Effective dissipation

Transcription

1 Effective dissipation Stefan Flörchinger (Heidelberg U.) Functional Renormalization, Heidelberg, March 7, 2017.

2 Effective dissipation Dissipation is generation of entropy von Neumann definition S = Trρ ln ρ Entropy measures information we have about a state maximal information for pure state with S = 0 minimal information for thermal state S = max. E, p,n Unitary evolution conserves entropy! What information is really accessible and relevant? 1 / 29

3 Entanglement entropy Consider splitting of system into two parts A + B Reduced density matrix ρ A = Tr B ρ Entanglement entropy between A and B S A = Tr A ρ A ln ρ A Spatial splitting: entanglement entropy of ground state C-theorem & A-theorem 2 / 29

4 Dissipation and effective field theory What are the RG equations for the dissipative terms? Is there universality in the effective dissipative sector? What dissipative terms are relevant for dynamics close to (quantum) phase transitions? 3 / 29

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 / 29

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 / 29

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 / 29

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 / 29

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 / 29

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 / 29

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 / 29

12 Sign operator in position space [Floerchinger, JHEP 1609 (2016) 099] 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 / 29

13 Analytically continued 1 PI effective action [Floerchinger, JHEP 1609 (2016) 099] 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 12 / 29

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 / 29

15 Retarded functional derivative [Floerchinger, JHEP 1609 (2016) 099] 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 Leads to causal equations of motion 14 / 29

16 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 15 / 29

17 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 16 / 29

18 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? 17 / 29

19 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 18 / 29

20 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 ] 19 / 29

21 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 20 / 29

22 Situation with dissipation [Floerchinger, JHEP 1609 (2016) 099] 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 21 / 29

23 Entropy production [Floerchinger, JHEP 1609 (2016) 099] 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 22 / 29

24 Could dissipation affect the cosmological expansion?

25 Backreaction: General idea for dimensional, non-linear dynamics ϕ = f(ϕ) = f 0 + f 1 ϕ f2 ϕ one has for expectation values ϕ = ϕ ϕ = f 0 + f 1 ϕ f2 ϕ f2 (ϕ ϕ) evolution equation for expectation value ϕ depends on two-point correlation function or spectrum P 2 = (ϕ ϕ) 2 evolution equation for spectrum depends on bispectrum and so on more complicated for higher dimensional theories more complicated for gauge theories such as gravity 23 / 29

26 Backreaction in gravity Einstein s equations are non-linear. Important question [G. F. R. Ellis (1984)]: If Einstein s field equations describe small scales, including inhomogeneities, do they also hold on large scales? Is there a sizable backreaction from inhomogeneities to the cosmological expansion? Difficult question, has been studied by many people [Ellis & Stoeger (1987); Mukhanov, Abramo & Brandenberger (1997); Unruh (1998); Buchert (2000); Geshnzjani & Brandenberger (2002); Schwarz (2002); Wetterich (2003); Räsänen (2004); Kolb, Matarrese & Riotto (2006); Brown, Behrend, Malik (2009); Gasperini, Marozzi & Veneziano (2009); Clarkson & Umeh (2011); Green & Wald (2011);...] Recent reviews: [Buchert & Räsänen, Ann. Rev. Nucl. Part. Sci. 62, 57 (2012); Green & Wald, Class. Quant. Grav. 31, (2014)] No general consensus but most people believe now that gravitational backreaction is rather small. In the following we look at a new backreaction on the matter side of Einstein s equations. 24 / 29

27 Fluid equation for energy density First order viscous fluid dynamics u µ µɛ + (ɛ + p) µu µ ζθ 2 2ησ µν σ µν = 0 For v 2 c 2 and Newtonian potentials Φ, Ψ 1 ɛ + v ɛ ( + (ɛ + p) 3 ȧ + ) v a [ = ζ 3 ȧ + ] 2 v a a + η [ a iv j iv j + iv j jv i 23 ( v) 2] 25 / 29

28 Fluid dynamic backreaction in Cosmology [Floerchinger, Tetradis & Wiedemann, PRL 114, (2015)] Expectation value of energy density ɛ = ɛ with dissipative backreaction term 1 a ɛ + 3H ( ɛ + p 3 ζh) = D D = 1 a 2 η [ iv j iv j + iv j jv i 2 3 ivi jvj ] + 1 a 2 ζ[ v] a v (p 6ζH) D vanishes for unperturbed homogeneous and isotropic universe D has contribution from shear & bulk viscous dissipation and thermodynamic work done by contraction against pressure gradients dissipative terms in D are positive semi-definite for spatially constant viscosities and scalar perturbations only D = ζ+ 4 3 η d 3 q P a 2 θθ (q) 26 / 29

29 Dissipation of perturbations [Floerchinger, Tetradis & Wiedemann, PRL 114, (2015)] Dissipative backreaction does not need negative effective pressure 1 a ɛ + 3H ( ɛ + p eff ) = D D is an integral over perturbations, could become large at late times. Can it potentially accelerate the universe? Need additional equation for scale parameter a Use trace of Einstein s equations R = 8πG N T µ µ 1 Ḣ + a 2H2 = 4πG N ( ɛ 3 p 3 eff ) does not depend on unknown quantities like (ɛ + p eff )u µ u ν To close the equations one needs equation of state p eff = p eff ( ɛ) and dissipation parameter D 27 / 29

30 Deceleration parameter [Floerchinger, Tetradis & Wiedemann, PRL 114, (2015)] assume now vanishing effective pressure p eff = 0 obtain for deceleration parameter q = 1 Ḣ ah 2 dq + 2(q 1) ( ) q 1 d ln a 2 = 4πG N D 3H 3 for D = 0 attractive fixed point at q = 1 2 (deceleration) for D > 0 fixed point shifted towards q < 0 (acceleration) dq + 4 G ND d ln a 3H deceleration parameter q 28 / 29

31 Conclusions Effective dissipation can arise in quantum field theories due to effective loss of information. Equations of motion for close-to-equilibrium theories can be obtained from analytic continuation. 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]. Interesting applications in cosmology and condensed matter physics. 29 / 29

32 Backup

33 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 difficult to recover thermal equilibrium, in particular non-perturbatively formalism algebraically somewhat involved

34 Causality [Floerchinger, JHEP 1609 (2016) 099] 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

35 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)

36 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. Example: Reheating after inflation.

37 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

38 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

39 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.

40 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 [ 2ησµνσ µν + ζ( ρu ρ ) 2] T

41 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

42 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

43 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/ )

44 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 ]

45 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

46 Bulk viscosity Bulk viscous pressure is negative for expanding universe π bulk = ζ µu µ = ζ 3H < 0 Negative effective pressure p eff = p + π bulk < 0 would act similar to dark energy in Friedmann s equations [Murphy (1973), Padmanabhan & Chitre (1987), Fabris, Goncalves & de Sa Ribeiro (2006), Li & Barrow (2009), Velten & Schwarz (2011), Gagnon & Lesgourgues (2011),...] Is negative effective pressure physical? In context of heavy ion physics: instability for p eff < 0 ( cavitation ) [Torrieri & Mishustin (2008), Rajagopal & Tripuraneni (2010), Buchel, Camanho & Edelstein (2014), Habich & Romatschke (2015), Denicol, Gale & Jeon (2015)] What precisely happens at the instability?

47 Is negative effective pressure physical? Kinetic theory p eff (x) = d 3 p p 2 f(x, p) 0 (2π) 3 3E p Stability argument p eff Ε p eff Ε p eff Ε Ε p eff Ε 1 2 Ε 2 Ε Ε 2 If there is a vacuum with ɛ = p eff = 0, phases with p eff < 0 cannot be mechanically stable. (But could be metastable.)