniquely as a product of a boost B(Λ) anda rotation (Λ), but the boosts are not a subgroup.

(e) The Poincare group turns out to be the semi-direct product of the Lorentz group and thetranslation group R4. It is a ten-dimensional non-compact Lie group.

We reconsider the geometry of Minkowski space:Definition 3.2.4

(i) A non-zero vector x in Minkowski spaces is called

time like, if x2 = (x, x) < 0

light like, if x2 = (x, x) = 0

space like, if x2 = (s, s) > 0

(ii) A time like or light like vector is called future directed, if x0 > 0 and past directed ifx0 < 0. The light cone decomposes into a forward and backward light cone.

(iii) The future I+(x) and the past I−(x) of a point x ∈M are the sets

I±(x) := y ∈M|(y − x)2 ≤ 0, y − x future / past directed .

The future of a subset S ⊂M is defined by

I±(S) := ∪x∈SI±(x)

We next introduce the notion of an observer. A basic postulate requires massive particlesand thus observers to move, in the absence of external forces (and gravitational forces) on affinelines in Minkowski space with time like velocity vectors that are future directed. Light rays arerequired to have lightlike velocity vectors.

Definition 3.2.5

50

1. A velocity unit vector w is a future directed timelike vector normalized to (w, w) = −1.

2. For any a ∈M4, the affine line a+Rw is the worldline of an observer which we abbreviatewith Oa,w or sometimes even with Ow, when the basepoint a does not matter.

3. The time interval elapsed between two events x, y ∈ R4 as observed by observer Ow isgiven by the observer-dependent time function

∆tw(x, y) := (w, x− y) .

Two events are simultaneous, if ∆tw(x, y) = 0. We say that x happens before y, if∆tw(x, y) < 0 and x happens after y if ∆tw(x, y) > 0.

Remarks 3.2.6.

(a) The time interval between two events x, y ∈M depends on the observer.

(b) The set of events simultaneous to x for the observer w is the affine hyperplane perpen-dicular to w containing x. This gives a foliation of Minkowski space into space-like affinehyperplanes that depends on the velocity unit vector w.

(c) More generally, a massive of massless point particle is described by its trajectory

x : I → Mπ 7→ x (τ)

where we require for physical motions the velocity vector to be future directed and timelikeor lightlike, respectively,

dx0

dσ> 0

(dx

dσ

)2

≤ 0 .

The Lorentz invariant quantity

τ12 =1

c

∫ σ2

σ1

dσ

√−(

dx

dσ

)2

is called eigen time of the particle elapsed between the two points x(σ1) and x(σ2) on itstrajectory.

We can now formulate the postulate of causality:an event x ∈ M can influence an event y ∈ M (slang: a signal can be sent from x to y) if andonly if y lies in the future of x.If x 6∈ I+(y) and y 6∈ I+(x), the the events are called causally disconnected.

The following lemma guarantees that causality is reasonably related to the time differencesobserved by all observers:

Lemma 3.2.7.For any observer Ow and for any events x, y ∈ M simultaneous for Ow, the relative positiony − x is space like.

51

The following lemma ensures that all observer dependent time functions are consistent withcausality:

Lemma 3.2.8.

1. For any two observers Ow and Ov with different velocity unit vectors w and v, thereexist events x, y ∈ M such that x happens before y for w and y happens before x for v.(“Relativity of simultaneity”). However, such events are always causally disconnected andcannot influence each other so that causal paradoxes are excluded.

2. The future I+(x) of an event x ∈M equals

I+(x) = y ∈M|(v, y − x) < 0 for all observersOv

i.e. the set of events that happens after x for all initial observers.

Observation 3.2.9.We discuss some famous relativistic phenomena:

a) Time dilatation:Consider two observers w and v and two events x, y ∈ M. Assume that they occur at restfor the observer w, i.e. x− y = tw with some t ∈ R. One could think about a clock travellingwith observer w on which this observer looks twice; he measures a time interval

∆tw(x, y) = |x− y| =√−(x− y)2 .

In contrast, observer Ov measures a time difference

∆tv(x, y) = |(v, x− y)|

Introducea := (x− y) + (x− y, v) v

which is a space-like vector orthogonal to v. We have thus

x− y = −∆tv(x, y) v + a

which implies∆tw(x, y)2 = |(v, x− y)|2 + a2 = ∆tv(x, y)

2 + a2

Hence|∆tv(x, y)| > |∆tw(x, y)|

Thus the moving observer Ov measures a longer time interval than the observer Ow at rest.This effect has been measure with very high precision in observations of the life time ofinstable particles in motion.

b) Similarly, lengths in motion see to be contracted: one has the phenomenon of(length contraction).

52

c) We finally discuss the twin paradoxon as the geometry of triangles given by future-directedtimelike vectors in M. Consider three future directed timelike vectors in M that obey

z = x+ y

with the interpretation that z is the twin at rest, x describes the way forth and y the wayback. One has

(z, z) = (x, x) + 2(x, y) + (y, y)

and thus in the rest system of x where x takes the form x = (t, 0, 0, 0):

2(x, y) = 2x0y0 = 2√x2√y2 + ~y 2 > 2

√x2√y2 .

One should appreciate that this has essentially the structure of the Cauchy-Schwarz inequal-ity, but with the opposite inequality. It implies

z2 >(√

x2 +√y2)2

√z2 >

√x2 +

√y2

so that the twin at rest is older.

3.4 Electrodynamics as a gauge theory

We have seen that the homogeneous Maxwell equation dF = 0 suggests to apply the Poincarelemma on a star-shaped subset U ⊂ R4 (or more generally on a Lorentzian manifold modellingspace time). and to write the two-form F as the exterior derivative dA of a one-form A ∈ Ω1(U).

• On a general manifold, this is possible only locally. Hence, we fix a good open coverU = (Uα) of our (Lorentz-)manifold. This means that arbitrary intersections

Uα1...αn := Uα1 ∩ . . . ∩ Uαn

of coordinate neighbourhoods are either empty or contractible.

• Applying the Poincare lemma to F |Uα ∈ Ω2(U) we find Aα ∈ Ω1(Uα)such that F

∣∣Uα

= dAα.

• On Uαβ = Uα ∩ Uβ we find a one-form

d (Aα − Aβ) = F∣∣Uα∩Uβ

− F∣∣Uβ∩Uα

= 0.

and thus, again by the Poincare lemma a function λαβ ∈ C∞(Uαβ,R) with

(δA)αβ := Aα − Aβ = dλαβ =1

ie−iλαβ d eiλαβ =

1

id log eiλαβ

We therefore put gαβ := ei λαβ ∈ Ω0(Uαβ) := C∞ (Uαβ, U(1)) .On triple overlaps Uαβγ, we find the so-called cocycle relation more

(δg)αβγ := gαβ g−1γ gβγ = 1.

53

• DefineV n,m :=

⊕α1,...,αm+1

Ωn(Uα1,...,αm+1

)and form the double complex

↑ ... ↑V 0,1 →

dV 1,1

↑δ ↑δ0 → V 0,0 →

dV 1,0

with d2 = 0, δ2 = 0, d δ + δ d = 0. Then D = d + (−1)nδ is the differential on thetotal complex

V n :=⊕k+l=n

V k,l

• Hence a class in H1D is represented by an element in

V 1,0 ⊕ V 0,1 = ⊕α Ω1(Uα)⊕α,β Ω0(Uαβ),

and hence by a pair (Aα, gαβ) as above, for which the identity holds

0 = D (Aα, gαβ) = ((δ A)αβ − dlog gαβ, (δ g)αβγ)

∈ Ω1,1(Uαβ) ⊕ Ω0,2(Uαβγ).

The class is determined up to the imppage of

D : V 00 ⊕α Ω0(Uα) → V 1,0 ⊕ V 0,1

D(λα) =(dλα, λαλ

−1β

).

• For a geometric interpretation, we consider on the disjoint union

L =⊔α

(Uα × C ) =⋃α

(Uα × α × C)

the equivalence relation(x, α, gαβ z) ∼ (x, β, z) .

Then the quotientL := L

/∼

comes with a natural projection to M which provides a complex line bundle π : L→M .

• This bundle comes with a linear connections

5α = d +1

iAα

which furnishes a curvatures which is just the electromagnetic field F by

F∣∣Uα

:= dAα.

54

Because ofdAα − dAβ = d (δ A)αβ = d dlog gαβ = 0

this is globally defined. One should appreciate that this imposes on the cohomology classof the two-form F the condition to be integral,

[F ] ∈ H2(M,Z) ⊆ H2dR(M,R) .

• One should also note that if the fundamental group π1(M) 6= 0 is non-trivial, there arenon-isomorphic line bundles with the same field strength F . They can be dinstinguishedby quantum mechanical measurements in the Aharonov-Bohm effect.

4 Hamiltonian mechanics

4.1 (Pre-) symplectic manifolds

We will now get to learn a different formulation of mechanical systems that turns out to bea good starting point for quantum mechanics as well. To present it, we need a few geometricnotions.

Definition 4.1.1 Let M be a smooth manifold.

(i) A 2-form ω ∈ Ω2(M) is called a presymplectic form, if it is closed and of constant rank.

(ii) A non-degenerate pre-symplectic form is called a symplectic form.

(iii) A smooth manifold M together with a (pre-)symplectic form ω is called a(pre-)symplectic manifold.

(iv) A morphism f : (M,ωM)→ (N,ωN) of (pre-)symplectic manifolds is a differentiable mapf : M → N that preserves the (pre-)symplectic form, f ∗ωN = ωM . Such morphisms arealso called canonical transformations.

Remarks 4.1.2.

1. Since for any point p of a symplectic manifold M the tangent space TpM is a symplecticvector space, the dimension of a symplectic manifold is necessarily even. A presymplecticmanifold, in contrast, can have odd or even dimension.

2. One checks readily that if M is a symplectic manifold of dimension dimM = 2n, then-the power ωn of the symplectic form ω is a volume form on M .

Example 4.1.3.For any smooth manifold M of dimension n, the cotangent bundle T ∗M has a natural structureof a symplectic manifold of dimension 2n.

Indeed, local coordinates (xi)i=1...n for M give local coordinates (xi, pi) for the cotangentbundle T ∗M , by describing local one-forms as

λ(x) =n∑i=1

pi(x)dxi .

55

Under changes of coordinates, we have

dxi =n∑j=1

∂xi

∂xjdxj.

Consider the locally defined two-form on the total space T ∗M of the cotangent bundle

ω0 :=n∑i=1

dpi ∧ dxi .

Locally, it can be written as dΘ with

Θ :=n∑i=1

pidxi .

Indeed,

dΘ =n∑i=1

dpi ∧ dqi = ω0 .

Hence the form ω0 is closed; one checks that it is non-degenerate. Moreover, it is independentof local coordinates: if xi are different coordinates, we find

n∑i=1

dpi ∧ dxi =n∑

i,j,k=1

∂xi

∂xk∂xj

∂xidpj ∧ dxk = ω0 .

As a consequence, the locally defined two-forms patch together to a globally defined sym-plectic form on the cotangent bundle T ∗M . The symplectic manifold (T ∗M,ω0) is called thecanonical phase space associated to the manifold M .

The following result is central to symplectic geometry:

Theorem 4.1.4 (Darboux’ theorem).Let (M,ω) be a (2n+ k)-dimensional presymplectic manifold with rank rankω = 2n.

Then we can find for any point m ∈M a neighborhood U and a local coordinate chart

ψ : U → R2n+k

ψ(u) =(q1, ..., qn, p1, ..., pn, η

1, ..., ηk)

such that ω∣∣U

=∑n

i=1 dpi∧dqi. Such coordinates are called Darboux coordinates. One can evenchoose a covering such that the induced coordinate changes are canonical transformations.

Darboux’ theorem implies that symplectic geometry is locally trivial - in contrast to Rie-mannian geometry where curvature provides a local invariant.

Observation 4.1.5.Let (M,ω) by a symplectic manifold and f ∈ C∞(M,R) be a smooth function. Since ω is notdegerate, we can find a unique vector field Xf on M such that

df(Y ) = ω(Xf , Y )

56

holds for all local vector fields Y on M holds. The vector field Xf is called the symplectic gradientof f . This way, we obtain an embedding

C∞(M,R)/

R → VectM .

We compute the symplectic gradient in local Darboux coordinates defined on U ⊂ M . Wemake the ansatz

Xf =n∑i=1

ξi∂

∂qi+ ξi

∂

∂pi∈ vect(U)

with local coordinate functions ξi and ξi. We evaluate both sides on a local vector field

Y =n∑i=1

ηi∂

∂qi+ ηi

∂

∂pi∈ vect(U) .

Since we work in Darboux coordinates, we find

ω(Xf , Y ) =n∑i=1

(ξiη

i − ξiηi)

which has to equal

df(Y ) =n∑i=1

∂f

∂qiηi +

∂f

∂piηi

The comparison of the coefficients yields for the symplectic gradient the local expression

Xf =n∑i=1

− ∂f∂pi

∂

∂qi+∂f

∂qi∂

∂pi∈ vect(U) .

The (commutative) algebra of smooth functions on a symplectic manifolds carries additionalalgebraic structure.

Definition 4.1.6Let k be a field of characteristic different from zero.

(i) A Poisson algebra is a k vector space P together with two bilinear products · and −,−with the following properties

• (P, ·) is an associative algebra.

• (P, −,−) is a Lie algebra.

• The bracket provides derivations for the associative product:

x, y · z = x, yz + yx, z.

(ii) A Poisson manifold is a smooth manifold M with a Lie bracket

−,− : C∞(M)× C∞(M)→ C∞(M)

such that (C∞(M), −,−) together with the pointwise product of functions is a (com-mutative) Poisson algebra.

57

Examples 4.1.7.

(i) Every associative algebra (A, ·) together with the commutator

[x, y] := x · y − y · x

has the structure of a Poisson algebra. One should notice that this commutator is trivial,if the algebra is commutative.

(ii) Consider the associative commutative algebra A = C∞(M) of smooth functions on asmooth manifold M . If M has a Poisson structure, we can using the symplectic gradientto define the following Poisson bracket

F,G := 〈dG,Xf〉

for F,G ∈ A. From the last equation in observation 4.1.5, we find

f, g = dg(Xf ) =∂g

∂pi

∂f

∂qi− ∂g

∂qi∂f

∂pi.

It is an instructive exercise to verify antisymmetry and the Jacobi identity from thislocal formula. Also, the reader should check the following equation which asserts that thesymplectic gradient provides a morphism of Lie algebras from the Lie algebra structure onsmooth functions given by the Poisson structure to the Lie algebra of vector fields:

Xf,g = [Xf , Xg] .

(iii) For any smooth function h ∈ C∞(M) on a Poisson manifold M , the map

h,− : C∞(M)→ C∞(M)

f 7→ h, f

is a derivation and thus provides a global tangent vector field. We consider its integralcurves s : I → M which are solutions of the following first order ordinary differentialequations

d

dtf s(t) = h, f(s(t))

for which we introduce the short hand notation:

f = f, h .

In local Darboux coordinates, we find for the special case of a local coordinate functionf = qi the equation

qi =∂h

∂pi

and for f = pi

pi = − ∂h∂qi

.

These equations are called Hamilton’s equations for the Hamilton function h.

58

(iv) For a simple time independent mechanical system with configuration space M , the equa-tions of motion are second order on M and thus first order on TM . They correspond toa vector field on TM .

The cotangent bundle, on the other hand, is a symplectic manifold. This raises the questionon whether there is a function h on T ∗M such that the vector field h,− on T ∗M containsinformation equivalent to the mechanical system.

Observation 4.1.8.Before we explain this point, we wish to have another look at presymplectic manifolds. For everypoint p ∈M , consider the subspace

kerp ω = v ∈ TpM | ωp(v,−) = 0 = v ∈ TpM | ιvωp = 0 ⊂ TpM

which gives a distribution on M

kerω =⋃p∈M

kerp ω ⊂ TM

of constant rank. We use Cartan’s formulae to check that the commutator of two vector fieldsv, w with values in kerω is contained in kerω.To this end, we have to show that ιvω = ιwω = 0implies ι[v,w]ω = 0. We find

ι[v,w]ω = Lvιwω − ιwLvω = 0− iw(dιv + ιvd)ω = 0 .

This allows us to apply a theorem of Frobenius’ that asserts that there is a foliation of M ,

i.e. M can be written as a disjoint union of submanifoldsLαα∈A, called the leaves of the

foliation, whose tangent spaces are just kerω, i.e. for p ∈ Lα

TpLα = kerpω

or, more globally, ⋃α∈A

⋃p∈Lα

TpLα = kerω .

In general, the set of leaves UM = M/kerω is not a smooth manifold. If this happens to be

the case, then it is of dimension dimM − dim kerω . In this case, UM carries a symplecticstructure ω such that the projection

π : M →M/kerω = UM

becomes a morphism of presymplectic manifolds, π∗ω = ω.One sufficient condition is the existence of local slices: for every point p ∈M , one can find

a submanifold Σp that intersects every leaf at most once and whose tangent space complementskerω in every point q ∈ Σ,

TqM = kerqω ⊕ TqΣ .

59

4.2 Hamiltonian dynamics

Observation 4.2.1.

• The kinematical framework for Lagrangian mechanics is a fibre bundle π = pr1 : E =I × E× → I with ⊂ R and interval and a Lagrangian function

l : J∞π → R.

The dynamics is given by a differential equation R ⊂ J∞π given by the variational problemfor l.

• We now invoke the Newtonian principle that R is second order. This implies that theLagrangian function l factorizes to a function on J1π ∼= TE × I, cf. Lemma 2.3.6

• Our idea is to describe the motion of the mechanical system, i.e. solutions to the varia-tional problem given by l in terms of the integral curves of a vector field of the form h,−on T ∗E. Since the generalized momenta ∂l

∂sitare cotangent vectors to E, this amounts to a

transition from generalized velocities, i.e. tangent vectors to E, to generalized momenta.

• Hence we consider the symplectic manifold

T ∗E ∼= T ∗E × T ∗I

with local coordinates (q, p, t, h) and and the local canonical 1-form

Θ = p dq − h dt

which induces the global symplectic form ω = dΘ.

The submanifold relevant for the equations of motions is defined as the zero locus of afunction

µ : T ∗M → R

We will from now on assume that µ is of the form

µ(p, q, t, h) = pr∗1 H− h0

with a constant h0 and a function H : T ∗M → R called the Hamilton function.

• We could interpret µ as a moment map with values in the Lie algebra R. By a so-calledsymplectic reduction, we eliminate h and obtain the manifold

T ∗E × I,

called the extended phase space or evolution space. The local one-forms

Θ = pdq − pr∗1H dt = pr∗1 ΘE − pr∗1H dt

provide us the global closed 2-form

ω = dΘ = pr∗1 ωT ∗E − pr∗1 dH ∧ dt.

on T ∗E × I.

60

Lemma 4.2.2.(T ∗M × I, ω) is a presymplectic manifold with

ker(m,t)ω = span(((XH)m, 1)

)Hence the integral curves of the symplectic gradient XH on T ∗E are the images of the leaves ofT ∗E × I under projection on the first factor.

Proof:In local coordinates (pi, q

i, t) on T ∗E × I we find

dΘ = dp ∧ dq − ∂H

∂pidpi ∧ dt− ∂H

∂qidqidt .

The corresponding antisymmetric matrix reads

A =

0 −En×n ∂H∂pi

En×n 0 ∂H∂qi

− ∂H∂pi

− ∂H∂qi 0

It has obviously rank at least 2n. On the other hand, the vector(

−∂H

∂qi,∂H

∂pi, 1

)= (XH, 1)

is in the kernel of A, hence the rank equals 2n.

One can even show that the space of leaves

T ∗E × I/ker ω

is a symplectic manifold and isomorphic, as a symplectic manifold, to the cotangent space(T ∗E, ω). The cotangent bundle can thus be interpreted as the space of trajectories or classicalsolutions. Following Souriau, we call it the space of motions.

More generally, we define:

Definition 4.2.3

1. A Hamiltonian system consists of a presymplectic manifold (M,ω), called evolution spacewith one-dimensional leaves and a smooth space of leaves, called the phase space of thesystem, together with a smooth function

H : M → R ,

called the Hamiltonian function. A leaf of the foliation is also called a trajectory. The

images of trajectories in evolution space M in phase space M are called (Hamiltonian)trajectories.

2. A time-independent Hamiltonian system consists of a symplectic manifold (M, ω), to-gether with a smooth function

H : M → R ,

also called the Hamiltonian function.

61

Lemma 4.2.4.1. Let (M, ω, H) be a time-independent Hamiltonian system. Then for any interval I ⊂ R,

one has a Hamiltonian system with evolution space I × M , where the presymplectic formω := pr∗2ω is the pullback under the

pr2 : M → M ;

and the Hamiltonian is H := pr∗2 H.

2. Then the symplectic manifold (M, ω) is the phase space of the Hamiltonian system(M,ω,H). Hamiltonian motions are then integral curves of the symplectic gradient XH .

Let us discuss an important example of Hamiltonian systems:

Example 4.2.5.A (time-independent) natural Hamiltonian system consists of the cotangent space M = T ∗E

of a smooth manifold E with its canonical symplectic structure, together with a Hamiltonianfunction H : T ∗E → R that is the sum of a positive definite quadratic form 〈−,−〉 on thefibres of T ∗E and the pull back of a smooth function

V : E → R

under the projection π : T ∗E → E. In local Darboux coordinates:

h(p, q) =1

2〈p, p〉+ V (q) .

We still have to understand how to relate for natural systems a Lagrangian function

l : TE → R

to a Hamiltonian functionH : T ∗E → R .

Definition 4.2.6Let I ⊂ R be an interval and f : I → R be a piecewise smooth continuous convex function, i.e.

f(tx+ (1− t)y) 6 t f(x) + (1− t) f(y)

for all x, y ∈ I and all t ∈ [0, 1]. This implies that f ′′(x) > 0, wherever the second derivative isdefined.

Consider the functionF : R× I → R

(p, x) 7→ px− f(x) .

for a fixed p ∈ R. Suppose maxx∈IF (p, x) exists. The fact that f is convex implies that it isassumed for a unique x = x(p) ∈ I. In case f is differentiable in x, we find f ′(x) = p. Thisdefines implicitly x as a function of p.

This allows us to introduce a function of P by

g(p) := F (p, x(p)) = maxx∈IF (p, x).

The function g is called the Legendre transform of f .

62

Remarks 4.2.7.

(i) Consider as an example the function f(x) = mxα

αon R with m > 0 and α > 1. Then the

function

F (p, x) = px−mxα

α

has extrema for p = mxα−1. Hence we find for the Legendre transform

g(p) = m− 1α−1

pβ

βwith

1

α+

1

β= 1.

As special cases, we find for α = 2

f(x) =m

2x2 with Legendre transform g(p) =

p2

2m

and for m = 1

f(x) =xα

αwith Legendre transform g(p) =

pβ

β,

1

α+

1

β= 1

(ii) The definition via a maximum implies the inequality

F (x, p) = xp− f(x) 6 g(p)

for all values of x, p where the expression makes sense. We rewrite this as the so-calledYoung’s inequality

px 6 f(x) + g(p).

Applying this to the second example just discussed, we find the classical inequality

px 61

αxα +

1

βxβ with

1

α+

1

β= 1.

(iii) If f is a smooth function with f ′′ > 0, then its Legendre transform g is again a convexfunction. It turns out that the Legendre transform is an involution.

(iv) We mention the following interpretation: the enveloping curve of the family of lines y =px − g(p), with g(p) a convex function, has the equation y = f(x), where f and g arerelated by a Legendre transform.

(v) We finally mention the generalization to finite-dimension R-vector spaces. Let f : V → Rbe a convex (smooth) function, i.e. the Hessian ∂2f

∂xi∂xj is positive definite. Then defined afunction on the dual space

g : V ∗ → R

by

F : V ∗ × V → RF (p, q) = 〈p, q〉 − f(q)

and put g(p) = maxx∈V F (p, x). All results, including Young’s inequality, generalize.

63

(vi) It is instructive to specialize to a positive definite quadratic form

f : V → R

with f(q) = 12Aijq

iqj. Minimizing

F (p, q) = 〈p, q〉 − f(q)

for a fixed p yields pi = Aijqj and thus qi = (A−1)ijpi and thus

g(p) = pi(A−1)ijpj −

1

2(A−1)ijpipj =

1

2(A−1)ijpipj .

First explain 4.2.3 – 4.2.5.Next, we relate the Lagrangian and the Hamiltonian formulation for natural systems.

Observation 4.2.8.

1. Let E be a smooth manifold and I ⊂ R an interval. Consider a natural mechanicalLagrangian system with Lagrangian function

l : TE × I → R

leading to second order equations of motions. Fix local coordinates qi on E. Then, locally,we have

l = l(qi, qit, t)

2. On the other hand, consider the cotangent bundle T ∗E of E with its canonical symplecticstructure as the phase space of a natural Hamiltonian system and the manifold T ∗E × Iwith its canonical presymplectic structure as the evolution space. Local coordinates on Ethen give local Darboux coordinates (qi, pi) on the symplectic manifold T ∗E and on thepresymplectic manifold I × T ∗E A (time dependent) Hamiltonian function

H : T ∗E × I → R

is then given in local Darboux coordinates as

H = H(p, q, t) .

Theorem 4.2.9.Suppose that the Lagrangian function l is for fixed q ∈ M convex in its second argumentqt ∈ Tq(t)M . Suppose that the Lagrangian function l and the Hamiltonian function H are relatedfibrewise by a Legendre transform. In local coordinates:

H(p, q, t) = pqt − l(q, qt, t)

with qt = qt(p, q, t) as defined by the Legendre transform and

l(q, qt, t) = pqt − H(pi, qi, t)

with p = p(q, qt, t) respectively.

64

1. Ifs : I → E × I

is a motion for the Lagrangian system given by l, i.e. a solution of the Euler-Lagrangeequations, then

s : I → T ∗E × It 7→ (ds(t), t) ∈ T ∗E × I

is a motion for the Hamiltonian system given by H.

2. Conversely, ifs : I → T ∗E × I

is a motion for the Hamiltonian system given by H, then using the projection π : T ∗E → Eof the cotangent bundle, we obtain a solution

s := ((π × idI) s, id) : I → E × I

of the Lagrangian system given by l.

These statements are sometimes summarized that, if a Lagrangian function l and a Hamil-tonian function H are related by a Legendre transform, then the Euler-Lagrange equations

D∂l

∂qit=

∂l

∂qi

are equivalent to the Hamiltonian equations

pi = −∂H

∂qi, qi =

∂H

∂pi,

i.e. 2n ordinary differential equations of first order, where is the Legendre transform of l.

Proof:The total derivative of the time-dependent Hamiltonian H is the one-form

dH =∂H

∂pdp+

∂H

∂qdq +

∂H

∂tdt ∈ Ω1(T ∗E × I)

which we compute explicitly forH = pqt − l(q, qt, t)

where qt = qt(p, q, t) is defined implicitly by the Legendre transform as a function on evolutionspace T ∗E × I. Using the identity pi = ∂l

∂qit

of local functions on evolution space implied by the

Legendre transform, we find

d(pqt − l(q, qt, t)) =

= qtdp+(p∂qt∂p− ∂l

∂qt

∂qt∂p︸ ︷︷ ︸

=0

)dp− ∂l

∂qdq +

(p∂qt∂q− ∂l

∂qt

∂qt∂q︸ ︷︷ ︸

=0

)dq − ∂l

∂tdt

= qtdp−∂l

∂qdq − ∂l

∂tdt (10)

65

By comparison of coefficients, we find the following identities of locally defined functions onT ∗E:

qit =∂H

∂pi,

∂H

∂qi= − ∂l

∂qi,

∂H

∂t= −∂l

∂t

Suppose, s : I → E × I is a Lagrangian motion. Then the first equation becomes

qi =∂H

∂pi.

The Euler-Lagrange equations read

∂l

∂q s = D

∂l

∂qt s =

d

dt

(∂l

∂qt s)

.

Substituting the second equation yields

pi = −∂H

∂qi

so that we have obtained the Hamiltonian equations.From the Euler Lagrange equations ∂l

∂q= D ∂l

∂q= Dpt we now derive Hamilton’s equations.

The second part of the assertion is shown analogously.

Observation 4.2.10.

• The following observation is formulated for a Hamiltonian system with phase space (M, ω)and local Darboux coordinates (p, q) and evolution space M := M × I with local Darbouxcoordinates (p, q, t). On phase space M , we have the symplectic form which is locally

ω =∑i

dpi ∧ dqi = d

(∑i

pidqi

)= dΘ ;

on the evolution space M = I × M , we have the presymplectic form which is locally

ω = p∗1ω − p∗1dH ∧ p∗2dt

= d

(∑i

pidqi − Hdt

)= dΘ .

• Motions in M are solutions of the Hamiltonian equations

x = x,H with x(t1) = (p1, q1).

They lead to a phase flow which we assume to be globally defined. It gives us diffeomor-phisms of M

gt2t1 : M → M

for any pair t1, t2 ∈ I. On evolution space M , we obtain parametrizations of the leaves:

t 7→(gtt1(x), t

).

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Proposition 4.2.11.Let γ1 and γ2 be two curves in evolution space M that encircle the same leaves. Then we have∮

γ1

Θ =

∮γ2

Θ.

The one-form Θ is called Poincare-Cartan integral invariant.

Proof:Let Σ be the the surface formed by those parts of the leaves intersecting γ1 (and thus γ2) thatis bounded by the curves, ∂Σ = γ2− γ1. Such a surface is sometimes called a flux tube. Stokes’theorem implies ∮

γ2

Θ−∮γ1

Θ =

∫∂Σ

Θ =

∫Σ

dΘ =

∫Σ

ω = 0,

since Σ consists of leaves whose tangent space is by definition the kernel ker ω of the presym-plectic form.

Observation 4.2.12.We now specialize to curves in evolution space M at constant time, γi ∈ p−1

2 (ti), to find∮γ1

p dq =

∮γ2

p dq

We can obtain such a pair of curves from a closed curve γ in phase space M parametrizedby s ∈ [0, 1] and first lifting γ to a curve

γ1(s) =(γ(s), t1

)parametrized by s ∈ [0, 1] at fixed time t1 ∈ I and and then applying a phase flow to obtain aclosed curve

γ2(s) =(gt2t1γ(s), t2

)in evolution space at another fixed time t2.

Using the surface

Σ =(gtt1γ(s), t

), s ∈ [0, 1] , t ∈ [t1, t2]

formed by the leaves, we obtain by proposition 4.2.11 the equality∮

γ

p dq =

∮g

t2t1γ

p dq

of integrals along curves in phase space M .As a consequence, we find for any surface Σ ∈ M in phase space for the integral over the

symplectic form ∫Σ

ω =

∫g

t2t1

Σ

ω.

The symplectic form ω on phase space is thus invariant under the phase flow. We conclude thatthe maps gt2t1 : M →M describing phase flow are canonical transformations.

The phase flow has as further invariants the power ωk of the symplectic form. This includesas a special case the canonical volume form ωdimM/2 of the symplectic manifold (M, ω).

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We need the following simple

Lemma 4.2.13.Let M be a manifold with a volume form and g : M → M be a volume preserving map that ismapping a measurable subset D ⊂ M of finite volume to itself. Given a point p ∈ D, we canfind in any neighborhood U of p a point x ∈ U which comes back to the neighborhood U , i.e.there is n ∈ N such that gnx ∈ U .

Proof:The images U, gU, . . . , gnU, . . . have all the same finite volume and are contained in the subsetD of finite volume. Hence they cannot be disjoint. Hence there are integers k and l with k > lsuch that

gkU ∩ glU 6= ∅ .

We deduce gk−lU ∩ U 6= ∅.

We deduce the following famous statement:

Proposition 4.2.14 (Poincare’s recurrence theorem).Consider a (time-independent) natural Hamiltonian system consisting of a phase space(T ∗E, ω0) and a Hamiltonian function

h : M → R(p, q) 7→ 〈p,p〉

2m+ V (q)

Then the associated phase flowg : I × M → M

has the property that in any neighborhood U of any point p ∈ M we can find a point x ∈ Uwhich returns to U for sufficiently large time t, g(t, x) ∈ U .

Proof:Consider the subset

D = (p, q) | H(p, q) 6 const

of finite volume. Since phase flow preserves the symplectic volume, we can apply the lemma.

5 Quantum mechanics

5.1 Deformations

We start by considering the mathematical notion of a formal deformation of an associativealgebra.

Definition 5.1.1Let k be a field of characteristic 0 and A an associative k-algebra with unit.

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(i) An algebra homomorphismε : A→ k

is called an augmentation, if ε(λ · 1A) = λ holds for all λ ∈ k. An augmentation endowsthe ground field k with the structure of an A-bimodule by

A× k → k(a, λ) 7→ ε(a · λ)k × A → k(λ, a) 7→ ε(λ · α) .

For example, the ring k [[t]] of formal power series with coefficients in k admits the aug-mentation

ε : k [[t]]→ k

ε

(∑i∈N0

aiti

):= a0.

It can be shown that this ring is a local Noetherian ring.

(ii) A formal deformation of an associative algebra A is an associative k [[t]]-algebra B togetherwith an isomorphism of k-algebras

α : B := k ⊗k[[t]] B → A .

A morphism of formal deformations (B′, α′ : B′ → A) and (B′′, α′′ : B′′ → A) is amorphism

Φ : B′ → B′′

of k [[t]]-algebras such that(α′′)−1 α′ = idk ⊗k[[t]] Φ .

Isomorphic deformations are also called equivalent.

Theorem 5.1.2.A formal deformation of a k-algebra A is equivalent to a family of k-linear maps

µi : A⊗ A→ Ai∈N

with µ0(a, b) = a · b and ∑i+j=ki,j>0

µi (µj(a, b), c) =∑

i+j=ki,j>0

µi (a, µj(b, c))

for a, b ∈ A.

Indeed, we can use these maps to endow the free k [[t]]-module B := k [[t]] ⊗ A with theproduct

a ∗ b := a · b+ tµ1(a, b) + t2µ2(a, b) + ... for all a, b ∈ Awhich we extend in a k [[t]]-linear way.

The associative product on k [[t]] ⊗ A is also called a star product. The following lemmarelates Poisson algebra and formal deformations:

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Lemma 5.1.3.Let A be a commutative associative k-algebra. Then for any formal deformation of A, the k-bilinear map

a, b := µ1(a, b)− µ1(b, a) for all a, b ∈ A,endows A together with its commutative product with the structure of a Poisson algebra.

Proof:As we have seen in example 4.1.7 for any associative algebra, the commutator

[x, y] = x ∗ y − y ∗ x

endows the associative algebra k [[t]] ⊗ A with the structure of a Poisson algebra. A simplecalculation then shows due to commutativity of the product in A, the first non-vanishing termis the bracket −,−:

[x, y] = x, yt+O(t2) .

The Jacobi identity follows from the identity

[x, [y, z]] = x, y, zt2 +O(t3) .

Similarly, one shows that −,− is a derivation for the commutative product of A.

This motivates the following

Definition 5.1.4

(i) Given a formal deformation (k [[t]]⊗ A, ∗) of an associative commutative algebra A, thePoisson algebra (A, ·, ) = P is called the classical limit of (k [[t]]⊗ A, ∗).

(ii) Conversely, given a Poisson algebra P = (A, ·, ), a k [[t]]-algebra B is called adeformation quantization of P if the classical limit of B is isomorphic to the Poissonalgebra P .

(iii) A deformation quantization of a Poisson manifold M is a deformation quantization of thePoisson algebra (C∞(M), ·, −,−) in which all maps µi are differential operators.

Theorem 5.1.5. (Kontsevich 1997)Any Poisson manifold admits a deformation quantization that is essentially unique.

Observation 5.1.6.(a) The phase space of a Hamiltonian system is a Poisson manifold. The Poisson structure can

be interpreted as a hint to the existence of a family of associative algebras (which in generalare not commutative any longer). This idea is at the basis of quantum mechanics.

(b) The Poisson bracket on a phase space is in local Darboux coordinates

f, g =∂f

∂qi∂g

∂pi− ∂f

∂pi

∂g

∂qi.

Hence it has dimension [length×momentum]−1 = action−1. As a consequence, the de-formation parameter t has to be a dimensionfull quantity, t = ~ with the dimension of anaction, the Planck constant. It has the value of 1.054·10−34Js and is one of the fundamentalconstants of nature.

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(c) A dimensionfull quantity like ~ is neither big not small per se. Only its ratio with a quantityof the same dimension, i.e. a characteristic action of a system can be small or big.

(d) While ~ has a definite value, there are no evaluation homomorphisms for a formal defor-mations for other values of the deformation parameter than zero. Hence we have to look fora different framework that allows to formulate convergence properties: normed algebras.

5.2 C∗-algebras and states

Definition 5.2.1A (unital) C∗-algebra A is a complex (unital) associative algebra

• which is a Banach algebra, i.e. it is endowed with a norm ‖·‖ : A→ R>0 such that (A, ‖ · ‖)is a complete topological vector space and such that the norm is submultiplicative

‖a · b‖ ≤ ‖a‖ · ‖b‖ .

• with an involution generalizing complex conjugation and hermitian conjugation

∗ : A→ A

a 7→ a∗,

which is an antilinear antihomomorphism of algebras:

(λa+ µb)∗ = λa∗ + µb∗ fur alle λ, µ ∈ C, a, b ∈ A

(ab)∗ = b∗a∗ fur alle a, b ∈ A

• The following identity holds

‖a∗a‖ = ‖a‖2 for all a ∈ A.

Remarks 5.2.2.

(a) The identity ‖a∗a‖ = ‖a‖2 implies

‖1‖ = 1, ‖a∗‖ = ‖a‖, ‖a1 · a2‖ 6 ‖a1‖ · ‖a2‖.

(b) Elements such that a∗ = a are called self-adjoint. They form a real subspace, but in generalnot an associative subalgebra, but only a Lie subalgebra.

(c) Let H be a separable Hilbert space, i.e. a complete unitary vector space with a topologicalbasis that is at most countable. The scalar product 〈−,−〉 on a Hilbert space H will be inour conventions always linear in the first argument and antilinear A linear map

A : H → H

is called bounded, if‖A‖ := sup‖x‖=1‖Ax‖ <∞

The space of bounded operators on a Hilbert space is a Banach algebra B(H).

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(d) From Riesz representation theorem we deduce that for given A ∈ B(H) and v ∈ H thecontinuous linear function

〈A·, v〉 : H → C

can be represented by a unique vector A∗v ∈ H:

〈A·, v〉 = 〈·, A∗v〉 for all v ∈ H.

It is not hard to show that A∗ ∈ B(H) and that the Banach algebra B(H) is endowed bythe map A 7→ A∗ with the structure of a C∗-algebra.

(e) A theorem of Gelfand and Naimark asserts that every C∗-algebra is isomorphic to a closed∗-subalgebra of B(H).

The idea is that a C∗-algebra serves as an algebra of observables, as a “quantization” of thePoisson algebra of functions on (extended) phase space. Quantum mechanics is an inherentlyprobabilistic theory. Hence we want to associate to any observable at least an expectation value.This motivates the following definition:

Definition 5.2.3Let A be a C∗-algebra. A state ω on A is a normed positive linear functional

ω : A→ C,

i.e. we have ω ∈ A∗ and positivity and a normalization condition,

ω(a∗a) > 0 and ω(1) = 1 .

Remarks 5.2.4.

(a) The set of all states is convex: if ω1 and ω2 are states, then for all t ∈ [0, 1] the convexlinear combination

ωt = tω1 + (1− t)ω2

is a state.

(b) A point x of a convex subset X of a real vector space V is called extremal, if for everysegment yz ⊂ X containing x one has either y = x or z = x.

(c) Extremal states on a C∗-algebra are called pure states.

(d) Such states exist: according to the Krein-Mil’man theorem every compact convex subset ofa locally convex vector space equals the closed convex hull of its extremal points.

Examples 5.2.5.

1. The complex valued continuous functions on R

C0(R) =f : R→ C continuous | for all ε > 0 existsN = N(f),

such that |f(λ)| < ε for all |λ| > N

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form a commutative algebra without unit. Taking complex conjugation as an involutionand the supremum as a norm, we endow it with the structure of a C∗-algebra.

Since the algebra C0(R) is not unital, we need to modify the normalization condition. Astate µ is called normalizable, if there is k ∈ R such that

|µ(f)| 6 k‖f‖

for all f . We define ‖µ‖ as the infimum of such k. A state is called normalized, if ‖µ‖ = 1.Thus states are normalized Radon measures, i.e. normalized probability measures.

The pure states of C0(R) are Dirac measures, i.e. evaluations at some x ∈ R,

δx(f) = f(x) .

They are in bijection to points in R.

2. A second example is the C∗-algebra A = B(H) of bounded operators on a Hilbert spaceH: an endomorphism ρ of H is called a density matrix, if of all a ∈ A the endomorphismρ a ∈ B(H) is trace class. In this case,

ωρ(a) =Tr(ρa)

Tr(ρ)

is a state on B(H). Taking for ρ the orthogonal projection on a unit vector ψ ∈ H, weobtain a pure state

ωρ(a) = (aψ, ψ) .

We next wish to endow the continuous dual of a Banach space with a topology. Two topolo-gies are of particular importance:

Remark 5.2.6.

1. Let A be a complex Banach space and A∗ its continuous dual, i.e. the continuous linearfunctionals. It can be endowed with the structure of a Banach space by the operator norm

‖µ‖ := sup a∈A

‖a‖=1|µ(a)| .

2. There is a second important topology on the continuous dual A∗, the weak ∗ topology ortopology of pointwise convergence. It is generated by the following collection of subsets ofA∗: for µ0 ∈ A∗, a ∈ A and δ > 0 we set

Uδ(µ0; a) :=µ ∈ A

∣∣ |µ(a)− µ0(a)| < δ.

Definition 5.2.7Consider a C∗ algebra A.

(i) An element µ ∈ A∗ is called a character, if µ 6= 0 and

µ(ab) = µ(a)µ(b) for all a, b ∈ A .

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(ii) The subset M(A) ⊂ A∗ of all characters of A together with the topology induced by theweak ∗-topology on A∗ is called the Gelfand spectrum of the C∗-algebra A.

Example 5.2.8.Let X be a locally compact Hausdorff space and A = C0(X) be the algebra of continuousfunctions on X. Then for every point p ∈ X, the evaluation map f 7→ f(p) is a character.Moreover, the map

X → M(C0(X))p 7→ µp(f) = f(p)

that associates to every point p the character corresponding to evaluation in p is injective andcontinuous.

Proof:

• To see injectivity, consider two different points p 6= q. Since f is a locally compact Haus-dorff space, we can find a continuous function f ∈ C0(X) with f(p) 6= f(q). The inequality

µp(f) = f(p) 6= f(q) = µq(f)

implies that the characters are different, µp 6= µq.

• Consider a convergent sequence pi → p in X. Then for every f ∈ C0(X), continuity of fimplies

µpi(f) = f(pi)→ f(p) = µp(f) ,

which is just pointwise convergence µpi→ µp in A∗, i.e. convergence in the weak ∗-

topology.

Definition 5.2.9Let A be a C∗-algebra. For any element a ∈ A consider the following function on the Gelfandspectrum of A:

a : M(A) → Cµ 7→ µ(a)

The map that associates to an element of A a function on the spectrum

G : A → C0(M(A))a 7→ a

is called the Gelfand transform.

We quote the following theorem:

Theorem 5.2.10 (Gelfand-Naimark).Let A be a commutative C∗-algebra. Then the Gelfand transform G is an isometric ∗-algebraisomorphism.

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Remarks 5.2.11.

(a) A commutative C∗-algebra is thus in a natural way isomorphic to the algebra of coninuousfunctions on a topological space that is encoded in the algebra, the Gelfand spectrum. Thepoint of view to see C∗-algebras as generalizations algebras of functions is the starting pointfor non-commutative geometry.

(b) The pure states on a commutative C∗-algebra are its characters. Hence any commutativeC∗-algebra is in a natural way isomorphic to the C∗-algebra of continuous functions on itspure states.

(c) The Gelfand spectrum M(A) is compact if and only of the C∗ algebra is unital.

Definition 5.2.12

(i) A ∗-representation of a C∗-algebra A on a Hilbert space H is a ∗-preserving ring homo-morphism

π : A→ B(H) .

If the algebra A is unital, the map π is required to preserve the unit element. Otherwise,one requires the subspace

π(x)ξ∣∣ for all x ∈ A, ξ ∈ H

⊂ H

to be dense in H.

(ii) A vector ξ ∈ H in a ∗-representation π : A→ B(H) is called cyclic, if the subspace

π(x)ξ | x ∈ A

is dense in H. If a cyclic vector exists, the representation is called cyclic.

Remarks 5.2.13.

(a) Every non-zero vector in an irreducible representation is cyclic.

(b) In contrast, an arbitrary non-zero vector in a cyclic representation is not necessarily cyclic.

Definition 5.2.14Consider a C∗-algebra A and a ∗-representation π : A→ B(H).

(i) For every non-zero vector ψ ∈ H \ 0 the function

ωψ(a) :=〈π(a)ψ, ψ〉〈ψ, ψ〉

a ∈ A

is a state on A. Non-zero vectors in the same one-dimensional subspace of H give thesame state. Such a state is called a ray state.

(ii) For every density matrix ρ on H the function

ωρ(a) = Tr ρπ(a) a ∈ A

is a state on A. Such states are called normal states.

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We will see that all states on a C∗-algebra can be obtained this way:

Theorem 5.2.15 (Gelfand-Naimark-Segal).

(i) Let ω be a state on a unital C∗-algebra A. Then there is a ∗-representation πω of A witha cyclic vector ξ ∈ H such that

ω(a) = 〈πω(a)ξ, ξ〉

holds for all a ∈ A. This representation is unique up to unitary equivalence.

(ii) The representation πω is irreducible, if and only if ω is a pure state.

Proof:We only sketch the first part and restrict to the case of unital algebras. We first endow A witha degenerate hermitian product

〈x, y〉 := ω(x · y∗) for all x, y ∈ A.

ThenIω := a ∈ A | ω(a∗a) = 0

turns out to be a left ideal in A. The quotient A/Iω has a natural non-degenerate hermitianproduct which endows it with the structure of a pre-Hilbert space. We consider its completion

Hω := A/Iω .

This gives a ∗-representation of A with cyclic vector 1 ∈ Hω.

We need a final piece of mathematical theory to discuss quantum mechanics:

Definition 5.2.16Let A be a unital C∗-algebra. Denote by B(R) the sigma-algebra of Borel subsets of R. A(normalized) projector-valued measure on R with values in A is a mapping

P : B(R)→ A

satisfying the following axioms:

1. The measure is projector valued: For every Borel subset E ⊂ R, P (E) is an orthogonalprojection, i.e.

P (E) = P (E)2 and P (E) = P (E)∗ .

2. The measure is normalized: One has

P (∅) = 0 and P (R) = 1 .

3. Additivity: for every countable disjoint union E = t∞n=1En of Borel subsets with pairwiseempty intersection, one has

P (E) = limn→∞

∞∑i=1

P (Ei) .

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Remarks 5.2.17.

1. To every projector-valued measure one can associate a projector valued resolution of theidentity. This is the projector-valued function P : R→ A:

P (λ) := P ((−∞, λ))

with the properties

P (λ)P (µ) = P (minλ, µ)limλ→−∞ P (λ) = 0 and limλ→∞ P (λ) = 1

limµ→λ− P (µ) = P (λ)

2. Suppose, we are given a state ρ on A. Then ρ(P (λ) defines a probability measure.

In the special case when A is the algebra of bounded operators on a separable Hilbert spaceH, A = B(H), for any ψ ∈ H \ 0 the distribution function

Pψ(λ) = 〈P (λ)ψ, ψ〉

defines a bounded measure on R. It is a probability measure, if |ψ| = 1. More generally,for any pair of vectors ϕ, ψ ∈ H\0, the function 〈P (λ)ψ, ϕ〉 defines a complex measureon R.

In quantum mechanics, unbounded linear operators on a separable Hilbert space H areomnipresent. Their range of definition is only a dense subspace of H. There is a notion of self-adjointness for such operators. If A is bounded, it is an element of the C∗-algebra B(H), andself-adjointness means A∗ = A. One should be aware of the fact that the C∗ algebra B(H) doesnot contain unbounded operators.

We introduce the notion of the spectrum of an operator:

Definition 5.2.18

1. Let A be a linear operator on a separable Hilbert space H. The spectrum of A is definedas

Spec(A) =λ ∈ C

∣∣ A− λ1 has no bounded inverse.

2. Let a be a self-adjoint element of an abstract C∗ algebra A. The spectrum of a is definedas

Spec(a) =λ ∈ C

∣∣ a− λ1 not invertible in A.

Remarks 5.2.19.

1. Both definitions agree for elements of the C∗-algebra B(H) of bounded operators on aseparable Hilbert space H.

2. Any eigenvalue of an operator A on the Hilbert space H is an element of the spectrumspecA. The collection of eigenvalues is also called the point spectrum. If H is infinite-dimensional, the point spectrum is, in general, a proper subset of the spectrum.

We are now ready to formulate von Neumann’s spectral theorem for self-adjoint operatorson a separable Hilbert space.

77

Theorem 5.2.20.For every self-adjoint operator A on a Hilbert space H, there exists a unique projector-valuedresolution P (λ) of the identity satisfying the following properties:

• (Spectral decomposition)The range of definition of A can be characterized as

D(A) = ϕ ∈ H :

∫ ∞

−∞λ2d〈P (λ)ϕ, ϕ〉 <∞

and for every ϕ ∈ D(A)

Aϕ =

∫ ∞

−∞λdP (λ)ϕ

defined as a limit of Riemann-Stieltjes sums. The support of the spectral measure dP (λ)equals the spectrum Spec(A) of A.

• (Functional calculus)For every continuous function f on R, f(A) is a linear operator on H with dense domain

D(f(A)) = ϕ ∈ H :

∫ ∞

−∞|f(λ)|2d〈P (λ)ϕ, ϕ〉 <∞

and for every ϕ ∈ D(f(A))

f(A)ϕ =

∫ ∞

−∞f(λ)dP (λ)ϕ

One has f(A)∗ = f(A) with the complex conjugate function. The operator f(A) is bounded,iff the continuous function f is bounded on the spectrum Spec(A). On bounded functions,the map f 7→ f(A) is an isomorphism of commutative C∗-algebras. In a sense, an un-bounded self-adjoint operator A on a Hilbert space encodes such an isomorphism.

There is also a version of the theorem for a self-adjoint element a of an abstract C∗-algebraA. Again, there is a projector-valued measure P (λ) on Spec(a) such that for any continuousfunction f on Spec(a), one has

f(a) =

∫Spec(a)

f(λ)dP (λ) .

In particular,

a =

∫Spec(a)

λdP (λ) .

For any state ρ, one obtains

ρ(a) =

∫Spec(a)

λρ(dP (λ))

with a probability measure ρ(dP (λ)) on Spec(a).

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5.3 Strong quantizations

Definition 5.3.1Consider a unital C∗-algebra A ⊂ B(H). We endow A with a 1-parameter family of Lie brackets

1

i~[a, b] =: a, b~ ∈ R∗

parametrized by ~ ∈ R∗. These Lie brackets preserve the real subspace x |x = x∗ ⊂ A ofself-adjoint elements and endow it with the structure of a real Lie algebra.

Let (A, ·, ·) be a commutative unital real Poisson algebra. A full quantization of A is aC∗-algebra A ⊂ B(H), together with a homomorphism of real Lie algebras

Q : (A, ·, ·)→ (A, ·, ·~)

for some real value of ~ such that

(i) Q preserves the unit, Q(1) = 1.

(ii) The Lie-algebra representation of A on H induced by Q(A) ⊂ B(H) is irreducible.

Remarks 5.3.2.

(a) We only require compatibility of Q with the Lie algebra structure on A, not with the com-mutative product.

(b) It can be helpful to admit unbounded operators with dense domain on the Hilbert space H inthe image of Q. In this case, the image of A is required to be contained in the real subvectorspace in the symmetric operators on H.

In the case of a full quantization of the Poisson algebra of smooth functions on a symplecticmanifold M , the images of smooth function f ∈ C∞(M) whose symplectic gradient Xf isnot completely integrable as a vector field on M take values in unbounded operators.

In this case, one adds the following further technical requirement: suppose the vector fieldsfor two smooth functions f, g ∈ C∞(M) are completely integrable and one has f, g = 0.One says that the two functions f and g are “in involution”. Then the two operators Q(f)and Q(g) are required to commute in the strong sense that all spectral projectors commute.

We are now ready to formulate some axioms of a quantum mechanical system:

Definition 5.3.3

1. The first datum for a quantum system consists of a C∗-algebra A. The self-adjoint elementsof A are called observables. A state of a quantum system is a state of A.

2. Given a quantum mechanical system A in a state ρ, the result of a measurement of anobservable a ∈ A cannot be predicted. The possible results of the measurements at agiven time t are given by the spectrum Spec(a) ⊂ R and quantum mechanics predicts theprobability measure

µa,ρ = ρ(dPa(λ))

79

for the outcome. In particular, the expectation value equals

〈a〉ρ =

∫Spec(a)

λρ(dPa(λ)) = ρ(a)

and the variance

∆(a) = 〈(a− 〈a〉)2〉 = 〈a2〉 − (〈a〉)2 = ρ(a2)− ρ(a)2 .

3. The measurement process influences the state: if the value a0 ∈ spec(a) for an observablea has been measured, the system is in a state where the osbervable a has variance zero.

4. One says that a finite set of observables ~A := A1, A2, . . . , An can be measured simulta-neously, if they commute pairwise. In this case, there is a a projector valued measure onRn given by

P ~A(E1 × E2 × . . .× En) = PA1(E1) · PA2(E2) · . . . · PAn(En)

Then there can exist vectors in a ∗-representation and thus states which are simultaneouseigenstates so that the predictions with variance zero are possible for all observablessimultaneously.

Non-commuting observables a1, a2can be measured simultaneously; otherwise Heisen-berg’s uncertainty relation would be an unobservable statement. Every single measure-ment itself still produces a “sharp” results, i.e. a pair of values in (spec(a1), spec(a2)).

5. The dynamics of the quantum mechanical system is given by a strongly continuous one-parameter group U(t) of unitary operators acting on observables as

a(t) = U(t)−1aU(t) .

The idea is thus to implement time evolution by inner automorphisms of A. States aretime-independent. This induces a time-dependent family of probability measures for eachpair (a, ρ) with a an observable and ρ a state on A.

If we realize the C∗-algebra A as a subalgebra of B(H), there is an “infinitesi-mal description” in terms of an unbounded self-adjoint operator H on H, calledthe Hamiltonian or Hamilton-operator such that any observable a ∈ A obeys theHeisenberg equation of motion

d

dta(t) =

i

~[H, a] .

Heuristically, we see the observable H as the image of the Hamilton function h under astrong quantization, H = Q(h).

Proposition 5.3.4 (Heisenberg’s uncertainty relations).Given a quantum mechanical system A and observables a, b ∈ A and a state ρ of A, the followingrelation for the variance holds:

∆ρ(a)∆ρ(b) >1

2

∣∣ρ([a, b])∣∣ .80

In the special case when a = q, b = p with [a, b] = i~1, i.e. when a and b obey canonicalcommutation relations, we find

∆(q)∆(p) >~2.

Hence there is no state in which position and momentum of a free system have variance zero.

Proof:

We decompose the product ab into the sum of an anti-commutator a, b := ab+ ba and acommutator:

ab =1

2(ab+ ba) +

1

2(ab− ba) =

1

2a, b+

1

2[a, b] .

If a and b are self-adjoint, the anti-commutator is self-adjoint, the commutator is anti self-adjoint. We conclude for the absolute value

a, b = ab+ ba = a∗b∗ + b∗a∗ = (ba)∗ + (ab)∗ = a, b∗

[a, b] = − [b, a]∗ .

Hence a state ρ yields real values on the anticommutator and purely imaginary values on thecommutator: ∣∣ρ(ab)∣∣2 =

1

4ρ(a, b)2 +

1

4

∣∣ρ([a, b])∣∣2.We find an upper bound for this expression using a GNS represention:∣∣ρ(ab)∣∣2 =

∣∣〈abψ, ψ〉∣∣2 =∣∣〈bψ, aψ〉∣∣2 6

6 ‖bψ‖2 · ‖aψ‖2 = 〈b2ψ, ψ〉〈a2ψ, ψ〉 = ρ(b2)ρ(a2),

where we used the Cauchy-Schwarz inequality. Altogether we find

ρ(a2)ρ(b2) >1

4

∣∣ρ([a, b])∣∣2.By considering a := A− ρ(A)1 and b := B − ρ(B)1 and taking into account that

[a, b] = [A,B] ,

we deduce

∆(A)2∆(A)2 >1

4|ρ([A,B])|2.

Even when one admits unbounded operators, strong quantizations need not exist. We willsee that they do not exist for the Poisson algebra of smooth functions on the cotangent bundleM = T ∗Rn with its canonical symplectic structure. It is also known that the cotangent bundleT ∗S2 of the 2-sphere does not admit a strong quantization while the cotangent T ∗T 2 of the2-torus admits a strong quantization.

Observation 5.3.5.

81

1. Consider a symplectic manifold (M,ω) with local Darboux coordinates (pi, qi) with i =

1, . . . , n. A strong quantization

Q : (A, ·, ·)→ (A, ·, ·~)

provides (essentially) self-adjoint operators on H

Qi := Q(qi), Pi := Q(pi) .

They obey Heisenberg’s commutation relations or canonical commutation relations :[Qi, Qj

]= 0, [Pi, Pj] = 0

[Pj, Q

k]

= i~ δkj

Taking the trace of the last relation yields

0 = tr[Pj, Q

k]

= i~ δkj tridH = i~ δkj dimH

This implies dimH = 0; thus there are no strong quantizations on a finite-dimensionalHilbert space H.

2. It is convenient to consider the commutative n-parameter groups generated by Qi and Pjrespectively:

U(α) = eiαjPj and V (β) = eiβjQj

with α ∈ Rn. The Weyl algebra is the closed subalgebra of B(H) generated by

W (α, β) := U(α)V (β) .

For such operators Weyl’s rules read

W (α, β)W (α′, β′) = eiω((α,β),(α′,β′))W (α+ α′, β + β′)

with ω the standard symplectic form.

In particular, we obtain two unitary representations

α 7→ U(α)

β 7→ V (β)

of Rn on H.

Definition 5.3.6

1. The Schrodinger representation of the Weyl algebra is the representation defined on theHilbert space H = L2(Rn) by

[U(α)f ] (x) := f(x− α)

[V (β)f ] (x) := eiβxf(x) .

2. More generally, given a finite or infinite-dimensional Hilbert space H, we obtain ananalogous representation of the Weyl algebra on the infinite-dimensional Hilbert spaceHH = L2(Rn,H).

82

These are indeed all representations of the Weyl algebra.

Theorem 5.3.7 (Stone - von Neumann).

(i) For any continuous unitary representation

U : Rn → B(H)

V : Rn → B(H)

of the Weyl operators on a Hilbert space H, there is a Hilbert space H and a unitaryoperator

U : H → HH

providing an isomorphism of ∗-representations to the Schrodinger representation on HH

(ii) The representation of the Weyl algebra on H is irreducible, iff dimC H = 1.

Let us now consider the case of global Darboux coordinates, i.e. M = T ∗Rn with thecanonical symplectic structure. For a full quantization of the Poisson algebra for M = T ∗Rn,we can assume that the Hilbert space is HH with the Weyl operators acting on ψ ∈ L2(Rn,H)as

Q(qi)ψ(q) = qiψ(q) and Q(pi)ψ(q) = −i~ ∂qiψ(q) .

Lemma 5.3.8.For any full quantization Q of T ∗R, we have the quadratic identities

Q(q2) = (Q(q))2

Q(p2) = (Q(p))2

Q(qp) =1

2(Q(q)Q(p) +Q(p)Q(q)) .

Proof:

• We introduce the shorthand f := Q(f). We first note that for any f with f, q = 0, fhas to be of the form

fψ(q) = A(q)ψ(q) .

with A(q) a hermitian operator on H.

• In particular, q2, q = 0 implies q2 = A(q). From p, q2 = −2q, we deduce 1i~

[p, q2

]=

−2q which evaluation in the Schrodinger representation to

1

i~

[~i∂q, A(q)

]ψ(q) = −A′(q)ψ(q)

This implies A′(q) = 2q and thus

q2 = q2 − 2e−

with e− a hermitian endomorphism of H. Analogously, we obtain p2 = p2 + 2e+ with ahermitian endomorphism e+ of H.

83

• From the Poisson bracket 4pq = q2, p2, we deduce

qp =1

i~

[q2, p2

]=

1

i~[q2, p2

]− 1

i~[e−, e+] =

=1

2(qp+ pq) + h with h =

1

i~[e+, e−]

• Further calculations furnish the relations

1

i~[e+, e−] = h,

1

i~[h, e±] = ±2e±

which are just the commutation relations for three generators of the non-compact real Liealgebra sl(2,R). This Lie algebra is known not to have finite-dimensional representationswith these hermiteicity properties.

Theorem 5.3.9 (Groenewold, van Hove).No strong quantization can be extended from the Lie algebra of polynomials that are first orderin p and q to a Lie subalgebra of C∞(T ∗R) containing Polynomials in q and p which are in bothvariables of degree strictly bigger than two.

Proof:

• In general, one has for real polynomials f, g the identities

f(q) = f(q), f(p) = f(p)

and

g(q)p =1

2(g(q)p+ pg(q)) and P (g)q =

1

2(g(p)q + qg(p)) .

We only need these identities for f(x) = x3 and g(x) = x2 and restrict our proof to thesecases.

• Applying this result to the identity

1

9q3, p3 =

1

3q2p, p2q

in the Poisson algebra, we find that a strong quantization of the left hand side yields

1

9i~[q3, p3

]= q2p2 − 2i~qp− 2

3~21,

while for the right hand side we obtain

= q2p2 − 2i~qp− 1

3~21 .

The two results differ, hence we have obtained a contradiction.

84

• Since the two operators q3 and q3 commute with q, we conclude that their differenceequals a hermitian multiplication operator A(q),

q3 − q3 = A(q) .

The identity [q3, p

]= i~q3, p = 3i~q2 == 3i~q2 =

[q3, p

]implies that A(q) = A is constant,

q3 = q3 + A .

To determine the constant, we compute

q3 =1

3q3, qp =

1

3i~

[q3, qp

]==

1

3i~

[q3 + A,

1

2(qp+ pq)

]= q3 ,

hence A = 0. The identity p3 = p3 is derived analogously.

• A straightforward calculation shows

q2p =1

6q3, p2 =

1

6i~

[q3, p2

]==

1

6i~[q3, p2

]=

1

2

(q2p+ pq2

).

The expression for p2q is derived analogously.

Remarks 5.3.10.

• While the Lie subalgebra of all polynomials does not admit a strong quantization, itssubalgebra F(2) = span(1, p, q, p2, q2, pq) admits a strong quantization.

• On the other hand, the Lie subalgebra of all polynomials is larger than required for ourpurposes. Already the Lie subalgebra

F(1) = span(1, p, q) ,

the so-called Heisenberg Lie algebra contains sufficiently many variables to separate allpoints of the phase space T ∗Rn.

• Besides F(2) also the Lie subalgebra F(∞,1) = span(qi, pqi)i=0,1,... can be quantized. Thesetwo Lie algebras are the only maximal Lie subalgebras of Fpol containing the HeisenbergLie algebra F(1) .

This leads to the following definition:

Definition 5.3.11An imperfect quantization of a Poisson algebra (A, ·, ·) is an injective linear map Q : A→ A

into a unital C∗-algebra A ⊂ B(H) of bounded operators on a separable Hilbert space H suchthat

85

1. The unit of the Poisson algebra is mapped to the unit of the C∗-algebra.

2. The representation of the subalgebra generated by Q(A) on H is irreducible.

3. There exists a Lie subalgebra A′ of A such that the polynomials in A′ are dense in A andthat the restriction Q|A′ is an injection of Lie algebras.

Remarks 5.3.12.1. Imperfect quantizations exist for most physical systems.

2. For polynomials in A′ of higher order, the quantization condition only holds in the weak-ened form

Q(f, g) =1

i~[Q(f), Q(g)] +O(~2) .

5.4 Schrodinger picture and examples

Observation 5.4.1.1. So far, we have discussed the so called Heisenberg picture which we summarize as follows:

• The observables form a C∗-algebra A. The dynamics is given by a one-parametergroup U(t) of unitary operators. With a GNS construction, we can identify A with asubalgebra of the algebra B(H) of bounded operators on H. Then the time-evolutioncan be described in terms of an (in general unbounded) linear operator H by Heisen-berg’s equations

d

dta =

1

i~[a,H] +

∂a

∂t.

We will suppress explicit dependence on time in our discussion.

• States, in contrast, do not depend on time. With the GNS construction, we find aHilbert space H and a non-zero vector ψ ∈ H such that

ρ(a) = 〈a(t)ψ0, ψ0〉 .

2. The self-adjoint operator H gives a unitary 1-parameter group

U(t) = e−i~ Ht.

Consider the unitarily transformed operator

a(t) := e−i~ Ht · a(t) · e

i~ Ht .

It does not depend on time:

d

dta(t) = e−

i~ Ht

[− i

~H, a(t)

]e

i~ Ht

+ e−i~ Ht

[i

~H, a(t)

]e

i~ Ht = 0

We consider the expectation value in the state ρ at the time t and find

ρ(a(t)) = 〈a(t)ψ0, ψ0〉 = 〈U(t)a(t)U−1(t)U(t)ψ0, U(t)ψ0〉= 〈aψ(t), ψ(t)〉 .

In the last step, we introduce the time-dependent vector ψ(t) := U(t)ψ0 ∈ H.

86

3. This way, we obtain the Schrodinger picture which we summarize as follows:

• Observables are time-independent.

• (Pure) states are represented as vectors in a separable Hilbert space H. They aretime-dependent. They obey the (time-dependent) Schrodinger equation:

i~d

dtψ = Hψ.

The eigenvector equation for the Hamiltonian H

Hψ = Eψ

is also called time-independent Schrodinger equation. Indeed, every eigenvector ψE of H

HψE = EψE

provides a solution of the Schrodinger equation:

ψ(t) = e−i~ Et · ψE ,

since

i~d

dtψ(t) = i~

−i

~EψE = HψE .

The (generalized) eigenvectors of a self-adjoint operator being complete, one can describe ageneral state as a superposition of eigenstates and discuss time evolution componentwise.

We finally discuss two important examples: the harmonic oscillator and the hydrogen atom.

Example 5.4.2 (Harmonic oscillator).We discuss the harmonic oscillator in the Schrodinger picture using the Schrodinger represen-tation. The classical phase space is the cotangent space T ∗R with Hamilton function

hcl =p2

2m+

1

2mω2x2 .

The quantization in the Schrodinger representation uses the Hilbert space H = L2(R) and yieldsoperators

Q(x) = x Q(p) =~i

d

dxand Q(p2) = −~2 d2

dx2.

We thus obtain the Hamilton operator

H = − ~2

2m

d2

dx2+m

2ω2x2 =

~ω2

(−x2

0

d2

dx2+x2

x20

)where we have introduced the parameter

x0 :=

√~mω

which has the dimension of length.

87

It is helpful to introduce the operator

a :=1√2

(x0

d

dx+

x

x0

)together with its adjoint

a∗ =1√2

(−x0

d

dx+

x

x0

).

Heisenberg’s commutation relations imply the commutation relation

[a, a∗] = 1 .

The Hamiltonian now reads

H = ~ω(a∗a+

1

2

);

since the operator a∗a is positive definite, the spectrum of H is contained in Spec(H) ⊆[~ω

2,∞).

To investigate the spectrum, we first convince ourselves that the smallest possible value inthe spectrum is an eigenvalue: the eigenvalue equation

Hψ0 =1

2~ω ψ0

is equivalent to the equation aψ0 = 0 which in the Schrodinger representation gives the followingordinary differential equation

x0dψ0

dx= − x

x0

ψ0 .

It has Gaussian function

ψ0(x) = N0 exp

(− x2

2x20

)as a solution where N0 is a normalization factor.

To discuss the spectrum further, we introduce the number operator n := a∗ a which obeysthe commutation relations

[n, a∗] = a∗ and [n, a] = −a .

Suppose that ψν is an eigenvector of n to the eigenvalue ν. Then the commutation relationsimply the relations

n(a∗ψν) = a∗nψν + a∗ψν = (ν + 1)a∗ψν

n(aψ) = anψ − aψ = (ν − 1)aψ,

so that the two vectors a∗ψ and aψ, if they are non-zero, are eigenvectors of n as well andhence eigenvectors of H. This explains the name raising operator or creation operator for a∗

and lowering operator or annihilation operator for a.One now concludes that all eigenvectors of H are multiples of vectors of the form (a∗)nψ0

with ∈ Z>0. We compute them explicitly:

ψn(x) = Nn(x

x0

+d

dx

)nψ0 = NnHn

(x

x0

)exp

(− x2

2x20

)88

where Nn is a normalization factor and Hn are Hermite polynomials.One can show that the spectrum of H is a pure point spectrum so that

spec(H) =

~ω(n+

1

2

) ∣∣ n ∈ Z>0

.

Example 5.4.3 (Hydrogen atom).For a central potential V (~x) = V (r) with r := |~x| with classical Hamiltonian

hcl =1

2mp2 + V (r)

we introduce the generatorsLi = εijkxjPk

for angular momentum. From the classical Poisson bracket, we find the commutation relationsfor the operators

[Li,Lj] = i~ εijkLk .

We obtain a representation of the real Lie algebra su(2) by anti-self-adjoint operators iLi.These operators have commutation relations

[Li, pj] = i~ εijkpk[Li, xj] = i~ εijkxk

with the operators xj arising as the quantization of space coordinates and the momenta pj. Onecomputes that L2 = L2

x + L2y + L2

z is the quadratic Casimir operator in U(su(2)).Let us obtain unitary representations of the Lie algebra su(2) by considering the Hilbert space

H = L2(S2,C) of square integrable function on the two-dimensional sphere S2. The identity

S2 = SU(2)/U(1) = SO(3)

/SO(2)

shows that S2 has a natural left action. It factorizes to an action of SO(3) and is explicitlyobtained by restricting rotations of R3 to the two-sphere S2 ⊂ R3.

One can show that a Hilbert space basis is given by the countable set of functions

Yl,m with l ∈ 0, 1, 2, ..., m ∈ −l,−l + 1, ..., l − 1, l

which has eigenvalues

L2Yl,m = ~2l(l + 1)Yl,m

L3Yl,m = ~mYl,m

The functions Yl,m are called spherical harmonics. The non-negative integer l iscalled angular quantum number or orbital quantum number, the integer m is calledmagnetic quantum number.

To investigate the kinetic part of the Hamiltonian, we rewrite

p2 =L2

r2+ pr

89

where we have introduced the differential operator

pr :=~i

(∂r +

1

r

)for radial momentum. To solve the Schrodinger equation[

p2

2m+ V (r)

]ψ(r, θ, φ) = E ψ(r, θ, φ)

in radial coordinates (r, θ, φ), we make the ansatz

ψ(r, θ, φ) = R(r)Yl,m(θ, φ) ,

which yields for the function u(r) = rR(r) the ordinary differential equation[− ~2

2m

d2

dr2+

~2l(l + 1)

2mr2+ V (r)

]u(r) = Eu(r) .

It suggests to introduce the effective potential

Veff :=~2l(l + 1)

2mr2+ V (r) .

In the special case of the Coulomb potential V (r) = − e2

r, we find eigenvalues

E = −me4

2~2

z2

n2=E1

n2n = 1, 2, 3, . . .

The integer n is called principal quantum number. The range of the orbital quantum numberl$is restricted for given value n of the principal quantum number to l ∈ 0, 1, ..., n− 1.

We thus find the following eigenvectors:Principal Angular Magnetic Energy # statesquantum nr. quantum nr. quantum nr.n = 1 l = 0 m = 0 E1 1 (ground state)n = 2 l = 0 m = 0 E1/4

l = 1 m ∈ ±1, 0 E1/4 4...

...n l = 0, . . . n− 1 E1/n

2 n2

We finally present three comments:

1. The energy eigenvalue does not depend on the quantum numbers l and m, yielding adegeneracy of the eigenstates. This is specific for the Coulomb potential and can be ex-plained by the fact that for this potential the so-called Runge Lenz vector provides threemore quantities commuting with the Hamiltonian (but not with angular momentum). Inthe real hydrogen atom, this degeneracy is lifted by relativistic effects.

2. The spectrum has also a continuous part for E > 0; the corresponding generalized eigen-states are called scattering states.

3. When one measures radiative transitions in atomic spectra, one measures energy differ-ences. These are of the form ∆E = |E1|

(1n2 − 1

m2

), with integers n and m, in very good

agreement with experiment.

90

6 A glimpse to quantum field theory

We start with a very naive idea on what a quantum field is. In the simplest case, for a con-figuration space of the forms p1 : M × R → M , a classical field is a function on space time.In quantum mechanics, numbers are replaced by operators. So instead of ordinary functions,we study operator-valued functions on space time. It turns out that we also have to deal withsomething like a δ-function, we thus deal with operator-valued distributions, and they shouldbe mathematical formulations of quantum fields. An ordinary Schwartz distribution assigns anumber to each test function, by definition. An operator-valued distribution should assign anoperator, which may be unbounded, to each test function. There is a precise mathematicaldefinition of this notion of an operator-valued distribution, and we can further axiomatize aphysical idea of what a quantum field should be. Such an axiomatization is known as a set ofWightman axioms.

We work on a Minkowski space R4. The scalar product of x = (x0, x1, x2, x3) and y =(y0, y1, y2, y3) is x0y0 − x1y1 − x2y2 − x3y3. The linear maps R4 → R4 preserving the scalarproduct are the Lorentz transformations. The restricted Lorentz group is the subgroup suchthat Λ00 > 0 and det Λ = 1. The semi-direct product with translations is called the Poincaregroup; the subgroup of the Poincare group of elements x 7→ Λx + b such that Λ is in therestricted Lorentz group is called the restricted Poincare group.

The universal cover of the restricted Poincare group is naturally identified with

(A, a)|A ∈ SL(2,C), a ∈ R4 .

We say that two regions O1, O2 of Minkowski space are spacelike separated, if for any x =(x0, x1, x2, x3) ∈ O1 and y = (y0, y1, y2, y3) ∈ O2, we have (x0 − y0)

2 − (x1 − y1)2 − (x2 − y2)

2 −(x3 − y3)

2 < 0.We are now ready to state the Wightman axioms.

1. We have closed operators φ1(f), φ2(f), . . . φn(f) on a Hilbert space H for each smoothfunction f on R4 with compact support.

2. There exists a dense subspace D ⊂ H that is contained in the domains of all φi(f), φ∗(f)for all f ∈ C∞

C (R4). For all such f , we have φ(f)D ⊂ D and φ∗i (f)D ⊂ D. For all v, s ∈ D,the function f 7→ 〈v, φ(f)w〉 is a Schwartz distribution.

3. We require thatH carries a unitary representation U of the universal cover of the restrictedPoincare group and of an n-dimensional representation of SL(2,C) such that

U(A, a)D = DU(A, a)φi(f)U(A, a)∗ =

∑j S(A−1)ijφj(f(A, a)), with f(A, a) = f(Λ(A)−1(x− a)).

Here the last identity is to be understood as identities on U .

4. If the supports of two smooth functions f and g are compact and spacelike separated,then we [φi(f), φj(g)] = 0 and [φi(f), φ∗j(g)] = 0

5. There is a non-zero vector Ω ∈ D, the vacuum vector such that

• U(A, a)Ω = Ω for all elements of the restricted Lorentz group

91

• the spectrum of the four-parameter unitary group U(I, a), a ∈ R4, is in the positivecone

(x0, x1, x2, x3)|x0 > 0, x20 − x2

1 − x22 − x2

3 ≥ 0

• The subspace generated by finitely many applications of φi(f) and φ∗j(g) on Ω isdense in H.

Moreover, we require the subspace of such vectors to be one-dimensional.

Distributions and unbounded operators cause various technical difficulties in a rigoroustreatment. Bounded linear operators are much more convenient for algebraic handling, and weseek for a mathematical framework using only bounded linear operators. There has been sucha framework pursued by Araki, Haag and Kastler, and such an approach is called algebraicquantum field theory today. The basic reference is Haag’s book and we now explain its basicideas.

Suppose we have a family of operator-valued distributions φ subject to the Wightmanaxioms. For an operator valued distribution φ and a smooth test function f ∈ C∞

c (R4) withsupport in a bounded region O the expression φ(f) gives an unbounded operator operator whichwe assume to be self-adjoint.

In quantum mechanics, observables are represented as (possibly unbounded) self-adjointoperators. We regard φ(f) as an observable in the spacetime region O. We have a family ofsuch quantum fields and many test functions with supports contained in O, so we have manyunbounded operators for each O. Applying spectral projections of these unbounded operators,we can consider the von Neumann algebra A(O) generated by these projections. In this way, weobtain a family of von Neumann algebras A(O) on the same Hilbert space H parameterizedby bounded space time regions O. We axiomatize properties of this family.

We list the axioms for space time being Minkowski space R4. In this case, it is enough toconsider as bounded regions double cones of the form (x+ V+)∩ (y+ V−), where x, y ∈ R4 andwe introduce the forward and backward cones

V± = z = (z0, z1, z2, z3) ∈ R4|z20 − z2

1 − z22 − z2

3 >,±z0 > 0 .

1. (Isotony) For a larger double cone O2 ⊃ O1, we have more test functions, and operators:A(O1) ⊂ A(O2).

2. (Locality) Suppose two double cones O1 and O2 are spacelike separated so that no inter-action between them is possible, even at the speed of light. Then the observables in thetwo regions should commute, leading to the requirement that the elements in A(O1) andA(O2) commute.

3. (Poincare Covariance) The (universal cover of the) natural symmetry group of Minkowskispace, the restricted Poincare group, is required to be unitarily represented on H suchthat A(gO) = UgA(O)U∗

g .

4. (Vacuum) There is a distinguished unit vector Ω ∈ H, the vacuum vector such thatUgΩ = Ω for all elements g in the restricted Poincare group.

5. (Irreducibility) We require that ∪OA(O)Ω is dense in H.

6. (Spectrum Condition) If we restrict the representation U to the translation subgroup, itsspectrum is contained in the closure of V+.

92

It is clear that the above set of axioms is very similar to the Wightman axioms. Theassignment of 0 7→ A(O) is called a net (of von Neumann algebras). It is very hard to constructan example satisfying the above axioms. In the 4-dimensional Minkowski space, we have onlyone example known and this is known under the name of free fields.

93

A Differentiable Manifolds

A.1 Definition of differentiable manifolds

It is important to describe spaces which locally “look like” an open subset of Rn. Examplesare abundant and most easily visualized in the case of two-dimensions: a sphere or a doughnutlocally look like R2. Indeed, we describe the earth in an atlas as follows:

• we have a collection of charts which are pieces of R2 on which the earth is represented.Admittedly with simplifications, but this is inessential for our purposes. What matters isthat every point appears in at least one chart.

• At the boundary of each chart, we have a prescription on how to glue together the charts.In fact, the points close to the boundary of any chart have to appear in at least one morechart.

We formalize this in the following definition:

Definition A.1.1

(i) A topological manifold M is a Hausdorff topological space with a countable basis of opensets which is locally homeomorphic to Rn. The natural number n is called the dimensionof the manifold. We sometimes write Mn or dimM = n.

In more detail, this means: for every point p ∈ M there exists an open neighborhoodp ∈ U ⊂M and a homeomorphism xU : U → V to some open subset V ⊂ Rn.

(ii) A chart of a topological manifold is a homeomorphism x : U → V from an open subset ofU ⊂M to an open subset of V ⊂ Rn.

(iii) An atlas is a family I of charts xα : Uα → Rn such that the map

tα∈Ix−1α : tα∈IUα →M

given by the collection x−1α is a surjection.

(iv) An atlas is called differentiable, if for all pairs x, y of coordinate charts the map

y x−1 : Rn → Rn

is differentiable bijection everywhere where it is defined.

(v) A topological manifold, together with the choice of a differentiable atlas is called adifferentiable manifold or, synonymously a smooth manifold.

Remarks A.1.2.

1. A Hausdorff space is a topological space M in which for any two disjoint points p, q ∈Mthere exist disjoint open sets U, V in M with p ∈ U and q ∈ V . It is necessary to imposethe Hausdorff property: identify all points with y = y′ < 0 on two copies of the real axisR with coordinates y and y′ respectively. Then each point is contained in a coordinateneighbourhood homeomorphic to an open interval in R, but there are no disjoint openneighbourhoods of the two distinct points y = 0 and y′ = 0.

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2. The requirement of M to have a countable basis implies that M is paracompact: for everyatlas (Uα), there exists a locally finite atlas (Vβ)β∈I with every Vβ contained in some Uα.Locally finite means that for every point p ∈M , there exists an open neighborhood whichintersects only a finite number of the sets Vβ.

In this situation, one can find a partition of unity: this is a set of smooth real-valuedfunctions gβ such that

(a) 0 ≤ gβ ≤ 1 in M for each β ∈ I(b) the support of gβ is contained in Vβ

(c)∑

β∈I gβ(p) = 1 for each p

Using a partition of unity, a smooth function f ∈ C∞(M) can be decomposed

f = f · 1 =∑β

(f · gβ)

into a locally finite sum of functions f · gβ with support contained in the coordinate chartVβ. This is essential for setting up an integration theory on manifolds.

Examples A.1.3.

1. An affine space modelled over Rn is an n-dimensional manifold.

2. Any discrete topological space is a zero-dimensional manifold.

3. The circle S1 is a one-dimensional manifold. More generally, the n-sphere in an n + 1-dimensional Euclidean vector space V

Sn := x ∈ V |〈x, x〉 = 1

is an n-dimensional (compact) manifold.

4. A torus is a two-dimensional manifold. The Klein bottle is a two-dimensional manifold.

5. Let L ⊂ Rn be a lattice, i.e. L is the integral span of a basis of Rn. A lattice is also adiscrete subgroup of the additive group of Rn, and the quotient Rn/L is an n-dimensionalmanifold, the n-dimensional torus T n.

Definition A.1.4

(i) Let M,N be differentiable manifolds of dimensions m,n respectively. Let G be open inM . A map

f : G→ N

is called differentiable or smooth , if for all charts the expression

y f x−1 : Rm → Rn

is differentiable as soon as it is defined.

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(ii) Differentiable maps are the morphisms in the category of differentiable or smooth man-ifolds. This category is also called the category of smooth manifolds. The isomorphismsare called diffeomorphisms. (Note that at this point, we do not know how to differentiatediffeomorphisms!)

(iii) For any open subset U ⊂ M , we consider the real-valued differentiable functionsC∞(U,R) =: F(U). They form an R-algebra. For any inclusion U ⊂ V we have a resctric-tion

resVU : F∞(V,R)→ F∞(U,R)

which is a morphism of R-algebras. Obviously, resUU = id and for U ⊂ V ⊂ W we have

resWU = resVU resWV .

This means that we have a presheaf of R-algebras.

(iv) We have two more properties:

• if (Uα)α∈I is an open covering of an open subset U ⊂ M and if for f, g ∈ F(U)we have resUUα

(f) = resUUα(g) for all α ∈ I, then f = g. This just formalizes that a

function is uniquely determined once we know it everywhere locally.

• if (Uα)α∈I is an open covering of an open subset U ⊂ M and if we have a collectionof functions fα ∈ F(Uα) such that the restrictions agree on all intersections:

resUαUα∩Uβ

(fα) = resUβ

Uα∩Uβ(fβ)

for all α, β ∈ I, there is f ∈ F(U) such that fα = resUUα(f). This formalizes the idea

that we can define functions by patching together locally defined functions, if theycoincide on overlaps.

This means that we have a sheaf of R-algebras.

(v) In our case, the restriction morphisms are surjective. One says that the sheaf is flabby.

(vi) Let U ⊂ M be open and let for p ∈ U be F0p (U) the ideal of functions that vanish in a

neighborhood of p. The quotient algebra

Fp(U) := F(U)/F0p (U)

is called the algebra of germs of differentiable functions on U in P .

Lemma A.1.5.Given an inclusion U ⊂ V , the restriction

F(V )→ F(U)

induces a canonical isomorphismFp(M) → Fp(G).

We call this algebra the germs of differentiable functions in p ∈M and denote it by Fp.

One should be aware that this lemma does not hold, if one works with analytic rather thandifferentiable functions. In this case, one has to consider a direct limit.

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A.2 Tangent vectors and differentiation

Differentiation requires only a local knowledge of the function. We should thus define differen-tiation on germs of functions. We consider all operations on germs of functions that obey thealgebraic rules for differentiation: linearity and the Leibniz rule.

Definition A.2.1

(i) A tangent vector at M in the point p is a derivation

v : FpM → R

i.e. an R-linear map for which the Leibniz rule

v(fg) = v(f)g(p) + f(p)v(g)

holds.

(ii) The set of all derivation in a point p forms an R-vector space, the tangent space TpM .

Remarks A.2.2.

(a) Examples for tangent vectors of Rn are provided by partial derivatives:

∂

∂xi

∣∣∣p

: f 7→ ∂f

∂xi(p) .

(b) More generally, a coordinate chart x around p ∈M defines for all i = 1, . . . , n

∂∂xi

∣∣∣p

: FpM → R,

f 7→ ∂∂xi (f x−1)

∣∣∣x(p)

a tangent vector.

Lemma A.2.3.

1. For any coordinate system, the family

∂∂xi |p

i=1,...,n

is a basis of TpM .

2. This implies in particular dimR TpM = dimM and the statement that any tangent vectorv ∈ TpM can be written uniquely as a linear combination

v =n∑i=1

αi∂

∂xi

∣∣∣p.

We claim that αi = v(xi), i.e. the coefficients are obtained by evaluating the derivationon the germ of the coordinate function xi.

Proof:

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1. For an open ball U around 0 ∈ Rn and a function ϕ ∈ C∞(U,R) with ϕ(0) = 0 considerthe function

ψi(u) =

∫ 1

0

∂

∂xiϕ(tu) dt ∈ C∞(U,R).

A brief calculation shows

ψi(0) =∂

∂uiϕ∣∣0

and

ϕ(u) =

∫ 1

0

d

dtϕ(tu) =

n∑i=1

uiψi(u).

2. Given a coordinate x around p with x(p) = 0 we use the preceding remark to deduce thatfor any function f the expression f x−1 can be written as

f x−1(u) =n∑i=1

uiψi(u)

and thus

f =n∑i=1

xiψi x∣∣x−1(U)

.

We conclude

v(f) = v(f |x−1(U)) =n∑i=1

v(xi)ψi(0) =n∑i=1

v(xi)∂

∂xi

∣∣∣pf

Lemma A.2.4.If x and y are coordinate charts around p ∈M , we have

∂

∂yi

∣∣∣p

=n∑j=1

αji∂

∂xj

∣∣∣p

with α a square matrix given by the Jacobian matrix for locally defined function describeing thechange of coordinates x y−1,

αki =∂xk

∂yi.

Proof:The existence of the coefficients α follows from the fact that we have two bases of tangent spaceTpM . Applying both sides to the function on U ⊂M given by the k-th coordinate

xk : U → R

gives the concrete form of the coefficients.

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Example A.2.5.Let (A, V ) be an affine space modelled over an R-vector space V . For p ∈ A consider on FpAthe derivation

f 7→ d

dt

∣∣∣t=0f(p+ tv) for v ∈ V,

which is, of course, just differentiation in the direction of the affine line p+ tv, t ∈ R. For anaffine space, this provides a natural isomorphism of the tangent space TpA and the differencespace V :

V → TpA .

Definition A.2.6Let f : M → N be a differentiable mapping. Then f induces for any point p ∈ M a map ongerms of functions

f ∗ : Ff(p)N → FpMϕ 7→ ϕ f

a linear mapf∗p : TpM → Tf(p)N

by(f∗pv) (ϕ) = v(ϕ f) for v ∈ TpM, ϕ ∈ Ff(p)N.

For this linear mapping, we also use the notation f ′p, Df |p, Tpf

Remarks A.2.7.

(a) One shows thatid∗p = idTpM

and that the chain rule holds:

(g f)∗p = g∗f(p) f∗p.

(b) If x is a chart around p and y a chart around f(p), then f∗p is given with respect to the

bases

∂∂xi

∣∣p

and

∂∂yj

∣∣f(p)

by the Jacobian matrix of y f x−1 in the point x(p).

Definition A.2.8

(i) A differentiable application f : M → N is called an immersion, if the linear map

f∗p : TpM → Tf(p)N

is injective for all p ∈M . It is called a submersion, if the maps are surjective for all p ∈M .

(ii) An embedding is an immersion that is a homeomorphism of M to a subspace of N .(Exercise: find an immersion that is not an embedding.)

(iii) Let N be a smooth manifold. If the inclusion M ⊆ N of a subset is an embedding, thenM is called a submanifold of N .

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(iv) Let f : M → N be a differentiable map. Then p ∈ M is called a regular point of f if thelinear map f∗p is surjective; otherwise it is called a critical point of f .

(v) A point q ∈ N is called a regular value of f , if all p ∈ f−1(q) are regular points of f .Otherwise, it is called a critical value.

The proof of the following statement uses the theorem of implicit functions. We leave it asan exercise to the reader:

Lemma A.2.9.Let M be a differentiable manifold of dimension n and N be a differentiable manifold of dimen-sion k. The preimage of a regular value under a differentiable map

f : Mn → Nk

is a submanifold of dimension n− k.

A.3 Fibre bundles

Definition A.3.1

(i) The Cartesian product M ×N of two differentiable manifolds M,N of dimension n andk respectively, is endowed with the structure of a differentiable manifold by products ofcoordinate charts. One has dim M ×N = n+ k.

The projections pr1, pr2

M ×N

M N

are surjective submersions. To this end, note that, in local coordinates, the projectionsare the projections Rn+k → Rn and Rn+k → Rk.)

(ii) A surjective submersion π : P → M is called a differentiable fibration or fibre bundlewith total space P , base space M and fibre N , if for each point p ∈ M there exists aneighborhood U and a diffeomorphism

x : π−1(U) → U ×N

such that the diagramπ−1(U)

∼−−−→ U ×N

π

y yp1U U

commutes. The diffeomorphism x is called a local trivialization or bundle chart. The man-ifold π−1(q) is called the fibre over q ∈M .

(iii) A bundle is called trivial, if there exists global bundle chart P∼→M ×N .

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(iv) Let U be an open subset of M . A local section on U of a bundle π : P → M is adifferentiable mapping

s : U → P

such that π s = idU . The sets F(U) of sections on all open subsets U ⊂M forms a sheaf.The elements of F(M) are called global sections.

The following lemma shows why surjective submersions are particularly important:

Lemma A.3.2.Consider a surjective map π : M → N of smooth manifold. For any open subset U ⊂ N , alocal section of π is a map s : U →M such that π s = idU , i.e. a local one-sided inverse of π.

A surjection π : M → Nof smooth manifolds has local smooth sections, iff π is a surjectivesubmersion.

Proof:If π is a surjective submersion, the existence of local sections is a consequence of the theoremof implicitly defined functions. If π admits a smooth section, the surjectivity of π follows fromdifferentiating π s = idU to find dπ ds = id.

Remark A.3.3.If we describe a mechanical system of N mass points moving in a Galilei space A in termsof world lines parametrized by an eigentime in an interval I, we start with the fibre bundleπ : I × An → I given by projection on the first factor.

Trajectories are then sections of π. If we impose constraints – e.g. to describe a solid body,we impose that the spacial distances between the mass points are constant in time –, we restrictourselves to a submanifold M ⊂ An which still provides a fibre bundle over I. The manifold Mis then called the (extended) configuration space of the system. To describe (extended) configu-ration spaces that depend on time, we have to consider general surjective submersions.

We finally need manifolds with additional structure.

Definition A.3.4An n-dimensional smooth manifold G with a group structure such that the map

µ : G×G→ G

(g, h) 7→ gh−1 ,

is smooth, is called a Lie group.

Remarks A.3.5.

1. Note that this implies that both the inverse g 7→ g−1 and the multiplication (g1, g2) 7→ g1 ·g2

are smooth maps.

2. Examples:

• The general linear group GL(n,R) is an n2-dimensional Lie group. It is non-abelianfor n > 1.

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• The orthogonal group SO(n), defined as the structure group of a Euclidian vector

space of dimension n is an n(n−1)2

-dimensional Lie group. It is non-abelian for n > 2.

• The unitary group U(n), defined as the structure group of a unitary vector space ofdimension n is an n2-dimensional real Lie group. It is non-abelian for n > 2.

• The unitary group SU(n), defined as the structure group of a unitary vector spaceof dimension n consisting of maps of determinant 1 is an n2 − 1-dimensional Liegroup. It is non-abelian for n > 2.

• The Galilei group is a ten-dimensional non-compact Lie group.

Observation A.3.6.Let M be a differentiable manifold of dimension n.

• The set⋃p TpM of all tangent vectors can be endowed with the structure of a differentiable

manifold as follows. For a local coordinate chart defined on U ⊂M

x : U → Rn

we consider the local bijection, called the associated bundle chart,

x :⋃p∈U

TpM → Rn × Rn

defined byx(v) =

(x(p), v(x1), ..., v(xn)

)for v ∈ TpM

One deduces from Lemma A.2.4 the following identity for a change of local coordinatesx, y:

x y−1(a, b) =

(x y−1(a),

n∑i=1

∂yi

∂xibi, ...

),

so that we have differentiable maps which endow

TM :=⋃p∈M

TpM

with the structure of a manifold of dimension 2n.

• In local coordinates x and x, the map

π : TM →M with π(v) = p for v ∈ TpM

is the projection on the first n components and thus a submersion. This way, TM be-comes a bundle over M with fibre an n-dimensional vector space. This bundle is calledthe tangent bundle.

• Any smooth map f : M → N gives rise to a map

Tf : TM → TN(v, p) 7→ f∗p(v) ∈ Tf(p)N

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This map is differentiable and we have a commuting diagram

TMTf−−−→ TN

π

y yπM −−−→

fN

One says that Tf covers f . Thus T is a (covariant) functor in the category of (smooth)manifolds.

• An important aspect of bundles is the fact that they can be pulled back : given a smoothmap of manifolds f : M → N and a smooth bundle π : E → N , one defines

f ∗E := M ×N E ∼= (m, e)|m ∈M, e ∈ E such that f(m) = π(e)

The bundle projection f ∗E → M is induced from the projection on the first factor. Inother words, the fibre of f ∗E in the point m ∈M is the fibre of E in the point f(m) ∈ N .We then have a map such that the following diagram commutes:

M ×N Ef∗ //

E

π

M

f // N

A.4 Vector fields and Lie algebras

Definition A.4.1Let M be a differentiable manifold. A differentiable vector field on M is a global section Xof the tangent bundle. A local vector field on an open subset U ⊂ M is a local section of thetangent bundle.

In other words, for every point p ∈M we have Xp ∈ TpM such that for all smooth functionsf ∈ C∞(M) the function

X(f) : M → Rp 7→ Xp(f)

is smooth.

Remarks A.4.2.Let M be an n-dimensional smooth manifold.

(a) The set vect(M) of all vector fields is an infinite-dimensional real vector space. One caneven define a scalar multiplication of smooth real-valued functions and vector fields. Thusvect(M) is a module over the algebra C∞(M,R) of smooth functions.

(b) For a local coordinate chart x with domain U ⊆M , we define n local vector fields on U :

∂

∂xi(p) =

∂

∂xi

∣∣∣p

with i = 1, 2, . . . n

They form a basis of the C∞(U)-module vect(U). Local vector fields thus form a free C∞(U)-module; one says that vector fields form a locally free C∞(M)-module.

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(c) One has a Leibniz rule for the action of vector fields on smooth functions f, g ∈ C∞(M)

X(fg) = X(f)g + fX(g).

Vector fields are thus globally defined R-linear differential operators on C∞(M).

Definition A.4.3A Lie algebra over a ring R is an R-module L together with an antisymmetric R-bilinearmapping, called the Lie bracket or commutator

[ · , · ] : L× L→ L

such that the Jacobi identity

[x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0

holds for all x, y, z ∈ L.

Vector fields over any smooth manifold M have the structure of an (infinite-dimensional)real Lie algebra:

Observation A.4.4.

(a) Given two vector fields X, Y ∈ Vect(M), we consider the following map on germs of func-tions:

XpY : FpM → Rwith (XpY )(f) := Xp(Y (f))

Because of

XpY (fg) = Xp(Y (f)g + fY (g))

= XpY (f)g + Yp(f)Xp(g) +Xp(f)Yp(g) + fXpY (g)

the map XpY is not a derivation on germs of smooth functions and thus does not give riseto a vector field. However,

[X, Y ]p := XpY − YpX

is a derivation and thus provides a vector field. One checks that this way Vect(M) becomesa Lie algebra over R. One should be aware that one does not obtain a Lie algebra over thering C∞(M), because we find for any germ ϕ of a function:

[X, fY ]ϕ = X(fY ϕ)− fY (Xϕ)

= X(f)Y (ϕ) + fXY (ϕ)− fY X(ϕ)

= f [X, Y ]ϕ+X(f)Y (ϕ)

hence [X, fY ] = f [X, Y ] +X(f)Y

Similarly, we find [fX, Y ] = f [X, Y ]− Y (f)X.

(b) The local basis fields ∂∂xi commute, since the partial derivatives commute.

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(c) In local coordinates, we write

X =n∑i=1

ξi∂

∂xi

Y =n∑i=1

ηi∂

∂xi

Note that ξi = X(xi) with xi the i-th coordinate function. We compute for the Lie bracket,using observation A.4.4 (a):

[X, Y ] =∑

i,j

[ξi ∂∂xi , η

j ∂∂xj

]= ξi ∂

∂xi (ηj) ∂∂xj + ηj[ξi ∂

∂xi ,∂∂xj ]

= ξi ∂ηj

∂xi∂∂xj + ηjξi[ ∂

∂xi ,∂∂xj ]− ηj ∂ξ

i

∂xj∂∂xi

=∑

k,l

(ηl ∂ξ

k

∂xl − ξl ∂ηk

∂xl

)∂∂xk

If the manifold carries the additional structure of a Lie group G, we can single out a subclassof vector fields:

Definition A.4.5Let G be a smooth Lie group.

1. For any element g ∈ G, we call the smooth map

Lg : G → Gh 7→ gh

the left translation by g. This defines a smooth left action of G on itself. Similarly for anyg ∈ G , the map h 7→ h · g defines a right translation.

2. A global vector field V ∈ Γ(TG) is called left invariant, if (Lg)∗V = V . Right invariantvector fields are defined analogously.

3. One can verify that left invariant and right invariant vector fields form a Lie subalgebraof the Lie algebra of all vector fields, called the Lie algebra Lie(G) of the Lie group G.Left translation acts transitively and freely on the Lie group G. Hence, a left invariantvector field is determined by its value in the neutral element e ∈ G in TeG As a vectorspace, the Lie algebra Lie(G) can be identified with the tangent space TeG. In particular,it is finite-dimensional of dimension dimG.

Remark A.4.6.

1. We extend the notion of a trajectory to an arbitrary smooth manifold M . Consider aninterval I ⊂ R and a smooth map

ϕ : I →M

Since I is a subset of an affine space, we can identify the derivative with a linear functionon the difference vector space R

Dϕ : R ∼= TtI → Tϕ(t)M

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which we describe byϕ(t) := Dϕ(1) ∈ Tϕ(t)M ,

i.e. by a tangent vector.

Given a vector field V on M , we call a trajectory such that

dϕ

dt t=t0= V (ϕ(t0))

the integral curve of the vector field V . In local coordinates

x : U ⊂M → Rn

the vector field and the trajectory read

V =n∑i=1

V i ∂

∂xiand ξ = x ϕ : R→ Rn

Then the condition that the trajectory is an integral curve amounts to the system ofordinaty differential equations

dξi

dt(t) = V i(ξ(t))

This justifies the point of view that integral curves of vector fields are a geometric expres-sion for solutions of a system of ordinary differential equations of first order.

2. Standard theorems assert the existence of local solutions of ordinary differential equations.As a consequence, for each point q ∈ M , there is an open neighborhood U and ε > 0such that the vector field X defines a family φt : U → M of diffeomorphisms for all|t| < ε, obtained by taking each point p ∈ U a parameter distance t along the integralcurves of X. In fact, the φt form a one-parameter local group of diffeomorphisms, sinceφt+s = φt φs = φs φt for |t| < ε, |s| < ε and |s+ t| < ε.

A.5 Differential forms and the de Rham complex

Definition A.5.1Let k be a field of characteristic different from two and V be a k-vector space.

(i) A multilinear map f : V p = V × . . .× V → k is called alternating, if f(v1, ..., vp) = 0, assoon as vi = vj for a pair i 6= j.

(ii) The vector space of alternating p-forms on V will be denoted by ΛpV . We have Λ1V = V ∗,the dual vector space, and we set Λ0V := k.

Lemma A.5.2.

(i) Given p one-forms η1, ..., ηp ∈ Λ1V = V ∗ we consider the multilinear map

η1 ∧ ... ∧ ηp : V ×p → k(v1, ..., vp) 7→ det(ηi(vj))

This defines an alternating p-form, η1 ∧ . . . ∧ ηp ∈ ΛpV.

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(ii) If b1, . . . bn is a basis of V and b1, . . . bn the corresponding dual basis of V ∗, thenbi1 ∧ ... ∧ bip

∣∣∣ 1 6 i1 < i2 < ... < ip 6 n

is a basis of ΛpV.

(iii) For a finite-dimensional vector space V of dimension n, the dimension of the space of pforms is dim ΛpV =

(np

).

Definition A.5.3On the graded vector space

Λ•V :=n⊕p=0

ΛpV

we define the product on the basis vectors:

(bi1 ∧ . . . ∧ bik) ∧ (bj1 ∧ . . . ∧ bjl):= bi1 ∧ . . . ∧ bik ∧ bj1 ∧ . . . ∧ bjl

This product is called exterior product, wedge product or Grassmann product.

Remarks A.5.4.

(a) The wedge product is independent of the choice of basis bi of V . More invariantly, wedefine the exterior algebra Λ•(V ) as the quotient of the tensor algebra T •V ∗ = ⊕∞p=0(V

∗)⊗n

of the dual space modulo a two-sided ideal

Λ•(V ) = T •V ∗/〈ω ⊗ ω′ + ω′ ⊗ ω〉

generated by all pairs ω, ω′ ∈ V ∗.

(b) The wedge product is associative.

(c) The wedge product is graded commutative: for ω ∈ ΛiV and ω′ ∈ ΛjV , we have

ω ∧ ω′ = (−1)ijω′ ∧ ω

(d) Alternating forms can be pulled back along a linear map f : V → W : we define a linearmap

Λpf : ΛpW → ΛpV

by(Λpf(ω)) (v1, ..., vp) = ω(fv1, ..., fvp)

for vi ∈ V and ω ∈ ΛpW . For p = 1, this is just the dual map f ∗ : W ∗ → V ∗. Forp = n, ΛnV is one-dimensional and Λnf is a 1 × 1-matrix which we canonically identifywith a scalar. From the axiomatic definition of the determinant as a normalized alternat-ing multilinear form on square matrices, one easily derives that the scalar Λnf equals thedeterminant.

107

(e) Given another linear map g : W → U , we find

Λp(g f) = Λp(f) Λp(g).

We also use the short hand notation f ∗ instead of Λp(f). This way, we have for each p acontravariant functor

Λp : vect(k)→ vect(k) .

We now apply this construction of linear algebra to the tangent bundle fibrewise:

Definition A.5.5

(i) Let M be a smooth n-dimensional manifold. The set

ΛpTM :=⋃x∈M

ΛpTxM

can be endowed with the structure of a smooth manifold as in observation A.3.6. Theevident projection toM turns this into a fibre bundle with fibre a vector space of dimension(dimMp

).

(ii) In particular, for p = n, we get a line bundle, the determinant line bundle.

(iii) The local sections of the bundle ΛpT ∗M on an open subset U ⊂ M are called (local)differential forms Ωp(U). They form a C∞(U)-module. The collection Ωp(U) for all opensubsets U ⊂M forms again a sheaf. The elements of Ωp(M) are called global differentialforms.

(iv) The exterior product of differential forms is defined fiberwise and gives for every opensubset U ⊂M an infinite-dimensional graded commutative algebra

∧ : Ωk(U)× Ωl(U)→ Ωk+l(U) .

We consider the situation also in local coordinates on the differentiable manifold M :

Remarks A.5.6.

(a) Let x : M ⊃ U → Rn be a coordinate chart of a smooth manifold M of dimension n. We

have seen that

∂∂xi

∣∣∣p

i=1,...,n

is a basis of the tangent space TpM . We denote the dual basis

of the cotangent space T ∗pM by

dxi∣∣p

i=1...n

.

Recall that ∂∂xi is a local section in the tangent bundle TM , i.e. a local vector field. Similarly,

dxi is a local section in the cotangent bundle T ∗M , i.e. a local one-form.

(b) Correspondingly, local p-forms can be written as

ω(x) =∑

16i1<...<ip6n

ωi1...ip(x)dxi1 ∧ . . . ∧ dxip

with smooth functions ωi1...ik(x) on U ⊂M . This implies that a local p-form defined on thedomain of definition of the coordinate chart x is determined by its values on the local vectorfields ∂

∂xi . The C∞(U)-module Ωp(U) is thus locally free.

108

(c) Under change of local coordinates, we deduce from

∂

∂xi=

∂yj

∂xi∂

∂yj

that

dxi =∑j

∂xi

∂yjdyj

As a consequence, we obtain

dx1 ∧ . . . ∧ dxn = det

(∂xi

∂yj

)dy1 ∧ . . . ∧ dyn

which justifies the name determinant line bundle for the highest non-vanishing exteriorpower ΛnTM of the cotangent bundle.

In several contexts, it is helpful to have antisymmetrized expressions in derivatives: forexample, the curl of a vector field on R3 is given by (rotv)i := εijk∂jvk (with ε totally anti-symmetric in the indices) and the electromagnetic field strength is given in terms of the vectorpotential as Fµν := ∂νAµ − ∂µAν . This leads us to the following

Definition A.5.7For any open subset U of a smooth manifold M , we define a map

d : Ωp(U)→ Ωp+1(U)

by defining dω for ω ∈ Ωp(U) on p+ 1 (local) vector fields v1, ..., vp+1 as

dω(v1, ..., vp+1) =

=

p+1∑i=1

(−1)i+1viω(v1, ..., vi, ..., vp+1) +

+∑i<j

(−1)i+jω([vi, vj] , v1, ..., vi, vj, ..., vp+1

)Here the hat denotes expressions to be left out.

One now checks the following claims:

Remarks A.5.8.

(a) The map d is R-linear.

(b) If ω ∈ Ωp(M), then we have for all differential forms η

d(ω ∧ η) = dω ∧ η + (−1)pω ∧ dη .

(c) For any smooth map f : M → N , we have as a consequence of the chain rule:

df ∗ω = f ∗dω.

109

(d) We find d2ω = 0 for all ω ∈ Ω•M .

We are now ready to introduce the Lie derivative:

Definition A.5.9Let M be a smooth manifold and X be a smooth vector field on M . Recall from RemarkA.4.6 the local one-parameter group φt of diffeomorphisms given by X. Since these maps areinvertible, we can push for q := φ−t(p) vector fields

Φt := φt∗ : TqM → TpM

and pull back differential forms

Φt := φ∗−t : Ωk(M)q → Ωk(M)p

The Lie derivative with respect to the vector field X is defined for T being either a differentialform or a vector field as

LXT |p := limt→0

1

t(Tp − (ΦtT )|p) .

Remarks A.5.10.From the properties of Φt, it follows:

1. The Lie derivative LXv of a vector field v is a vector field; the Lie derivative LXω of ak-form ω is a k-form.

2. On vector fields, this operation reduces to the Lie bracket of vector fields, LXY = [X, Y ].

3. The Lie derivative LX is linear and preserves operations from tensor calculus like con-tractions.

4. There is a Leibniz rule for the tensor product:

LX(S ⊗ T ) = LXS ⊗ T + S ⊗ LXT .

5. For a one-form ω = ωidxi, one finds LXω = (LXω)idx

i with

(LXω)i =∂ωi∂xj

Xj + ωj(∂Xj

∂xi) .

For a k-form, the last term is replaced by k terms of similar form with insertions ofderivatives of the vector field X at any position.

6. One hasd(LXω) = LX(dω)

Observation A.5.11.Let M be a smooth manifold and X be a smooth vector field on M .

110

1. We define a linear map, called the interior product

ιX : Ωk+1(M)→ Ωk(M)

by(ιXω)(X1, X2, . . . , Xk) = ω(X,X1, X2, . . . , Xk)

for any k-tuple of local vector fields X1, . . . , Xk.

2. ιX is R-linear and an antiderivation,

ιX(ω ∧ η) = ιXω ∧ η + (−1)kω ∧ (ιXη)

for ω ∈ Ωk(M).

3. Cartan’s magic formula relates the Lie derivative to the interior product and the exteriordifferential d:

LXω = ιXdω + d(ιXω) .

4. For any smooth function f , one has the relation

LfX = fLXω + df ∧ ιXω .

The equality d2 = 0 leads us to the following

Definition A.5.12Let M be a smooth manifold of dimension n and U ⊂M open.

1. A differential form ω ∈ Ωp(U) such that dω = 0 is called closed.

2. A differential form ω ∈ Ωp(U) is called exact , if there exists η ∈ Ωp−1(U) such thatω = dη.

3. Because of d2ω = 0 for all ω, exact differential forms are closed. Hence we consider thequotient space. The real vector space for global differential forms

HpdR(M) := ker

(d : ΩpM → Ωp+1M

)/im(d : Ωp−1M → ΩpM

)is called the p-th de Rham cohomology group of M . For convenience, we have adoptedhere the convention Ωn+1(M) = 0, Ω−1(M) = 0.

4. Given a smooth map f : M → N , the pull back f ∗ : Ωp(N)→ Ωp(M) of differential formscommutes with the exterior derivative and thus gives rise to a pull back

f ∗ : HpdR(N)→ Hp

dR(M)

of de Rham cohomology groups.

Lemma A.5.13.

111

1. For any smooth manifold M , the number of connected components of M equalsdimR H0

dR(M).

2. We have HkdR(Rn) = 0 for all k > 1.

Example A.5.14.Consider on the punctured plane R2 \ 0 the one-form

ω :=x

x2 + y2dy − y

x2 + y2dx ∈ Ω1(R2 \ 0) .

A simple computation shows that ω is closed:

dω =∂

∂x

(x

x2 + y2

)dx ∧ dy − ∂

∂y

(y

x2 + y2

)dy ∧ dx = 0 .

In radial coordinates, we find

ω =1

r2(r cosϕ(dr sinϕ+ r cosϕdϕ)− r sinϕ(cosϕdr − r sinϕdϕ))

= dϕ.

If ω would be the exterior derivative of a function f ∈ C∞(R2 \ 0), i.e. ω = df we wouldnecessarily have f = ϕ+ const. on R2 \ half axis. Such an f cannot exist on R2 \ 0; hence

H1dR(R2 \ 0) 6= 0 .

We conclude in particular that the punctured plane R2 \ 0 is not diffeomorphic to the planeR2.

Definition A.5.15

1. Let M be a differentiable manifold of dimension n. A nowhere vanishing n-form, i.e.ω(p) 6= 0 for all p ∈M , is called a volume form on M .

2. A manifold that admits a volume form is called orientable.

3. An orientation of M is the an equivalence class of volume forms that differ by a positivefunction.

Proposition A.5.16.A smooth manifold M is orientable if and only if there exists an atlas (xi)i∈I such that thedeterminant of the Jacobian of all diffeomorphisms xi x−1

j is positive.

Facts A.5.17.

(a) An n-form can be integrated over any oriented n-dimensional manifold M with boundary:there is a linear map ∫

: Ωn(M)→ R

ω 7→∫M

ω

112

If φ : M → N is a diffeomorphism of smooth manifolds of dimension n and if ω ∈ Ωn(N),then ∫

M

φ∗ω =

∫N

ω .

(b) Let M be a smooth n-dimensional manifold with boundary and let i : ∂M → M be theembedding of the boundary. For any (n − 1)-form ω ∈ Ωn−1(M) with compact support,Stokes’ theorem asserts the following equality:∫

∂M

i∗ω =

∫M

dω

(c) A smooth manifold M is said to be contractible to a point p0 ∈M , if there exists a smoothmap

H : M × [0, 1]→M ,

called a homotopy, such that

H(p, 0) = p and H(p, 1) = p0 for all p ∈M .

(d) Poincare lemma asserts that any closed p-form ω on a contractible manifold M is exact.

For proofs and details, we refer to the book by Boot and Tu and by Madsen and Tornehave.

A.6 Riemannian manifolds and the Hodge dual

We start again with linear algebra: for any finite dimensional k-vector space V , the dual vectorspace V ∗ has the same dimension as V . The two vector spaces are thus isomorphic, but thereis no canonical isomorphism. However, a non-degenerate symmetric bilinear form

g : V × V → k

provides an isomorphism

ιgV : V → V ∗

v 7→ (g(v, ·) : w 7→ g(v, w))

To get a geometric notion derived from this notion of linear algebra, we endow the tangentspaces in a smooth way with non-degenerate symmetric bilinear forms:

Definition A.6.11. A Riemannian manifold (M, g) is a smooth manifold M such that for every point p ∈M

the tangent space TpM has the structure of a Euclidean vector space, i.e. there is a positivesymmetric definite bilinear form

gp : TpM × TpM → R ,

called the metric of M , such that for all local coordinate charts x of M the locally definedfunctions

gij : U → R

gij(p) := gp

(∂

∂xi

∣∣∣p,∂

∂xj

∣∣∣p

)are smooth for all 1 ≤ i, j,≤ dimM .

113

2. More generally, we also consider non-degenerate symmetric bilinear forms

gp : TpM × TpM → R

of any signature s and obtain the notion of a pseudo-Riemannian manifold. For the caseof signature (1, n− 1), the manifold (M, g) is called a Lorentzian manifold.

3. For any open subset U ⊂ M of a (pseudo-)Riemannian manifold, a canonical isomor-phism of bundles between the tangent bundle TM and the cotangent bundle is obtainedfibrewise:

ιgTpM: TpM → T ∗pM .

This gives a bijection between local one-forms (which are just local sections of the cotan-gent bundle) and local vector fields (which are just local sections of the tangent bundle).

ιgU : vect(U)∼→ Ω1(U) .

of one-forms and vector fields induced fibrewise by the isomorphism ιV above.

An important aspect of Riemannian manifolds is the fact that we can define lengths ofcurves on them (and also angles between intersecting curves). We can also define volumes:

Lemma A.6.2.Let (M, g) be an oriented (pseudo-)Riemannian manifold of dimension n. If (dx1, dx2, . . . dxn)is an oriented local basis of the cotangent bundle, then the n-forms

ωg(U) :=√| det gij|dx1 ∧ dx2 ∧ . . . ∧ dxn ∈ Ωn(U)

patch together to a globally defined n-form ωg ∈ Ωn(M) which is nowhere vanishing. It is calledthe normalized volume form for the metric g.

If V is a finite-dimensional vector space of dimension n, the two vector spaces ΛpV andΛn−pV have the same dimension and are isomorphic. Again, there is no canonical isomorphism.

Lemma A.6.3.

1. Let V be an orientable finite-dimensional R-vector space of dimension n. Assume furtherthat there is a non-degenerate symmetric bilinear form g on V . Fix an orthonormal basis(b1, b2, . . . , bn) on V . We extend the bilinear form g to a non-degenerate symmetric bilinearform g on ΛpV for all p by setting for ω, ω′ ∈ Λp V

〈ω, ω′〉 :=∑

i1,i2,...ip

ω(bi1 , . . . , bip)ω′(bi1 , . . . , bip)g(bi1 , bi1) . . . g(bip , bip)

2. If the vector space is even oriented, there exists a unique isomorphism of vector spaces

∗g : ΛpV → Λn−pV

such that for all α, β ∈ ∧pV the equality

α ∧ ∗β = g(α, β)ωg

holds. The isomorphism ∗ is called the Hodge operator.

114

Remarks A.6.4.

1. For V an oriented Euclidean vector space with oriented orthonormal basis (e1, . . . en), theHodge operator acts as

∗(e1 ∧ e2 ∧ . . . ∧ ep) = ep+1 ∧ ek+1 ∧ . . . ∧ en .

2. In particular, for R3 with the standard orientation and the standard Euclidean structure,one finds

∗dx = dy ∧ dz ∗ dy = dz ∧ dx ∗ dz = dx ∧ dy

Indeed, the three equations on ∗dx

dx ∧ (∗dx) = g(dx, dx)dx ∧ dy ∧ dz = dx ∧ dy ∧ dzdy ∧ (∗dx) = g(dy, dx)dx ∧ dy ∧ dz = 0dz ∧ (∗dx) = g(dz, dx)dx ∧ dy ∧ dz = 0

have the unique solution dy ∧ dz.

This allows us to formulate the cross product of vectors in R3 with the standard structureas an oriented Euclidean vector space:

R3 × R3 → R3

(v, v′) 7→ ι−1 (ι(v) ∧ ι(v′))

3. Another important example is R4 with coordinates (t, x, y, z) and a metric of signature(+,−,−,−). We find, e.g. for one-forms

∗dt = dx ∧ dy ∧ dz ∗ dx = dt ∧ dy ∧ dz ∗ dy = dt ∧ dz ∧ dx ∗ dz = dt ∧ dx ∧ dy .

and for 2-forms

∗dt ∧ dx = −dy ∧ dz ∗dt ∧ dy = −dz ∧ dx ∗dt ∧ dz = −dx ∧ dy∗dx ∧ dy = dt ∧ dz ∗dy ∧ dz = dt ∧ dx ∗dz ∧ dx = dt ∧ dy

4. If g is a non-degenerate bilinear symmetric form on a vector space V , then we have forthe component functions

(∗ω)i1i2...in−p =1

p!ωj1...jp

√| det g|εj1...jpi1...in−p

where the indices of ω are raised and lowered with g and its inverse and where ε is totallyantisymmetric in the indices with normalization ε1,2,...n = 1.

Proposition A.6.5.The Hodge operator has the following properties:

(i) ∗1 = ωg, ∗ωg = (−1)s

(ii) For α ∈∧p V, β ∈

∧n−p V , we have

g(α, ∗β) = (−1)p(n−p)g(∗α, β)

115

(iii) On∧p V , we find for the square of the Hodge operator

(∗)2 = (−1)p(n−p)+s idVp V

Observation A.6.6.Let (M, g) be an oriented (pseudo-)Riemannian manifold.

1. In this case, we use the canonical volume form ωg on the manifold and extend the Hodgeoperator fibrewise to a smooth map

Ωp(U)→ Ωn−p(U)

for any open subset U ⊂M .

2. In this way, we obtain an L2-norm on a suitable subspace of Ωp(M):

〈ω, ω′〉 :=

∫M

ω ∧ ∗ω′ .

3. In particular, we can define the adjoint

δ : Ωp(U)→ Ωp−1(U)

of the exterior derivative d, called the codifferential by the equation

〈ω, δω′〉 = 〈dω, ω′〉 .

On p-forms, it reads explicitlyδ = (−1)p ∗−1 d ∗ .

Indeed, for ω ∈ Ωp−1(M) and ω′ ∈ Ωp(M) we deduce from

d(ω ∧ ∗ω′) = dω ∧ ∗ω′ + (−1)p−1ω ∧ d ∗ ω′

that

(−1)p〈ω, ∗−1d ∗ ω′〉 = (−1)p∫Mω ∧ ∗(∗−1d ∗ ω′〉 = (−1)p

∫Mω ∧ d ∗ ω′〉

= −∫M

d(ω ∧ ∗ω′) +∫M

dω ∧ ∗ω′= 〈dω, ω′〉

it satisfies δ2 = 0.

4. The Laplace operator on differential forms is given by

∆ := (δ + d)2 = δ d + d δ .

It is symmetric〈∆ω, ω′〉 = 〈ω,∆ω′〉

and positive definite〈∆ω, ω〉 ≥ 0 .

The elements in its kernel are called harmonic forms. The vector space of harmonic formscan be shown to be naturally isomorphic to de Rham cohomology. The Hodge operatorinduces an isomorphism of harmonic forms

∗ : Hp∆(M)→ Hn−p

∆ (M)

which implements Poincare duality of de Rham cohomology.

116

We finally comment on the relation of the operations we just introduced to the classicaloperations of gradient, curl and divergence in three-dimensional vector calculus. To this end,we consider the special case of M being R3 with the standard scalar product and the standardorientation.

Observation A.6.7.

1. For a function f ∈ Ω0(U) = C∞(U,R), the derivative dω is a one-form. I. The vectorfield corresponding to df is called the gradient of the function f ∈ C∞(U):

grad(f) = ι−1(df) ∈ vect(U) .

2. For a one-form ω = Adx+Bdy + Cdz ∈ Ω1(R3), we obtain a 2-form

dω = (∂C

∂y− ∂B

∂z)dy ∧ dz + (

∂A

∂z− ∂C

∂x)dz ∧ dx+ (

∂B

∂x− ∂A

∂y)dx ∧ dy

whose coefficient functions we recognize as the components of the curl.

We therefore define the curl operator as

curl : vect(U)→ vect(U)

by

vect(U) ι// Ω1(U)

d// Ω2(U) ∗

// Ω1(U)ι−1

// vect(U)

3. Finally, for a 1-form, we compute for ω = Adx+Bdy + Cdz ∈ Ω1(R3)

∗d ∗ ω = ∗d (Ady ∧ dz +Bdz ∧ dx+ Cdx ∧ dy)

= ∗(∂A∂x

+ ∂B∂y

+ ∂C∂z

)dx ∧ dy ∧ dz = ∂A

∂x+ ∂B

∂y+ ∂C

∂z.

We therefore define the standard divergence operator on vector fields as

div(v) : vect(U)→ C∞(U,R)

by

vect(U) ι// Ω1(U) ∗

// Ω2(U)d

// Ω3(U) ∗// C∞(U,R)

4. The relation d2 = 0 now implies the two classical identities

curl gradf = 0 and div curl v = 0 .

117

Index

∗-representation, 75

acceleration, 6action of a group, 1action of a Lagragian system, 33affine map, 1affine space, 1Aharonov-Bohm effect, 55alternating form, 106angular quantum number, 89annihilation operator, 88atlas, 94augmentation, 69augmented variational bicomplex, 35

boost, 50bundle chart, 100

canonical commutation relations, 82canonical phase space, 56canonical transformations, 55Cartan connection, 30Cartan’s magic formula, 111causality, 51character, 73chart, 94choice of gauge, 45classical limit, 70closed differential form, 111codifferential, 116commutator, 104configuration space, 22, 33, 101connector, 29contact form, 30contact ideal, 31contractible manifold, 113Coulomb potential, 90Coulomb’s law, 45covariant derivative, 30covariant derivative, 39creation operator, 88critical point, 100critical value of a function, 100curl operator, 117curvature of a connection, 29

cyclic coordinate, 17cyclic representation, 75cyclic vector, 75

d’Alembert operator, 47Darboux coordinates, 56Darboux’ theorem, 56de Rham cohomology, 111deformation quantization, 70density matrix, 73dependent coordinate, 24determinant line bundle, 108diffeomorphisms, 96difference space, 1differentiable fibration, 100differentiable manifold, 94differentiable vector field, 103differential forms, 108dispersion relation, 47divergence operator, 117dual field strength, 44

Ehresmann connection, 28eigen time, 51eigentime, 6electric field, 43electromagnetic field strength, 43embedding of manifolds, 99Euclidean group, 3Euclidean space, 3Euler-Lagrange complex, 35Euler-Lagrange form, 33Euler-Lagrange operator, 13evolution space, 60, 61evolutionary vector field, 35exact differential form, 111extended phase space, 60exterior product, 107extremal point, 72

fibre, 100fibre bundle, 100field configurations, 22flabby sheaf, 96flat connection, 29

118

foliation, 59formal deformation, 69formal power series, 69free action of a group, 1full quantization, 79future of a point, 50

Galilean coordinate system, 5Galilean structure, 5Galilei group, 4Galilei space, 3Gauß’ law, 44gauge conditions, 45gauge freedom, 45gauge potential, 45gauge symmetry, 38Gelfand spectrum, 74Gelfand transform, 74Gelfand-Naimark theorem, 74generalized coordinate, 17generalized momentum, 17generalized velocity, 17germ of a function, 96germ of a local section, 23global section, 101gradient, 117gtime like vector, 50

Hamilton function, 60Hamilton’s equations, 58Hamiltonian, 80Hamiltonian function, 61Hamiltonian system, 61harmonic forms, 116harmonic oscillator, 87Heisenberg equation of motion, 80Heisenberg Lie algebra, 85Heisenberg picture, 86Heisenberg’s commutation relations, 82Heisenberg’s uncertainty relations, 80Helmholtz operator, 35Hilbert space, 71Hodge operator, 114homogeneous Maxwell equations, 44homotopy, 113horizontal bundle, 28horizontal lift, 29Hydrogen atom, 89

immersion, 99independent coordinate, 24inertial frame, 5inertial mass, 9inertial system, 5inhomogeneous Maxwell equations, 44inner Euler operators, 34integral curve, 106interior product, 111invariant vector field, 105

Jacobi identity, 104jet, 23jet bundle, 24jet prolongation, 25

Kepler’s problem, 16kinetic energy, 33

Lagrange function, 33Lagrangian, 14Lagrangian action, 17Lagrangian density, 33Lagrangian multiplier, 18Lagrangian system, 33Laplace operator, 116left invariant vector field, 105left translation, 105Legendre transform , 62length contraction, 52Lie algebra, 104, 105Lie bracket, 104Lie derivative, 110Lie group, 101lift of a curve, 29light cone, 49light like vector, 50linear connection, 29local differential form, 30local section, 101local trivialization, 100Lorentz force, 43Lorentz gauge, 45Lorentz system, 48Lorentzian manifold, 114

magnetic field, 43magnetic quantum number, 89

119

metric on a Riemannian manifold, 113Minkowski space, 48

natural bundle, 40natural Hamiltonian system, 62natural system, 14, 34Newton’s law of gravity, 8Newtonian equation, 7Newtonian trajectory, 7Noether identities, 39

observable, 79observer, 51operator norm, 73orbital quantum number, 89orientable manifold, 112

parity transformation, 49partial differential equation, 27partition of unity, 95past, 50past of a point, 50phase curves, 9, 10phase flow, 10phase space, 10, 61physical motion, 7Planck constant, 70Poincare lemma, 113Poincare transformation, 49Poincare’s recurrence theorem, 68Poincare-Cartan integral invariant, 67point spectrum, 77Poisson algebra, 57Poisson equation, 46Poisson manifold, 57potential energy, 8, 34presheaf, 96presymplectic form, 55principal homogenous space, 1principal quantum number, 90projector-valued measure, 76prolongation, 27pseudo-Riemannian manifold, 114pull back of bundles, 103pure states, 72

rapidity, 50ray, 2

ray state, 75regular point of a function, 100regular value of a function, 100relative uniform motion, 5Riemannian manifold, 113rigid body, 37

scalar potential, 46scattering states, 90Schrodinger equation, 87Schrodinger picture, 87Schrodinger representation, 82self-adjoint element, 71semi-direct product of groups, 2sheaf, 96shell, 27simultaneous events, 4slice of a foliation, 59smooth manifold, 94source forms, 34source of a jet, 24space like vector, 50space of motions, 61spectrum, 77spherical harmonics, 89star product, 69state, 72Stokes’ theorem, 113Stone - von Neumann theorem, 83submanifold, 99submersion, 99symplectic form, 55symplectic gradient, 57symplectic manifold, 55

tangent bundle, 102tangent space, 97tangent vector, 97target of a jet, 24Time dilatation, 52time reversal, 49topological manifold, 94trajectory, 6, 61transitive action of a group, 1trivial bundle, 100twin paradoxon, 53

vacuum vector, 91

120

variational bicomplex, 33variational symmetry, 35vector potential, 46velocity, 6, 50vertical bundle, 28vertical lift, 29volume form, 112

weak ∗ topology, 73wedge product, 107Weyl algebra, 82Wightman axioms, 91world line, 6

Young’s inequality, 63

121