Path-integral formula for local thermal equilibrium

≃ x Masaru Hongo RIKEN, iTHES research group Nuclear Physics Seminar at Stony Brook University, 2017 2/27 Based on Hayata-Hidaka-MH-Noumi PRD(2015), MH arXiv: 1611:07074 , and My Ph. D thesis

Microscopic

Neutron Star (Magnetar) Quark-Gluon Plasma (QGP)

Macroscopic

LQCD http://newsoffice.mjitugenn.edu/2012/model-bursting-star-0302

http://www.bnl.gov/rhic/news2/news.asp?a=1403&t=pr

QFT

?

Hydrodynamics

Question. d.o.f. Quark, Gluon

How to bridge the gap between micro and macro?

d.o.f

- Banerjee et al.(2012) - Jensen et al.(2012) - Haehl et al. (2015)

T (x), !v (x), µ(x)

Nakajima (1957), Mori (1958), McLennan (1960) Zubarev et al. (1979), Becattini et al. (2015) Hayata-Hidaka-MH-Noumi (2015)

Thermal QFT (Matsubara formalism)

Global equil. β0

T = const.

[ Matsubara, 1955 ]

dτ

Path int.

QFT in the flat spacetime with size β0

β0

x e Gibbs dist.:

ˆ ˆ) −β(H−µ N

ρˆG =

Z

ˆ

ˆ

= e−β(H−µN )−Ψ[β,ν]

Thermodynamic potential with Euclidean action ˆ

ˆ

!

ˆ

ˆ

Ψ[β, ν] = log Tr e−β(H−µN ) = log dϕ!±ϕ|e−β(H−µN ) |ϕ" ! β ! ! dτ d3 x LE (ϕ, ∂µ ϕ) = log Dϕ e+SE [ϕ] , SE [ϕ] = ϕ(β)=±ϕ(0)

0

Local Thermal QFT

Local equil. {β(x), "v (x)}

Path int.

Local thermal QFT can describe anomaly-induced transport µR != µL S

N

!j ∝ ω !

! !j ∝ B µR != µL Chiral Magnetic Effect

Chiral Vortical Effect

　 Motivation: Quantum field theory under local thermal equilibrium?

　 Approach: QFT for Local Gibbs distribution

　 Application:

Derivation of Anomalous hydrodynamics

µ5 != 0 N

S

! !j ∝ B

Determined only by local temperature, local velocity… at that time

Global thermal equilibrium: =const. const. TT =

Gibbs distribution:

ρˆG = e

ˆ −β H−Ψ[β]

,

Ψ[β] ≡ log Tre

ˆ −β H

Localize

Local thermal equilibrium: Local Gibbs (LG) distribution:

ρˆLG = e ˆ =− K {β(x), "v (x)}

!

µ ˆ −K−Ψ[β (x),ν(x)]

"

d3 x β µ (x)Tˆ0µ (x) + ν(x)Jˆ0 (x)

#

Gibbs distribution

Local Gibbs distribution

What is the state with maximizing ρ) = −Trˆ ρ log ρˆ information entropy: S(ˆ

What is the state with maximizing ρ) = −Trˆ ρ log ρˆ information entropy: S(ˆ

under constraints: ˆ = E = const., !N ˆ " = N = const. !H"

under constraints: !Tˆ0µ (x)" = pµ (x), !Jˆ0 (x)" = n(x)

Answer: ˆ ˆ ρˆG = e−β H−ν N −Ψ[β,ν]

Answer: ! − dd−1 x(β µ Tˆ 0µ +ν Jˆ0 )−Ψ[β µ ,ν] ρˆLG = e

Lagrange multipliers: Λa = {β, ν = βµ}

Lagrange multipliers: λa (x) = {β µ (x), ν(x)}

Flat spacetime

Curved spacetime

t

ηµν

Ex. Bjorken coord. t Σt¯

t

gµ¯ν¯

t¯ = τ =

!

t2 − z 2

dΣµ ∝ ∂µ t¯

dΣµ ∝ (1, 0)

z

Σt¯

t = const.

t¯(x) = const.

x ˆ =− K

!

3

"

d x β

µ

(x)Tˆ0µ (x)

ˆ0

+ ν(x)J (x)

#

x ˆ =− K

!

"

dΣt¯ν β

µ

(x)Tˆνµ (x)

ˆν

+ ν(x)J (x)

① Formulation becomes manifestly covariant { ② Background metric plays a role as external field coupled to T

µν

#

LG distribution w/ λ = {β µ , ν}

t dΣµ ∝ ∂µ t¯

Σt¯ t¯(x) = const.

x Masseiu-Planck functional !" # $% Ψ[t¯; λ] ≡ log Tr exp dΣt¯ν β µ (x)Tˆνµ (x) + ν(x)Jˆν (x) % ! " # $ ¯ ¯ 3 √ = log Tr exp − d x ¯ −g β µ¯ (¯ x)Tˆ0µ¯ (¯ x) + ν(¯ x)Jˆ0 (¯ x)

[ Banerjee et al.(2012), Jensen et al.(2012) , Haehl et al. (2015), MH(2016)]

Variation formula in “hydrostatic gauge” 2 1 δ δ µν LG µ LG ˆ ˆ ¯ !T (x)"t¯ = √ Ψ[t; λ], !J (x)"t¯ = √ Ψ[t¯; λ] −g δgµν (x) −g δAµ (x)

t¯ dΣµ ∝ ∂µ t¯

LG distribution w/ λ = {β µ , ν}

t tµ ≡ ∂t¯xµ

Σt¯ t¯(x) = const.

x Masseiu-Planck functional !" # $% Ψ[t¯; λ] ≡ log Tr exp dΣt¯ν β µ (x)Tˆνµ (x) + ν(x)Jˆν (x) % ! " # $ ¯ ¯ 3 √ = log Tr exp − d x ¯ −g β µ¯ (¯ x)Tˆ0µ¯ (¯ x) + ν(¯ x)Jˆ0 (¯ x)

Picture before gauge fixing Future time direction

Picture in hydrostatic gauge

¯ β µ (t¯2 , x)

β µ¯ (x) = β0 δ¯0µ¯

t¯

tµ t¯(x) = t¯2

¯ β (t¯1 , x) µ

Gauge fixing

t¯2

tµ = e σ u µ (eσ ≡ β/β0 )

x ¯j

tµ t¯(x) = t¯1

t x j xi

x ¯i = const.

t¯1

x ¯i

We can choose the time direction vector tµ (x) ≡ ∂t¯xµ Hydrostatic gauge fixing Let us choose tµ (x) = β µ (x)/β0 , A¯0 (x) = ν(x)

[ Banerjee et al.(2012), Jensen et al.(2012) , Haehl et al. (2015), MH(2016)]

Variation formula in “hydrostatic gauge” 2 1 δ δ µν LG µ LG ˆ ˆ ¯ !T (x)"t¯ = √ Ψ[t; λ], !J (x)"t¯ = √ Ψ[t¯; λ] −g δgµν (x) −g δAµ (x)

Proof. Consider time derivative of Ψ[λ] ∂t¯Ψ[t¯; λ] =

!

d

d−1

√

"

√

"

x ¯ −g

∇µ βν $Tˆµν %LG t¯

ˆµ

#

+ (∇µ ν + Fνµ β )$J %t¯ ν

# 1 ν ν ˆµ (∇µ βν + ∇ν βµ )$Tˆµν %LG = dd−1 x ¯ −g t¯ + (β ∇ν Aµ + Aν ∇µ β )$J %t¯ 2 # " ! √ 1 ˆµ £β gµν #Tˆµν $LG = dd−1 x ¯ −g t¯ + £β Aµ #J $t¯ 2 !

On the other hand, since tµ = β µ , we can express the LHS as ∂t¯Ψ[t¯; λ] =

!

"

dd−1 x ¯ £β gµν

δΨ δΨ + £ β Aµ δgµν δAµ

#

Matching them gives the above variation formula!

g µ¯ν¯ L=− ∂µ¯ φ∂ν¯ φ − V (φ) 2 2 δS µν ˆ ∂ρ φ) ˆ ν φˆ + g µν L(φ, ˆ ˆ T ≡√ = ∂ µ φ∂ −g δgµν # ! " ¯ d−1 √ ¯ Ψ[t; λ] = log Tr exp − d x ¯ −gβ µ (x)Tˆ0µ (x) ! ! = log Dφ exp (SE [φ, β µ ]) = log Dφ exp (SE [φ, g˜]) S[φ, β µ ] =

=

!

β0

dτ

!

dτ

!

0

!

β0 0

"

−σ ¯i

√ e ¯ ˙ 2 − −e u (iφ)∂ ˙ ¯i φ − 1 d3 x ¯ −geσ u0 − ¯0 (iφ) 2 2u u¯0 u¯0 u¯0 −2σ

g˜µ¯ν¯ 3 d x ¯ −˜ g − ∂µ¯ φ∂ν¯ φ − V (φ) 2 "

#

$

#

¯¯

γ ij +

¯i ¯ j

uu u¯0 u¯0

$

∂¯i φ∂¯j φ − V (φ)

(eσ(¯x) ≡ β(¯ x)/β0 )

%

Ψ[t¯; λ] = log

!

Dφ exp (SE [φ, ; g˜])

Thermal metric

g˜µ¯ν¯ =

!

2σ

−e eσ u¯i

σ

e u¯j γ¯i¯j

Inverse thermal metric

"

(eσ(¯x) ≡ β(¯ x)/β0 )

g˜µ¯ν¯



−2σ

e  u¯0 u¯ 0 =  e−σ u¯i − ¯0 u u¯0

j −σ ¯

e u − ¯0 u u¯0   ¯i ¯ j  u u ¯i¯ γ j + ¯0 u u¯0

Interpretation of above result Ψ[t¯; λ] is described by QFT in ”curved spacetime” s. t. ¯i ¯ ¯i 2 ′ ˜ d˜ s = −e (dt + a¯i dx ) + γ¯i¯j dx dxj 2

2σ

(a¯i ≡ e−σ u¯i ,

γ¯i′¯j ≡ γ¯i¯j + u¯i u¯j ,



dt˜ = −idτ )

− ← − a µ¯ " 1 ¯ ! a µ¯ → ¯ L = − ψ γ ea D µ¯ − D µ¯ γ ea ψ − mψψ 2

Symmetric energy-momentum tensor T µ¯ν¯

=

−δνµ¯¯ L

→ − µ¯ ← − µ¯ ← −µ¯ − 1 ¯ µ¯ → − ψ(γ D ν¯ + γν¯ D − D ν¯ γ − D γν¯ )ψ 4

Result of path integral !" $% # dΣt¯ν β µ (x)Tˆνµ (x) + ν(x)Jˆν (x) Ψ[t¯; λ] ≡ log Tr exp

= log

!

¯ ¯ e˜] DψDψ exp SE [ψ, ψ; "

#

Ψ[t¯; λ] = log

!

¯ ¯ e˜] DψDψ exp SE [ψ, ψ; "

#

Euclidean action with thermal vielbein ¯ e˜] = SE [ψ, ψ;

!

β0

dτ 0

!

"

% # $ → ← − a µ¯ 1 ¯ a µ¯ − 3 ¯ d x ¯e˜ − ψ γ e˜a D µ¯ − D µ¯ γ e˜a ψ − mψψ 2

： e˜¯0a = eσ ua ,

Thermal vielbein

e˜¯i a = e¯i a (eσ ≡ β(x)/β0 )

Interpretation of above result Ψ[t¯; λ] is described by QFT in ”curved spacetime” s. t. 2

d˜ s =

¯i 2 ¯i ¯ a b µ ¯ ν ¯ 2σ ˜ ′ e˜µ¯ e˜ν¯ ηab dx dx = −e (dt + a¯i dx ) + γ¯i¯j dx dxj (a¯i ≡ e−σ u¯i , γ¯i′¯j ≡ γ¯i¯j + u¯i u¯j , dt˜ = −idτ )

Thermal QFT (Matsubara formalism)

Global equil. β0

T = const.

[ Matsubara, 1955 ]

Path int.

dτ

QFT in the flat spacetime with size β0

β0

x Local Thermal QFT

Local equil. {β(x), "v (x)}

Path int.

[ Hayata-Hidaka-MH-Noumi PRD(2015) ] [ MH (2016) ]

dτ

QFT in the “curved spacetime” with “line element”

β(x)

d˜ s2 = d˜ s2 (β, "v )

x

¯i ¯ ¯i 2 ′ ˜ d˜ s = −e (dt + a¯i dx ) + γ¯i¯j dx dxj 2

2σ

dτ

β(x)

(a¯i ≡ −e−σ u¯i , γ¯i′¯j ≡ γ¯i¯j + u¯i u¯j , dt˜ ≡ −idτ )

x

Symmetry of “curved spacetime” ( 1 ) Spatial diffeomorphism

：


　　Physics does not depend on the choice of Spatial coordinate system!! ( 2 ) Imaginary time tran. & Kaluza-Klein gauge：
 　　Parameters　λ (e.g. T) does not depend on imaginary time!! Ψ[λ] = log

!

Partition function method Banerjee et al.(2012), Jensen et al.(2012)

¯ e] S[ψ,ψ,˜ ¯ DψDψe should respect the above symmetries!!

¯i 2 ¯i ¯ ′ j ˜ d˜ s = −e (dt + a¯i dx ) + γ¯i¯j dx dx (dt˜ = −idτ ) 2

2σ

e˜µ¯µ (β µ )

Ψ[λ] = log

!

τ

“Kaluza-Klein” gauge tr.

β(x)

dτ

　

Parameters λ don’t depend on imaginary time .

x ¯ e] S[ψ,ψ,˜ ¯ DψDψe ∋

(f¯i¯j ≡ ∂¯i a¯j − ∂¯j a¯i )

  ¯ t˜ → t˜ + χ(x) ¯→x ¯ x   ¯ → a¯i (x) ¯ − ∂¯i χ(x) ¯ a¯i (x) ¯i¯ j

f f¯i¯j , · · · ¯

a¯i , a¯i ai , · · ·

Local Thermal QFT

Local equil. {β(x), "v (x)}

[ Hayata-Hidaka-MH-Noumi PRD(2015) ] [ MH (2016) ]

Path int.

dτ

QFT in the “curved spacetime” with “line element”

β(x)

d˜ s2 = d˜ s2 (β, "v )

x Ψ[t¯; λ] ≡ log Tr exp

!"

#

dΣt¯ν β µ (x)Tˆνµ (x) + ν(x)Jˆν (x)



%$① Ψ[λ] ② Ψ[λ] is written in terms of QFT in curved spacetime

2 δ µν LG ˆ Ψ[λ] plays a role as the generating functional: !T (x)" = √ −g δgµν (x) ¯ ¯ ¯ d˜ s2 = −e2σ (dt˜ + a¯i dxi )2 + γ¯i′¯j dxi dxj

Symmetry = Spatial diffeomorphism + Kaluza-Klein gauge

　 Motivation:

µ5 != 0

Quantum field theory under local thermal equilibrium?

N

S

! !j ∝ B

　 Approach: QFT for Local Gibbs distribution

dτ

β(x)

① ② Ψ[λ] is written in terms of QFT in “curved¯ spacetime” 2 δ µν LG ˆ √ ! T (x)" = Ψ[λ] Variation formula: −g δgµν (x)

¯i ¯ i 2 ′ ˜ d˜ s = −e (dt + a¯i dx ) + γ¯i¯j dx dxj 2

2σ

Symmetry = Spatial diffeomorphism + Kaluza-Klein gauge

　 Application:

Derivation of Anomalous hydrodynamics

x

µR = µL

Derivative expansion of ψ Ψ[β µ , ν] = Ψ(0) [β µ , ν] + Ψ(1) [β µ , ν, ∂] + O(∂ 2 ) + · · ·

≃ βp

= 0 Parity-even system

Symmetry property

Non-dissipative constitutive relation 2 δ µν µν ¯ = T [λ(x)] + T √ !Tˆµν (x)"LG = Ψ[ t ; λ] (0) (1) [λ(x), ∇λ(x)] + · · · t¯ −g δgµν (x) 2 δ µ µ µ LG ˆ !J (x)"t¯ = √ Ψ[t¯; λ] = J(0) [λ(x)] + J(1) [λ(x), ∇λ(x)] + · · · −g δAµ (x)

=0

[ Banerjee et al.(2012), Jensen et al.(2012) ]

Masseiu-Planck functional ! S[φ,˜ g ] = Ψ(0) [λ] + Ψ(1) [λ, ∂] + O(∂ 2 ) Ψ[λ] = log Dφe O(p0 )

O(p1 )

：

- Building blocks λ = {eσ , a¯i , µ, A¯i }

：Spatial diffeo，Kaluza-Klein，Gauge

- Symmetry A¯i

：not Kaluza-Klein inv.

A¯¯i ≡ A¯i − µa¯i

： λ = O(p )

- Power counting scheme

f¯i¯j ≡ ∂¯i a¯j − ∂¯j a¯i = O(p1 )

0

f f = O(p2 )

0

O(p ) Masseiu-Planck functional ! S[φ,˜ g ] = Ψ(0) [λ] + Ψ(1) [λ, ∂] + O(∂ 2 ) Ψ[λ] = log Dφe O(p0 )

O(p1 )

：

- Building blocks λ = {eσ , a¯i , µ, A¯¯i } ! ! β0 " dτ d3 x ¯ γ ′ eσ p(β, µ) Ψ(0) [λ] = 0

Perfect fluid µ ν µν !Tˆµν (x)"LG = (e + p)u u + pη ¯ t µ !Jˆµ (x)"LG = nu t¯

µR != µL

◆ Chiral Magnetic Effect (CME)

eµ ! !j = 5 B 2π 2 ◆ Chiral Vortical Effect (CVE)

µµ 5 !j = ω ! 2π 2

[ Fukushima et al.2008, Vilenkin 1980 ]

µR != µL S

N

! !j ∝ B

[ Erdmenger et al. 2008, Son-Surowka 2009 ]

!j ∝ ω !

µR != µL

Derivative expansion of ψ Ψ[β µ , ν] = Ψ(0) [β µ , ν] + Ψ(1) [β µ , ν, ∂] + O(∂ 2 ) + · · ·

≃ βp Symmetry property

= 0 Parity-even system != 0 Parity-odd system

Non-dissipative constitutive relation 2 δ µν µν ¯ = T [λ(x)] + T √ !Tˆµν (x)"LG = Ψ[ t ; λ] (0) (1) [λ(x), ∇λ(x)] + · · · t¯ −g δgµν (x) 2 δ µ µ µ LG ˆ !J (x)"t¯ = √ Ψ[t¯; λ] = J(0) [λ(x)] + J(1) [λ(x), ∇λ(x)] + · · · −g δAµ (x)

=0

!= 0

：

→ ← − m µ" i † ! µ m− Weyl fermion L = ξ em σ D µ − D µ σ em ξ 2 ! † S[ξ,ξ † ,A,˜ e] = Ψ(0) [λ] + Ψ(1) [λ, ∂] + O(∂ 2 ) Ψ[λ] = log Dξ Dξe O(p0 ) O(p1 )

：

- Building blocks λ = {eσ , a¯i , µR , A¯¯i }

：Spatial diffeo，Kaluza-Klein，Gauge

- Symmetry A¯i

：not Kaluza-Klein inv.

A¯¯i ≡ A¯i − µR a¯i

： λ = O(p )

- Power counting scheme

f¯i¯j ≡ ∂¯i a¯j − ∂¯j a¯i = O(p1 )

0

f f = O(p2 )

0

O(p )

：

→ ← − m µ" i † ! µ m− Weyl fermion L = ξ em σ D µ − D µ σ em ξ 2 ! † S[ξ,ξ † ,A,˜ e] = Ψ(0) [λ] + Ψ(1) [λ, ∂] + O(∂ 2 ) Ψ[λ] = log Dξ Dξe O(p0 ) O(p1 )

： !

- Building blocks λ = {eσ , a¯i , µR , A¯¯i } β0

Ψ(0) [λ] =

0

dτ

!

d3 x ¯

"

γ ′ eσ p(β, µR )

Perfect fluid µ ν µν !Tˆµν (x)"LG = (e + p)u u + pη t¯ µ µ (x)"LG !JˆR = n u R t¯

O(p)

：

→ ← − m µ" i † ! µ m− Weyl fermion L = ξ em σ D µ − D µ σ em ξ 2 ! † S[ξ,ξ † ,A,˜ e] = Ψ(0) [λ] + Ψ(1) [λ, ∂] + O(∂ 2 ) Ψ[λ] = log Dξ Dξe O(p0 ) O(p1 )

：

- Building blocks λ = {eσ , a¯i , µR , A¯¯i } ! " ¯ ¯ ¯i¯ 3 jk ′ d x ¯ γ C1 (β, µR )ǫ A¯i ∂¯j A¯k¯

!

" ¯ ¯ ¯i¯ 3 jk ′ d x ¯ γ C2 (β, µR )ǫ A¯i ∂¯j ak¯

µR != µL S

! !j ∝ B

!j ∝ ω !

µR != µL

N

① Non-perturbative way (WZ consistency condition …) ② Perturbative evaluation of ψ in external field [ Banerjee et al.(2012), Jensen et al.(2012) , Haehl et al. (2015) ]

[ MH-Hidaka in preparation (2016) ]

P +Q

δ2Ψ = δAµ δAν

Aµ

Aν

Q

Q

≃ −iε

0µρν

P

µR ˜ Qρ 2 4π

P +Q

Aα Q

P

Ψ

(1)

[λ] =

!

3

d xε

0ijk

"

˜ ρ C(η ν0 ερµ0α + δij η νi ǫρµjα ) ≃ iQ {

δ˜ gµν δ2Ψ = δ˜ gµν δAα Q

µ2R T2 = 2+ 8π 24

νR Ai ∂j Ak + 2 8π

#

ν R µR T + 2 8π 24

$

Ai ∂j g˜0k

%

Ψ

(1)

[λ] =

!

3

d xε

0ijk

"

νR Ai ∂j Ak + 2 8π (1)

#

ν R µR T + 2 8π 24

µR i 1 δΨ i LG ˆ = !JR (x)"(0,1) = √ B + 2 −g δAi (x) 4π

!

µ2R 8π 2

$

N

S

! !j ∝ B

!j ∝ ω !

µR != µL

2

T + 24

µ5 i µµ5 i i LG ˆ !JV (x)"(0,1) = B + ω 2 2 2π 2π ! " 2 2 2 µ + µ5 µ i T i LG i ˆ !JA (x)"(0,1) = B + + ω 2π 2 4π 2 12 µR != µL

Ai ∂j g˜0k "

ωi

%

　 Motivation:

µ5 != 0

Quantum field theory under local thermal equilibrium?

N

S

! !j ∝ B

　 Approach: QFT for Local Gibbs distribution

dτ

β(x)

① ② Ψ[λ] is written in terms of QFT in “curved¯ spacetime” 2 δ µν LG ˆ √ ! T (x)" = Ψ[λ] Variation formula: −g δgµν (x)

x

¯i ¯ i 2 ′ ˜ d˜ s = −e (dt + a¯i dx ) + γ¯i¯j dx dxj 2

2σ

Symmetry = Spatial diffeomorphism + Kaluza-Klein gauge

　 Application:

Derivation of Anomalous hydrodynamics

Ψ(1) → !j =

eµ5 ! B 2 2π

　 Dissipation and Fluctuation: How to implement dissipation and fluctuation based on QFT? - Haehl, Loganayagam, Rangamani (2015-) - Harder, Kovtun, Ritz (2015) - Crossley, Giorioso, Liu (2015)

- Zubarev et al. (1979) - Becattini et al. (2015) - Hayata, Hidaka, MH, Noumi (2015)

　 Non-dissipative transport: Evaluation of Masseiu-Planck fcn. in several situations s.t. in the presence of magnetic field/vorticity … - Hattori, Yin(2016)

- Becattinil et al. (2015)

　 superfluid/ Magneto-hydrodynamics: Extension to cases with other zero modes ( NG-mode / photon )

t¯

Time evolution

dτ

β(x)

β0 t¯f

g˜µν Σt¯ × S 1

gµν t¯0 x ¯

λa (x) on Σt¯