Dissipation from the analytically continued 1 PI effective action

Transcription

1 Dissipation from the analytically continued 1 PI effective action Stefan Flörchinger (Heidelberg U.) 8th International Conference on the Exact Renormalization Group - ERG2016, ICTP Trieste, September 21, [based on , to appear in JHEP]

2 Dissipation in quantum field theory Dissipation is generation of entropy Unitary evolution conserves entropy in practice often only incomplete information available expectation values of fundamental quantum fields and some composite operators quantum states with minimal information given some constraints use truncation of 1 PI effective action 1 / 30

3 Dissipation from integrating out fields Example 1 consider muon with decay µ e + ν e + ν µ full electroweak theory is unitary consider now effective action where fields for e, ν e and ν µ have been integrated out effective action for µ contains decay width: appears as dissipative term Example 2 consider electromagnetic field A µ field strength above Schwinger threshold: electron-positron pair production electron / positron field can be integrated out: dissipative term for electromagnetic field 2 / 30

4 Double time path formalism formalism for general, far-from-equilibrium situations: Schwinger-Keldysh double time path can be formulated with two fields Φ = 1 (φ+ + φ ), χ = φ+ φ 2 in principle for arbitrary initial density matrices, in praxis mainly Gaussian initial states allows to treat also dissipation useful also to treat initial state fluctuations or forced noise in classical statistical theories [talks by N. Tetradis, L. Canet] difficult to recover thermal equilibrium, in particular non-perturbatively formalism algebraically somewhat involved 3 / 30

5 Close-to-equilibrium situations out-of-equilibrium situations close-to-equilibrium: description by field expectation values and thermodynamic fields more complete description by following more fields explicitly example: Viscous fluid dynamics plus additional fields usually discussed in terms of phenomenological constitutive relations as a limit of kinetic theory in AdS/CFT want non-perturbative formulation in terms of QFT concepts Analytic continuation as an alternative to Schwinger-Keldysh direct generalization of equilibrium formalism 4 / 30

6 Local equilibrium states Dissipation: energy and momentum get transferred to a heat bath Even if one starts with pure state T = 0 initially, dissipation will generate nonzero temperature Close-to-equilibrium situations: dissipation is local Convenient to use general coordinates with metric g µν(x) Need approximate local equilibrium description with temperature T (x) and fluid velocity u µ (x), will appear in combination β µ (x) = uµ (x) T (x) Global thermal equilibrium corresponds to β µ Killing vector µβ ν(x) + νβ µ(x) = 0 5 / 30

7 Local equilibrium Use similarity between local density matrix and translation operator e βµ (x)p µ e i x µ P µ to represent partition function as functional integral with periodicity in imaginary direction such that φ(x µ iβ µ (x)) = ±φ(x µ ) Partition function Z[J], Schwinger functional W [J] in Euclidean domain Z[J] = e W E [J] = Dφ e S E [φ]+ x Jφ First defined on Euclidean manifold Σ M at constant time Approximate local equilibrium at all times: Hypersurface Σ can be shifted (a) Global thermal equilibrium (b) Local thermal equilibrium d 0 d (x) x x FIG. 2. Comparison between the global thermal equilibrium (a) and local thermal equilibrium 6 / 30

8 Effective action Defined in euclidean domain by Legendre transform Γ E[Φ] = J a(x)φ a(x) W E[J] with expectation values Φ a(x) = 1 δ g(x) δj WE[J] a(x) x Euclidean field equation δ δφ a(x) ΓE[Φ] = g(x) J a(x) resembles classical equation of motion for J = 0. Need analytic continuation to obtain a viable equation of motion 7 / 30

9 Two-point functions Consider homogeneous background fields and global equilibrium ( ) β µ 1 =, 0, 0, 0 T Propagator and inverse propagator δ 2 δj a( p)δj b (q) WE[J] = G ab(iω n, p) δ(p q) δ 2 δφ a( p)δφ b (q) ΓE[Φ] = P ab(iω n, p) δ(p q) From definition of effective action G ab (p)p bc (p) = δ ac b 8 / 30

10 Spectral representation Källen-Lehmann spectral representation with ρ ab R G ab (ω, p) = dz ρ ab(z 2 p 2, z) z ω correlation functions can be analytically continued in ω = u µ p µ branch cut or poles on real frequency axis ω R but nowhere else different propagators follow by evaluation of G ab in different regions Im(ω) retarded advanced Matsubara Feynman Re(ω) M ab(p) =G ab (iω n, p ) R ab(p) =G ab ( p 0 + iɛ, p ) A ab(p) =G ab ( p 0 iɛ, p ) F ab(p) =G ab ( p 0 + iɛ sign ( p 0), p ) 9 / 30

11 Inverse propagator spectral representation for G ab implies that inverse propagator P ab (ω, p) can have zero-crossings for ω = p 0 R has in general branch-cut for ω = p 0 R so far reference frame with u µ = (1, 0, 0, 0) more general: analytic continuation with respect to ω = u µ p µ use decomposition P ab (p) = P 1,ab (p) is I ( u µ p µ) P 2,ab (p) with sign function s I (ω) = sign(im ω) both functions P 1,ab (p) and P 2,ab (p) are regular (no discontinuities) 10 / 30

12 Sign operator in position space In position space, sign function becomes operator s I ( u µ p µ) = sign (Im( u µ p µ)) sign ( Im ( iu µ )) ( ( x = sign Re u µ )) ( µ x = µ sr u µ Geometric representation in terms of Lie derivative s R (L u) or s R (L β ) ) x µ Sign operator appears also in analytically continued quantum effective action Γ[Φ] 11 / 30

13 Analytically continued 1 PI effective action Analytically continued quantum effective action defined by analytic continuation of correlation functions Quadratic part Γ 2[Φ] = 1 2 x,y ( Φ a(x) [P 1,ab (x y) + P 2,ab (x y)s R u µ y µ )] Φ b (y) Higher orders correlation functions less understood: no spectral representation Use inverse Hubbard-Stratonovich trick: terms quadratic in auxiliary field can be integrated out Allows to understand analytic structures of higher order terms [Floerchinger, ] 12 / 30

14 Equations of motion Can one obtain causal and real renormalized equations of motion from the 1 PI effective action? naively: time-ordered action / Feynman iɛ prescription: δ δφ a(x) Γ time ordered[φ] = g J a(x) This does not lead to causal and real equations of motion! [e.g. Calzetta & Hu: Non-equilibrium Quantum Field Theory (2008)] 13 / 30

15 Retarded functional derivative [Floerchinger, ] Real and causal dissipative field equations follow from analytically continued effective action δγ[φ] = gj(x) δφ a(x) ret to calculate retarded variational derivative determine δγ[φ] by varying the fields δφ(x) including dissipative terms set signs according to s R (u µ µ) δφ(x) δφ(x), δφ(x) s R (u µ µ) +δφ(x) proceed as usual opposite choice of sign: field equations for backward time evolution 14 / 30

16 Causality consider derivative of field equation (in flat space with g = 1) δ δγ δ δφ b (y) δφ a(x) = ret δφ b (y) Ja(x) inverting this equation gives retarded Green s function δ δj b (y) Φa(x) = R ab(x, y) only non-zero for x future or null to y Causality: Field expectation value Φ a(x) can only be influenced by the source J b (y) in or on the past light cone 15 / 30

17 Damped harmonic oscillator 1 Equation of motion or with ω 0 = k/m and ζ = c/ 4mk mẍ + cẋ + kx = 0 ẍ + 2ζω 0ẋ + ω 2 0x = 0 What is action for damped oscillator? This does not work: dω m 2π 2 x (ω) [ ω 2 + 2i ω ζω 0 ω0 2 ] x(ω) Consider inverse propagator ω 2 + 2i s I (ω) ω ζω 0 ω0 2 with s I (ω) = sign (Im ω) zero crossings (poles in the eff. propagator) are broadened to branch cut 16 / 30

18 Damped harmonic oscillator 2 Take for effective action dω m Γ[x] = 2π 2 x (ω) [ ω 2 2i s I (ω) ω ζω 0 + ω0 2 ] x(ω) = dt { 12 mẋ c x s R( t)ẋ + 12 } kx2 where the second line uses s I (ω) = sign(im ω) sign(im i t) = sign(re t) = s R ( t) Variation gives up to boundary terms { δγ = dt mẍ δx c δx s R( t)ẋ 1 } 2 c ẋ s R( t)δx + kx δx Set now s R ( t)δx δx and δx s R ( t) δx. Defines δγ ret. δx Equation of motion for forward time evolution δγ = mẍ + cẋ + kx = 0 δx ret 17 / 30

19 Scalar field with O(N) symmetry Consider effective action (with ρ = 1 ϕjϕj) 2 Γ[ϕ, g µν, β µ ] = d d x { 1 g 2 Z(ρ, T )gµν µϕ j νϕ j + U(ρ, T ) + 1 } 2 C(ρ, T ) [ϕj, s R(u µ µ)] β ν νϕ j Variation at fixed metric g µν and β µ gives δγ = d d x { g Z(ρ, T )g µν µδϕ j νϕ j Z (ρ, T )ϕ mδϕ m g µν µϕ j νϕ j + U (ρ, T )ϕ mδϕ m C(ρ, T ) [δϕ j, s R (u µ µ)] β ν νϕ j C(ρ, T ) [ϕ j, s R (u µ µ)] β ν νδϕ j C (ρ, T )ϕ mδϕ m [ϕ j, s R (u µ µ)] β ν νϕ j } set now δϕ j s R (u µ µ) δϕ j and s R (u µ µ) δϕ j δϕ j 18 / 30

20 Scalar field with O(N) symmetry Field equation becomes µ [Z(ρ, T ) µ ϕ j] Z (ρ, T )ϕ j µϕ m µ ϕ m +U (ρ, T )ϕ j + C(ρ, T )β µ µϕ j = 0 Generalized Klein-Gordon equation with additional damping term 19 / 30

21 Where do energy & momentum go? Modified variational principle leads to equations of motion with dissipation. But what happens to the dissipated energy and momentum? And other conserved quantum numbers? What about entropy production? 20 / 30

22 Energy-momentum tensor expectation value Analogous to field equation, obtain by retarded variation δγ[φ, g µν, β µ ] δg µν(x) = 1 g T µν (x) ret 2 Leads to Einstein s field equation when Γ[Φ, g µν, β µ ] contains Einstein-Hilbert term Useful to decompose Γ[Φ, g µν, β µ ] = Γ R[Φ, g µν, β µ ] + Γ D[Φ, g µν, β µ ] where reduced action Γ R contains no dissipative / discontinuous terms and Γ D only dissipative terms Energy-momentum tensor has two parts T µν = ( T R) µν + ( T D) µν 21 / 30

23 General covariance Infinitesimal general coordinate transformations as a gauge transformation of the metric δg G µν(x) = g µλ (x) ɛλ (x) x ν + g νλ (x) ɛλ (x) x µ + gµν(x) ɛ λ (x) x λ Temperature / fluid velocity field transforms as vector δβ µ G (x) = βν (x) ɛµ (x) x ν + βµ (x) ɛ ν (x) x ν Also fields Φ a transform in some representation, e. g. as scalars δφ G a (x) = ɛ λ (x) x λ Φa(x) Reduced action is invariant Γ R[Φ + δφ G, g µν + δg G µν, β µ + β µ G ] = ΓR[Φ, gµν, βµ ] 22 / 30

24 Situation without dissipation Consider first situation without dissipation Γ[Φ, g µν, β µ ] = Γ R[Φ, g µν] Field equation implies (for J = 0) δ ΓR[Φ, gµν] = 0 δφ a(x) Gauge variation of the metric δγ R = d d x g ɛ λ (x) µ T µ λ (x) General covariance δγ R = 0 and field equations imply covariant energy-momentum conservation µ T µ λ (x) = 0 23 / 30

25 Situation with dissipation Consider now situation with dissipation. General covariance of Γ R: { δγ R = d d δγr x δφ G a δφ + g ɛ λ µ( T R ) µ λ + δγ } R a δβ µ δβµ G = 0 Reduced action not stationary with respect to field variations δγ R δφ = δγ D a(x) δφ =: g(x) M a(x) a(x) ret Reduced energy-momentum tensor not conserved µ( T R ) µ λ (x) = µ( T D ) µ λ (x) Dependence on β µ (x) cannot be dropped δγ R δβ µ (x) =: g(x) K µ(x) General covariance implies four additional differential equations that determine β µ M a λ Φ a + µ( T D ) µ λ = µ [βµ K λ ] + K µ λ β µ 24 / 30

26 Entropy production Contraction of previous equation with β λ gives M aβ λ λ Φ a + β λ µ( T D) µ λ = µ [ β µ β λ K λ ] Consider special case g Kµ(x) = δγr δβ µ (x) = δ δβ µ (x) d d x g U(T ) with grand canonical potential density U(T ) = p(t ) and temperature T = 1 gµνβ µ β ν Using s = p/ T gives entropy current β µ β λ K λ = s µ = su µ Local form of second law of thermodynamics µs µ = M aβ λ λ Φ a + β λ µ( T D) µ λ 0 25 / 30

27 Energy-momentum tensor for scalar field Analytic action Γ[ϕ, g µν, β µ ] = d d x { 1 g 2 Z(ρ, T )gµν µϕ j νϕ j + U(ρ, T ) C(ρ, T ) [ϕ j, s R (u µ µ)] β ν νϕ j } Energy-momentum tensor T µν (x) =Z(ρ, T ) µ ϕ j ν ϕ j ( g µν + u µ u ν T ) { } 1 T 2 Z(ρ, T )gµν µϕ j νϕ j + U(ρ, T ) Generalizes T µν for scalar field and T µν = (ɛ + p)u µ u ν + g µν p for ideal fluid with pressure p = U and enthalpy density ɛ + p = st = T T U. General covariance and covariant conservation law imply µ T µν (x) = 0 = Differential eqs. for β µ (x) 26 / 30

28 Entropy production for scalar field Entropy current s µ = β µ β λ K λ = β µ T T { } 1 2 Z(ρ, T )gαβ αϕ j β ϕ j + U(ρ, T ) Generalized entropy density s G = T { 1 2 Z(ρ, T )gαβ αϕ j β ϕ j + U(ρ, T ) } Entropy generation positive semi-definite for C(ρ, T ) 0 µs µ = C(ρ, T ) (β µ µϕ j) (β ν νϕ j) 0 For fluid at rest u µ = (1, 0, 0, 0) µs µ = ṡ G = C(ρ, T ) T 2 ϕ j ϕ j entropy increases when ϕ j oscillates. For example reheating after inflation. 27 / 30

29 Ideal fluid Consider effective action Γ[g µν, β µ ] = Γ R[g µν, β µ ] = d d x g U(T ) with effective potential U(T ) = p(t ) and temperature T = 1 gµνβ µ β ν Variation of g µν at fixed β µ leads to T µν = (ɛ + p)u µ u ν + pg µν where ɛ + p = T s = T p is the enthalpy density T Describes ideal fluid. General covariance of covariant conservation µt µν = 0 leads to ideal fluid equations u µ µɛ + (ɛ + p) µu µ = 0 (ɛ + p)u µ µu ν + νµ µp = 0 28 / 30

30 Viscous fluid Analytic action Γ[g µν, β µ ] = with projector and leads to T µν = 2 x { U(T ) + 1 } 4 [gµν, s R(L u)] (2η(T )σ µν + ζ(t ) µν ρu ρ ) µν = u µ u ν + g µν ( 1 σ µν = 2 µα µβ µβ µα 1 ) d 1 µν αβ αu β δγ[g µν,β µ ] g δg µν ret = (ɛ + p)u µ u ν + pg µν 2ησ µν ζ µν ρu ρ Describes viscous fluid with shear viscosity η(t ) and bulk viscosity ζ(t ) Entropy production µs µ = 1 T [ 2ησµνσ µν + ζ( ρu ρ ) 2] 29 / 30

31 Conclusions A variational principle for theories with dissipation can be based on analytic continuation. Needs a local equilibrium setup: Generalized Gibbs ensemble with T (x) and u µ (x). Works at least for close-to-equilibrium situations, e. g. fluid dynamics coupled to additional fields. General covariance and energy-momentum conservation lead to equations for fluid velocity and entropy production. Local form of second law of thermodynamics is implemented on the level of the effective action Γ[Φ]. Many potential applications. 30 / 30

32 Backup

33 Equations of motion from the Feynman action? Consider damped harmonic oscillator as example. Time-ordered or Feynman action is obtained from analytic action by replacing s I (ω) sign(ω) Γ time ordered [x] = dω 2π m 2 x (ω) [ ω 2 2i ω ζω 0 + ω0 2 ] x(ω) Field equation δ Γ δx(t) time ordered[x] = J(t) would give [ ω 2 2i ω ζω 0 + ω0 2 ] x(ω) = J(ω) Violates reality constraint x (ω) = x( ω) for J (ω) = J( ω) Solution not causal x(t) = F (t t )J(t ) t because Feynman propagator F (t t ) not causal. In contrast, retarded variation of analytic action leads to real and causal equation of motion

34 Tree-like structures Discontinuous terms in analytic action could be of the form Γ Disc [Φ] = d d x g { ( f[φ](x) s R u µ ) } (x) x g[φ](x) µ More general, tree-like structure are possible such as { ( Γ Disc [Φ] = f[φ](x) s R u µ ) ( (x) x g[φ](x, y) µ sr u µ ) } (y) y h[φ](y) µ or Γ Disc [Φ] = x,y x,y,z { ( f[φ](x) s R u µ ) ( (x) x g[φ](x, y, z) µ sr u µ ) (y) y h[φ](y) µ ( s R u µ ) } (z) z j[φ](z) µ For retarded variation calculate δγ and set s R (u µ µ) 1 if derivative operator points towards node that is varied and s R (u µ µ) 1 if derivative operator points in opposite direction

35 Analytic continuation of FRG equations [Floerchinger, JHEP 1205 (2012) 021] Consider a point p 2 0 p 2 = m 2 where P 1(m 2 ) = 0. One can expand around this point P 1 = Z( p p 2 + m 2 ) + P 2 = Zγ 2 + Leads to Breit-Wigner form of propagator (with γ 2 = mγ) G(p) = 1 Z p p 2 + m 2 + i s(p 0) mγ ( p p2 + m 2 ) 2 + m 2 Γ 2. A few flowing parameters describe efficiently the singular structure of the propagator. Flow of the discontinuity coe cient / ln(k/ )

36 Truncation for relativistic scalar O(N) theory Γ k = t, x { N j=1 1 2 φ j Pφ (i t, i ) φ j ρ P ρ(i t, i ) ρ + Ūk( ρ) } with ρ = 1 2 N j=1 φ 2 j. Goldstone propagator massless, expanded around p 0 p 2 = 0 P φ (p 0, p) Z φ ( p p 2 ) Radial mode is massive, expanded around p 2 0 p 2 = m 2 1 P φ (p 0, p) + ρ 0 Pρ(p 0, p) + Ū k + 2 ρū k Z φ Z 1 [( p p 2 + m 2 1) is(p 0) γ 2 1 ]

37 Flow of the effective potential tu k (ρ) = 1 { (N 1) ρ 2 p 2 p U + 1 Zφ R k p 0 =iω n, p } 1 + Z 1 [( p 2 p 2 0 ) i s(p0)γ2 1 ] + U tr + 2ρU + 1 Zφ k. R k 1 Z φ Summation over Matsubara frequencies p 0 = i2πt n can be done using contour integrals. Radial mode has non-zero decay width since it can decay into Goldstone excitations. Use Taylor expansion for numerical calculations U k (ρ) = U k (ρ 0,k ) + m 2 k(ρ ρ 0,k ) λ k(ρ ρ 0,k ) 2