Effective dissipation Stefan Fl¨ orchinger (Heidelberg U.) Functional Renormalization, Heidelberg, March 7, 2017.
Effective dissipation Dissipation is generation of entropy von Neumann definition S = − Tr ρ 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
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
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
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
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 ) = u µ ( x ) T ( x ) Global thermal equilibrium corresponds to β µ Killing vector ∇ µ β ν ( x ) + ∇ ν β µ ( x ) = 0 5 / 29
Local equilibrium Use similarity between local density matrix and translation operator e β µ ( 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 β 0 β ( x ) d τ d τ x x 6 / 29
Effective action Defined in euclidean domain by Legendre transform � Γ E [Φ] = J a ( x )Φ a ( x ) − W E [ J ] x with expectation values 1 δ Φ a ( x ) = √ g ( x ) δJ a ( x ) W E [ J ] Euclidean field equation δ Φ a ( x )Γ E [Φ] = √ g ( x ) J a ( x ) δ resembles classical equation of motion for J = 0 . Need analytic continuation to obtain a viable equation of motion 7 / 29
Two-point functions Consider homogeneous background fields and global equilibrium � 1 � β µ = T , 0 , 0 , 0 Propagator and inverse propagator δ 2 δJ a ( − p ) δJ b ( q ) W E [ J ] = G ab ( iω n , p ) δ ( p − q ) δ 2 δ Φ a ( − p ) δ Φ b ( q )Γ E [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
Spectral representation K¨ allen-Lehmann spectral representation dz ρ ab ( z 2 − p 2 , z ) � ∞ G ab ( ω, p ) = z − ω −∞ with ρ ab ∈ R correlation functions can be analytically continued in ω = − u µ p µ branch cut or poles on real frequency axis ω ∈ ❘ but nowhere else different propagators follow by evaluation of G ab in different regions Im ( ω ) Matsubara ∆ M ab ( p ) = G ab ( iω n , p ) p 0 + iǫ, p ∆ R � � retarded Feynman ab ( p ) = G ab Re ( ω ) p 0 − iǫ, p ∆ A � � ab ( p ) = G ab advanced p 0 + iǫ sign ∆ F p 0 � � � � ab ( p ) = G ab , p 9 / 29
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
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 )) iu µ u µ u µ � � ∂ �� � � ∂ �� � ∂ � → sign Im = sign Re = s R ∂x µ ∂x µ ∂x µ Geometric representation in terms of Lie derivative s R ( L u ) or s R ( L β ) Sign operator appears also in analytically continued quantum effective action Γ[Φ] 11 / 29
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 � � � �� u µ ∂ Φ a ( x ) P 1 ,ab ( x − y ) + P 2 ,ab ( x − y ) s R Φ b ( y ) ∂y µ 2 x,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
Equations of motion Can one obtain causal and real renormalized equations of motion from the 1 PI effective action? naively: time-ordered action / Feynman iǫ prescription: δ Φ a ( x )Γ time ordered [Φ] = √ g J a ( x ) δ This does not lead to causal and real equations of motion ! [e.g. Calzetta & Hu: Non-equilibrium Quantum Field Theory (2008)] 13 / 29
Retarded functional derivative [Floerchinger, JHEP 1609 (2016) 099] Real and causal dissipative field equations follow from analytically continued effective action δ Γ[Φ] ret = √ gJ ( x ) � � δ Φ a ( x ) � to calculate retarded variational derivative determine δ Γ[Φ] by varying the fields δ Φ( x ) including dissipative terms set signs according to s R ( u µ ∂ µ ) δ Φ( x ) → − δ Φ( x ) , δ Φ( x ) s R ( u µ ∂ µ ) → + δ Φ( x ) proceed as usual opposite choice of sign: field equations for backward time evolution Leads to causal equations of motion 14 / 29
Scalar field with O ( N ) symmetry Consider effective action (with ρ = 1 2 ϕ j ϕ j ) � 1 � d d x √ g Γ[ ϕ, g µν , β µ ] = 2 Z ( ρ, T ) g µν ∂ µ ϕ j ∂ ν ϕ j + U ( ρ, T ) + 1 � 2 C ( ρ, T ) [ ϕ j , s R ( u µ ∂ µ )] β ν ∂ ν ϕ j Variation at fixed metric g µν and β µ gives � Z ( ρ, T ) g µν ∂ µ δϕ j ∂ ν ϕ j + 1 � d d x √ g 2 Z ′ ( ρ, T ) ϕ m δϕ m g µν ∂ µ ϕ j ∂ ν ϕ j δ Γ = + U ′ ( ρ, T ) ϕ m δϕ m + 1 2 C ( ρ, T ) [ δϕ j , s R ( u µ ∂ µ )] β ν ∂ ν ϕ j + 1 2 C ( ρ, T ) [ ϕ j , s R ( u µ ∂ µ )] β ν ∂ ν δϕ j + 1 � 2 C ′ ( ρ, T ) ϕ m δϕ m [ ϕ j , s R ( u µ ∂ µ )] β ν ∂ ν ϕ j set now δϕ j s R ( u µ ∂ µ ) → δϕ j and s R ( u µ ∂ µ ) δϕ j → − δϕ j 15 / 29
Scalar field with O ( N ) symmetry Field equation becomes −∇ µ [ Z ( ρ, T ) ∂ µ ϕ j ] + 1 2 Z ′ ( ρ, T ) ϕ j ∂ µ ϕ m ∂ µ ϕ m + U ′ ( ρ, T ) ϕ j + C ( ρ, T ) β µ ∂ µ ϕ j = 0 Generalized Klein-Gordon equation with additional damping term 16 / 29
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
Energy-momentum tensor expectation value Analogous to field equation, obtain by retarded variation δ Γ[Φ , g µν , β µ ] � √ g � T µν ( x ) � = − 1 � � δg µν ( x ) 2 � ret 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 R ) µν + ( ¯ � T µν � = ( ¯ T D ) µν 18 / 29
General covariance Infinitesimal general coordinate transformations as a “gauge transformation” of the metric µν ( x ) = g µλ ( x ) ∂ǫ λ ( x ) + g νλ ( x ) ∂ǫ λ ( x ) + ∂g µν ( x ) δg G ǫ λ ( x ) ∂x ν ∂x µ ∂x λ Temperature / fluid velocity field transforms as vector G ( x ) = − β ν ( x ) ∂ǫ µ ( x ) + ∂β µ ( x ) δβ µ ǫ ν ( x ) ∂x ν ∂x ν Also fields Φ a transform in some representation, e. g. as scalars a ( x ) = ǫ λ ( x ) ∂ δ Φ G ∂x λ Φ a ( x ) Reduced action is invariant µν , β µ + β µ Γ R [Φ + δ Φ G , g µν + δg G G ] = Γ R [Φ , g µν , β µ ] 19 / 29
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