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General Relativity in a Nutshell
(And Beyond)
Federico Faldino
Dipartimento di Matematica
Università degli Studi di Genova
27/04/2016
1 Gravity and General Relativity
2 Quantum Mechanics, Quantum Field Theory and All That...
3 An insight into QFT on Curved Backgrounds
1 Gravity and General Relativity
2 Quantum Mechanics, Quantum Field Theory and All That...
3 An insight into QFT on Curved Backgrounds
Newtonian Gravity
Newton's Law of Gravitation
F = GmGM
r3r G = 6.67 · 10−11 N ·m
2
kg2
This formulation provides problems (e.g. precession of Mercury's orbitperihelion, wrong deviation of light rays, instantaneous propagation)
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 1 / 27
Newtonian Gravity
Newton's Law of Gravitation
F = GmGM
r3r G = 6.67 · 10−11 N ·m
2
kg2
This formulation provides problems (e.g. precession of Mercury's orbitperihelion, wrong deviation of light rays, instantaneous propagation)
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 1 / 27
Galilean Relativity
Principle (Galilean Relativity)
The laws of Mechanics are invariant under a change of inertial frame (IF).
Galileo's Transformationsx ′ = x − vt
y ′ = y
z ′ = z
t ′ = t
Velocity Transormation Law: u′ = u − v
Maxwell Equations are not invariant under Galilean Relativity
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 2 / 27
Galilean Relativity
Principle (Galilean Relativity)
The laws of Mechanics are invariant under a change of inertial frame (IF).
Galileo's Transformationsx ′ = x − vt
y ′ = y
z ′ = z
t ′ = t
Velocity Transormation Law: u′ = u − v
Maxwell Equations are not invariant under Galilean Relativity
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 2 / 27
Galilean Relativity
Principle (Galilean Relativity)
The laws of Mechanics are invariant under a change of inertial frame (IF).
Galileo's Transformationsx ′ = x − vt
y ′ = y
z ′ = z
t ′ = t
Velocity Transormation Law: u′ = u − v
Maxwell Equations are not invariant under Galilean Relativity
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 2 / 27
Special Relativity
Principle (Einstein's Relativity)
The laws of Physics are invariant under a change of inertial frame.
Lorentz Transformations
For two IF R e R ′ in x-standard con�guration, assuming space and timeisotropy and homogeneity:
x ′ = x−vt√1− v
2
c2
y ′ = y
z ′ = z
t ′ =t− v
cx√
1− v2
c2
Velocity Transformation Law: u′ =u − v
1− uvc2
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 3 / 27
Special Relativity
Principle (Einstein's Relativity)
The laws of Physics are invariant under a change of inertial frame.
Lorentz Transformations
For two IF R e R ′ in x-standard con�guration, assuming space and timeisotropy and homogeneity:
x ′ = x−vt√1− v
2
c2
y ′ = y
z ′ = z
t ′ =t− v
cx√
1− v2
c2
Velocity Transformation Law: u′ =u − v
1− uvc2
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 3 / 27
Minkowski Spacetime M
De�nition
We call Minkowski Spacetime M the vector space R4 endowed with
Orientation;
Metric η with signature (+−−−) (or (−+ ++));
Time Orientation.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 4 / 27
Causal Structure on M
Timelike η(u, u) > 0;
Spacelike η(u, u) < 0;
Lightlike η(u, u) = 0;
Causal η(u, u) ≥ 0.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 5 / 27
Causal Structure on M
Timelike η(u, u) > 0;
Spacelike η(u, u) < 0;
Lightlike η(u, u) = 0;
Causal η(u, u) ≥ 0.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 5 / 27
Equivalence Principle
Let us turn on gravity and remember Newton's Second Law
F = mIa F = GMmG
r3r
In principle mI 6= mG since they correspond to two di�erent physicalproperties!
Weak Equivalence Principle ⇒ Existence of Local Inertial Frames
Principle (Equivalence Principle)
In small enough regions of space-time, the laws of physics reduce to thoseof special relativity; it is impossible to detect the existence of agravitational �eld by means of local experiments.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 6 / 27
Equivalence Principle
Let us turn on gravity and remember Newton's Second Law
F = mIa F = GMmG
r3r
In principle mI 6= mG since they correspond to two di�erent physicalproperties!
Weak Equivalence Principle ⇒ Existence of Local Inertial Frames
Principle (Equivalence Principle)
In small enough regions of space-time, the laws of physics reduce to thoseof special relativity; it is impossible to detect the existence of agravitational �eld by means of local experiments.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 6 / 27
Equivalence Principle
Let us turn on gravity and remember Newton's Second Law
F = mIa F = GMmG
r3r
In principle mI 6= mG since they correspond to two di�erent physicalproperties!
Weak Equivalence Principle ⇒ Existence of Local Inertial Frames
Principle (Equivalence Principle)
In small enough regions of space-time, the laws of physics reduce to thoseof special relativity; it is impossible to detect the existence of agravitational �eld by means of local experiments.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 6 / 27
The Einstein' way to General Relativity
We want to obtain an analogue for the Poisson Equation for the �classic�gravitation potential:
∆ϕG = 4πGρ.
We look for:
2nd order tensorial equations, linear in the derivatives of greater order;
We need to reach the Newtonian theory in a suitable limit;
The equations must grant the condition ∇iTik = 0 (freely gravitating
mass-energy).
Einstein Equations: Rµν −1
2gµνR + (Λgµν) = −4πG
c4Tµν
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 7 / 27
The Einstein' way to General Relativity
We want to obtain an analogue for the Poisson Equation for the �classic�gravitation potential:
∆ϕG = 4πGρ.
We look for:
2nd order tensorial equations, linear in the derivatives of greater order;
We need to reach the Newtonian theory in a suitable limit;
The equations must grant the condition ∇iTik = 0 (freely gravitating
mass-energy).
Einstein Equations: Rµν −1
2gµνR + (Λgµν) = −4πG
c4Tµν
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 7 / 27
A Pictorial Viewpoint
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 8 / 27
Some well-known solutions to the Einstein Equations
Schwarzschild:
ds2 = −(1− 2M
r
)dt2 +
(1− 2M
r
)−1dr2 + r2dθ2 + r2 sin2 θdφ2
Friedmann - Lemaitre - Robertson - Walker (FLRW):
ds2 = a2(t)
[− dt2
a2(t)+ dx2 + dy2 + dz2
]Kerr
de Sitter
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 9 / 27
Quantum Gravity?
Rµν −1
2gµνR = −8πTµν
Possible Quantum Gravity Theory:
String Theory
Loop Quantum Gravity
Other Approaches (Twistors, Non-Commutative Geometry, etc.)
Up to now we are far from the solution...
Semi-classical theory of Gravity and QFT over Curved Backgrounds
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 10 / 27
Quantum Gravity?
Rµν −1
2gµνR = −8πTµν
Possible Quantum Gravity Theory:
String Theory
Loop Quantum Gravity
Other Approaches (Twistors, Non-Commutative Geometry, etc.)
Up to now we are far from the solution...
Semi-classical theory of Gravity and QFT over Curved Backgrounds
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 10 / 27
Quantum Gravity?
Rµν −1
2gµνR = −8πTµν
Possible Quantum Gravity Theory:
String Theory
Loop Quantum Gravity
Other Approaches (Twistors, Non-Commutative Geometry, etc.)
Up to now we are far from the solution...
Semi-classical theory of Gravity and QFT over Curved Backgrounds
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 10 / 27
1 Gravity and General Relativity
2 Quantum Mechanics, Quantum Field Theory and All That...
3 An insight into QFT on Curved Backgrounds
What a �Quantum Theory� is?
Oxford Dictionary: a �Quantum� is �a discrete amount of any physicalquantity, such as energy, momentum or electric charge.�
States: Vectors |ψ〉 in a suitable Hilbert spaces H;Observables: Self-adjoint operators on H;Time Evolution: Two ways for implementing it:
Schrödinger Picture Evolution equation for vector states;Heisenberg Picture Evolution equation for the observables.
Everything can be made rigorous using the Algebraic Formulation!
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27
What a �Quantum Theory� is?
Oxford Dictionary: a �Quantum� is �a discrete amount of any physicalquantity, such as energy, momentum or electric charge.�
States: Vectors |ψ〉 in a suitable Hilbert spaces H;
Observables: Self-adjoint operators on H;Time Evolution: Two ways for implementing it:
Schrödinger Picture Evolution equation for vector states;Heisenberg Picture Evolution equation for the observables.
Everything can be made rigorous using the Algebraic Formulation!
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27
What a �Quantum Theory� is?
Oxford Dictionary: a �Quantum� is �a discrete amount of any physicalquantity, such as energy, momentum or electric charge.�
States: Vectors |ψ〉 in a suitable Hilbert spaces H;Observables: Self-adjoint operators on H;
Time Evolution: Two ways for implementing it:
Schrödinger Picture Evolution equation for vector states;Heisenberg Picture Evolution equation for the observables.
Everything can be made rigorous using the Algebraic Formulation!
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27
What a �Quantum Theory� is?
Oxford Dictionary: a �Quantum� is �a discrete amount of any physicalquantity, such as energy, momentum or electric charge.�
States: Vectors |ψ〉 in a suitable Hilbert spaces H;Observables: Self-adjoint operators on H;Time Evolution: Two ways for implementing it:
Schrödinger Picture Evolution equation for vector states;Heisenberg Picture Evolution equation for the observables.
Everything can be made rigorous using the Algebraic Formulation!
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27
What a �Quantum Theory� is?
Oxford Dictionary: a �Quantum� is �a discrete amount of any physicalquantity, such as energy, momentum or electric charge.�
States: Vectors |ψ〉 in a suitable Hilbert spaces H;Observables: Self-adjoint operators on H;Time Evolution: Two ways for implementing it:
Schrödinger Picture Evolution equation for vector states;
Heisenberg Picture Evolution equation for the observables.
Everything can be made rigorous using the Algebraic Formulation!
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27
What a �Quantum Theory� is?
Oxford Dictionary: a �Quantum� is �a discrete amount of any physicalquantity, such as energy, momentum or electric charge.�
States: Vectors |ψ〉 in a suitable Hilbert spaces H;Observables: Self-adjoint operators on H;Time Evolution: Two ways for implementing it:
Schrödinger Picture Evolution equation for vector states;Heisenberg Picture Evolution equation for the observables.
Everything can be made rigorous using the Algebraic Formulation!
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27
What a �Quantum Theory� is?
Oxford Dictionary: a �Quantum� is �a discrete amount of any physicalquantity, such as energy, momentum or electric charge.�
States: Vectors |ψ〉 in a suitable Hilbert spaces H;Observables: Self-adjoint operators on H;Time Evolution: Two ways for implementing it:
Schrödinger Picture Evolution equation for vector states;Heisenberg Picture Evolution equation for the observables.
Everything can be made rigorous using the Algebraic Formulation!
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27
Canonical Quantisation
In Schrödinger picture the vector states are represented byψ(x ; t) ∈ L2(R3), which are the components of
|ψ(t)〉 =
∫dx ψ(x ; t) |x〉 .
We associate a quantum observable to every classical one via the�promotion to operator� prescription. In Schrödinger picture:
x → x p → p.
= −i∂x E → E.
= i∂t .
x and p satisfy the commutation relation
[x , p] = i .
From the classical energy-dispersion relation we obtain the SchrödingerEquation:
E =p2
2m+ V (x)⇒ −i∂t |ψ〉 = − 1
2m
d2
dx2|ψ〉+ V (x).
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 12 / 27
Canonical Quantisation
In Schrödinger picture the vector states are represented byψ(x ; t) ∈ L2(R3), which are the components of
|ψ(t)〉 =
∫dx ψ(x ; t) |x〉 .
We associate a quantum observable to every classical one via the�promotion to operator� prescription. In Schrödinger picture:
x → x p → p.
= −i∂x E → E.
= i∂t .
x and p satisfy the commutation relation
[x , p] = i .
From the classical energy-dispersion relation we obtain the SchrödingerEquation:
E =p2
2m+ V (x)⇒ −i∂t |ψ〉 = − 1
2m
d2
dx2|ψ〉+ V (x).
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 12 / 27
Canonical Quantisation
In Schrödinger picture the vector states are represented byψ(x ; t) ∈ L2(R3), which are the components of
|ψ(t)〉 =
∫dx ψ(x ; t) |x〉 .
We associate a quantum observable to every classical one via the�promotion to operator� prescription. In Schrödinger picture:
x → x p → p.
= −i∂x E → E.
= i∂t .
x and p satisfy the commutation relation
[x , p] = i .
From the classical energy-dispersion relation we obtain the SchrödingerEquation:
E =p2
2m+ V (x)⇒ −i∂t |ψ〉 = − 1
2m
d2
dx2|ψ〉+ V (x).
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 12 / 27
Canonical Quantisation
In Schrödinger picture the vector states are represented byψ(x ; t) ∈ L2(R3), which are the components of
|ψ(t)〉 =
∫dx ψ(x ; t) |x〉 .
We associate a quantum observable to every classical one via the�promotion to operator� prescription. In Schrödinger picture:
x → x p → p.
= −i∂x E → E.
= i∂t .
x and p satisfy the commutation relation
[x , p] = i .
From the classical energy-dispersion relation we obtain the SchrödingerEquation:
E =p2
2m+ V (x)⇒ −i∂t |ψ〉 = − 1
2m
d2
dx2|ψ〉+ V (x).
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 12 / 27
Quantisation of the Harmonic Oscillator
The Schrödinger equation for a 1-dimensional harmonic oscillator reads
−i∂tψ(x ; t) = − 1
2m
d2
dx2ψ(x ; t) +
1
2ω2x2
Its solution is
ψn(x ; t) = e−12ωx2Hn(x
√ω)e−iEnt , En =
(n +
1
2
)ω.
Important features of Quantum Mechanics:
Discrete spectrum of energy eigenstates;
E0 = 12ω: there is a ground state with non-zero energy ( Heisenberg
Principle)
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 13 / 27
Quantisation of the Harmonic Oscillator
The Schrödinger equation for a 1-dimensional harmonic oscillator reads
−i∂tψ(x ; t) = − 1
2m
d2
dx2ψ(x ; t) +
1
2ω2x2
Its solution is
ψn(x ; t) = e−12ωx2Hn(x
√ω)e−iEnt , En =
(n +
1
2
)ω.
Important features of Quantum Mechanics:
Discrete spectrum of energy eigenstates;
E0 = 12ω: there is a ground state with non-zero energy ( Heisenberg
Principle)
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 13 / 27
Quantisation of the Harmonic Oscillator
The Schrödinger equation for a 1-dimensional harmonic oscillator reads
−i∂tψ(x ; t) = − 1
2m
d2
dx2ψ(x ; t) +
1
2ω2x2
Its solution is
ψn(x ; t) = e−12ωx2Hn(x
√ω)e−iEnt , En =
(n +
1
2
)ω.
Important features of Quantum Mechanics:
Discrete spectrum of energy eigenstates;
E0 = 12ω: there is a ground state with non-zero energy ( Heisenberg
Principle)
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 13 / 27
Second Quantisation
Let us introduce the Lowering and Raising Operators:
a =1√2ω
(ωx + i p) a† =1√2ω
(ωx − i p)[a, a†
]= 1.
They allow us to reformulate the Hamiltonian as
H =
(N +
1
2
)ω N
.= a†a = Number Operator.
Calling |n〉 the eigenstates of N we obtain that a generic state vector isgiven by:
|ψ(t)〉 =∑n
cne−iEnt |n〉 .
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 14 / 27
Second Quantisation
Let us introduce the Lowering and Raising Operators:
a =1√2ω
(ωx + i p) a† =1√2ω
(ωx − i p)[a, a†
]= 1.
They allow us to reformulate the Hamiltonian as
H =
(N +
1
2
)ω N
.= a†a = Number Operator.
Calling |n〉 the eigenstates of N we obtain that a generic state vector isgiven by:
|ψ(t)〉 =∑n
cne−iEnt |n〉 .
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 14 / 27
Second Quantisation
Let us introduce the Lowering and Raising Operators:
a =1√2ω
(ωx + i p) a† =1√2ω
(ωx − i p)[a, a†
]= 1.
They allow us to reformulate the Hamiltonian as
H =
(N +
1
2
)ω N
.= a†a = Number Operator.
Calling |n〉 the eigenstates of N we obtain that a generic state vector isgiven by:
|ψ(t)〉 =∑n
cne−iEnt |n〉 .
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 14 / 27
Special Relativity and Quantum Mechanics: the
Klein-Gordon Equation
Switching to the relativistic energy dispersion relation E 2 = k2 + m2
(c = 1) we get the the Klein-Gordon Equation:
(�+ m2) |ψ〉 = 0.
One can derive it computing the Euler-Lagrange Equations of theKlein-Gordon Lagrangian:
SKG =
∫d4xLKG , LKG = −1
2ηµν∂µφ∂νφ−
1
2m2φ2
Problems:
Possible negative-energy solutions (Fourier transform);
Violation of causality;
||ψ〉|2 can not be interpreted as a probability amplitude.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 15 / 27
Special Relativity and Quantum Mechanics: the
Klein-Gordon Equation
Switching to the relativistic energy dispersion relation E 2 = k2 + m2
(c = 1) we get the the Klein-Gordon Equation:
(�+ m2) |ψ〉 = 0.
One can derive it computing the Euler-Lagrange Equations of theKlein-Gordon Lagrangian:
SKG =
∫d4xLKG , LKG = −1
2ηµν∂µφ∂νφ−
1
2m2φ2
Problems:
Possible negative-energy solutions (Fourier transform);
Violation of causality;
||ψ〉|2 can not be interpreted as a probability amplitude.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 15 / 27
A World Made of Fields
A solution to these problems can be found considering a system within�nitely many degrees of freedom, i.e. a �eld (Dirac sea).
The KGHamiltonian is (π = φ)
H =1
2π2 +
1
2(∇φ)2 +
1
2m2φ2 Harmonic Oscillator!
(x ; p) 7→ (φ(xµ);π(xµ))
φ(xµ) has no more to be read as a wave function, but �xed-time initialvalue of the KG equation.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 16 / 27
A World Made of Fields
A solution to these problems can be found considering a system within�nitely many degrees of freedom, i.e. a �eld (Dirac sea).The KGHamiltonian is (π = φ)
H =1
2π2 +
1
2(∇φ)2 +
1
2m2φ2 Harmonic Oscillator!
(x ; p) 7→ (φ(xµ);π(xµ))
φ(xµ) has no more to be read as a wave function, but �xed-time initialvalue of the KG equation.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 16 / 27
Space of Solutions
The KG Equation admits plane-wave solutions:
φ(xµ) = φ0eikµx
µ= φ0e
iωt−ik·x ω2 = k2 + m2.
We look for a complete o.n. set of solutions, hence we need a scalarproduct on the solutions' space:
(φ1, φ2) = −i∫
Σt
(φ1∂tφ∗2 − φ∗2∂tφ1) d3x
so that we get the set (kµ : ω2 = k2 + m2):
fk(xµ) =e ikµx
µ
2π√2ω, (fk1 , fk2) = δ(3)(k1 − k2).
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 17 / 27
Space of Solutions
The KG Equation admits plane-wave solutions:
φ(xµ) = φ0eikµx
µ= φ0e
iωt−ik·x ω2 = k2 + m2.
We look for a complete o.n. set of solutions, hence we need a scalarproduct on the solutions' space:
(φ1, φ2) = −i∫
Σt
(φ1∂tφ∗2 − φ∗2∂tφ1) d3x
so that we get the set (kµ : ω2 = k2 + m2):
fk(xµ) =e ikµx
µ
2π√2ω, (fk1 , fk2) = δ(3)(k1 − k2).
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 17 / 27
Positive- and Negative-Frequencies Solutions
The solution are labelled by the continuous parameter k and are determinedup to the sign of ω. Since energy (E = hω) is a positive-de�nite quantitywe would like to consider only solutions with positive frequency.
This is done by introducing, for all ω > 0, positive-frequency solutions
∂t fk = −iωfk
and negative-frequency solutions
∂t f∗k
= iωf ∗k.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 18 / 27
Positive- and Negative-Frequencies Solutions
The solution are labelled by the continuous parameter k and are determinedup to the sign of ω. Since energy (E = hω) is a positive-de�nite quantitywe would like to consider only solutions with positive frequency.This is done by introducing, for all ω > 0, positive-frequency solutions
∂t fk = −iωfk
and negative-frequency solutions
∂t f∗k
= iωf ∗k.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 18 / 27
Canonical Quantisation
In total analogy with the harmonic oscillator in QM we promote φ and π tooperator, imposing the equal-time commutation relations (HeisenbergPicture) [
φ(x), φ(x′)]t
= iδ(3)(x− x′).
Expanding as a function of the modes
φ(t; x) =
∫d3x
[akfk(t; x) + a†
kf ∗k
(t; x)]
which leads to the Canonical Commutation Relations[ak, a
†k′
]= δ(3)(k− k
′).
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 19 / 27
Canonical Quantisation
In total analogy with the harmonic oscillator in QM we promote φ and π tooperator, imposing the equal-time commutation relations (HeisenbergPicture) [
φ(x), φ(x′)]t
= iδ(3)(x− x′).
Expanding as a function of the modes
φ(t; x) =
∫d3x
[akfk(t; x) + a†
kf ∗k
(t; x)]
which leads to the Canonical Commutation Relations[ak, a
†k′
]= δ(3)(k− k
′).
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 19 / 27
Particles and Anti-Particles
Positive-frequencies modes are the coe�cients of the Annihilationoperator ak;
Negative-frequencies modes are the coe�cients of the Creationoperator a†
k.
|nk〉 =1√nk!
(a†k)nk |0〉
We are creating n particles with momentum k. |0〉 is the Vacuum State.
Likewise in QM, we can introduce the Number Operator Nk = a†kak, whose
eigenvectors constitute the Fock Basis.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 20 / 27
Particles and Anti-Particles
Positive-frequencies modes are the coe�cients of the Annihilationoperator ak;
Negative-frequencies modes are the coe�cients of the Creationoperator a†
k.
|nk〉 =1√nk!
(a†k)nk |0〉
We are creating n particles with momentum k. |0〉 is the Vacuum State.
Likewise in QM, we can introduce the Number Operator Nk = a†kak, whose
eigenvectors constitute the Fock Basis.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 20 / 27
Particles and Anti-Particles
Positive-frequencies modes are the coe�cients of the Annihilationoperator ak;
Negative-frequencies modes are the coe�cients of the Creationoperator a†
k.
|nk〉 =1√nk!
(a†k)nk |0〉
We are creating n particles with momentum k. |0〉 is the Vacuum State.
Likewise in QM, we can introduce the Number Operator Nk = a†kak, whose
eigenvectors constitute the Fock Basis.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 20 / 27
What's Missing?
Renormalization Subtraction of (in�nite) the point-zero energy
Interactions and Scattering Theory
Other kind of Fields (QED, Gauge Theories)
....
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 21 / 27
1 Gravity and General Relativity
2 Quantum Mechanics, Quantum Field Theory and All That...
3 An insight into QFT on Curved Backgrounds
Fundamental Ideas
The interactions do not in�uence the �xed background;
Investigation of the back-reaction;
Black Holes Physics;
States (Vacua, Thermal...) and Renormalization;
Making the theory mathematically rigorous.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 22 / 27
Fundamental Ideas
The interactions do not in�uence the �xed background;
Investigation of the back-reaction;
Black Holes Physics;
States (Vacua, Thermal...) and Renormalization;
Making the theory mathematically rigorous.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 22 / 27
Fundamental Ideas
The interactions do not in�uence the �xed background;
Investigation of the back-reaction;
Black Holes Physics;
States (Vacua, Thermal...) and Renormalization;
Making the theory mathematically rigorous.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 22 / 27
Fundamental Ideas
The interactions do not in�uence the �xed background;
Investigation of the back-reaction;
Black Holes Physics;
States (Vacua, Thermal...) and Renormalization;
Making the theory mathematically rigorous.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 22 / 27
Fundamental Ideas
The interactions do not in�uence the �xed background;
Investigation of the back-reaction;
Black Holes Physics;
States (Vacua, Thermal...) and Renormalization;
Making the theory mathematically rigorous.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 22 / 27
Choice of the background
L =√−g(−1
2∇µφ∇µφ−
1
2m2φ2 − ξRφ2
)⇒ �gφ−m2φ− ξRφ = 0.
For consistency of the Causchy problem and for causality issues we need to�x ourselves on a Globally Hyperbolic Space-Time:
Theorem (Bernal - Sanchez)
Let (M, g) be a 4-dimensional, time-oriented space-time. Then the
following statements are equivalent:
1 (M, g) is globally hyperbolic;
2 (M, g) is hysometric to R× Σ with ds2 = +βdt2 − hijdxidx j . Here
(t, xi )3i=1 is a suitable coordinate system s.t. β ∈ C∞(M; (0,∞)), h
is a Riemannian metric on Σ depending smoothly on t and each locus
{t = const} ×Σ is a smooth spacelike Cauchy hypersurface embedded
inM.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 23 / 27
Choice of the background
L =√−g(−1
2∇µφ∇µφ−
1
2m2φ2 − ξRφ2
)⇒ �gφ−m2φ− ξRφ = 0.
For consistency of the Causchy problem and for causality issues we need to�x ourselves on a Globally Hyperbolic Space-Time:
Theorem (Bernal - Sanchez)
Let (M, g) be a 4-dimensional, time-oriented space-time. Then the
following statements are equivalent:
1 (M, g) is globally hyperbolic;
2 (M, g) is hysometric to R× Σ with ds2 = +βdt2 − hijdxidx j . Here
(t, xi )3i=1 is a suitable coordinate system s.t. β ∈ C∞(M; (0,∞)), h
is a Riemannian metric on Σ depending smoothly on t and each locus
{t = const} ×Σ is a smooth spacelike Cauchy hypersurface embedded
inM.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 23 / 27
Choice of the background
L =√−g(−1
2∇µφ∇µφ−
1
2m2φ2 − ξRφ2
)⇒ �gφ−m2φ− ξRφ = 0.
For consistency of the Causchy problem and for causality issues we need to�x ourselves on a Globally Hyperbolic Space-Time:
Theorem (Bernal - Sanchez)
Let (M, g) be a 4-dimensional, time-oriented space-time. Then the
following statements are equivalent:
1 (M, g) is globally hyperbolic;
2 (M, g) is hysometric to R× Σ with ds2 = +βdt2 − hijdxidx j . Here
(t, xi )3i=1 is a suitable coordinate system s.t. β ∈ C∞(M; (0,∞)), h
is a Riemannian metric on Σ depending smoothly on t and each locus
{t = const} ×Σ is a smooth spacelike Cauchy hypersurface embedded
inM.
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 23 / 27
What about particles?
We de�ne the conjugate momentum π =√−g∇0φ and de�ne the scalar
product on a spacelike 3-surface Σ and take over the quantisation imposingimpose the CCR:
(φ1, φ2).
= −i∫
Σ(φ1∇µφ∗2 − φ∗2∇µφ1) nµ
√γd3x[
φ(t, x), φ(t, x′)]
=i√−g
δ(3)(x− x′)
Problem
In a general space-time it is impossible to de�ne positive-frequencysolutions because there is no unique notion of time. Hence the concept ofparticle is ill-de�ned!
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 24 / 27
What about particles?
We de�ne the conjugate momentum π =√−g∇0φ and de�ne the scalar
product on a spacelike 3-surface Σ and take over the quantisation imposingimpose the CCR:
(φ1, φ2).
= −i∫
Σ(φ1∇µφ∗2 − φ∗2∇µφ1) nµ
√γd3x[
φ(t, x), φ(t, x′)]
=i√−g
δ(3)(x− x′)
Problem
In a general space-time it is impossible to de�ne positive-frequencysolutions because there is no unique notion of time. Hence the concept ofparticle is ill-de�ned!
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 24 / 27
Hadamard States
We are dealing with semiclassical Einstein equations:
Rµν −1
2gµνR = −8π < Tµν >, < Tµν >
.= 〈0|Tµν |0〉 =?
The lack of the modes expansion implies that there is no de�nite notion ofvacuum state.This lead to the introduction of Hadamard States:
H(x , x ′) =U(x , x ′)
(2π)2σε+V (x , x ′) log(σε) +W (x , x ′), σε
.= σ+2iε(t− t ′) + ε2
< Tµν > well-de�ned
Expectation values with �nite �uctuations
Compatible with the unique Minkowski vacuum state
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 25 / 27
Hadamard States
We are dealing with semiclassical Einstein equations:
Rµν −1
2gµνR = −8π < Tµν >, < Tµν >
.= 〈0|Tµν |0〉 =?
The lack of the modes expansion implies that there is no de�nite notion ofvacuum state.
This lead to the introduction of Hadamard States:
H(x , x ′) =U(x , x ′)
(2π)2σε+V (x , x ′) log(σε) +W (x , x ′), σε
.= σ+2iε(t− t ′) + ε2
< Tµν > well-de�ned
Expectation values with �nite �uctuations
Compatible with the unique Minkowski vacuum state
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 25 / 27
Hadamard States
We are dealing with semiclassical Einstein equations:
Rµν −1
2gµνR = −8π < Tµν >, < Tµν >
.= 〈0|Tµν |0〉 =?
The lack of the modes expansion implies that there is no de�nite notion ofvacuum state.This lead to the introduction of Hadamard States:
H(x , x ′) =U(x , x ′)
(2π)2σε+V (x , x ′) log(σε) +W (x , x ′), σε
.= σ+2iε(t− t ′) + ε2
< Tµν > well-de�ned
Expectation values with �nite �uctuations
Compatible with the unique Minkowski vacuum state
Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 25 / 27