Quantum field theory on curved Spacetime

Transcription

1 Quantum field theory on curved Spacetime YOUSSEF Ahmed Director : J. Iliopoulos LPTENS Paris Laboratoire de physique théorique de l école normale supérieure

2 Why QFT on curved In its usual formulation QFT simply ignores gravity No one knows how to write a full quantum gravity theory But we expect the existence a semi-classical regime where one can only quantize matter fields and keep gravity classical (cf a theory of quantum matter interacting with a classical electromagnetic field) ψ matter coupled to classical g µν ψ matter lives on curved

3 of the coupling to an external field The coupling to an external field furnish energy that can create particles : Schwinger effet in QED coupled to an external E field ) P e + e pair creation exp ( m2 ee More importantly, the notion of particle is ambiguous. Remember that origin of the particle concept in QFT is an asymptotic one Free QFT E = space of stationnary solutions E has a Fock space structure = particle interpretation of theory

4 of the coupling to an external field The same reasoning doesn t hold in an interacting theory of QCD : Free theory Quarks Interacting theory hadrons Particle notion defined asymptotically in a free theory Particle notion defined asymptotically in a flat

5 Quantum driven harmonic oscillator equation of motion q + ω 2 q = J(t) { q = p ṗ = ω 2 q + J(t) a ± ω = [q(t) iω ] 2 and a (t = 0) = a in Solution a (t) = a ine iωt + i t 2ω dτj(τ)e iω(τ t) 0

6 Quantum driven harmonic oscillator 2 asymptotic regions { a a in (t) = e iωt if t 0 a oute iωt if t T 2 vacuum a out = a in + J 0 J 0 = i 2ω T 0 in and 0 out 0 dτj(τ)e iωτ a in 0in = 0 a out 0 out = 0 a out 0 in = J 0 0 in Particle (exitation) creation 0 in N(t) 0 in = 0 in = 0 out N(t) = a + (t)a (t) { 0 if t 0 J 0 2 if t T

7 Real scalar field Minimal coupling S = d 4 x g {g µν µφ νφ 12 } m2 φ 2 ( + m 2 ) φ = 0 φ = 1 g µ ( g µν g νφ ) Mode decomposition Minkowski : Poincaré invariance gives a privileged coordinate system (t, x, y, z) { φ(t, x) = k a k u k (t, x) + a k u k (t, x) u k e i k. x e iωt Curved : many different mode decomposition φ(x) = i a iu i(x) + a i u i (x) = i ā iū i(x) + ā i ū i (x)

8 Bogoliubov transformation 2 different vacuum { ai 0 = 0 i ā i 0 = 0 i {u i} and {ū i} complete bases of states = { ū j = i a i ā j α jiu i + β jiu i = j αjiāj + β jiā j = i α jia i βjia j a i 0 = j β ji 1 j = 0 if a β ji 0 Created particle number N i = a i ai 0 N i 0 = j β ji 2

9 in spacially flat FRW Conformally flat ds 2 = dt 2 a 2 (t)dx 2 dη = dt ds 2 = a 2 (η) [ dη 2 dx 2] a(t) g µν = a 2 (η)η µν g = a 4 (η) field equation u k (η, x) = 1 2π e i k. x χ k (η) χ k + [ k 2 + a 2 (η)m 2] χ k = 0 Exact soution in terms of hypergeometric functions if a 2 (η) = A + B tanh(ρ η)

10 in spacially flat FRW We impose Minkowskian modes as η ± { u in k = 1 4πωin e i(kx ω inη) as u out k = 1 4πωout e i(kx ω outη) as One can then find analytically α k and β k such that u in k = α k u out k + β k u out k η η Number of created particles N = k β k 2

11 de Sitter : maximally symmetric with isotropic and homogeneous spacial sections, positive scalar curvature : Flat spatial sections ds 2 = dt 2 + e 2Ht d x 2 ds d may be realized in M d,1 as the hyperboloid X X X 2 d = l 2 Here the O(d, 1) symmetry is manifest

12 2 point function Free field theory, so all the information is in the 2 point function. For instance the Wightman function ( G(X, Y ) = 0 φ(x)φ(y ) 0 dsd m 2) G = 0 G(X, Y ) = G (P (X, Y )) where P (X, Y ) is the de Sitter invariant length. Hypergeometric equation : ( ) z(1 z) G d + 2 zd Ġ m 2 G = 0 with z = 1 + P 2 Solutions : a one parameter family of de Sitter invariant Green functions G α corresponding to a one parameter family of de Sitter invariant vacuum states α G α(x, Y ) = α φ(x)φ(y ) α

13 Thermal radiation A geodesic observer x(τ) equipped with a detector of Hamiltonian and energy eigenstates H E j = E j E j The geodesic observer measures a thermal bath of particles when the field φ is in the vacuum state 0 : the field-detector coupling induces a thermally populated energy levels N i e βe i The de Sitter temperature is T = 1 2πl

14 One can compute the vectorial and spinorial propagator too Instead of a matter field we can consider linearized gravity itself : de Sitter quantum gravity g µν = η µν + h µν h µν 1 Perturbative quantum gravity is non renormalizable. This is a short distance property that is independent of the large scale shape of But one can still treat it as an effective field theory and get the first quantum corrections The graviton h µν propagator on de Sitter has an infrared pathology even at the tree level

15 One can try to consider the back reaction of quantum fields (matter and gravitons) on G µν = 8πG T µν constant problem : gravity couples to any form of energy. So a naive renormalization of the vacuum energy is not possible and E 0 V = ΛPlanck d 3 k 1 k2 + m 2 2 Λ 4 Planck g.cm 3! The Stone von Neumann theorem breaks down in infinite dimensional context (field theory). Infinitley many inequivalent representations of the quantum algebra exists and no Poincaré invariance to pick one algebraic approach to QFT on curved space time

16 Probably profound link between gravity, the quantum and thermodynamics ds/cft correspondence Finally one more reason to think that QFT is the quantum theory of fields and not a quantum theory of particles. To mention also : Rovelli s global/local particles in QFT