On the effective action in hydrodynamics

Transcription

1 On the effective action in hydrodynamics Pavel Kovtun, University of Victoria arxiv: August 2015, Seattle

2 Outline Introduction Effective action: Phenomenology Effective action: Bottom-up Effective action: Top-down Questions

3 all here What is hydrodynamics?

4 Basic hydrodynamics Symmetry conservation laws conserved density ρ t = J spatial current ρ = number density, momentum density, energy density

5 Cooking ingredients Variables: velocity, temperature, B fields, superfluids, liquid crystals,... Symmetries: Galilean, Lorentz, anomalies, external sources... Derivative expansion: J=J(u, u, 2 u,...) Constraints: Onsager relations, entropy current,...

6 Predictions of hydro can be tested hydrodynamics fails at long times

7 Want an incredible machine state all orderings: partially retarded etc knows about FDT good for ω,k 0 knows about long-time tails

8 Want an incredible machine W[g]= DΦ exp(iseff[φ,g]) Q: what are Φ, g, and Seff? state all orderings: partially retarded etc knows about FDT good for ω,k 0 knows about long-time tails

9 Main question How do you build the incredible machine? The point of the action is to provide a weight in the path integral. The point of the path integral is to build the generating functional. In a classical thermal system, path integral = thermal fluctuations. What is the action for that path integral?

10 Main question How do you do low-energy effective field theory in a thermal state?

11 Outline Introduction Effective action: Phenomenology Effective action: Bottom-up Effective action: Top-down Questions

12 Linear hydro fluctuations all here T ij = T cl ij + τ ij Gaussian noise τ ij(x)τ kl(y) =2TG ijkl δ(x y) Solve µ T µν =0 δv i = δv i [τ], δt = δt[τ] Get δt δt, δv i δv j, T ij T kl, Kubo formulas...

13 Non-linear hydro fluctuations T ij = +(+p)v i v j + + τ ij linear leading non-linear, no derivatives term higher order δv δv Txy Txy = 2Tη + (ε+p) (ε+p) δv δv y " x!

14 Example: viscosity in 3+1 dim T xy T xy R = p + O(Λ 3 T ) iω η + 17T 2 Λ 120π 2 η/s + O ω 3/2 (η/s) 3/2 + O(ω 2 ) 0-th order classical 1-st order classical 2-nd order classical correction to p correction to η cutoff-independent Total physical viscosity includes all such corrections Function η + 1/η has a minimum, lower bound on ηtot? Small viscosity implies large corrections Quark-gluon plasma at T Tc : corrections are large for η/s=0.08

15 Analogy with gravity Classical hydro: T µν cl = T µν (0) + T µν (1) + T µν (2) +... G xy,xy (ω) =O(1) + O(ω)+O(ω 3/2 )+O(ω 2 )+... Classical gravity: S = d 4 x 1 16πG R + c 1R 2 + c 2 R µν R µν +... V (r) = Gm 1m 2 r 1+O Gm r + O G r 2 + O(e m0r ) m 0 (c i G) 1/2 Classical: Stelle 1978 Quantum: Bjerrum-Bohr, Donoghue, Holstein, 2002

16 Example: charge conductivity in 2+1 dim 2+1 dim: η=η0+o(lnω), σ=σ0+o(lnω) Running viscosity: η(λ) η(ω=λ), σ(λ) σ(ω=λ), g η η/s, g σ σt/χ λ g η λ = 1 16πc λ g σ λ = 1 8πc s/t 2, counts d.o.f. 1 g η, 1 g σ + g η Σ T Χ Η s IR fixed line as ω 0 : η s = σt χ

17 These were bits and pieces of the machine How do we build the full incredible machine?

18 Building machine: just add the noise µ (T µν cl + τ µν )=0 T µν cl = u µ u ν + p µν G µναβ α u β viscosities & projectors, standard Landau-Lifshitz hydro τ µν (x)τ αβ (y) =2TG µναβ δ(x y) same Gμναβ for FDT in equilibrium Get stochastic differential eq-s with multiplicative noise Discretization ambiguities...

19 Effective action for dissipative relativistic fluids Integrate out the noise: S eff = dt d d x i µ φν T µν cl + T µ φν G µνλσ λ φσ + exponentiating the Jacobian Where is the derivative expansion? Where is the frame invariance? How to get all types of correlation functions? Why is the noise Gaussian? UV divergences?

20 Outline Introduction Effective action: Phenomenology Effective action: Bottom-up Effective action: Top-down Questions

21 FInite-temperature field theory 0 ϕ r = 1 2 (ϕ 1 + ϕ 2 ) -iβ ϕ a = ϕ 1 ϕ 2 G Ret = iϕ r ϕ a G Adv = iϕ a ϕ r G Symm =2ϕ r ϕ r

22 Example: diffusion G Ret nn = Dχk2 iω Dk 2, GAdv nn = Dχk2 iω + Dk 2, GSymm nn = 4TDχk2 ω 2 +(Dk 2 ) 2 iω-dk 2 from diffusion eqn tn-d 2 n=0 χ=( n/ μ)μ=0 susceptibility, from coupling to the source T=temperature, from FDT G Ret nn = iϕ r ϕ a Q: What is Seff[φr,φa] which gives in DφrDφa exp[iseff]...? G Adv nn G Symm nn = iϕ a ϕ r =2ϕ r ϕ r A: Such Seff is easy to find, but it is non-local.

23 Diffusion: effective action Add auxiliary fields Add external sources Make Lorentz-invariant Make r- and a-gauge invariant Z[A r,a a ]= Dφ r Dφ a e is[φ r,φ a,a r,a a ] S = dt d d x (J µ cl [φ r,a r ]D µ φ a + it σ µν D µ φ a D ν φ a ) hydro current n or μ physical gauge field µ φ a + A a µ set to zero at the end This gives all <J μ J ν > R,A,S in equilibrium

24 Linear hydro: effective action Z[h r,h a ]= Dφ r µdφ a µ e is[φr,φ a,h r,h a ] S = dt d d x g r T µν cl [φ r,g r ]D µ φ a ν + it D µ φ a ν G µναβ D α φ a β hydro E-M tensor β μ physical metric 1 2 h a µν µφ r a ν ν r φ a µ set to zero at the end This gives all <T μν T αβ > R,A,S in equilibrium Whatever Seff is, it must reduce to this near equilibrium

25 Full hydro effective action? S = dt d d x g r T µν cl [φ r,g r ]D µ φ a ν + it D µ φ a ν G µναβ D α φ a β Small fluctuations only (2-point functions) Derivative expansion not obvious Need to understand the symmetries

26 Outline Introduction Effective action: Phenomenology Effective action: Bottom-up Effective action: Top-down Questions

27 Effective action: first pass Near equilibrium β µ = β µ + β µ Choose T=1/(-β β) 1/2, u μ =Tβ μ as variables Derivative expansion: Γ= dt d d x g p(t )+a(t ) u + b(t ) T +... Vary w.r.t. g, keep β fixed: T µν = pg µν + Tp u µ u ν +(b a ) µν T Tu µ u ν u normal perfect fluid can be set to zero by a frame choice No dissipation here

28 Symmetries CTP generating functional: Z ρ [j 1,j 2 ]= d q 1 d q 2 dq f q 1 ρ q 2 qf q 1 Dq 1 qf q 2 Dq 2 e i t f t 0 L(q 1,j 1 ) i t f t 0 L(q 2,j 2 ) 1. Local symmetries: Z ρ [j 1,j 2 ]=Z ρ [j 1 + δ ξ1 j 1,j 2 + δ ξ2 j 2 ] 2. Can use r,a variables: q r =(q 1 +q 2 )/2, q a =q 1 -q 2 S[q 1,j 1 ] S[q 2,j 2 ]= q a EOM(q r,j r )+O(j a,q 2 a) 3. Normalization: Z ρ [j 1,j 1 ]=1

29 Conservation laws: 1 and 2 δ g W [g 1,g 2 ]= 1 2 g1 T µν 1 δg1 µν 1 2 g2 T µν 2 δg2 µν δg 1 µν = g 1 µλ ν ξ λ 1 + g 1 νλ µ ξ λ 1 + λ g 1 µν ξ λ 1, δg 2 µν = g 2 µλ ν ξ λ 2 + g 2 νλ µ ξ λ 2 + λ g 2 µν ξ λ 2 Diffeo invariance of W[g 1,g 2 ] gives 1 µt µν 1 =0, 2 µt µν 2 =0 not useful b/c hydro variables are r-fields

30 Conservation laws: r and a Define r- and a-sources: g r 1 2 g 1 + g 2, g a g 1 g 2 physical metric set to zero at the end gr T r µν 1 2 g1 T µν g2 T µν 2 physical stress tensor gr T µν a g 1 T µν 1 g 2 T µν 2 Diffeo invariance of W[g 1,g 2 ] gives g r νλ µ T µν r g a νλ µ T a µν +Γ a λµνt a µν + gνλγ a ρr ρµ T a µν =0, gνλ r µ T a µν + gνλ a µ T r µν +Γ a λµνt r µν + gνλγ a ρr ρµ T r µν =0 Γ λµν = 1 2 ( µg νλ + ν g µλ λ g µν ), µ r µ

31 Main problem To manifest the symmetries, need 1- and 2-fields To write down hydro e.o.m., need r- and a-fields

32 A proposal Realize the symmetries as if D 1 D 2 D r D r manifest in hydro a-type d.o.f. ξa φa look like Goldstone bosons

33 Structure of the effective action S eff = I r + J r Dϕ a + K r (Dϕ a ) O(a 0 ) O(a) O(a 2 ) I r : contains thermodynamics J r : classical hydro equations K r : responsible for FDT Different orders in the a-expansion are not independent

34 Effective action for the neutral fluid Up to O(a) can work with r- and a-diffeos instead of 1- and 2-diffeos Variables: β μ vector under r, singlet under a ϕ μ a vector under r, shifts under a Data to construct Seff : T =1/ β β u µ = β µ / β β g r µν ξ a µν g a µν µ ϕ a ν ν ϕ a µ

35 Effective action for the neutral fluid Scalars up to O(a) and O( 0 ): Scalars up to O(a) and O( ): α 1 T, α 2 u µ u ν ξ a µν, α 3 g µν r ξ a µν α i, µ u µ, u µ ξ a µν ν T, ξ a µν µ u ν, u µ ξ a µν u ν, u µ u ν ξa µν, u µ ν ξ a µν Effective action up to O(a) and up to O( ): integrate by parts S = gr F + f 1 T + f 2 µ u µ + f 3 ξ a µνu (µ ν) T + f 4 ξ a µν (µ u ν) + f 5 ξ a µνu (µ u ν) + O(a 2 ) This does describe the most general dissipative fluid

36 Energy-momentum tensor from the effective action S eff = O(a 0 )+ 1 2 gr T r µν g a µν µ ϕ a ν ν ϕ a µ + O(a 2 ) T µν r = Eu µ u ν + P µν +(q µ u ν + q ν u µ )+t µν E, P, q μ, t μν made out of F(α j ), f i (α j ) This is the standard dissipative fluid in a general frame: (T )=2 F α 2 2 F α 3, p(t )=2 F α 3 η= f 4, ς=(combination of f 1,f 2,f 3,f 4 )

37 To sum up S eff = I r + J r Dϕ a + K r (Dϕ a ) O(a 0 ) O(a) O(a 2 ) Seff up to O(a), up to O( ) gives the standard dissipative hydro Can generalize to charged fluids, with external gauge fields Can presumably repeat up to O( 2 ), get 2 nd order hydro ta-da!!!

38 Outline Introduction Effective action: Phenomenology Effective action: Bottom-up Effective action: Top-down Questions

39 Where is the FDT? We only went to O(a), using r- and a- diffeo invariance FDT appears at O(a 2 ) But it s not legit to use r- and a-diffeo invariance beyond O(a) So need Seff in terms of 1- and 2-variables... what are they?

40 Example: U(1) 1 U(1) 2 U(1) r Break to the diagonal subgroup, one Goldstone: L = v 2 ( µ ϕ + Z µ )( µ ϕ + Z µ ) + (massive) A 1 µ A 2 µ Naive derivative expansion for φ does not know about U(1) U(1) Realizing U(1) U(1) requires the Higgs, a non-hydro mode

41 Other questions S eff = I r + J r Dϕ a + K r (Dϕ a ) I r, J r, K r appear at different orders in the derivative expansion, but they are not independent Constraints coming from the entropy current or from the existence of equilibrium are not obvious Integration measure for r-fields is not clear: remember?

42 Related work The eightfold way Haehl, Loganayagam, Rangamani, arxiv: Classify transport coefficients in classical hydro Non-dissipative transport follows from an effective action Impose extra U(1)T gauge invariance for adiabaticity and 2nd law Double the d.o.f. similar to the CTP formalism Coupling between 1- and 2-sectors unclear

43 Conclusions Effective action for near-equilibrium states is not obvious: are thermal fluctuations really harder than quantum fluctuations? Double the d.o.f. Non-linear realization of the doubled symmetry May need high-energy fields (1 and 2) in the effective theory If you know how to construct the effective theory, please tell me!

44 Thank you!