Quantum Fields in Curved Spacetime


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1 Quantum Fields in Curved Spacetime Lecture 1: Introduction Finn Larsen Michigan Center for Theoretical Physics Yerevan, August 20, 2016.
2 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. 2
3 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. 3
4 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. 4
5 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: Smatrix (scattering amplitudes)  Another observable: Correlation functions (response to slowly varying external sources). These issues present many technical and conceptual challenges. 5
6 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. 6
7 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... 7
8 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. 8
9 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. 9
10 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. 10
11 Quantum Fluctuations: Strategy Some set of matter fields, collectively denoted φ (but in fact may contain multiple scalar fields φ i, vector fields A µ etc). Schematic of the Euclidean Path Integral: e W [g µν] = Dφ e S[φ] Functional integration over all matter fields. They are quantum. No functional integral over metric g µν : Quantum fields IN classical curved space. Quantum gravity: integrate over g µν as well. We do not do that. Observables: study dependence on g µν. 11
12 Free Matter Fields Example: Lagrangian quadratic in fields gives a Gaussian integral e W [g µν] = Dφ e φλφ = 1 detλ. Notation: Λ = + m 2 where Laplacian depends on g µν. The determinant is divergent but we can study its dependence on g µν. Formally: W = 1 2 ln detλ = 1 2 ln λ i where λ i are eigenvalues of the operator Λ. i 12
13 The Heat Kernel Convenient to rewrite W = 1 2 W = 1 2 where i i ln λ i as ( α ) α 0 λ α i = ( α ) α 0 1 Γ( α) D(t) = Tr e tλ = i 0 dt 2t α+1d(t) = e tλ i 0 dt 2t D(t) To compute D(t) we introduce the (bilocal) heat kernel density K(x, x ; t) = e tλ = e tλ i fi (x)f i (x ) i where f i are a basis of normalized eigenfunctions f i. The spacetime integral of the (equal point) heat kernel density D(t) = d 4 xk(x, x; t) 13
14 Computing the Heat Kernel Density The heat kernel satisfies the heat kernel equation ( t + Λ x )K(x, x ; t) = 0 With boundary condition as t 0 K(x, x ; t) = i e tλ i f i (x)f i (x ) δ(x x ) 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). 14
15 A Free Boson in D Dimensions The heat kernel equation: In Fourier space: In real space: ( t + ( 2 + m 2 ))K(x, x ; t) = 0 K( k; t) = K(x, x ; t) = The integrated equal point heat kernel D(t) = ( )D 1 2 e ( k 2 +m 2 )t 2π ( )D 1 2 e 4t 1 ( x x ) 2 m 2 t 4πt d 4 xk(x, x; t) = ( )D 1 2 e m 2t Vol 4πt 15
16 Regularization The quantum effective action W = 0 dt 2t D(t) 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 λ i ). The UV details do not matter so we can modify them at will. We introduce a small length ɛ that cuts the integral off at small t: dt W = 2t D(t) Note: this is totally arbitrary. Final results will only be physical if they are independent of ɛ (and other details of regularization). ɛ 2 16
17 The Cosmological Constant All spacetimes appear flat at short distances so the small s behavior is universal D(t 0) = It corresponds to the effective action W = ɛ 2 dt 2t D(t) = 1 D ( )D 1 2 Vol 4πt ( )D π ɛ DVol Interpretation: this is a cosmological constant of order Λ ɛ D. The result depends on ɛ so our arbitrary regularization enters and therefore the result is not meaningful. 17
18 Free Boson on a 2D Sphere The spectrum of a boson on S 2 (with radius a) is well known 2 S 2 l(l + 1) a 2 (degeneracy : 2l + 1) l = 0, 1, 2,... The (equal point) heat kernel density: K(t) = 1 4πa 2 l=0 (2l+1)e l(l+1)t/a2 = 1 4πt ( 1 + t 3a 2 + ) t2 15a Precise computations: use EulerMacLauren 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. 18
19 Heat Kernel Structure The general heat kernel equation ( t + Λ x )K(x, x ; t) = 0 Perturbation theory: Λ x 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: K(x, x; t) = where R is curvature of spacetime. For scalar field on S 2 : R = 2 a 2, c 1 = ( 1 + c1 Rt + c (4πt) D/2 2 R 2 t ) Schematic: the curvature symbol R 2 could be contracted in multiple ways. 19
20 Renormalization Divergences persist up to the constant term in D(t). The structure of divergences in 4D: Cosmological constant, Einstein term, Curvature 2 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. 20
21 Return to the 2D Sphere Subtraction of divergences renders the 2D effective action finite: W Vol = 0 dt 2t But: this is wrong. ( 1 4πa 2 l=0 1 (2l + 1)e l(l+1)t/a2 a2 t 1 3 ) = finite Interpreting subtraction of 4πt : the cosmological constant at low energy already includes this correction. Subtracting this divergence from the quantum correction prevents overcounting. It is correct. The next order subtracts the constant in D 0 : 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. 21
22 Scale Dependence We recognize that the field theory description is valid only in the IR, at large distances L > L cutoff. So UV contributions due to distances L < L cutoff were already included and must be subtracted to avoid overcounting. The logarithmic term due to large distances L > L cutoff must not be subtracted, it is a genuine contribution to the quantum action. It depends on the arbitrary (or at least negotiable) distance L cutoff. The physical contribution to the S 2 effective action: W Vol = 1 ( ) 24πa ln LIR = R 2 48π ln L cutoff ( LIR L cutoff ) 22
23 Trace Anomaly The arbitrary scale L cutoff is a physical quantity applied consistently everywhere, despite variations in the metric. The trace of the energy momentum tensor (in some conventions): T µ µ = 2π µν δs g g δg µν acquires a special contribution from the constant piece in the heat kernel ( T µ µ )an = 1 Vol D 0 Example: for the free scalar on S 2 T µ µ = 1 12 R In CFT terminology: the central charge is c = 1. 23
24 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 cutoff even when there is a classical contribution. The terminology is not universally accepted. 24
25 Heat Kernel Terminology General perturbative expansion of the (equal point) heat kernel density: K(x, x; t) = t n 2 a 2n (x) n=0 Terminology: a 2n is the n th SeeleydeWitt coefficient. It is formed from the n th powers of the curvature R (with various contractions). In D dimensions (with D even) a D is (up to a numerical constant) the integrated trace anomaly. The a D is equivalent to D 0, the constant term in the meromorphic function D(t). 25
26 Heat Kernel Computations So far: minimally coupled scalars. A general operator Λ: Λ n m = I n m(d µ D µ ) (2ω µ D µ ) n m P n m General result for a 4 (essentially the trace anomaly in D = 4): (4π) 2 a 4 (x) = Tr [ 1 2 E Ω µνω µν + 1 ] 180 (R µνρσr µνρσ R µν R µν )I. Notation: E = P ω µ ω µ (D µ ω µ ), D µ = D µ +ω µ, Ω µν [D µ, D ν ]. Formalism also applies to fermions: take the square of the (generalized) Dirac operator and apply an overall minus sign. 26
27 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). 27
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