PROJECTIVE AND CONFORMAL STRUCTURES IN GENERAL RELATIVITY John Stachel Loops ’07, Morelia June 25-30, 2007

PROJECTIVE AND CONFORMAL STRUCTURES IN GENERAL RELATIVITY

John StachelLoops ’07, Morelia

June 25-30, 2007

Work being done in collaboration with

Mihaela Iftime

Quantum Gravity– The Great Challenge

• How can we combine the background-independent, dynamical approach to space-time of General Relativity with

• Quantum theory, which is based on fixed, absolute background space-time structures?

• That is the fundamental problem of quantum gravity

Goal

Our goal is to contribute to the development of a background-independent, non-perturbative approach to quantization of the gravitational field based on the conformal and projective structures of space-time.

Outline of the Talk

1) What quantization is and is not

2) Measurement analysis and quantization

3) Processes are primary

4) Space-time structures

5) Conformal and projective structures

6) Variational principle based on them

7) Hopes for the future

Outline of the Talk








What is NOT Being Claimed

Quantization only makes sense when applied to “fundamental” structures or entities.

The Mystique Surrounding Quantum Mechanics

“Anything touched by this formalism thereby seems to be elevated– or should it be lowered?– to a fundamental ontological status. The very words ‘quantum mechanics’ conjure up visions of electrons, photons, baryons, mesons, neutrinos, quarks and other exotic building blocks of the universe.”

The Mystique Surrounding Quantum Mechanics (cont’d)

“But the scope of the quantum mechanical formalism is by no means limited to such (presumed) fundamental particles. There is no restriction of principle on its application to any physical system. One could apply the formalism to sewing machines if there were any reason to do so!” (Stachel 1986)

What IS Quantization?

Quantization is just a way accounting for the effects of h, the quantum of action, on any process undergone by some system: “fundamental” or

“composite”

What IS Quantization?

Quantization is just a way accounting for the effects of h, the quantum of action, on any process involving some system,–

or rather on theoretical models of such a system-- “fundamental” or “composite”, in which the collective behavior of a set of more fundamental entities is quantized

The Quantum of Action

"Anyone who is not dizzy after his first acquaintance with the quantum of action has not understood a word."

Niels Bohr

“Atoms and Human Knowledge”--Niels Bohr 1957

“..an element of wholeness, so to speak, in the physical processes, a feature going far beyond the old doctrine of the restricted divisibility of matter. This element is called the universal quantum of action. It was discovered by Max Planck in the first year of this century and came to inaugurate a whole new epoch in physics and natural philosophy. We came to understand that the ordinary laws of physics, i.e., classical mechanics and electrodynamics, are idealizations that can only be applied in the analysis of phenomena in which the action involved at every stage is so large compared to the quantum that the latter can be completely disregarded.

Some Non-fundamental Quanta

1) quasi-particles: particle-like entities arising in certain systems of interacting particles, such as phonons and rotons in hydrodynamics (Landau 1941)

2) phenomenological photons: quantized electromagnetic waves in a homogeneous, isotropic dielectric (Ginzburg 1940)

Solid-state physics: A polariton laser

Leonid V. Butov (Nature 447, 31 May 2007)

At the foundation of modern quantum physics, waves in nature were divided into electromagnetic waves, such as the photon, and matter waves, such as the electron. Both can form a coherent state in which individual waves synchronize and combine. A coherent state of electromagnetic waves is known as a laser; a coherent state of matter waves is termed a Bose–Einstein condensate. But what if a particle is a mixture of an electromagnetic wave and matter? Can such particles form a coherent state? What does it look like?

Solid-state physics: A polariton laser (cont’d)

If the microcavity is the right width, the energies of the cavity photon and the exciton can be made to match up. When this happens the two mix, forming a new particle. This is a combination of matter and electromagnetic waves — an exciton-polariton, or simply 'polariton'. These polaritons inherit some of the lightness of the cavity photons, and have masses much smaller than me.

Successful Quantization

Successful quantization of some classical formalism does not mean that one has achieved a deeper understanding of reality– or better, an understanding of a deeper level of reality. It means that one has successfully understood the effects of the quantum of action on the phenomena (processes) described by the formalism

Peaceful Coexistence in QG

Having passed beyond the quantum mystique, one is free to explore how to apply quantization techniques to various formulations of a theory without the need to single one out as the unique “right” one. One might say: “Let a hundred flowers blossom, let a hundred schools contend”

Three Morals of This Tale (1)

If two such quantizations at different levels are carried out, one may then investigate the relation between them

Example: Crenshaw demonstrates: “A limited equivalence between microscopic and macroscopic quantizations of the electromagntic field in a dielectric”

[Phys. Rev. A 67 033805 (2003)]

Three Morals of This Tale (1 cont’d)

If two such quantizations at the same level are carried out, one may also investigate the relation between them

Example: the relation between loop quantization and usual field quantization of the electromagnetic field: If you “thicken” the loops, they are equivalent (Ashtekar and Rovelli 1992)


The search for a method of quantizing space-time structures associated with the Einstein equations is distinct from:

The search for an underlying theory of all “fundamental” interactions

Carlo Rovelli, 2004 I see no reason why a quantum theory of

gravity should not be sought within a standard interpretation of quantum mechanics (whatever one prefers). …A common [argument] is that in the Copen-hagen interpretation the observer must be external, but it is not possible to be external from the gravitational field. I think that this argument is wrong; if it was correct it would apply to the Max-well field as well (Quantum Gravity, 370).

Carlo Rovelli, 2004 We can consistently use the Copenhagen

interpretation to describe the interaction between a macroscopic classical apparatus and a quantum-gravitational phenomenon happening, say, in a small region of (macroscopic) spacetime. The fact that the notion of spacetime breaks down at short scale within this region does not prevent us from having the region interacting with an external Copenhagen observer (ibid.)


An attempt to quantize the conformal and projective structures does not negate, and need not replace, attempts to quantize other space-time structures. Everything depends on the utility of the results in explaining some physical processes

Outline of the Talk


2) Measureability analysis and quantization






Commutation Relations

One central method of taking into account the quantum of action is by means of introducing commutation relations between various particle (non- rel QM) or field (SR QFT) quantities into the formalism.

But these commutation relations have more than a purely formal significance

Peter G. Bergmann

Collaborator of Einstein

Pioneer in study of quantization of “generally covariant” theories, including GR

Bergmann and Smith 1982 Measurability Analysis for the Linearized

Gravitational Field

“Measurability analysis identifies those dynamic field variables that are susceptible to observation and measurement (“observables”), and investigates to what extent limitations inherent in their experimental determination are consistent with the uncertainties predicted by the formal theory.”

Bergmann & Smith 1982 (cont’d)

Measurability analysis of linearized GR, treated as massless spin 2 field, showed limits of co-measurability of “electric” and “magnetic” components of the linearized Riemann tensor coincide with limits of co-definability imposed by the covariant commutation relations.

Bergmann & Smith 1982 (cont’d)

As in Bohr-Rosenfeld analysis of the co-measurability of electric and magnetic field components, rather than a test point particle, they had to use averages over test bodies occupying space-time regions, and with a similar treatment of the commutation relations.

Louis Crane”Categorical

Geometry and the Mathema-tical Founda-tions of Quantum Gravity” (2006)

Louis Crane:”Categorical Geometry and the Mathematical Foundations

of Quantum Gravity” (2006)

“The ideal foundation for a quantum theory of gravity would begin with a description of a quantum mechanical measurement of some part of the geometry of some region; proceed to an analysis of the commutation relations between different observations, and then hypothesize a mathematical structure for space-time which would contain these relations and give general relativity in a classical limit.

We do not know how to do this at present.”

Stachel”Prolegomena to any future Quantum

Gravity” (2007)

“ ‘measurability analysis’… is based on ‘the relation between formalism and observation’; its aim is to shed light on the physical implications of any formalism: the possibility of formally defining any physically significant quantity should coincide with the possibility of measuring it in principle; i.e., by means of some idealized measurement procedure that is consistent with that formalism.

Non-relativistic QM and special relativistic quantum electrodynamics, have both passed this test ; and its use in QG is discussed in Section 4.

Amelino-Camelia and Stachel”Measurement of the space-time interval between two events …” (2007)

We share the point of view emphasized by Heisenberg and Bohr and Rosenfeld, that the limits of definability of a quantity within any formalism should coincide with the limits of measurability of that quantity for all conceivable (ideal) measurement procedures. For well-established theories, this criterion can be tested. For example, in spite of a serious challenge, source-free quantum electro-dynamics was shown to pass this test.

Amelino-Camelia and Stachel (cont’d)

In the case of quantum gravity, our situation is rather the opposite. In the absence of a fully accepted, rigorous theory, exploration of the limits of measurability of various quantities can serve as a tool to provide clues in the search for such a theory: If we are fairly certain of the results of our measurability analysis, the proposed theory must be fully consistent with these results.”

Outline of the Talk








Lee Smolin “[R]elativity theory and quantum theory each ...

tell us-- no, better, they scream at us- that our world is a history of processes. Motion and change are primary. Nothing is, except in a very approximate and temporary sense. How something is, or what its state is, is an illusion. It may be a useful illusion for some purposes, but if we want to think fundamentally we must not lose sight of the essential fact that 'is' is an illusion. So to speak the language of the new physics we must learn a vocabulary in which process is more important than, and prior to, stasis.“ (p. 53).

Carlo Rovelli, 2004 “The data from a local experiment

(measurements, preparation, or just assumptions) must in fact refer to the state of the system on the entire bound-ary of a finite spacetime region. The field theoretical space ... is therefore the space of surfaces Σ [where Σ is a 3d surface bounding a finite spacetime region] and field configurations φ on Σ . Quantum dynamics can be expressed in terms of an [probability] amplitude W[Σ , φ].

Rovelli, 2004 (cont’d)Following Feynman’s intuition, we can formally define W[Σ , φ] in terms of a sum over bulk field configurations that take the value φ on Σ. … Notice that the dependence of W[Σ, φ] on the geometry of Σ codes the spacetime position of the measuring apparatus. In fact, the relative position of the components of the apparatus is determined by their physical distance and the physical time elapsed between measurements, and these data are contained in the metric of Σ.” (p. 23).

Rovelli, 2004 (cont’d)

Consider now a background independent theory. Diffeomorphism invariance implies immediately that W[Σ , φ] is independent of Σ ... Therefore in gravity W depends only on the boundary value of the fields. However, the fields include the gravitational field, and the gravitational field determines the spacetime geometry. Therefore the dependence of W on the fields is still sufficient to code the relative distance and time separation of the components of the measuring apparatus! (p. 23)

Rovelli, 2004 (cont’d)

What is happening is that in background-dependent QFT we have two kinds of measurements: those that determine the distances of the parts of the apparatus and the time elapsed between measurements, and the actual measurements of the fields’ dynamical variables. In quantum gravity, instead, distances and time separations are on an equal footing with the dynamical fields. This is the core of the general relativistic revolution, and the key for back-ground- independent QFT

Outline of the Talk



3) Processes are primary:

Loops– but what kind?





Loops and Holonomies

Loop integrals and holonomies may well be the most suitable set of observables for quantization. But:

Loop integrals of a one-form are independent of all other space-time structures. Why confine the loops to space-like hypersurfaces?

Example: The Two Aharonov-Bohm Effects

There are two Aharonov-Bohm effects:

1) Magnetic: involving space-like loops

2) Electric: involving time-like loops

Aharonov-Bohm Effect (cont’d) In the terms of modern differential geometry,

the Aharonov-Bohm effect can be understood to be the holonomy of the complex-valued line bundle representing the electromagnetic field. The connection on the line bundle is given by the electromagnetic potential A, and thus the electromagnetic field strength is the curvature of the line bundle F=dA. The integral of A around a closed loop is the holonomy…

Electric Aharonov-Bohm effect Just as the phase of the wave function de-

ends upon the magnetic vector potential, it also depends upon the scalar electric potential. By constructing a situation in which the electrostatic potential varies for two paths of a particle, through reg-ions of zero electric field, an observable Aharonov-Bohm interference phenomen-on from the phase shift has been predict-ed; again, the absence of an electric field means that, classically, there would be no effect.

Outline of the Talk








Space-time Structures

There are a number of space-time structures that play an important role in the general theory of relativity.

Pseudo-Metric tensor field

The chrono-geometry is represented mathematically by a pseudo-Riemannian metric tensor field on a four-dimensional manifold

Assume that the spatial metric is locally Euclidean, but globally non flat: there is a curvature that varies from point to point. What kind of curvature?

Bernhard Riemann- From Globally to Locally Euclidean

Karl Friedrich Gauss & Surface Curvature

What are the radii R1, R2, .. of the circles that best fit the normal cross sections of the surface?

The product of the inverses 1/(RmaxRmin) is the Gaussian curvature

Higher Dimensions

Riemann extended Gauss’ ideas about curvature of a 2-D surface to higher dimensions, and related this curvature to the (metric) Riemann tensor:

R[μν][κλ]

Another Kind of Curvature

But there is another kind of curvature, more important for geometrizing gravitation

To understand it, we must turn back to Hermann Grassmann and affine spaces

Affine Space – Parallelism is all!

Forget about distance, keep concept of parallel lines;

Only the ratio of parallel intervals has meaning;

Affine Curvature

Tullio Levi Civita did for affine space what Riemann had done for Euclidean space:

He went from global to local, and this enabled a new interpretation of curvature

Affine Curvature

Geodesic Deviation

Geodesic Deviation (cont’d)

Is there any relative acceleration between two nearby freely-falling test bodies (i.e., each following a geodesic)

The amount of such relative accelerations in various directions is a measure of the components of the affine curvature tensor

Hermann Weyl and Affine Connections

The inertio-gravitational field is represented by a symmetric affine connection Γκ

μν on the space-time manifold

But Geodesic Deviation

Physically = Tidal Forces !!

Tidal Force “The tidal force is a secondary

effect of the force of gravity and is responsible for the tides. It arises because the gravitational field is not constant across a body's diameter” (Wikipedia).

Parallel Displacement: A Post-Mature Concept

Riemann went from global Euclidean to local Euclidean with Gaussian curvature (1854)

Grassmann developed global affine geometry (1844, 1862)

Someone should have gone from global affine to locally affine with Riemannian curvature by 1880 at the latest– but no one did (poor AE!)

Compatibility Conditions

In GR there are compatibility conditions between the chrono-geometry and the inertio-gravitational field.

Mathematically:

The covariant derivative of the metric with respect to the connection must vanish:

g;k = 0.

Physical Interpretations Pseudo-metric tensor field g: Chrono-geometry governs behavior of

rods and clocks, proper spatial, temporal and null intervals between events;

Affine connection field Γκμν :

Inertio-gravitational field governs behavior of (structureless) particles, geodesic curves (paths) together with preferred parameterization =proper tiime);

Physical Interpret’ns (cont’d)

Compatibility conditions Dg = 0 :

1) proper spatial and temporal intervals are preferred parameters along affine parallel (geodesic) paths

2) rods and clocks still read proper spatial and temporal intervals as they move in inertio-gravitational field

Original 2nd Order Formalism Start with just the metric field g and derive

from it the unique symmetric connection (the Christoffel symbols {...}) that identically satisfies the compatibility conditions.

Variation of a second order Lagrangian (integrand = the densitized curvature scalar

√-g R ) leads to the field equations of second order in

the metric. This is the route first taken by Hilbert, and still followed by most textbooks.

First Order Formalism It is also possible to treat metric and

connection initially as independent structures, and then derive the compatibility conditions between them, together with the first order field equations (written initially in terms of the connection), from a so-called Palatini variational principle (not really what Palatini did, but AE called it so).

Metric-Connection Duality In this approach, metric and connection are in

a sense dual, like coordinates and momenta in mechanics.

Major technical advances in the canonical quantization program came when:

rather than taking the three-metric as coordinates (geometrodynamics)

A three-connection was taken as coordinates (loop quantum gravity)

Tetrads, Forms

Either the second or first order method may be combined with a tetrad formalism for the metric, introducing tetrad vectors eμ

A (& dual eμA) and tetrad

metric ηAB , with corresponding local

symmetry group SO(3,1) or. if spinor formalism is used, SU(2), together with

Tetrads, Forms (cont’d)

a corresponding mathematical representation of the connection, e.g., matrix of connection one-forms Γμ

AB , or tetrad components of the

connection ΓCA

B , or corresponding

spin connections with local symmetry group GL(4, R).

Outline of the Talk








Further Decomposition But neither metric nor connection is irreducible.

One can go a step further and decompose both metric and affine connection themselves.

In homogeneous (flat) geometries:

metric geometry is a special case of conformal geometry;

affine geometry is a special case of projective geometry

.

Felix Klein’s Erlangen Program

"The remarkable power of this program is revealed in its applicability to situations which Klein himself had not yet envisaged."

H. S. M. Coxeter

Erlangen Program (cont’d) Different geometries can

co-exist, because they dealt with different types of propositions and invariances related to different types of symmetry transform’ns. By abstracting the underlying symmetry groups from the geometries, the relations between them can be re-established at the group level.

Erlangen Program (example) The distinction between affine geometry and

projective geometry lies just in the fact that affine-invariant notions such as parallelism are the proper subject matter of the first, while not being principal notions in the second. Since the group of affine geometry is a (non-empty) subset of the group of projective geometry, any notion invariant in projective geometry is a priori meaningful in affine geometry; but not the other way round. If you add required symmetries, you have a more powerful theory (which is deeper and more general) but with fewer concepts and theorems .

Elie Cartan takes conformal and

projective from global to local Sur les variétés à

connexion projective (1922)

Sur les espaces a connexion conforme (1923)

G-Structures

G-structures are the continuation of the Klein program in spaces that are only locally flat in the appropriate sense of the word.

Keep the local group but drop the translations, replacing them with some non-flat Cartan connection

Conformal Structure

If one abstracts from the four- volume-defining property of the metric, one gets the conformal structure on the manifold.

Physically, this conformal structure represents the causal structure of space-time. One can measure the conformal structure of space-time by using massless test entities

The Eikonal Equation for Light

gμνΦ,μΦ,ν = 0 ,

where Φ is the phase of the light wave,

is invariant under conformal transformations

Light rays are the bi-characteristics of this equation

Projective Structure Similarly, if one abstracts from the

preferred parametrization (proper length along spacelike, proper time along timelike) of the affine parallel curves associated with an affine connection, one gets a projective structure on the manifold.

Physically the projective structure picks out the class of preferred paths in space-time. One can measure the projective structure by use of massive test bodies

Projective Transformations Γ’κ

μν= Γκμν + δκ

μ pν+ δκν pμ, ,

where pν is an arbitrary vector, preserve the affine parallel paths.

Since they affect only the trace of the affine connection, we can form projectively invariant quantities by removing the trace:

Projective parameters:

Πκμν= Γκ

μν–1/5(δκμΓν+δκ

νΓμ),

Γμ= Γκμκ

Parallel paths:

d2xκ/dλ2 + Πκμν (dxμ/dλ)(dxν/dλ) = 0

where λ is the projective parameter, invariant under all projective

transformations

E-P-S: From Conformal and Projective Strs to Metric Str

Ehlers, J., Pirani, F., and Schild, A. (1973). "The Geometry of Freefall and Light Propagation", in L. O' Raifeartaigh (ed.), General Relativity: Papers in Honour of J.L. Synge. (Oxford: Clarendon Press), 63-84.

The geometry of free fall and light propagation J. Ehlers, F.A.E. Pirani and A. Schild

The geometry of the space-time has a conformal Lorentzian structure that explains the phenomenon of light propagation.

In order to obtain the Lorentzian metric structure of the space-time, just as in General Relativity theory, these authors carry out the following steps.

E-P-S ( cont’d)

Firstly, they introduce a projective structure that represents the movement of free test particles, suffering gravitational effects only.

Then, a compatibility condition is necessary to link the projective and conformal structures in order to obtain a symmetric linear connection, the so called Weyl connection. This mathematical condition is physically justified by the experimental fact that a particle with mass can be made to chase a photon arbitrarily closely .

E-P-S ( concl’n) Finally, they introduce the axiom which affirms

that the Ricci tensor of the connection must be symmetric so that the Weyl connection becomes the linear connection associated with a Lorentzian metric.

Once this axiom at hand has been physically justified, they deduce the existence of a Lorentzian metric structure, uniquely determined except for a constant factor.

2 Compatibility Conditions

A compatibility condition between the causal and projective structures can be defined, leading to a Weyl connection and a further condition that guarantees the existence of a metric compatible with the affine connection

First Compatibility Condition

If it is obeyed, the affine connection is fixed uniquely. So concept of proper time along a timelike world line emerges from this condition.

But this proper time subject to a second clock effect: two initially synchronous clocks will not run at the same rate when reunited.

Second Compatibility Condition

If it is obeyed, the second clock effect disappears.

Moreover, the conformal factor is fixed, so a unique four-volume element is defined

Volume

StructureProjectiveStructure

ConformalStructure

Lorentzmetric

Linear Symmetricconnection

Weyl connection

R[ab]=0 <=>

Summary

• Conformal structure: measured by massless test entities

• Projective structure: measured by massive test entities

• First compatibility condition: Concept of proper time emerges

• Second compatibility condition: Concept of proper four-volume emerges

Outline of the Talk








New Palatini-Type Variational Principle

In a manifold with metric, its conformal and affine structures suffice to determine that metric up to an overall constant factor (Weyl1921).

This circumstance enables us to set up a new first order, Palatini-type variational principle for general relativity.

Further Breakup Break up the metric into its

volume-determining part (essentially, the determinant of the metric) and its conformal, causal structure-determining part (with unit determinant);

Further Breakup (cont’d) Break up the affine connection

into its projective, preferred path-determining part (the trace-free part of the connection) and its preferred parameter-determining part ( the trace of the connection).

Each of these four parts may then be varied independently

Pseudo-Metric The pseudo-metric gν can be built up out

of two parts gν and γ.

The first part gν

is a tensor density of weight -1/2, with determinant = -1, invariant under conformal transformations.

Pseudo-Metric (cont’d)The second part γ is a scalar density of weight +1/2 and transforms under conformal transformations as:

γ → 'γ = φ γ . Then, we may define

gν= γ gν , so it is a pseudo-metric tensor; with det g= -γ4

and inverse gν = γ-1gν

Pseudo-Metric (cont’d) Since det gν = -1, the components of its

inverse gν will be cubic functions of the components of gν.

Thus, the Lagrangian will be cubic in these variables– important for quantization?

The Connection The connection Γκ

μν can be built up out of

two parts, Pκμν and Γμ . The first part Pκ

μν

is traceless, Pκμκ = 0, and invariant under

projective transformations.

The second part Γμ transforms under a

projective transformation as:

Γμ→ Γμ+Tμ .

The Connection (cont’d) The connection is then:

Γκμν = Pκ

μν +(1/5) (δκμΓν + δκ

νΓμ) .

Since Pκμν is traceless,

Γμ = Γκμκ ,

Conformal Curvature Tensor

There is a decomposition of the Riemann tensor in terms of the Weyl conformal curvature tensor C[μ] [κλ] ,

the traceless part of the Ricci tensor Sν

Sν = Rμ -1/4 Rgν ,

and the Ricci scalar R:

R[ν][κλ] = C[ν][κλ] + E[ν][κλ] + (1/12) R G[ν][κλ] ,

where:

E[ν][κλ] = 1/2(Sκgλ -Sλgκν +Sλgκ -Sκ gλ) ,

G[ν][κλ] = gκgλ -gλgκν

Conformal Curvature Tensor (cont’d)

The conformal curvature tensor with one raised index C

[κλ] = γ-1gκ C[κ][κλ] is

invariant under conformal transforma-tions, so it cannot depend on γ but only on g ν.

This means that C[][κλ] must be γ times a

function of gν. and its derivatives.

Projective Curvature Tensor

The affine curvature tensor A [κλ] , formed from

the affine connection and its first derivatives, can be written in terms of Pκ

μν and Γμ , and their

derivatives. There is a decompo-sition of the affine curvature tensor in terms of the projective curvature tensor Pν

[κλ] , which is also traceless,

and the Ricci tensor Aν of the affine curvature

tensor:

A [κλ] = P

[κλ]+ 2 (δνμP[κλ])+ 2(δν

[κPλ]μ) ,

Projective Curvature Tensor(cont’d)

where:

Pκλ = (4/15)Aκλ - (1/15)Aλκ .

This definition of Pκλ simplifies the breakup of the affine curvature tensor, and it has a very simple transformation law under projective transformations.

The projective curvature tensor P [κλ] is

invariant under projective transformations, so it cannot depend on Γ .

Lagrangian The Lagrangian density is then:

L = γgνAμν (Pκ

μν, Γμ),

and variations are to be taken with respect to

γ , gν, Pκμν and Γμ.

Compatibility Conditions Variations with respect to Pκ

μν and Γν give

2 compatibility conditions between the conformal and projective structures:

Dλ{[γgν]-(1/5)(δμλDκ[γg

νκ] + δνλDκ[ γg

κ]) = 0. Dν[γgν]= 0 ,

where Dλ stands for the covariant

derivative w.r.t. the affine connection

Weyl Space Dλ{[γgν]-(1/5)(δμ

λDκ[γgνκ] + δν

λDκ[γgκ]) = 0

implies that the conformal and projective structures together define a Weyl space.

If we define a tensor density Qλν:

Dλ[γgν] = Qλ

ν,

then the equation above can be written:

Qλν-1/5(δμ

λQκνκ

+ δνλ Qκ

κ) = 0 .

These are 36 equations for 40 unknowns

Weyl Space (cont’d) Setting:

Qλν = - γgν Qλ ,,

where Qλ is an arbitrary one-form, it is

easily shown it is the general solution.

But the condition:

Dλ[γgν] = - γgνQλ

defines a Weyl space .

Weyl Space (cont’d) The affine connection so defined is actually

unique. If one carries out a conformal transformation on the metric gν →φ gν , then

a corresponding transformation:

Qλ → Qλ - ∂λ log φ

leaves the connection invariant. Thus, in a Weyl space, the connection is fixed, but the metric is only fixed up to a conformal factor.

From Weyl to Pseudo-Riemannian Space

Imposing the second equation:

Dν[γgν]= 0

forces Qλ to vanish. We then have a

pseudo-Riemannian metric and connection

Relation to Causal Sets This breakup is of

special interest because the causal set theory approach to quantum gravity, developed by Rafael Sorkin (but cf. Kronheimer and Penrose), is based on taking the conformal structure and the four-volume structure as the primary constituents of the classical theory, and then replacing them with discretized versions

Rel’n to Causal Sets (cont’d) The causal set approach

combines the two ideas of: discreteness (four-volume

structure) and order (conformal structure)

to produce a structure, on which a theory of quantum gravity can be based.

.

Problems With Causal Sets However, in causal set theory: 1) quantization of four-volume is

simply inserted by hand; 2) no attention has been paid to the

affine connection, and hence the possibility of finding quantum analogues of the traceless (projective) and trace parts of the connection;

3) no real quantization has been achieved: where is h?

Outline of the Talk








“Zukunftsmusik” (Promissory Notes)

Program of work:

Program

Extend the classical measurability analysis of the conformal and projective structures from point particles to extended massless test entities and massive test bodies.

Program

Investigate effects of quantum of action on the measurability and co-measurability of these structures

See what this implies about possible commutation relations between them

Program

Investigate the effect of the quantum of action on the compatibility conditions.

In particular, what are the implications for possible quantization of the proper time and proper four-volume?

Program

• Reformulate the theory in terms of Cartan connections and curvatures for the conformal and projective structures

• Investigate the conformal and projective holonomies, and:

• Their relation to LQG based on metric and affine connection

Program

The conformal structure singles out null hypersurfaces, which suggest application of null-hypersurface quantization techniques.

Null hypersurface breaks up (2+1) naturally into spacelike 2-surfaces plus null congruence, suggesting use of conformal 2-structure as the dynamical degrees of freedom of the field.

Program

Carry out the quantization program for the cylindrical wave models (midi-superspaces) with both degrees of polarization, and compare with existing results

Final Summary

Classically, the conformal and projective structures provide a sort of minimal basis for constructing Lorentzian space-times based on the behavior of massless and massive entities.

Study of their quantization seems a worthwhile avenue of approach to the problem of quantum gravity

Decomposition of Pseudo-Metric

Pseudo-metric Structure: gμν = ηAB eμA

eνB

Conformal Structure: {k eμA } , where k is any

non-vanishing scalar field

Volume Structure: {│eμA│} , where the value of

the determinant remains fixed

Decomposition of Affine Structure

Affine Structure: ΓμA

B , matrix-valued one-form

Due to antisymmetry, ΓμA

A vanishes, so we have to

construct the trace

The physical components of the connection are:

ΓBA

C = ΓμA

C eμB

Decomposition of Affine Structure (cont’d)

Their trace is:

ΓC = ΓAA

C = ΓμA

C eμA

We convert it into a one-form by con-tracting it with eν

C:

Γν = eν

C ΓC

Cartan-Maurer Equations

Insert this decomposition into the Cartan-Maurer structure equations to determine the Riemann two-form, etc.

Intersection of G-structures of first or secondorder

I. Sanchez-Rodrıguez

Abstract. The concept of G-structure of first or second order includes the majorityof geometrical structures that can be defined on a manifold. We analyse the intersection of G-structures with different kinds of G. In particular, we discuss the relations among conformal, (semi) Riemannian, volume and projective geometrical structures on a fixed manifold. Some consequences are pointed out relative to geometrical aspects of General Relativity.

Sanchez-Rodriguez

The results, that I will show here, prove that given a volume structure and a conformal structure, we obtain a uniquely determined metric structure.

This suggests that we must investigate the physical motivation in order to introduce a volume structure as a component of the space-time geometry.

Reductions of a Principal Bundle

Let G be a Lie group and let P(M,G) be a principal bundle over M with G as structure group.

If we have a closed subgroup H of G, we define a reduction Q(M,H) of P, or an H-reduction, as a subbundle Q of P with H as structure group.

Semi-Riemannian structures Let η be the standard scalar product on Rn of a

fixed signature. Oη(n) -structures or (semi) Riemannian

structures are the reductions of LM with subgroup

Oη(n) :={a ε GL(n,R) : a†a = In } where

In is the identity matrix in GL(n,R) a† is the unique matrix such that

η(v, a†w) = η(av,w), v,w ε Rn.

Conformal Structures

The COη(n)-structures or conformal structures are the reductions of LM with subgroup

COη(n) := {a GL(n,R) : a†a = k In, k > 0}

A COη(n)-structure on M determines and is determined by a set {g} of metric tensors which are proportional by a positive factor to a given metric tensor g on M (i.e., g {g} if and only if

g = ωg, with ω: M → R+).

Volume Structures The SL±(n)-structures or volume structures are the

reductions of LM with subgroup

SL± (n) := {a GL(n,R) : |det(a)| = 1} An SL± (n)-structure on M can be understood as a

selection, for every point x M, of a maximal set of basis of TmM with the same unoriented volume, in the sense of linear algebra.

It determines a family of local volume forms, which when applied to a basis belonging to the SL± (n)-structure, gives +1 or −1.

The existence of volume structures does not depend on the orientability of M as in the case of SL(n)-structures, but if M is an orientable manifold a SL± (n)-structure on M determines and is determined by a global volume form, unique except for the sign.

Semi-Riemannian Structures

It follows that:

CO(n) ∩ SL±(n) = O(n) And

GL(n,R) = CO(n) SL± (n) Theorem: The (semi) Riemannian

structures on M are the intersections of conformal and volume structures on M.

In other words:

Given a volume structure and a conformal structure, we obtain a uniquely determined metric structure.

Application to geometrical structures of second order

The second order frame bundle of M, F2M(M,G2(n)) is the bundle of 2-jets at 0 Rn of inverse of charts of M

The group G2(n) is the group of 2-jets at 0 Rn of (local) diffeomorphisms φ of Rn, with φ(0) = 0

It is described as the semi-direct product of Lie groups GL(n,R) S2(n), with the product law

(a, t)•(a’, t’):= (aa’, a−1 t(a’, a’) + t’),

Affine and Projective structures

We can identify F2M with the first prolongation (LM)1 of LM.

A G-structure of second order on M is a G-reduction of F2(M), with G being a subgroup of

G2(n) ≡ GL(n,R) Sym2(n).

The symmetric linear connections of M are examples of G-structures of second order, when

G = GL(n,R) {0}

The projective structures on M are also examples of it, when G = GL(n,R)x p, where

p := {t ε S2(n) : tijk = δij pk + δikpj }

for some (pi ) ε Rn.

Projective & volume structures

Theorem The intersection of a projective structure on M and the first prolongation of a volume structure on M gives a symmetric linear connection on M.

Theorem The intersection of the first prolongation of a conformal structure on M and the first prolongation of a

volume structure on M gives a symmetric linear connection on M.

Space-Time Structures Physically the conformal structure is

determined by the local behavior of null wave-fronts and rays; and the projective structure by the local behavior of freely falling massive test particles, respectively. In general relativity, these structures may be taken as fundamental, and the pseudo-Riemannian metric and affine connection derived from them.

Two Degrees of Freedom Various initial value problems in GR

may be reformulated on the basis of the conformal-projective breakup-- in particular, null-initial value problems and the 2+2 decomposition of the field equations-- with the aim of investigating how best to isolate the two degrees of freedom of the gravitational field, a question of crucial importance for their quantization.

Measurability Analysis The quantum of action sets limits on the

co-measurability of various physically measurable quantities, and thereby determines their commutation relations. Hence, the co-measurability of quantities derived from the conformal and projective structures, such as the conformal 2-structure, will be analyzed as a heuristic guide to their quantization.

Phonons A phonon is a quantized mode of vibration

occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because phonons play an important role in many of the physical properties of solids, such as the thermal conductivity and the electrical conductivity. In particular, the properties of long-wavelength phonons gives rise to sound in solids -- hence the name phonon. In insulating solids, phonons are also the primary mechan-ism by which heat conduction takes place.

Phonons (cont’d) Phonons are a quantum mechanical version of a special

type of vibrational motion, known as normal modes in classical mechanics, in which each part of a lattice oscillates with the same frequency. These normal modes are important because, according to a well-known result in classical mechanics, any arbitrary vibrational motion of a lattice can be considered as a superposition of normal modes with various frequencies; in this sense, the normal modes are the elementary vibrations of the lattice. Although normal modes are wave-like phenomena in classical mechanics, they acquire certain particle-like properties when the lattice is analysed using quantum mechanics. They are then known as phonons. Phonons are bosons possessing zero spin.

Documents

PROJECTIVE AND CONFORMAL STRUCTURES IN GENERAL RELATIVITY John Stachel Loops ’07, Morelia June 25-30, 2007