Download Quantum Fields in Curved Spacetime

Quantum Fields in Curved Spacetime
Lecture 1: Introduction
Finn Larsen
Michigan Center for Theoretical Physics
Yerevan, August 20, 2016 .
Intro: Fields
• Setting: many microscopic degrees of freedom interacting locally
with each other.
• Long distance behavior described by fields that parametrize
microscopic conditions as they vary slowly over space and time.
• Example: properties of fluids encoded in pressure, temperature,
flow velocity,... Few details of the atomic structure needed to
describe variation over space and time,
• Another example: magnetic degrees of freedom that
energetically favor alignment with neighbors described by a slowly
varying magnetization.
• The point: field theory is (almost) always ”just” a long distance
description of some microscopic theory.
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• The power: all details of the microscopic theory are encoded
in a few field theory parameters (speed of sound,....).
• Terminology:
- the UltraViolet=UV theory is the true microscopic description
- the InfraRed=IR theory is the long distance behavior described
by field theory.
• More terminology: the term effective field theory emphasizes
that field theory describes the IR.
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Effective Field Theory
• Effective field theory systematically computes the leading
corrections at higher energy (relative to typical energies in the IR).
• It applies fields at low energy.
• Some help: symmetries determine many feature of the field theory.
• These lectures: always assume Lorentz invariance (special
relativity) and covariance (general relativity).
• Comment: it is generally difficult to determine the IR theory even
when the UV theory is known.
• In effective field theory we do not even try, we just care about IR.
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Intro: Quantum Fields
• One issue: quantum fluctuations of fields are large (many d.o.f.).
- If we employ field theory to arbitrarily short distance, we
encounter divergences.
- These UV divergences remind us that field theory does not
describe the UV.
• Another issue: “observation” is subtle in quantum theory; fields
are usually not observable themselves.
- Typical observable: S-matrix (scattering amplitudes)
- Another observable: Correlation functions (response to slowly
varying external sources).
• These issues present many technical and conceptual challenges.
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Curved Spacetime
• General relativity identifies gravity with curved space.
• Gravity acts on matter: matter experiences relative motion due to
curved space.
• Matter acts on gravity: matter creates curved space.
• Example: the evolution of fields changes due to curved space.
• Also: fields create curved space.
• Special case: spectator fields experience curved space but we
neglect their ability to curve space, the curvature of space is due
to other matter.
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Quantum Gravity
• The description of gravity as curved space is an IR description.
• The underlying UV description must involve novel concepts (eg
String Theory or a holographic dual)
• Evidence: fluctuations of quantum gravity exhibit UV divergences.
• UV divergences are more severe in gravity than in other QFTs.
• Moreover: it is confusing what the quantum observables are even
in the IR theory.
• There are conceptual issues even in the IR: black hole
information paradox...
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Quantum Fields IN Curved Space
• Quantum Field Theory is an IR description: the UV completion
is unknown even in principle.
• Key observation: curved space appears flat locally (at short
distance=UV).
• So: the error made by employing QFT all the way to the UV in
curved space is the same as the error made when employing
QFT all the way to the UV in flat space.
• Upshot: effects of curved space on the IR properties of QFT are
determined unambiguously even though we do not know the UV
QFT.
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Quantum Observables
• Another perspective: divergences remind us that the UV is
ambiguous.
• But: small changes in the spacetime geometry do not change the
divergences.
• So the response to such changes is unambiguous, it does not
depend on the unknown UV.
• Interpretation: we can determine how gravity responds to quantum
fields, including their virtual effects.
• Gravitational observables are unambiguous in this setting.
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Plan
• After all these conceptual comments: take a “practical” approach:
”how to do things”.
• This illustrates conceptual issues concretely.
• Lecture 1: General Structure.
• Lecture 2: Maximally symmetric spaces: applications to AdS/CFT
correspondence.
• Lecture 3: Index theorems, Zero modes, and Boundary States.
• Lecture 4: Logarithmic corrections to black hole entropy.
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Quantum Fluctuations: Strategy
• Some set of matter fields, collectively denoted φ (but in fact may
contain multiple scalar fields φi, vector fields Aµ etc).
• Schematic of the Euclidean Path Integral:
Z
e−W [gµν ] = Dφ e−S[φ]
• Functional integration over all matter fields. They are quantum.
• No functional integral over metric gµν : Quantum fields IN
classical curved space.
• Quantum gravity: integrate over gµν as well. We do not do that.
• Observables: study dependence on gµν .
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Free Matter Fields
• Example: Lagrangian quadratic in fields gives a Gaussian
integral
Z
1
−W [gµν ]
−φΛφ
e
= Dφ e
=√
.
detΛ
• Notation: Λ = −∆ + m2 where Laplacian −∆ depends on gµν .
• The determinant is divergent but we can study its dependence
on gµν .
• Formally:
1X
1
ln λi
W = ln detΛ =
2
2 i
where λi are eigenvalues of the operator Λ.
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The Heat Kernel
• Convenient to rewrite W =
1
2
P
i ln λi
1X
1
W =
(∂α )α→0λαi = (∂α )α→0
2 i
Γ(−α)
D(t) = Tr e
∞
Z
where
−tΛ
as
=
0
dt
D(t) = −
2tα+1
X
Z
0
∞
dt
D(t)
2t
e−tλi
i
• To compute D(t) we introduce the (bilocal) heat kernel density
X
0
−tΛ
K(x, x ; t) = e
=
e−tλi fi∗(x)fi(x0)
i
where fi are a basis of normalized eigenfunctions fi.
• The spacetime integral of the (equal point) heat kernel density
Z
D(t) = d4xK(x, x; t)
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Computing the Heat Kernel Density
• The heat kernel satisfies the heat kernel equation
(∂t + Λx)K(x, x0; t) = 0
• With boundary condition as t → 0
X
0
e−tλi fi∗(x)fi(x0) → δ(x − x0)
K(x, x ; t) =
i
• Intuitive interpretation: diffusion in a D dimensional space with
the parameter t the diffusion time.
• Practical advantage: The heat kernel equation is a manageable
partial differential equation that can be solved (at least
approximately).
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A Free Boson in D Dimensions
• The heat kernel equation:
(∂t + (−∇2 + m2))K(x, x0; t) = 0
• In Fourier space:
K(~k; t) =
1
2π
D2
1
4πt
D2
e
−(~k 2 +m2 )t
• In real space:
K(x, x0; t) =
1
~0
2
2
e− 4t (~x−x ) −m t
• The integrated equal point heat kernel
D2
Z
1
−m2 t
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D(t) = d xK(x, x; t) =
e
Vol
4πt
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Regularization
• The quantum effective action
Z
W =−
0
∞
dt
D(t)
2t
diverges (as promised) since D(t) is singular for small t.
• Small values of the Feynman parameter t corresponds to the
UV (and to large eigenvalues λi).
• The UV details do not matter so we can modify them at will.
• We introduce a small length that cuts the integral off at small t:
Z ∞
dt
W =−
D(t)
2 2t
• Note: this is totally arbitrary. Final results will only be physical if
they are independent of (and other details of regularization).
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The Cosmological Constant
• All spacetimes appear flat at short distances so the small s
behavior is universal
D(t → 0) =
1
4πt
D2
Vol
• It corresponds to the effective action
D2
Z ∞
dt
1
1
1
W =−
D(t) =
Vol
D
D 4π
2 2t
• Interpretation: this is a cosmological constant of order Λ ∼ −D .
• The result depends on so our arbitrary regularization enters and
therefore the result is not meaningful.
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Free Boson on a 2D Sphere
• The spectrum of a boson on S 2 (with radius a) is well known
−∇2S 2 →
l(l + 1)
a2
(degeneracy : 2l + 1) l = 0, 1, 2, . . .
• The (equal point) heat kernel density:
∞
2
1 X
1
t
t
−l(l+1)t/a2
K(t) =
(2l+1)e
=
1+ 2 +
+ ...
4πa2
4πt
3a
15a4
l=0
Precise computations: use Euler-MacLauren formula or (better) a
modular transformation.
• The leading short distance divergence agrees with the flat space
result in D = 2, higher orders are sensitive to curvature.
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Heat Kernel Structure
• The general heat kernel equation
(∂t + Λx)K(x, x0; t) = 0
• Perturbation theory: Λx is the flat space Laplacian + a correction.
• In practical computations: work out corrections in Riemann normal
coordinates around a reference point.
• Final result must be diffeomorphism invariant so the heat kernel
will depend on the curvature only.
• By dimensional analysis the equal point heat kernel takes the form:
1
2 2
K(x, x; t) =
1 + c1Rt + c2R t + . . .
(4πt)D/2
where R is curvature of spacetime.
• For scalar field on S 2: R = a22 , c1 = 61 .
• Schematic: the curvature symbol R2 could be contracted in
multiple ways.
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Renormalization
• Divergences persist up to the constant term in D(t).
• The structure of divergences in 4D: Cosmological constant,
Einstein term, Curvature2 corrections.
• We can eliminate divergences by renormalization.
• Idea: low energy observables must be expressed in terms of other
low energy observables.
• In practice: the “original” term in the Lagrangian (Cosmological
constant,...) is afflicted by divergences.
• The correct IR observable is the sum of the original term and the
(divergent) quantum contribution.
• The principle: divergences can be absorbed by any local operator
in the low energy theory.
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Return to the 2D Sphere
• Subtraction of divergences renders the 2D effective action finite:
!
Z ∞
∞
X
W
a2 1
dt 1
−l(l+1)t/a2
=−
(2l + 1)e
− −
= finite
2
Vol
2t 4πa
t
3
0
l=0
• But: this is wrong.
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: the cosmological constant at low
• Interpreting subtraction of 4πt
energy already includes this correction.
• Subtracting this divergence from the quantum correction prevents
over-counting. It is correct.
• The next order subtracts the constant in D0: the contribution to
the effective action diverges at both small s and large s
• This term modifies the IR so it is not a local term in the IR.
• It is wrong to subtract it.
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Scale Dependence
• We recognize that the field theory description is valid only in the
IR, at large distances L > Lcutoff .
• So UV contributions due to distances L < Lcutoff were already
included and must be subtracted to avoid overcounting.
• The logarithmic term due to large distances L > Lcutoff must not
be subtracted, it is a genuine contribution to the quantum action.
• It depends on the arbitrary (or at least negotiable) distance Lcutoff .
• The physical contribution to the S 2 effective action:
1
LIR
W
LIR
R
=
ln
=
ln
Vol 24πa2
Lcutoff
48π
Lcutoff
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Trace Anomaly
• The arbitrary scale Lcutoff is a physical quantity applied
consistently everywhere, despite variations in the metric.
• The trace of the energy momentum tensor (in some conventions):
2π
δS
Tµµ = − √ g µν µν
−g δg
acquires a special contribution from the constant piece in the heat
kernel
Tµµ an
1
=
D0
Vol
• Example: for the free scalar on S 2
Tµµ
1
= R
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• In CFT terminology: the central charge is c = 1.
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Conformal Symmetry
• Many interesting theories have conformal symmetry (and so scale
independence).
• For such theories the trace of the energy momentum tensor
vanishes classically but not in the quantum theory.
• This phenomenon is a universally referred to as an anomaly.
• We consider also theories that are not classically scale invariant.
• Here (and in much literature): trace anomaly = “special
dependence on cut-off” even when there is a classical contribution.
• The terminology is not universally accepted.
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Heat Kernel Terminology
• General perturbative expansion of the (equal point) heat kernel
density:
K(x, x; t) =
∞
X
tn−2a2n(x)
n=0
• Terminology: a2n is the n’th Seeley-deWitt coefficient.
• It is formed from the n’th powers of the curvature R (with
various contractions).
• In D dimensions (with D even) aD is (up to a numerical constant)
the integrated trace anomaly.
• The aD is equivalent to D0, the constant term in the meromorphic
function D(t).
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Heat Kernel Computations
• So far: minimally coupled scalars.
• A general operator Λ:
n
Λnm = −Im
(DµDµ) − (2ω µDµ)nm − Pmn
• General result for a4 (essentially the trace anomaly in D = 4):
(4π)2a4(x) = Tr
1 2 1
1
E + Ωµν Ωµν +
(Rµνρσ Rµνρσ − Rµν Rµν )I
2
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180
.
Notation:
E = P − ω µωµ − (Dµωµ) ,
Dµ = Dµ + ωµ ,
Ωµν ≡ [Dµ, Dν ] .
• Formalism also applies to fermions: take the square of the
(generalized) Dirac operator and apply an overall minus sign.
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Summary
• We discussed conceptual structure of quantum field theory in
curved space.
• There are UV divergences.
• The divergence depends only weakly on the geometry.
• So they can be renormalized (subtracted away).
• The remaining result is physical (independent of UV).
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