Effective field theory of dissipative fluids

We develop an effective field theory for dissipative fluids which governs the dynamics of long-lived gapless modes associated with conserved quantities. The resulting theory gives a path integral formulation of fluctuating hydrodynamics which systematically incorporates nonlinear interactions of noises. The dynamical variables are mappings between a “fluid spacetime” and the physical spacetime and an essential aspect of our formulation is to identify the appropriate symmetries in the fluid spacetime. The theory applies to nonlinear disturbances around a general density matrix. For a thermal density matrix, we require an additional Z2 symmetry, to which we refer as the local KMS condition. This leads to the standard constraints of hydrodynamics, as well as a nonlinear generalization of the Onsager relations. It also leads to an emergent supersymmetry in the classical statistical regime, and a higher derivative deformation of supersymmetry in the full quantum regime.


Motivations
Hydrodynamical phenomena are ubiquitous in nature, governing essentially all aspects of life. Hydrodynamics has also found important applications in many areas of modern physics, from evolution of galaxies, to heavy ion collisions, to classical and quantum phase transitions. More recently, deep connections have also emerged between hydrodynamics and the Einstein equations around black holes in holographic duality (see e.g. [1][2][3]). Despite its long and glorious history, hydrodynamics has so far been formulated only at the level of the equations of motion (except for the case of ideal fluids), which cannot capture effects of fluctuations. In a fluid, however, fluctuations occur spontaneously and continuously, at both the quantum and statistical levels, the understanding of which is important for a wide variety of physical problems, including equilibrium time correlation functions (see e.g. [4,5]), dynamical critical phenomena in classical and quantum phase transitions (see e.g. [6,7]), non-equilibrium steady states (see e.g. [8]), and possibly turbulence (see e.g. [9]). In holographic duality, hydrodynamical fluctuations can help probe quantum gravitational fluctuations of a black hole. Currently, the framework for dealing with hydrodynamical fluctuations is to add fluctuating dissipative fluxes with local Gaussian distributions to the stress tensor and other conserved currents [10,11] (see e.g. [8,12] for recent reviews). Such a formulation does not capture nonlinear interactions among noises, nor nonlinear interactions between dynamical variables and noises, nor fluctuations of dynamical variables. The situation becomes more acute for fluctuations around nonequilibrium steady states or dynamical flows, where the presence of nontrivial backgrounds of dynamical variables could induce new couplings and long-range correlations [8].
Another unsatisfactory aspect of the current formulation of hydrodynamics is that it is phenomenological in nature. While it works well in practice, the underlying theoretical structure is obscure. More explicitly, the equations of motion are constrained by various phenomenological conditions on the solutions. One is that the second law of thermodynamics should be satisfied locally [11], namely, there should exist an entropy current whose divergence is non-negative when evaluated on any solutions. The entropy current constraint imposes inequalities on various transport parameters such as the non-negativity of viscosities and conductivities. It also gives rise to equalities relating transport coefficients. For example, for a charged fluid at first derivative order, one of the transport coefficients is required to vanish, even though the corresponding term respects all symmetries. Another condition is the existence of a stationary equilibrium in the presence of stationary external sources, which again imposes various equalities among transport coefficients. A third condition is that the linear response matrix should be symmetric as a consequence of microscopic time reversal invariance, the so-called Onsager relations. While these constraints appear to be enough to first order in the derivative expansion, it is not clear whether they are the complete set of constraints at higher orders. Clearly a systematic formulation of the constraints from symmetry principles would be desirable. Recently, an interesting observation was made in [13][14][15][16] that the equality constraints from the entropy current appear to be equivalent to those from requiring that in a stationary equilibrium, the stress JHEP09(2017)095 tensor and conserved currents can be derived from an equilibrium partition function. The physical origin of the coincidence, however, appeared mysterious.
In this paper, assuming that a general quantum statistical system has a liquid phase, we develop a path integral formulation for dissipative fluids as a low energy effective field theory from symmetry principles. This formulation provides a systematic treatment of statistical and quantum hydrodynamical fluctuations at the full nonlinear level. With noises suppressed, it recovers the standard equations of motion for hydrodynamics with all the phenomenological constraints incorporated. Furthermore, we find a new set of constraints on the hydrodynamical equations of motion, which may be considered as nonlinear generalizations of Onsager relations. Truncating to quadratic order in noises in the action, we recover the previous formulation of fluctuating hydrodynamics based on Gaussian noises. As illustrations, we derive actions which generalize (a variation of) the stochastic Kardar-Parisi-Zhang equation and the relativistic stochastic Navier-Stokes equations to include nonlinear interactions of noises.
Interestingly, we also find unitarity of time evolution requires introducing in the low energy effective action additional anti-commuting fields and a BRST-type symmetry, which also survive in the classical limit. Thus even incorporating classical statistical fluctuations consistently requires anti-commuting fields.
Our formulation also reveals connections between thermal equilibrium and supersymmetry at a level much more general than that in the context of the Langevin equation. 1 In particular, we find hints of the existence of a "quantum deformed" supersymmetry involving an infinite number of time derivatives. Connections between supersymmetry and hydrodynamics have also been conjectured recently in [22].
We will restrict our discussion to a charged fluid with a single global symmetry in the absence of anomalies. Generalizations to more than one conserved current or non-Abelian global symmetries are immediate. Anomalies, the non-relativistic formulation, superfluids, as well as study of physical effects of the theory proposed here will be given elsewhere. When a system is near a phase transition or has a Fermi surface, there are additional gapless modes, which will also be left for future work.
In the rest of this section, we outline the basic structure of our theory.

Dynamical degrees of freedom
We are interested in formulating a low energy effective field theory for a quantum manybody system in a state described by some density matrix ρ 0 . As usual, to describe the time evolution of a density matrix and expectation values in it, we need to double the degrees of freedom and use the so-called closed time path integral (CTP) or the Schwinger-Keldysh

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formalism Tr (ρ 0 · · ·) = ρ 0 Dψ 1 Dψ 2 e iS[ψ 1 ]−iS[ψ 2 ] · · · , (1.1) where ψ 1,2 collectively denote dynamical fields for the two legs of the path, S[ψ] is the microscopic action of the system, and · · · denotes possible operator insertions. In this formalism, both dissipation and fluctuations can be incorporated in an action form, which is thus ideal for formulating an effective field theory for dissipative fluids. Aspects of the CTP formalism important for this paper will be reviewed in section 2. Now, let us assume that the density matrix ρ 0 is such that the system is in a liquid phase, and that the only long-lived gapless modes of the system in ρ 0 are hydrodynamical modes, i.e. those associated with conserved quantities such as the stress tensor and conserved currents for some global symmetries. 2 We will be interested in the behavior of the system at scales much larger than typical microscopic relaxation distance and time scales. In such a regime, each spatial point represents a very large number of microscopic constituents which interact very fast, and the system can be considered as in local equilibrium.
Let us imagine integrating out all other modes in (1.1), and obtain a low energy effective theory for hydrodynamical modes only: Tr (ρ 0 · · ·) = Dχ 1 Dχ 2 e iS hydro [χ 1 ,χ 2 ;ρ 0 ] · · · , (1.2) where χ 1,2 collectively denote hydrodynamical fields for the two legs of the path, and S hydro is the low energy effective action (hydrodynamical action) for them. Note that in the CTP formalism, there are two sets of hydrodynamical modes χ 1,2 , which will be important for incorporating dissipative effects and noises in an action principle. Note that S hydro no longer has the factorized form of (1.1), and ρ 0 is encoded in the coefficients of the action. The standard formulation of hydrodynamics arises as the saddle point equation of the path integral (1.2). While such an integrating-out procedure cannot be performed explicitly, following the usual philosophy of effective field theories, we should be able to write down S hydro in a derivative expansion based on general symmetry principles. The challenges are basic ones: (i) what the hydrodynamical modes χ 1,2 are, as it is clear that the standard hydrodynamical variables such as the velocity field and local chemical potential are not suited for writing down an action; (ii) what the symmetries are.
To answer the first question, a powerful tool is to put the system in a curved spacetime and to turn on external sources for the conserved currents. Due to (covariant) conservation of the stress tensor and currents, the corresponding generating functional should be invariant under diffeomorphisms of the curved spacetime, and gauge symmetries of the external sources. These symmetries then suggest a natural definition of hydrodynamical modes as Stueckelberg-like fields associated to diffeomorphisms and gauge transformations. 2 It is a very interesting question whether other continuous media such as solids or liquid crystals can also be formulated in terms of conservation laws using the formalism developed here. We will leave this for future research (see also footnote 1.4 in section 1.4).

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To illustrate the basic idea, let us consider the generating functional for a single conserved current J µ in a state described by some density matrix ρ 0 , (1. 3) where P denote the path orderings. Given that J µ 1,2 are conserved, we have for arbitrary functions λ 1 , λ 2 , i.e. W is invariant under independent gauge transformations of A 1µ and A 2µ . Since we do expect presence of terms in W at zero derivative order, this implies that W [A 1µ , A 2µ ] can not be written as a local functional of A 1µ , A 2µ . We interpret the non-locality as coming from integrating out certain gapless modes, which are identified with the hydrodynamic modes associated with conserved currents J 1,2 . In order to obtain a local action we need to un-integrate them. where and I is a local action for B 1µ , B 2µ . The integrations over Stueckelberg-like fields ϕ 1,2 remove the longitudinal part of A 1,2µ , and by definition, W obtained from (1.5) satisfies (1.4). We thus identify ϕ 1,2 as the hydrodynamical modes associated with J µ 1,2 . This discussion can be generalized immediately to also include the stress tensor T µν , turning on the source of which corresponds to putting the system in a curved spacetime. The generating functional now becomes where U 1 is the evolution operator for the system in a curved spacetime with metric g 1µν and external field A 1µ , and similarly with U 2 . Due to (covariant) conservation of the stress tensor and the current, W is invariant under independent diffeomorphisms of g 1,2 and "gauge transformations" of A 1,2 : and y σ 1,2 (x), λ 1,2 are arbitrary functions. Due to (1.8), for the same reason as in the vector case, W can not be a local functional of g 1,2 and A 1,2 . Again interpreting the non-locality as coming from integrating out hydrodynamical modes, we can write W as a path integral of a local action over gapless modes JHEP09(2017)095 obtained from promoting the symmetry transformation parameters of (1.9) to dynamical fields, i.e. (1.10) where (s = 1, 2 and no summation over s) and I is a local action of h 1,2 , τ, B 1,2 . As in the earlier example, integrations over the Stueckelberg-like fields X µ 1,2 (σ a ) and ϕ 1,2 guarantee that W as obtained from (1.10) will automatically satisfy (1.8). Note that, except in the implicit dependence of background fields, X µ s , ϕ s always come with derivatives and thus describe gapless modes. We have also introduced a new scalar field τ (σ) which will be interpreted as describing local temperatures.
The low energy effective field theory on the right hand side of (1.10) is unusual as the arguments X µ 1 , X µ 2 of background fields g 1 (X 1 ), A 1 (X 1 ) and g 2 (X 2 ), A 2 (X 2 ) are dynamical variables. 3 In particular, the spacetime σ a where h ab (σ) and B a (σ a ) are defined is not the physical spacetime, as the physical spacetime is where background fields g µν and A µ live. The spacetime represented by σ a is an "emergent" one arising from promoting the arguments of background fields to dynamical variables.
Despite the original microscopic theory (1.1) being formulated on a closed time path integral in the physical spacetime, the effective field theory (1.10) is defined on a single "emergent" spacetime, not on a Schwinger-Keldysh contour. The CTP nature of the microscopic formulation is reflected in the doubled degrees of freedom and in various features of the generating functional W which we will impose below.
We will interpret the spacetime spanned by σ a as that associated with fluid elements: the spatial part σ i of σ a labels fluid elements, while the time component σ 0 serves as an "internal clock" carried by a fluid element. In this interpretation, X µ 1,2 (σ a ) then corresponds to the Lagrange description of fluid flows. With a fixed σ i , X µ 1,2 (σ 0 , σ i ) describes how a fluid element labeled by σ i moves in (two copies of) physical spacetime as the internal clock σ 0 changes. This construction generalizes the standard Lagrange description, where σ 0 coincides with the physical time. In our current general relativistic context, it is more natural for a fluid element to be equipped with an internal time. The relation between σ a and X µ 1,2 (σ) is summarized in figure 1. Below, we will refer to σ a as the fluid coordinates and the corresponding spacetime as the fluid spacetime.
While in hindsight, one could have directly started with a doubled version of the standard Lagrange description, the "integration-in" procedure described above shows that such a phenomenological description does arise naturally as dynamical variables characterizing low energy gapless degrees of freedom of a general quantum many-body system.
Parts of these variables also have been considered in the literature, although the starting points were different. For example, the fields X µ (σ) already appeared in [24,25]. In

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Fluid spacetime Physical spacetime Physical spacetime Horizon UV UV Complexified bulk direction Figure 1. Relations between the fluid spacetime and two copies of physical spacetimes. The red straight line in the fluid spacetime with constant σ i is mapped by X µ 1,2 (σ 0 , σ i ) to physical spacetime trajectories (also in red) of the corresponding fluid element. In the holographic context, the fluid spacetime corresponds to the horizon hypersurface, and the two copies of physical spacetimes correspond to two asymptotic boundaries of AdS. X µ 1,2 describe relative embeddings of these hypersurfaces. the recent ideal fluid formulation of [36][37][38][39][40][41][42][43][44], a single set of σ i (X µ ) is used, which was subsequently generalized to the doubled version in the closed time path formalism in an attempt to include dissipation [31,37]. The set X µ (σ), ϕ(σ) for a single side arises naturally in the holographic context as first pointed out in [55], which along with [39,40] has been an important inspiration for our study. The doubled version of X µ 1,2 (σ a ), ϕ(σ a ) in the closed time path formalism first appeared in [32][33][34] (see also [56]). In the holographic context, X µ 1,2 (σ a ), correspond to the relative embeddings between the horizon hypersurface, which can be identified with the fluid spacetime, and the two asymptotic boundaries of AdS, which correspond to the physical spacetimes [55][56][57]. Similar variables were also employed in [22,35,45,46].
The interpretation of σ a as the fluid spacetime immediately leads to an identification of the standard hydrodynamical variables in terms of our variables X µ s , τ, ϕ s . With X µ s (σ 0 , σ i ) corresponding to the trajectory of a fluid element σ i moving in physical spacetime, then is the proper time square of the motion, and the fluid velocity is given by

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Similarly, interpreting B sa (σ) as the "external sources" for the currents of fluid elements in fluid space, we can define the local chemical potential µ(σ) (recall that for an equilibrium system, the chemical potential for a conserved charge is defined as the zeroth component of the external source for the corresponding current) (1.14) The reason for the 1/b s prefactor in (1.14) is the same as that in (1.13): to convert from dt to the local proper time d s . Finally we define the local proper temperature in fluid space as where T 0 = 1 β 0 is a reference scale. 4 Note that there is only one τ field rather than two copies. In contrast to other fields, it is defined only in the fluid spacetime. It should be considered as an intrinsic property associated with each fluid element.

Equations of motion
Given an action I in (1.10), we define the "off-shell hydrodynamical" stress tensors and currents as In (1.16)-(1.17), x µ denotes the physical spacetime location at whichT µν s ,Ĵ µ s (s = 1, 2) are evaluated, and should be distinguished from either σ or X, as X's are dynamical variables and σ a labels fluid elements.T µν s andĴ µ s are operators in the quantum effective field theory (1.10) of X µ s , τ and ϕ s . They are the low energy counterpart of the stress tensor T µν and current J µ of the microscopic theory (1.1). By definition, correlation functions of (1.16)-(1.17) in (1.10) should reproduce those of the microscopic theory in the long distance and time limit with choices of a finite number of parameters in (1.10).
By construction, h sab and B sa , and so the action, are invariant under physical spacetime diffeomorphisms, which have the infinitesimal form where for notational simplicity we have suppressed the index s = 1, 2 for each quantity in the above equation, i.e. there are two identical copies of them. Similarly, B sa is invariant under a gauge transformation of A sµ with a shift in ϕ s : Note that in the above equations, ∇ sµ are covariant derivatives in physical spacetimes.

Symmetry principles
We now consider the symmetries which should be satisfied by the hydrodynamical action I in (1.10). Let us start with diffeomorphisms of σ a and possible gauge symmetries of B sa . We require that I should be invariant under: 1. Time-independent reparameterizations of spatial manifolds of σ a , i.e.
Equation (1.22) corresponds to a (time-independent) relabeling of fluid elements, while (1.23) can be interpreted as reparameterizations of the internal time associated with fluid elements. Note that in (1.23) we allow time reparameterization to have arbitrary dependence on σ i , which physically can be interpreted as each fluid element having its own choice of time. In contrast, we do not allow (1.22) to depend on σ 0 . Requiring invariance under means allowing different labelings of fluid elements at different times. This would be too strong, as it would treat some physical fluid motions as relabelings. The same conclusion can also be reached from the combination of (1.26) with (1.23) amounting to full diffeomorphism invariance of σ a , under which one of the X µ 's can then be gauged away completely, which would be too strong. The origin of (1.24) can be understood as follows. In a charged fluid, each fluid element should have the freedom of making a phase rotation. As we are considering a global JHEP09(2017)095 symmetry, the phase cannot depend on time σ 0 , but since fluid elements are independent of one another, they should have the freedom of making independent phase rotations, i.e. we should allow phase rotations of the form e iλ(σ i ) , with λ(σ i ) an arbitrary function of σ i only. As B sa are the "gauge fields" coupled to charged fluid elements in the fluid space, we thus have the gauge symmetry (1.24) of B sa . This consideration also makes it natural that in a superfluid, when the U(1) symmetry is spontaneously broken, (1.24) should be dropped.
We emphasize that (1.22)-(1.24) are distinct from the physical spacetime diffeomorphisms (1.18) and gauge transformations (1.19). They are "emergent" gauge symmetries which arise from the freedom of relabeling fluid elements, choosing their clocks, and acting with independent phase rotations. 5 These symmetries "define" what we mean by a fluid. 6 Indeed we will see later they are responsible for recovering the standard hydrodynamical constitutive relations including all dissipations. These symmetries should be considered as gauge symmetries, as configurations related by these transformation are physically equivalent. Thus one should take care in treating them when "quantizing" the theory.
The local symmetries (1.22)- (1.24) are not yet enough to fix the action I. By definition, the generating functional (1.7) also has the following properties (see section 2 for their derivation) Reflectivity condition : (1.29) Equation (1.29) implies that terms in the action I which are even under 1↔2 must be pure imaginary. Since we expect such terms to be generically generated when integrating out modes, the action I is in general complex. For the path integral (1.10) to be well defined, we should also require that 5. The imaginary part of I is non-negative.
We will see later that this condition requires that noises have exponentially decaying distributions and leads to the non-negativity of various transport coefficients when combined with the local KMS conditions to be discussed below. Now consider the unitarity condition (1.28), which implies that when setting for a single patch. 6 It is interesting to speculate that by modifying these symmetries, one may be able to describe other continuous media such as solids or liquid crystals. We will leave this for future research.

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the path integral (1.10) becomes "topological", as W is independent of A µ and g µν . In terms of correlation functions in the absence of sources, equation (1.28) implies that all correlation functions ofT µν a andĴ µ a vanish among themselves, wherê To see this, let us adopt a simplified set of notation denoting the background fields (i.e. g sµν and A sµ ) collectively as φ s and dynamical variables as χ s , with χ r,a , φ r,a respectively symmetric and anti-symmetric combinations of various quantities, i.e. 7 Similarly the currents associated with φ s (i.e.T µν s andĴ µ s ) will be collectively denoted as J s . We then have (schematically) should not depend on φ = (g µν , A µ ) at all. Thus, from (1.33), all correlation functions of J a must be zero. We now show that at tree level of (1.10) (or (1.34)), this can be achieved by requiring that: 6. The action is zero when we set all the sources and dynamical fields of the two legs to be equal, i.e. I[χ r , χ a = 0; φ r , φ a = 0] = 0, (1.36) or, in our original notation, At tree-level, we have where χ cl a,r [φ r , φ a ] denote solutions to the equations of motion. Given (1.36), when φ a = 0, any term in I must contain at least one power of χ a . Thus, χ cl a = 0 must always be a solution to the resulting equations of motion. With the standard boundary conditions that χ a must vanish at spatial and temporal infinities, this is the unique solution. It 7 There is only one τ which should be considered as a r-field.
It can readily be seen, however, that beyond the tree level (1.37) is not enough to ensure (1.28). We will give a detailed discussion in the next subsection and here just state the result. To ensure (1.28) at the level of full path integrals, in addition to (1.37) we need to 7. Introduce a fermionic ("ghost") partner c r,a for each of the dynamical fields χ r,a , and add a "ghost" action I gh to the original action: so that when φ a = 0, the full action I B is invariant under the following BRST-type transformation (to which below we will simply refer as BRST transformation): Here, is a fermionic constant and i labels different fields. Note that the currents J r,a will now also depend on the ghost fields.
As will be discussed in the next subsection, given a bosonic action I the condition of BRST invariance does not fix the ghost action I gh and the symmetric current J r uniquely, i.e. there is freedom to parameterize them. For a general density matrix ρ 0 , we believe items 1 − 7 listed above are the minimal set of symmetries needed to be imposed to describe a fluid. For specific ρ 0 , there can be more symmetries. We will describe the example of thermal ensemble in section 1.6.
Recent works [32,33,45,46] also share some elements with our discussion here. In particular, ref. [45] started from the CTP formulation of the generating functional to deduce a hydrodynamical action at quadratic level. Refs. [32,33] proposed a classification of transports from entropy current using similar variables and also considered doubling degrees of freedom as in the CTP formulation. While this paper was being finalized, reference [22] (see also [47,48]) appeared which also pointed out that the path integral for hydrodynamical effective field theory should possess a topological sector and BRST invariance to ensure (1.28). See also [12,45,46].

Ghost fields and BRST symmetry
We now elaborate on how to ensure the unitarity condition (1.28) beyond the tree level. To gain some intuition, let us first look at how to do this at one loop. With φ a = 0, from (1.37), I can be expanded in powers of χ a as

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where indices i, j now collectively denote both field species and momenta. At one loop order, only the terms linear in χ a contribute, and we find 8 (1.43) Clearly the above expression depends nontrivially on φ from the determinant in evaluating the delta functions. To cancel the determinant, we can add to the action an additional term I 1 of the following form so that the path integral from the full action is independent of φ at one-loop level. Now using a standard trick we can introduce "ghost" partners c i r , c i a for χ i a , χ i r to write c i r,a have the same quantum numbers as χ i a,r , except that they are anti-commuting variables. The full path integral at one-loop order can then be written as with (1.48) Notice that I B has a BRST-type of symmetry δχ i r = c i r , δc i r = 0, δc i a = χ i a , δχ i a = 0, (1.49) with an anti-commuting constant. We can write (1.49) in terms of the action of a nilpotent differential operator (1.51) Now it can be readily seen that if we can make the full action to be BRST invariant, and variation with respect to φ to be BRST exact, then W will be independent of φ to where in the second equality we have used that I B is BRST invariant and in the third equality we have used that Q can be written as a total derivative under the path integration.
To make the full action I[χ r , χ a ; φ] BRST invariant, note that from (1.36) it contains at least one factor of χ a , i.e. we can write it as We can then construct a BRST invariant action: Note that the choice of F i is not unique, as (1.54) is invariant under the following redefinition of F i : Under (1.56), Ψ and I B change as Clearly there is much more freedom in writing down a BRST invariant action than (1.57). For example, in the construction above we set φ a = 0 at the beginning. But we could have kept the φ a dependence, which could lead to a different BRST invariant action. More explicitly, from (1.36) we can write the full action as where J (0) r does not contain any factors of χ a . We can then construct another action: which is again BRST invariant for φ a = 0. Note that in the absence of any background fields, (1.59) is equivalent to (1.55) up to the freedom (1.57) already noted, and they have the same current J a . But J r will in general differ by ghost dependent terms. To summarize, with the requirements that the action be invariant under BRST-type symmetry (1.49) and that currents J a be BRST exact, the unitarity condition (1.28) is satisfied at the level of full path integral. We also saw that the BRST symmetry does not fix the ghost action uniquely from the bosonic action, and there is freedom in choosing ghost dependent terms in the definition of J r .
We should also emphasize that here the BRST symmetry is a global symmetry; we do not require either physical operators or physical states to be BRST invariant. For example, J r is not BRST invariant. JHEP09(2017)095

Thermal ensemble and KMS conditions
Now let us take ρ 0 to be the thermal density matrix at some temperature T 0 = 1 β 0 and chemical potential µ 0 for Q = d d−1 x J 0 , i.e.
In this case, the generating functional W of (1.7) additionally satisfies the so-called KMS condition [58][59][60]. The KMS condition can be considered as a Z 2 operation which relates the generating functional W to the corresponding W T for a time-reversed process: where we have again used the simplified notation of (1.34) and x = (t, x) denote the coordinates in physical spacetime. See section 2 for the precise definition of W T and derivation of (1.61). In deriving (1.61), we also used that the stress tensor and current operators are neutral under Q. At quadratic order in φ's, (1.61) gives the familiar fluctuation-dissipation theorem (FDT) between retarded and symmetric Green functions At higher orders, W T cannot be expressed in terms of W , and the KMS condition (1.61) by itself does not impose constraints on W . However, in essentially all physical contexts, the Hamiltonian H is CPT invariant, for which ρ 0 (β 0 , µ 0 ) is mapped to ρ 0 (β 0 , −µ 0 ) and W T (µ 0 ) is related to W (−µ 0 ) by CPT . While our discussion can be applied to the most general cases, for simplicity here we will restrict to Hamiltonians invariant under PT . 9 With PT symmetry, W T is related to W as (see section 2 for a derivation, here for notational simplicity we have set free parameter θ = 0) and (1.61) can therefore be written as 64) and in terms of our original notation, (1.65) In the form of (1.65), the KMS condition is now a Z 2 symmetry of W . Now let us consider what symmetry to impose on the total action (1.39) so as to ensure the KMS condition (1.65). For this purpose, first note that the bosonic action I[χ r , χ a ; φ r , φ a ] can be split as (1.66)

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where I s [φ r , φ a ] is obtained by setting all the dynamical fields to zero, I d [χ r , χ a ] is obtained by setting all the background fields to zero, 10 and I sd is the collection of remaining cross terms of χ's and φ's. I d [χ r , χ a ] is the dynamical action for hydrodynamical modes χ in the absence of sources, while I sd describes the coupling of dynamical modes to sources from which our off-shell hydrodynamical stress tensors and currents (1.16)-(1.17) are extracted. Given that χ's are gapless, path integrals of I d + I sd generate nonlocal contributions to W , i.e. contributions which become singular in the zero momentum/frequency limit.
The source action I s [φ r , φ a ] gives local terms in the generating functional W . After differentiation, they give contributions to correlation functions of the stress tensor and current which are analytic in momentum and frequency, i.e. contact terms in coordinate space. In contrast to contact terms in vacuum correlation functions which are often discarded, these contact terms are due to medium effects from finite temperature/chemical potential and contain important physical information. For example, viscosities and conductivity can be extracted from them.
A remarkable fact of the structure of (1.10)-(1.11) is that once the couplings of the source action I s are specified, those of the dynamical action I d and the cross term action I sd are fully determined. In other words, once the local terms in W are fixed, the nonlocal parts are also fully determined.
Our proposal to ensure (1.65) consists of two parts. The first part concerns the bosonic action I: 8(a). We require that the contact term action I s satisfies the KMS conditions (1.64), i.e. I s should satisfy the following Z 2 symmetry: 67) or in terms of our original variables, 11 The proof requires introducing more specifics than the broad level at which we have been discussing so far, and will be left to appendix C. While we strongly suspect that the proof in appendix C can be generalized to a full charged fluid, the presence of τ fields make the story more tricky and a full proof will not be given here.

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From now on, we will refer to (1.68) as the local KMS conditions. We will show in section 3 that the local KMS conditions (1.68) not only reproduce all the standard constraints on the hydrodynamical equations of motion (including the entropy condition constraints and those from linear Onsager relations), but also impose a new set of constraints which may be considered as nonlinear generalizations of Onsager relations.
To conclude let us remark that for general non-equilibrium situations β 0 in (1.67)-(1.68) should be considered as the inverse temperature at spatial infinity, i.e. all dynamical modes including τ are assumed to fall off sufficiently rapidly approaching spatial infinities.
The importance of understanding macroscopic manifestations of the KMS condition has been emphasized in [22,32,33]. There a different approach based on a U(1) T symmetry was proposed.

KMS conditions and supersymmetry
We now consider how to ensure the KMS conditions (1.65) beyond the tree level, for which the situation becomes less clear. Currently we have a concrete proposal only for the classical statistical limit of (1.41).
Our understanding is mostly developed from the example of the hydrodynamics of a single vector current (1.5), which we summarize here using the notation of (1.32)-(1.34). Details are given in section 4. We believe the discussion below should apply, with small changes, to full charged fluids (1.10) in the small amplitude expansion. But the expressions become quite long and tedious, which we will leave for future investigation. Note that in both (1.5) and the small amplitude expansion of (1.10), the physical and fluid spacetimes coincide, so we will not make this distinction below.
Consider the small amplitude expansion of external sources and dynamical modes, i.e.
where I m contains altogether m factors of sources and dynamical fields (but can be kept to all derivative orders). We find that at quadratic order I 2 , the ghost action is uniquely determined from the requirement of BRST invariance for φ a = 0, and there is no freedom in J r . After imposing the local KMS conditions (1.68), with all external sources turned off, in addition to (1.49), the full action has an emergent fermonic symmetry, which can be written in a formδ (1.71) The appearance of Λ has its origin in the FDT relation (1.62). It can be readily checked that δ of (1.49) andδ satisfy the following supersymmetric algebra

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In addition, the currents J r,a , being linear in the dynamical fields, satisfy the following relations under δ andδ: where ξ a,r are some fermonic operators which may be interpreted as fermionic partners of J a,r . In other words, the current operators, (J a , J r , ξ a , ξ r ), transform in the same representation under (1.72) as the fundamental multiplet (χ a , χ r , c a , c r ). At cubic order I 3 , there are a few new elements. Firstly, BRST invariance no longer fixes the ghost action or the ghost part of J r . Secondly, the algebra (1.70) cannot remain a symmetry at nonlinear orders as there is a fundamental obstruction in applying the algebra (1.72) to a nonlinear action. By definition, acting on a product of fields, both δ andδ are derivations, i.e. they satisfy the Leibniz rule, and so does their commutator. But on the right hand side of (1.72), Λ does not satisfy the Leibniz rule. The contradiction does not cause a problem at quadratic level as where Λ 1 (Λ 2 ) denotes that Λ is acting on the first (second) field of L 2 . But this is no longer true at nonlinear orders. Both of the above issues can be addressed in the classical statistical limit → 0, which we will explain in more detail in next subsection. For now it is enough to note that in this limit, the path integrals (1.10) survive due to statistical fluctuations.
In the → 0 limit (restoring ), and equations (1.72) become the standard supersymmetric algebra, after a rescaling of¯ , and thus (1.76) could persist to all nonlinear orders. Indeed, we find that at cubic order in the → 0 limit, the local KMS conditions gives a bosonic action which is supersymmetrizable, and in addition invariance under (1.76) uniquely fixes the ghost action. Furthermore, we find that requiring that the currents J r,a satisfy the → 0 limit of (1.73), 12 i.e.
uniquely fixes J r . It is thus tempting to conjecture that in the → 0 limit, combined with local KMS conditions, supersymmetry will be able to uniquely determine the ghost action and J r to all nonlinear orders, and ensure the KMS conditions to all loops. One can immediately conclude from (1.77) that supersymmetry ensures one of the KMS conditions to be satisfied at the level of full path integral. From the fourth equation

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of (1.77), we find thatJ A ≡ J a + iβ 0 ∂ t J r =Qξ r whereQ is the operator which generates transformationδ. Given that the action is invariant underQ, then from manipulations exactly parallel to (1.53) (with Q replaced byQ) we conclude that correlation functions involving onlyJ A all vanish. As discussed around (B.17)-(B.21) in appendix B this is precisely one of the KMS conditions. In fact for two-point functions, it is the full KMS condition. Thus for two-point functions, supersymmetry (1.77) ensures KMS conditions at full path integral level. Perhaps not surprisingly, as we will see explicitly in section 4.2, it is exactly the local version of this particular KMS condition (i.e. this KMS condition applied to I s ) that leads to the invariance of the action underδ and the supermultiplet structure (1.77). It is still an open question at the moment for n-point functions with n ≥ 3 whether local KMS and SUSY are enough to ensure other KMS conditions and how.
To summarize, in the classical statistical limit we can now state the second part of the symmetries which need to imposed to ensure the KMS conditions (1. We believe these are the full set of symmetries which need to be imposed for a full classical statistical path integral. For finite , the story is more tantalizing and potentially more exciting, as some theoretical structure beyond the standard supersymmetry algebra should be in operation. The algebra (1.72) is reminiscent of higher spin symmetries and also possibly suggests a quantum group version of supersymmetry. 13 We have also only been looking at the situation where the fluid spacetime coincides with the physical spacetime. For (1.10) at full nonlinear level, supersymmetry (or whatever replaces it for finite ) should be formulated in the fluid spacetime. When combined with time diffeomorprhism (1.23), it should lead to a supergravity theory. We will leave this for future investigation.
We note that the emergence of supersymmetry in the classical statistical limit is in some sense anticipated from that for pure dissipative Langevin equation (see e.g. [19,20], and also [21] for a review). But even at the level of hydrodynamics for a single current (1.5), the interplay between local KMS conditions and supersymmetry already goes far beyond the scope of a Langevin equation whose corresponding action is quadratic and the distribution of noise is independent of dynamical variables. Here we have a full interacting theory between noises and dynamical variables.
At a philosophical level, the interplay between local KMS conditions and supersymmetry may be understood as follows. The thermal ensemble (1.60) is thermodynamically stable, i.e. any perturbations result in a higher free energy. Furthermore, KMS conditions have been known to be equivalent to the stability conditions. It appears reasonable that such thermodynamical stability conditions are reflected as supersymmetry in the closed time path formalism.

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While this paper was being finalized, reference [22] (see also [47,48]) appeared, which conjectures similar supersymmetric algebra for the hydrodynamical action based on the analogue with stochastic Langevin systems.

Various limits and expansion schemes
In this subsection we discuss various limits and expansion schemes of (1.41) which we copy here for convenience with reinstated In a usual quantum field theory controls the loop expansion. Here, however, the effective loop expansion constant eff is in general not , as the action I describes dynamics of macroscopic non-equilibrium configurations, which have both statistical and quantum fluctuations. In particular, statistical fluctuations should persist even in the → 0 limit, i.e. eff has a finite → 0 limit and the path integral in (1.78) survives. To emphasize the statistical aspect of it, from now on we will refer to the → 0 limit as the classical statistical limit.
More explicitly, we define the → 0 limit in (1.78) as and the coefficients of the action I B should be scaled in a way that the whole action has a well-defined limit. As an example, suppose I B contains the following terms: then G, H, K, f should scale in the → 0 limit as As will be seen in section 2.5, the above scalings are indeed those dictated by the small limit of various correlation functions. Below we will also use (1.78) to refer to its classical statistical limit. We also emphasize that while the "ghost" fields c r,a are introduced to satisfy the unitary condition (1.28) which is a quantum condition, they survive in the classical limit. Thus to describe (classical) thermal fluctuations consistently we still need anti-commuting fields! When eff is small, the path integral (1.78) can be evaluated using the saddle point approximation, with where the leading contribution is the tree-level term (1.38) discussed earlier. Note that the ghost action can be ignored at tree-level. The most convenient choice of the effective loop expansion parameter eff will in general depend on the specific system under consideration.

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On general grounds, we expect it to be proportional to the energy or entropy density of a macroscopic system. In particular, where N is the number of degrees of freedom. From now on we will refer to W tree as the thermodynamical limit of W . As usual in effective field theories, I B can contain an infinite number of terms, and for explicit calculations one needs to decide an expansion scheme to truncate it. In our current context, due to the doubled degrees of freedom and sources, there is also a new element. In this paper, the following expansions or their combinations will often be considered: a. Derivative expansion. As usual the UV cutoff scale for the derivative expansion is the mean free path mfp , whose explicit form of course depends on specific systems. For example, for a strongly interacting theory at a finite temperature T = 1 β , we expect mfp ∼ β. We always take the external sources to be slowly varying in spacetime, and vanishing at both spatial and temporal infinities.
b. Small amplitude expansion. One takes the external sources to be small and considers small perturbations of dynamical variables χ r,a around equilibrium values.
c. a-field expansion. We expand the action I B in terms of the number of a-fields, i.e.
contains altogether m factors of φ a , χ a and c a . The expansion starts with m = 1 due to (1.37). From (1.29), I (m) B is pure imaginary for even m and real for odd m. The a-field expansion is motivated from the structure of generating functional W [φ r , φ a ]. As will be discussed in section 2.2, the expansion of W in φ a gives rise to fluctuation functions of increasing orders. So if one is only interested in the fluctuation functions up to certain orders, one could truncate the expansion (1.84) to the appropriate order. In section 3.3 we also show χ a can be interpreted as noises. Thus a-field expansion essentially corresponds to expansion in terms of noises. For this reason, we will also refer to it as noise expansion.

Plan for the rest of the paper
In the next section, we review aspects of generating functionals in the CTP formalism, which will play an important role in our discussions. Of particular importance is the discussion of the KMS conditions at full nonlinear level as well as the constraints which the KMS conditions impose on response functions.
In section 3, we explain how the standard formulation of hydrodynamics arises in our formulation, and aspects of our theory going beyond it. We first discuss how to recover the standard hydrodynamical equations of motion and then constraints on the equations of motion following from our symmetry principles. In particular, in addition to recovering all the currently known constraints, we will find a set of new constraints to which we refer JHEP09(2017)095 as generalized Onsager conditions. We also discuss how to obtain the standard formulation of fluctuating hydrodynamics.
In the rest of the paper, we apply the formalism outlined in this introduction to two examples. In section 4, we consider the hydrodynamics associated with a conserved current (1.3)-(1.5). We discuss emergent supersymmetry in detail at quadratic and cubic level in the small amplitude expansion. We work to all orders in derivatives. We give an explicit example in which the generalized Onsager conditions give new constraints at second derivative order at cubic level (details in appendix D). We also derive a minimal truncation of our theory which provides a path integral formulation for a variation of stochastic Kardar-Parisi-Zhang equation.
In section 5, we apply the formalism to full dissipative charged fluids. We write the action in a double expansion of derivatives and a-fields. We prove that it reproduces the standard formulation of hydrodynamics as its equations of motion. We also use our formalism to derive the two-point functions of a neutral fluid, and provide a path integral formulation of the relativistic stochastic Navier-Stokes equations. Finally we show that a conserved entropy current arises at the ideal fluid level from an accidental symmetry.
We conclude in section 6 with future directions. We have also included a number of technical appendices. In particular, in appendix B we discuss constraints from the KMS condition at general orders and prove a generalized Onsager relation. In appendix C, we show how the local KMS condition leads to the KMS condition for full correlation functions at tree-level for the vector model. In appendix D we give an explicit example in the vector theory which shows that local KMS counterpart of the nonlinear Onsager relation gives new nontrivial constraints at second order in derivatives. In appendix F we prove that at O(a) level in the a-field expansion, the stress tensor and current can be solely expressed in terms of standard hydrodynamical variables.

Generating functional for closed time path integrals
Here we review aspects of the closed time path integral (CTP), or Schwinger-Keldysh formalism (see e.g. [61][62][63][64]), which will be used in this paper. At the end, we derive constraints on nonlinear response functions from KMS conditions, which will play an important role later in constraining hydrodynamics. This discussion is new.

Closed time path integrals
The evolution of a system with an initial state ρ 0 at some t i → −∞ can be written as where the evolution operator U (t, t i ) can be expressed as a path integral from t i to t. It then follows that ρ(t f ) with t f → ∞ is described by a path integral with two segments, one going forward in time from −∞ to +∞ and one going backward in time from +∞ to −∞ (see figure 2a), x |ρ(t f )|x = dx 0 dx 0 For notational simplicity, we have written the above equation for the quantum mechanics of a single degree of freedom x(t). Setting x = x = x and integrating over x, we then find that 3) where the path integrations on the right hand side are over arbitrary x 1,2 (t) with the only constraint x 1 (+∞) = x 2 (+∞) = x (see figure 2b). In (2.3) · · · denotes possible operator insertions, and P on the left hand side indicates that the inserted operators are path ordered: operators inserted on the first (i.e. upper) segment are time-ordered, while those on the second (i.e. lower) segment are anti-time-ordered, and the operators on the second segment always lie to the left of those on the first segment.
It is often convenient to consider the generating functional where i labels different operators, and the subscripts 1, 2 in O i denote whether the operators are inserted on the first or second segment of the contour (note O 1i and O 2i are the same operator), and φ 1i , φ 2i are independent sources for the operator O i along each segment. The − sign before terms with subscript 2 arises from reversed time integration. Taking functional derivatives of W gives path ordered connected correlation functions, for example where we have suppressed i, j indices. In the second line, T andT denote time and anti-time ordering respectively. In this notation, equation (2.4) can thus be written as We will take all operators O i under consideration to be Hermitian and bosonic. φ 1i , φ 2i are real. Taking the complex conjugate of (2.6), we then find that (2.7)

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Equation (2.4) can also be written as where U 1 is the evolution operator for the system obtained from the original system under the deformation dtφ 1i O i , and similarly for U 2 . From (2.8), we have It is convenient to introduce the so-called r − a variables with .
( 2.11) From (2.11), one obtains a set of correlation functions (in the absence of sources) with specific orderings (suppressing i, j indices for notational simplicity): (2.12) where α 1 , · · · , α n ∈ (a, r) andᾱ = r, a for α = a, r. n r,a are the number of r and a-index in {α 1 , · · · , α n } respectively (n a + n r = n). The r − a representation (2.10)-(2.12) is convenient as (2.12) is directly related to (nonlinear) response and fluctuation functions, which we will review momentarily.

Nonlinear response functions
In this subsection, for notational simplicity we will suppress i, j indices on O and φ's. To understand the physical meaning of correlation functions introduced in (2.12), let us first expand W in terms of φ a 's:

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For φ a = 0, we have φ 1 = φ 2 = φ r ≡ φ. Writing the last expression of (2.17) explicitly in terms of orderings of O's, we find that and D r···r (t 1 , · · · , t n ) is the fully symmetric n-point fluctuation functions of O, in the presence of external source φ. They are referred to as non-equilibrium fluctuation functions [65,66] (see also [63]). One can further expand these non-equilibrium fluctuations functions in the external source φ(t), for example, where G α 1 ···αn were introduced in (2.12). From (2.19), it follows that G r is the one-point function in the absence of source, and G ra , G raa , · · · are respectively linear, quadratic and high order response functions of O to the external source. Similarly, G rr is the symmetric two-point function in the absence of source, and G rra , G rraa , · · · are response functions for the second order fluctuations. Indeed, writing the last expression of (2.12) explicitly in terms of orderings of O's, one finds that G ra···a are the fully retarded n-point Green functions of [67], while G r···r is the symmetric n-point fluctuation function [65,66]. Other G α 1 ···αn involve some combinations of symmetrizations and antisymmetrizations. Note that, by definition, for hermitian operators, all of these functions are real in coordinate space. At the level of two-point functions, one has where G R , G A and G S are retarded, advanced and symmetric Green functions respectively. Explicit forms of various three-point functions are given in appendix A.

Time reversed process and discrete symmetries
Let us now consider constraints on the connected generating functional W when ρ 0 invariant under certain discrete symmetries. We will now restore spatial coordinates using the notation x = (t, x), and take spacetime dimension to be d.
Suppose that ρ 0 is invariant under parity P or charge conjugation C, i.e.

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where we have taken For even spacetime dimensions, Px changes the signs of all spatial directions, while for odd dimensions, it changes the sign of a single spatial direction. For time reversal, consider a process with ρ 0 the state at t = +∞ with the same external perturbations: It should be stressed that W T is a definition and we have not assumed time reversal symmetry. At quadratic order in φ's, we can write W as (2.27) with symmetric, retarded and advanced Green functions given respectively by From (2.26), W T can be written as (2.29) but for higher point functions, W T can no longer be directly obtained from W . Now let us suppose that ρ 0 is invariant under time-reversal symmetry, i.e.
For ρ 0 invariant under some products of C, P, T , the results can be readily obtained from (2.23)-(2.24) and (2.31). For example, suppose that ρ 0 is invariant under PT , i.e.

Thermal equilibrium and the KMS condition
Let us now specialize to a thermal density matrix We will restrict to our discussion to Hermitian operators O i which commute with charge Q. This is satisfied by the stress tensor T µν and the current J µ associated with Q which are the main interests of this paper. Then W satisfies the following KMS condition [58][59][60]: for arbitrary θ ∈ [0, β 0 ] whereĤ = H − µ 0 Q and we have used that .37) and (2.26). Similarly we have gives the standard fluctuation-dissipation theorem (FDT) for two-point functions: (2.40) For the stress tensor and conserved currents, which are our main interests of the paper, η P T i = 1 for all components. Below we will take η P T i = 1. For two point functions, with PT symmetry in addition to (2.39) we also have (2.34), which in momentum space becomes the second of which are Onsager relations. Recall that by definition, G ij is real in coordinate space and is Hermitian in momentum space. At cubic level in φ's, let us write W as

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where we have used a simplified notation, e.g. the first term should be understood in momentum space as (2.43) and similarly with others. Note that (suppressing ijk indices) By definition, the G ijk (k 1 , k 2 , k 3 ) are fully symmetric under simultaneous permutations of i, j, k and the corresponding momenta, and To write the KMS condition for three-point functions, it is convenient to introduce the following notation (suppressing all i, j indices): Then (2.40) applied to three-point level can be written in momentum space as [63]  Expressions of (2.40) in terms of correlation functions at general orders are reviewed in appendix B.

The classical statistical limit
Let us now consider the classical limit of the generating functional (2.11) for a density matrix ρ 0 which has a classical statistical mechanics description.
With restored, each term in (2.16) and (2.19)-(2.20) should have a factor −n with n equal to the number of φ r,a factors. As defined, the symmetric Green functions (2.17) should all have a well defined → 0 limit, and after taking the limit, they describe classical statistical fluctuations. G r···ra···a with n a a-indices should have the limiting behavior r···ra···a is defined exactly as G r···ra···a , but with all commutators replaced by Poisson brackets. From now on, to simplify notation, we will suppress the subscript "cl" and use the same notation to denote the quantum and classical correlation functions. Thus, for W [φ a , φ r ] to have a well-defined limit, the sources φ a , φ r should scale as Let us now look at the → 0 limit of the KMS conditions (2.40). With restored, β 0 in all expressions should be replaced by β 0 . At the level of two-point functions, equation (2.39) then becomes and H 2 , H 3 can be obtained from (2.55) by permutations.

Constraints on response functions from KMS conditions
The KMS conditions (2.40) not only relate various nonlinear response and fluctuation functions, they also imply conditions on correlation functions themselves. For example, at two point function level, (2.39), regularity of G ij in the limit ω → 0 requires that Im∆ ij → 0, ω → 0 . Of particular interest to us are consistency conditions involving only response functions G ra···a , which will play an important role in our discussion of hydrodynamics. For general n-point response functions, let us denote We can show that when taking any n − 2 frequencies to zero, e.g.
From equation (2.59) and permutations of it, it then follows that i.e. the familiar Onsager relations. From now on we will refer to (2.59) as generalized Onsager relations. It appears to us (2.59) and (2.60) are the only relations involving response functions alone. If one leaves more than two frequencies nonzero, then the KMS relations will necessary involve functions with more than one r-indices, as in n = 3 relations (2.47)-(2.50).
Equations (2.59)-(2.60) can be written in a compact way in terms of one-point function (2.19) in the presence of sources. For this purpose, it is convenient to define where again K ≡ K 1 , and the subscript S in the first line denotes the procedure that after taking the differentiation one should set all sources to be time-independent. The notation G(· · · ] highlights that it is a function of x 1 , x 2 , but a functional of φ i ( x). In the second line, φ( x) indicates that the sources only have spatial dependence. Then (2.59) can be written as 63) or in momentum space Now look at the first equation of (2.60), which implies that in the stationary limit there exists some functionalW [φ i ( x)] defined on the spatial part of the full spacetime, from which (2.65) The above equation implies that for stationary sources to first order in φ a , the generating functional (2.16) can be written in a "factorized" form: The second equation of (2.60) is the statement that K i 1 ···in ( k 1 , · · · k n ) are real in momentum space. By definition, K's are real in coordinate space. That they are also real in momentum space implies that (2.68)

JHEP09(2017)095 3 Relations with standard formulations
In this section we first explain how the standard hydrodynamical equations of motion arise in our framework. Then we consider constraints on hydrodynamical equations of motion following from our symmetry principles outlined in the introduction. In particular, the prescription [13,14] that in a stationary background the stress tensor and current should be obtainable from a stationary partition function will arise as a subset of our conditions. We will find a set of new constraints to which we refer as generalized Onsager conditions. Finally we discuss how to recover the standard formulation of fluctuating hydrodynamics and aspects of our theory going beyond it.

Recovering hydrodynamical equations of motion
Let us first explain how the standard hydrodynamical equations of motion arise in our formulation. To illustrate the basic idea, we again use the same simplified notation of (1.34). Since we are interested in the equations of motion (i.e. in the thermodynamical limit of section 1.8), it is enough to consider the bosonic theory, with all ghost dependence ignored.
Recall from section 1.3 that the equations of motion for the dynamical variables χ a,r correspond to the conservation of J a,r , which we can schematically write as 14 Let us now expand the bosonic action I in terms of the number of a-fields, as discussed around (1.84), where I (m) contains altogether m factors of φ a and χ a . From (1.33), the current operators J a,r can be similarly expanded as where m in the superscript (m) again denotes the number of a-fields in each expression. Note that J a starts with m = 1, i.e. J a | φa=0,χa=0 = 0, and J To make connection with the standard hydrodynamical equations, let us now take the background fields of the two segments of CTP to be the same, i.e.

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or in terms of our original fields, With φ a = 0, as already discussed after (1.38), the equations of motion give that In terms of our original dynamical variables, one then has With φ a = χ a = 0, J a vanishes identically and all terms in J r except for J (0) vanish. Thus, 10) and the remaining equations of motion are In terms of original variables, equation (3.10) corresponds tô and (3.11) to Furthermore, one can show from the symmetry requirements (1.22)-(1.24), as the zeroth order terms in the a-field expansion of currents,T µν hydro andĴ µ hydro can be expressed solely in terms of the velocity field (1.13), local chemical potential (1.14) and local temperature field (1.15) (which we will prove explicitly in section 5.3 and appendix F). Equations (3.13) then reproduce the standard hydrodynamical equations.
To summarize, the standard hydrodynamical equations of motion correspond to the zeroth order approximation in the a-field expansion in the thermodynamical limit.

Constraints on hydrodynamics
For ρ 0 given by the thermal ensemble (1.60), we also need to impose the local KMS conditions on the source action I s (1.68). As far as the hydrodynamical equations of motion (3.13) are concerned, we only need to look at constraints on I (1) s , which encode the contact contributions to all of the response functions.
In the standard formulation of hydrodynamics one needs to impose constraints from the local second law of thermodynamics, existence of stationary equilibrium, and the Onsager relations. In our formulation, these constraints are fully taken care of by the local KMS conditions (1.68). At an abstract level, this is a consequence of the facts that: (i) the local KMS conditions ensure that the full KMS conditions are satisfied in the thermodynamical limit; (ii) the full KMS conditions are known to imply the local second law (see e.g. [70]) as well as existence of stationary equilibrium; (iii) time reversal symmetry is encoded in JHEP09(2017)095 our formulation of local KMS conditions. In fact, from the discussion of section 2.6, local KMS conditions include not only the Onsager relations for linear responses, but also give full nonlinear generalizations.
More explicitly, restricted to I (1) s , the local KMS conditions give the following three types of constraints: (a) Relations between coefficients in I (1) s and higher order terms in a-expansion. For example, at first derivative order, (2.39) relates transport coefficients such as shear, bulk viscosities and conductivity in I (1) s to coefficients in I (2) s (FDT relations). From (1.29) I (2) , terms in the action are pure imaginary and their coefficients should satisfy certain non-negativity conditions in order for the path integral to be well defined. Altogether, this implies the non-negativity of various transport coefficients. As we shall see in section 5.8, while this works out easily for the shear viscosity, for conductivity and bulk viscosity it is highly nontrivial. At first derivative order, the non-negativity of shear, bulk viscosities and conductivity are all one gets. These are also the inequality constraints from the non-negative divergence of the entropy current. In fact it has been argued recently [15,16] these are the only inequality constraints from the entropy current to all orders in derivatives. It is conceivable, in our context at higher derivative orders the well-definedness of the integration measure combined with FDT relations may give additional inequality relations, thus predicting new relations going beyond those from the entropy current.
whereW [g, A] is a local functional of stationary metric g µν ( x) and gauge field A µ ( x) on the spatial manifold. Note that for stationary backgrounds, the dynamical modes will not be excited and thus I (1) s is the full contribution to the leading generating functional W (1) tree in the a-field expansion in the thermodynamical limit. We thus have derived the prescription [13,14] that in a stationary background the stress tensor and current should be derivable from a partition function. In [15,16] it has also been shown that this requirement is equivalent to equality-type constraints from the entropy current. Now this coincidence becomes completely natural.
s coefficients. In the next section (and appendix D), we will see that they lead to new constraints in the hydrodynamics of a single current starting at second order in derivative expansion. For a full charged fluid including the stress tensor, these new constraints will also start operating at the second derivative order, but we will not work them out explicitly in this paper.

Recovering stochastic hydrodynamics
Now we show how to recover the standard formulation of fluctuating hydrodynamics [10,11]. For this purpose, consider the first two terms in the a-field expansion (3.2): (3.15) From our discussion of section 3.1 we can write I (1) as which gives the equations of motion (3.11) when varied with respect to χ a . I (2) can be schematically written as where G is a local differential operator depending on χ r . Now, expanding G(∂, χ r ) in powers of χ r , where now G 0 is a local differential operator with no dependence on dynamical variables. Keeping only the G 0 term in I (2) , we can write the action schematically as Note that we are not doing any χ r expansion in I (1) . Now consider a Legendre transformation of the second term of (3.19), i.e. introducing ξ = − ∂Iaa ∂χa to rewrite I aa = i 2 χ a G 0 χ a as I can then be written as The path integral then becomes i.e. χ a is now a Lagrange multiplier, whose integration gives the stochastic diffusion equation where ξ is a stochastic force with local Gaussian distribution: Equations (3.23)-(3.24) recover the standard formulation of fluctuating hydrodynamics [10,11]. 15 We see that χ a is the conjugate variable for the noises, and thus the expansion in a-fields may be considered as an expansion in noises.

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The above discussion makes clear the aspects of our formulation that go beyond the traditional formulation of fluctuating hydrodynamics: (i) In addition to the G 0 term, the full I (2) also includes interactions between dynamical variables and the noises. (ii) I (n) with n ≥ 3 includes interactions among noises and higher order interactions among noises and dynamical variables. (iii) Beyond (3.21), dynamical variables can fluctuate on their own and are not constrained by fluctuations of noises as in (3.23). Furthermore, once we include interactions between χ r and χ a in I (2) , it is no longer convenient to perform the Legendre transform (3.20) from χ a to ξ which will result in a non-local and non-polynomial action. It is more sensible to simply work with χ a .
From the renormalization group perspective, the effective theory we are writing down is defined at a cutoff scale Λ, below which hydrodynamics is defined. 16 If one is interested in physics at some energy scale E Λ, then one should further integrate out hydrodynamical degrees of freedom with energies ω ∈ (E, Λ). It may happen for certain situations that the neglected interactions in (3.21) are all irrelevant. In such a case, the standard stochastic formulation (3.23)-(3.24) is already adequate for obtaining the leading physics at energies E Λ.

Correlation functions
We conclude the discussion of this section by making some comments on correlation functions. Let us use (J r ) cl in φ r from (2.19), one obtains the full set of nonlinear response functions G ra , G raa , · · · in the thermodynamical limit. This constitutes the standard hydrodynamical approach to response functions [60] (see also [12] for a recent review).
In the thermodynamical limit, we can go beyond the standard formulation by turning on φ a = 0. Then both equations (3.4)-(3.5) are nontrivial. Solving these equations to obtain (J a,r with n ≥ 1 cannot be expressed solely in terms of velocity-type variables u µ (σ), µ(σ), T (σ). Instead, the more fundamental fluid field variables, X µ s and ϕ s , must be used. Beyond the thermodynamical limit, we also need to include loop corrections from statistical or quantum fluctuations. Recall the expansion in eff discussed in section 1.8, which we copy here for convenience: Corrections from W 1 , W 2 , · · · will give rise to phenomena such as long time tails, as well as running transport coefficients with scales, and so on (see e.g. [12,68,69] for recent discussions). Such fluctuation effects may be particularly important near classical and quantum phase transitions and in non-equilibrium situations.

JHEP09(2017)095 4 A baby example: stochastic diffusion
As a baby example of the general formalism introduced earlier, we consider the hydrodynamical action associated with a conserved current discussed in (1.3)-(1.5), which we copy here for convenience This theory applies to situations where J µ either decouples from the stress tensor (as for example for a particle-hole symmetric neutral fluid) or the coupling of J µ to the stress tensor is small enough to be neglectable. In the stress tensor sector one takes the equilibrium solution X µ 1 = X µ 2 = x a δ µ a , τ = 0 with the metric backgrounds g 1µν = g 2µν = η µν . Thus in this case the fluid and physical spacetimes coincide. We will take ρ 0 to be the thermal ensemble (1.60).
It is convenient to introduce the r − a variables,  Writing we will expand L in powers of B r,a .

The quadratic action
At quadratic order in B r,a , the most general bosonic L consistent with rotational symmetries, (1.29) and (1.37) can be written as where the coefficients a, b, c, · · · should be understood as real scalar (under spatial rotations) local differential operators constructed out of ∂ t and ∂ i , and act on the second factor of a term. For example

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where in the second equality we have also written the expression in momentum space. All of the coefficients can be expanded in the number of derivatives, for example, in momentum space (q = | k|), a(k) = a 00 + a 20 ω 2 + a 02 q 2 + · · · , b(k) = b 00 + b 20 ω 2 + b 02 q 2 + · · · , g(k) = g 00 + ig 10 ω + g 20 ω 2 + g 02 q 2 + · · · , (4.8) and so on. Note that there is no term with odd powers of ω in the expansions of a, b, c as these correspond to total derivatives. Thus a, b, c are real in momentum space. Other coefficients can have odd powers in ω and are complex in momentum space with, e.g. (4.9) In coordinate space g * is the operator obtained from g by integration by parts i.e. g * (∂ t , ∂ i ) = g(−∂ t , −∂ i ). In the last term of (4.6), F ij = ∂ i A j − ∂ j A i and is independent of ϕ s . Due to (1.29), the aa terms in (4.6) are pure imaginary, and thus are real in the exponent of the path integral (4.1). This implies that the coefficients of the leading terms in the derivative expansion must be non-negative, for example, Equation (4.6) applies to general dimensions and is parity invariant. For a specific dimension, say d = 3, one can write down additional parity-breaking terms using fully antisymmetric -symbol.
We still need to impose the local KMS condition (1.68), which at quadratic level amounts to imposing (2.39) on the source action obtained by setting dynamical fields ϕ r,a to zero in (4.6). The source action is the same as (4.6) with B rµ and B aµ replaced by A rµ and A aµ . From (4.6) we can read where we have introducedb (4.14) Applying (2.39) we then have

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In particular, in (4.17), since the left hand side is regular as ω → 0, we need u to contain at least one power of ω, i.e.
u 00 = u 02 = u 04 = · · · = 0, (4.18) where various coefficients in the expansion of u are defined as in (4.8). Further imposing PT symmetry on the source action, i.e. requiring G and K to be symmetric (Onsager relations), we have additional constraints: The second equation above automatically implies (4.18), and one can check that equations (2.60) are also automatically satisfied. Equation (4.17) can now be written as (4.20)

Off-shell currents and constitutive relations
From (4.6), we find the corresponding off-shell currentŝ The equations of motion for ϕ r and ϕ a correspond to the conservation ofĴ µ a andĴ µ r respectively. To leading order in the a-field expansion, i.e. setting all the a-fields to zero (and dropping r-subscripts), we havê where from (1.14) µ = B 0 = A 0 + ∂ 0 ϕ is the chemical potential. That at leading order in the a-field expansionĴ µ can be expressed solely in terms of µ to all derivative orders is a consequence of fluid gauge symmetry (1.24). In fact, one can immediately see that this works at full nonlinear level, as the fluid gauge symmetry means that B ri can only appear either with a time derivative It is also clear from (4.21)-(4.23) that at higher orders in the a-field expansion,Ĵ µ r,a cannot be expressed in terms of µ r,a alone, and the more fundamental ϕ a has to be used.
It can also be readily checked from conservation of (4.24) that equation (4.18) is equivalent to the existence of a stationary equilibrium for a stationary background field A µ .

BRST invariance and supersymmetry
Let us now set A aµ = 0 in (4.6) and introduce ghost partners c a,r for φ a,r . Here the BRST transformation (1.49) becomes δϕ r = c r , δc a = ϕ a .
where (with P 0 , P z introduced in (4.25)) Note that the ghost action is uniquely determined and the currentsĴ µ a,r are not modified. Further setting A rµ = 0 in (4.31), we obtain the Lagrangian density for dynamical fields in the absence of external fields: In other words, for (4.33) to be invariant under (4.34), G and K should satisfy which follow from (4.15)-(4.17). It can readily be checked that δ andδ satisfy the following "supersymmetric" (SUSY) algebra: This is not the usual SUSY algebra, as Λ involves an infinite number of derivatives.

The full generating functional
For the quadratic action (4.6), the path integrals (4.1) can be evaluated exactly by solving the equations of motion for ϕ r,a . The ghost part does not contribute at quadratic order as it gives an overall constant (which cancels the determinant from the bosonic part). We can directly verify that the FDT (2.39) for the full correlation functions are satisfied given the local KMS conditions (4.15)-(4.17), although this is a special case of the general argument given in appendix C. We now restore the background fields A rµ , A aµ . To evaluate (4.1), it is convenient to work in momentum space. Taking k µ ≡ (k 0 , k z , k α ) = (−ω, q, 0), one can readily see that ϕ r,a only couples to A ≡ (A 0 , A z ), and B rα = A rα , B aα = A aα . We can then directly read from (4.6) the generating functional for A rα , A aα as By comparing with (2.21), we find that the corresponding components of the retarded and symmetric correlation functions in momentum space are The FDT relation (2.39) requires that which is satisfied as result of (4.16).
Integrating out ϕ r,a leads to a nonlocal generating functional for A r , A a ,

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where E a (ω, q) ≡ qA a0 (ω, q) + ωA az (ω, q), E r ≡ qA r0 + ωA rz , E a,r (−ω, −q) = −E * a,r (ω, q), (4.46) and As desired, there is no rr-type term in (4.45). That A appears only through the combinations in E a,r is a consequence of the gauge invariance of W . The nonlocality is reflected in the presence of a diffusion pole in Π L and G L .D can be considered as a diffusion function, which has also been discussed recently in [71] as well it holographic calculation. From (2.21), we can read various components of the symmetric and retarded Green functions where we have used (4.27)-(4.28), and D, which is the leading term ofD, is given by We see that the form of the diffusion constant D is consistent with the Einstein relations. Note that χ should be non-negative for a stable equilibrium state. Given (4.29), we then find that D is non-negative for a stable equilibrium state, and the pole of retarded Green functions (4.48) indeed lies in the lower half ω-plane. Note that the full generating functional given in (4.42) and (4.45) automatically satisfies time-reversal invariance (i.e. Onsager relations) without imposing conditions (4.19). This is an accident due to the simplicity of the system under consideration. This is no longer the case when including parity breaking terms or the stress tensor.

The cubic action
Let us now consider the bosonic action I of (4.1) at cubic order. We can write the corresponding Lagrangian as

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where G, H, K are real local differential operators acting on various fields. For example, the first term can be understood in momentum space as (4.53) where G µνρ (k 1 , k 2 , k 3 ) can be expressed as a power series of k 1,2,3 . By definition, G µνρ (k 1 , k 2 , k 3 ) is fully symmetric under simultaneous exchanges of subscripts µ, ν, ρ and k 1,2,3 . Similarly, G, H, K should be such that L 3 is rotationally invariant and satisfies (1.24). It is possible to write (4.52) more explicitly as in (4.6) to make these properties manifest, but the expression becomes quite long and we will not do it here. Imposing local KMS conditions amounts to requiring that G, H, K satisfy (2.47)-(2.50). H in (4.52) corresponds to H 3 , K corresponds to K 1 , and the other are obtained by permutations. For example, and similarly with the others.
As an illustration of implications of the local KMS conditions on (4.52), we consider a truncation of it in appendix D. In particular, we see that the generalized Onsager relations (2.63) lead to nontrivial relations on the transport coefficients at second order in derivative expansions at nonlinear level.
Setting the external fields to zero, we find the action for dynamical modes: where (note the i factor on left hand side of (4.56)) Also note that due to (1.24)

BRST invariance and supersymmetry
Setting A aµ to zero, and applying (1.54)-(1.55) to (4.52) we can obtain an BRST invariant action by adding to (4.52) the following fermionic action As noted in (1.56), the BRST invariant action is not unique (beginning at cubic order).
In (4.63), this non-uniqueness is parameterized by the term with coefficient f (k 1 , k 2 , k 3 ) which has the symmetry properties The full BRST invariant action in the absence of sources of can then be written as Following our earlier notations, below we will denote f as f 3 , and similarly introduce As already mentioned in section 1.7, the fermionic transformation (4.34) cannot remain a symmetry at nonlinear orders due to higher derivative nature of Λ. For example, were (4.34) a symmetry of our cubic Lagrangian, then from (4.37), Λ would also be a symmetry. However, this is not the case, as where Λ i ≡ 2 tanh β 0 ω i 2 , i = 1, 2, 3. There is a basic contradiction in (4.37): while the left hand side is a derivation by definition, the right hand side is not.
We will now show that in the → 0 limit (i.e. the classical statistical limit discussed in section 2.5 and section 1.8), in which the local KMS conditions satisfied by G, H, K ensure that (4.63) is supersymmetric. In particular, supersymmetry fixes uniquely the undetermined local operator f in (4.63) in terms of other quantities. As discussed in section 2.5 and section 1.8, in the → 0 limit, various quantities in (4.63) should scale as (c a , ϕ a ), c r , ϕ r → c r , ϕ r , (4.67) and the local KMS conditions in this limit are given by (2.55)-(2.56), which we copy here for convenience:

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Under (4.34), we find In the → 0 limit, C 3 and the symmetric part of C 2 are automatically zero, while the antisymmetric part of C 2 is equivalent to C 4 . Setting C 4 = 0, we can solve for f : Note that f 3 is regular as ω 3 → 0 due to (4.60). Thus, f 3 is a well-defined local differential operator. Plugging (4.75) into (4.71) we find that Now one can readily check from (4.68)-(4.69) that C 1 = 0.

Multiplet of currents
Now let us look at theĴ µ r,a in the absence of background fields. From (4.52) and (4.61), we find while expanding (4.52) to first order in A aµ , we find From the discussion around (1.59), there is freedom to add ghost terms to (4.78) of the form R µνρ ∂ ν c a ∂ ρ c r , with R µνρ a local differential operator. We thus now have We now show that requiring that J µ a and J µ r satisfy the → 0 limit of the transformations (4.40), i.e. δJ µ r = ξ µ r ,δJ µ r = ξ µ a¯ , δξ µ a = J µ a ,δξ µ r = (J µ a + iβ∂ 0 J µ r )¯ ,δJ µ a = −iβ∂ 0 ξ µ a¯ (4.80)

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uniquely fixes R. Note that the first two equations of (4.80) should be viewed as the definition for ξ µ r,a , while the last two equations follow from (4.66) once the thrid equation is satisfied. So we only need to check the third equation of (4.80).
From (4.79), we have where in the second equation for notational simplicities we have used a mixed coordinate and momentum representation. Now imposing the third equation of (4.80), we find One can now verify that equation (4.83) is equivalent to the symmetric part (in terms of the last two indices) of (4.84), if (4.76) vanishes. Thus we have a consistent set of equations.
R can now be solved as Note that R is local as due to (1.24), H 3 , K 1 , K 2 should all be proportional to ω 3 . To summarize, both the invariance of the action (4.61) under the supersymmetric transformation (4.34) and the existence of supermultiplet structure (4.80) can be attributed to the vanishing of equation (4.76). Now one can readily check that the combination of (4.68) and (4.69) which gives (4.76) precisely coincides with (B.17) for n = 3. Thus we conclude that in the current context, it is the local part of (B.17) (i.e. this KMS condition applied to I s ) that is responsible for the emergence of supersymmetry. As we already discussed in the paragraph after (1.77), supersymmetry in turn ensures that (B.17) is satisfied for full correlation functions at all loop orders.

A minimal model for stochastic diffusion
Let us now combine the quadratic and cubic actions and truncate them to the lowest nontrivial order in derivative expansions. From (2.55)-(2.56), the local KMS conditions imply that coefficients of O(a) terms with n derivatives are related to those of O(a 2 ) terms with n−1 derivatives, and those of O(a 3 ) terms with n−2 derivatives. Thus at lowest order in the derivative expansion, we will keep the first derivative in O(a) terms, zero derivatives in O(a 2 ) terms, and drop O(a 3 ) terms.

Linear stochastic diffusion
In (4.6), keeping zero derivative terms in O(a 2 ) terms and first derivative terms in O(a) terms, we find

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where we have used (4.27)-(4.28). In (4.87), we have dropped a zeroth derivative O(a 2 ) term a 00 B 2 a0 and a first derivative O(a) term g 10 ∂ 0 B a0 B r0 . The g 10 term is subleading compared to the term with coefficient χ. The a 00 term is dropped since it is related to g 10 by the local KMS conditions: In counting the relevance of terms we always drop terms which are related by local KMS conditions together. At this order, the off-shell currents arê Turning off the external fields, we get (4.33), with Now following the procedure outlined in (3.20)-(3.23) we obtain the stochastic diffusion equation where the noise force ξ is the Legendre conjugate of ϕ a and has a local Gaussian distribution given by

Action for a variation of stochastic Kardar-Parisi-Zhang equation
At cubic level, in (4.52) we keep first derivative terms in K, zero derivative terms in H, and drop all G terms. Then, after imposing local KMS conditions (see appendix D), we find where we have dropped ∂ 0 B a0 B 2 r0 and B 2 a0 B r0 . The former is subleading compared to B a0 B 2 r0 while the latter is related to the former by local KMS conditions. Now setting the background fields to zero, and combining (4.94) with the cubic fermionic action (4.61) and the quadratic action (4.87), we obtain the full action where we have used (4.75), which gives

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The off-shell currents arê with λ 1 = χ 1 χ 2 and λ = σ 1 χ 2 . Note that with nonlinear terms such as (∂ i ϕ a ) 2 ∂ 0 ϕ r ,L aa now contains interactions between ϕ r and ξ. In fact,L aa is neither local nor polynomial, thus it no longer makes sense to replace ϕ a by ξ via a Legendre transform. It could still happen that nonlinear terms such as (∂ i ϕ a ) 2 ∂ 0 ϕ r turn out to be irrelevant when going further into the IR, in which case the very low energy physics would still be governed by (4.99), with ξ a local Gaussian noise. Equation (4.99) is reminiscent of the Kardar-Parisi-Zhang (KPZ) equation [72]. They have similar nonlinear structure, but nonlinear terms are different, as the underlying symmetries used in deriving these equations are different. It is nevertheless tempting to ask whether they could be in the same universality class. We will leave understanding the renormalization group flow of (4.95) for future work.
Finally we should emphasize that in our framework, the forms of the action (4.95) and the equation (4.99) are completely determined by symmetries, with no other freedom.

Effective field theory for general charged fluids
In this section, we proceed to write down the bosonic part of the hydrodynamical action for a charged fluid.

Organization of variables
We first introduce a convenient set of variables which will make imposing (1. indices should be understood as a relation between variables pertaining to one segment of the CTP contour, and altogether there are two copies of the equations. Given the identification of the velocity field (1.13) and the form of the symmetries (1.22)-(1.23), it is convenient to decompose the matrix ∂ a X µ in (1.11) as and conversely, h ab in (1.11) can then be written as and we will denote its inverse as a ij . The inverse transformation can be written as It can be readily checked that The various quantities b, u µ , v i , λ i µ are not arbitrary. Following their definitions from ∂X µ ∂σ a and ∂σ a ∂X µ , they satisfy various integrability conditions, which are given in appendix E.1. Similar to (5.3) we can decompose B a as where the local chemical potential µ was introduced before in (1.14) and we have also introduced "covariant" derivatives: Also note that ϕ, τ transform as scalars under both diffeomorphisms.

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Now for r − a variables, we introduce µ r,a , v ai , v ri , b ai , b ri as usual (see (2.10)), while for b, a ij it is convenient to introduce instead the following definitions whereâ 1,2 denotes the unit determinant part of a 1,2 and thus Ξ is traceless. Under (1.22), a r transforms as tensor, E r,a , χ a , µ a , µ r , τ as scalars, v ai , v ri , b ai , b ri as vectors, while Ξ transform as Under (1.23), a r , χ a , Ξ, E a , τ, µ a , µ r , b ai , b ri transform as a scalar while which motivates us to further introduce Now V ai transforms as a scalar while V ri as Finally under (1.24), b ai is invariant while b ri transforms as

Covariant derivatives
Consider φ and φ i , which are a scalar and vector respectively under spatial diffeomorphisms (1.22), and are scalars under time diffeomorphisms (1.23). We would like to construct a covariant spatial derivative D i = ∂ i + · · · such that: 1. D i φ and D i φ j are tensors with respect to (1.22).
2. It is compatible with a rij , i.e. D i a rjk = 0 . The action of D i on higher rank and upper index tensors can be obtained using the Leibniz rule. Here and below, unless otherwise noted, all the indices are raised and lowered by a r . It can be readily verified the following definitions satisfy the above conditions jk the Christoffel symbol corresponding to a r . For the time derivative, one can check for a scalar φ under (1.23), is a scalar. One should be careful to note that the D 0 , D i introduced here are different from those in (5.9). E, v i in (5.9) should be understood to have subscripts s = 1, 2 and there are two copies of them. The D 0 , D i introduced here in a sense correspond to the r-version of the derivatives there.
E r and V ri do not transform as a scalar under (1.23). We can construct a combined object which allows us to do integration by part under the integrals: (5.28)

Torsion and curvature
Now consider the commutator of D i acting on a scalar: where we usedΓ k [ij] = 0. Clearly the torsion t ij has good transformation properties under both time and spatial diffeomorphisms as the left hand side of (5.29) does. Similarly, we can introduce the "Riemann tensor"R k lij by

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One can check that we still haveR ijk l +R kij l +R jki l = 0, (5.32) but due to the extra term on the right hand side of (5.30), As a result, there are two "Ricci tensors": neither of which is symmetric. It is convenient to consider where the second equality of the first equation follows from (5.33). Also note that

General structure
We are now ready to write down the bosonic part of the hydrodynamical action, Constraints from the local KMS condition (1.68) will be discussed later in section 5.5. Note that there is no separate dependence on ϕ in I other than that contained in µ and b i . From (1.29), and equation (1.37) implies that From (5.40), we cannot use any negative power of Ξ. In particular, while we start with two spatial metrics a 1 and a 2 , only a r can serve as a metric to raise and lower indices in constructing the action. We can write the action as

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where L is a function of Φ's and their derivatives, and should be a scalar under (1.22)-(1.23). We will write L as a double expansion in terms of the number of a-type fields in (5.38), and the number of derivatives. 17 More explicitly, where L (m) contains m factors of Φ a 's. From (5.39), L (m) is pure imaginary for even m and real for odd m. Each L (m) can then be further expanded in the number of derivatives. Let us first consider terms with only a single factor of Φ a . By using the covariant derivatives of section 5.1.2, we find to first order in derivatives the most general Lagrangian density can be written as where Ξ ij ≡ Ξ i k a kj r is symmetric and traceless, and for later convenience 18 we introduce where the local inverse temperature β(σ) was introduced in (1.15). In (5.43), η and λ's are all real functions of µ r and τ . f 1,2,3 can be further expanded in derivatives as f 1 = 0 + f 11 D 0 τ + f 12 D 0 log det a r + f 13 β −1 (σ)D 0μ + higher derivatives, (5.45) (5.46) f 3 = n 0 + f 31 D 0 τ + f 32 D 0 log det a r − f 33 β −1 (σ)D 0μ + higher derivatives, (5.47) with all coefficients f 11 , f 12 , · · · real functions of µ r and τ . Note that a rij was introduced in (5.13). Various signs are chosen for later convenience.
At O(a 2 ), to zeroth order in derivatives, we have −iL where again all coefficients are real and are functions of µ r , τ . It is straightforward to write down terms at higher order in the a-field expansion or with more derivatives, but the number of terms increases quickly. For the rest of this section, we will focus on analyzing (5.43)-(5.48). 17 Due to nonlinear relations in (5.12)-(5.13), this a-field expansion is slightly different from that outlined in section 1.8 and section 3.1, but qualitatively the same. 18 With these choices the coefficients of various terms of the stress tensor and current, e.g. those in (5.62), (5.64), (5.65), simplify.

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As usual, one has the freedom of making field redefinitions χ → χ + δχ =⇒ I → I + d d σ δI δχ δχ, (5.49) where χ collectively denotes all dynamical variables and δχ involves derivatives of χ. Equivalently, we could set to zero all terms in the action which are proportional to the equations of motion at lower derivative order.

Stress tensor and current operators
We now consider the stress tensor and current operators following from the action written above.

General discussion
The stress tensor and current operators are defined in (1.16) by varying the action with respect to g sµν (x), A sµ (x). Since both the action I and g sµν (x), A sµ (x) are invariant under (1.22)-(1.23) and (1.24), by definitionT µν s andĴ µ s are also invariant. As emphasized below (1.16), x denotes the spacetime location at whichT µν s ,Ĵ µ s (s = 1, 2) are evaluated and should be distinguished from either σ or X. Given the dependence of the action on g s and A s is of the form the stress tensor has the structure , (5.51) and similarly for the current. Note that since X µ s (σ) are dynamical variables, in the full "quantum" theory defined by the path integral (1.10), the delta function δ (d) (x − X s (σ)) on the right hand side of (5.51) is a quantum operator and should be understood as At the level of equations of motion, one can solve the delta function to find σ s (x) = X −1 s (x) and evaluate the integrals of (5.51). For example, the stress tensor for the first segment can be written as where Λ was introduced in (5.10) and we have suppressed the subscript 1 (all variables without an explicit subscript should be understood as with index 1). In obtaining (5.53),

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we have used expressions in appendix E.2, and it should be understood that the right hand side is evaluated at σ 1 (x) = X −1 1 (x). Similarly, from variation of A 1µ we find that T µν 2 andĴ µ 2 can be obtained from (5.53)-(5.54) by switching the signs of the terms involving derivatives with respect to the a-fields.
We can expand (5.53)- (5.54) in the number of a-fields. At zeroth order, as we discuss below and in more detail in appendix F, as a consequence of symmetries (1.22)-(1.23) and (1.24), the stress tensor and current can be expressed solely in terms of velocity-type variables u µ ,μ, τ and their derivatives to all derivative orders.
Going beyond zeroth order in the a-field expansion, other dependence on X µ 1,2 will be involved. For example, at O(a), the following quantities (which are invariant under (1.22)-(1.23) and (1.24)): will contribute to the stress tensor. These quantities cannot be written in terms of the velocity or chemical potential.

Lowest order in a-field expansion
Let us now look at the stress tensor and current at leading order in the a-field expansion, where we can take and thenT Setting all the a-fields to zero in (5.53)-(5.54) and dropping the r-indices, we find that they can be written aŝ It should be understood in (5.59)-(5.60) that after taking the derivative, one should set all the a-fields to zero. In appendix F, we show that all quantities of (5.59)-(5.60) can be expressed in terms of standard hydrodynamical variables.

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Applying (5.59)-(5.60) to (5.43), we find to first derivative order and As advertised in section 3.1, equations (5.58) and (5.61) are precisely the standard constitutive relations forT µν andĴ µ to first derivative order in a general frame (before one imposes entropy current constraints). In particular, 0 , p 0 , n 0 are the local energy, pressure and charge densities in the ideal fluid limit, with h , h p , h n their respective first order derivative corrections. η is the shear viscosity. We should emphasize that (5.61)- (5.65) are not yet the final form of the stress tensor and current, as we have not imposed the local KMS conditions in (5.43). In particular, at this stage, the energy density 0 , pressure p 0 , and charge density n 0 are completely independent. There are no relations among them. In the next subsection, we will discuss how thermodynamical relations emerge, along with other constraints on (5.43).

Formulation in the physical spacetime
The formulation of section 5.2 is convenient for writing down an action invariant under various fluid space diffeomorphisms. The resulting action is defined in the fluid spacetime. Here we discuss how to rewrite the action in the physical spacetime, which is more convenient for many questions.
For this purpose, consider We now invert X µ (σ a ) to obtain σ a (X µ ), and treat σ a (X) as dynamical variables. Other dynamical variables X µ a (σ), ϕ r,a (σ), τ (σ) are now all considered as functions of X µ through σ a (X). Since X µ are now simply the coordinates for the physical spacetime, there is no need to distinguish them from x µ . Thus the dynamical variables are now σ a (x), X µ a (x), ϕ r,a (x), τ (x). Below we will drop all r-subscripts. Now let us consider the actions (5.43) and (5.48) expressed in these variables. For simplicity, we will put all background fields to zero (except that corresponding to the chemical potential at infinity), i.e.

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So below all contractions between µ, ν, · · · indices are through η µν . Using σ a (x) we can define a velocity field as in (5.1): which can also be written as Note that in the form of (5.69), σ 0 is not needed to define u µ . Various quantities defined earlier can be straightforwardly converted into the new variables. For example, to first order in X a , ϕ a , we have Expanded in X µ a , ϕ a , the action can be written as Note that since the Φ a defined in section 5.2 depend nonlinearly on dynamical variables, the expansion (5.71) does not coincide with (5.42). For example, L (1) in (5.42) also contributes toĨ (3) ,Ĩ (5) , · · · . But noteĨ (1) is determined solely from L (1) andĨ (2) solely from L (2) . We then find from (5.43)Ĩ This form of (5.72) is of course expected since, as we discussed in section 1.3, the equations of motion for X µ a and ϕ a simply correspond to the conservation of the stress tensor and current respectively. For this reason, we expect (5.72) to apply to all derivative orders. Equation (5.72) was considered recently in [46] from exponentiating the hydrodynamical equations of motion.
At O(a 2 ), from (5.48) we find (5.73) In the above equations, the angular brackets denote the symmetric transverse traceless part of a tensor, i.e. for an arbitrary two-index tensor C µν

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We also follow the standard convention of using square brackets and parentheses to denote antisymmetrization and symmetrization respectively, i.e.
Note that in both (5.72) and (5.73), σ 0 has dropped out, which is a consequence of the time diffeomorphism (1.23). In fact, we expect σ 0 to completely decouple to all orders.
For a neutral fluid, we find that (5.76) Equation (5.73) contains three quadratic forms: one each for the tensor, vector, and scalar sectors. SinceĨ (2) is pure imaginary, for the path integral to be well defined the three quadratic forms should be separately non-negative, which implies that

The source action
We now discuss how to impose the local KMS conditions on the actions (5.41). For this purpose, we first need to obtain the corresponding action for sources only. Recall that from the prescription of section 1.6 we should first set all dynamical fields to zero. Here we have a complication regarding what should be the appropriate "background" values for τ . We propose the following prescription: and then Now σ a = δ a µ x µ spans the physical spacetime and we will simply use x µ . By definition, the resulting action obtained, I[g s , A s , τ ], is only invariant under (i) time diffeomorphisms, (ii) spatial diffeomorphisms, (iii) time-independent gauge transformations, of the physical spacetime.

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where T prop denotes the local proper temperature in the fluid space. In the absence of dynamics, it is natural to identify which then motivates us to set for τ the following background value τ = 1 2 log(−g r00 ) . (5.85) The resulting action I s [g 1 , A 2 ; g 2 , A 2 ] is then the one on which we will impose the local KMS conditions (1.68).

Constraints on constitutive relations from local KMS conditions
As outlined in section 3.2, the local KMS conditions include relations between coefficients of L (1) s and those of L (2) s , which will give rise to the non-negativity of various transport coefficients, as well as consistency conditions (2.59)-(2.60), which concern only L (1) s and give rise to constraints on constitutive relations. In this subsection, we focus on L (1) s and consider the latter type of constraints. Imposing (5.81) and (5.85) amounts to setting in (5.58) Let us now discuss (2.60) and (2.59) in turn.

Spatial partition function condition
Following the discussion (2.65)-(2.66), equation (2.60) says thatT µν hydro andĴ µ hydro in a stationary background should be obtainable from a partition function defined on the spatial manifold. This is precisely the prescription recently analyzed in detail in [13,14].
At zeroth order in derivatives, we havê where 0 = 0 (log b, A 0 /b), and similarly with p 0 and n 0 . For them to be obtainable from a single functional, we need to impose the integrability conditions but with rotational symmetry, there cannot be any first order derivative term in a partition function in general dimensions 20 and thus we need which gives (recallμ was introduced in (5.44)) To consider the implications of (5.92) for the constitutive relations for the stress tensor and current, let us consider the frame-independent vector Before imposing (5.92), upon using the thermodynamical relations (5.89)and the zeroderivative order equations of motion, µ has the form 21 where c 1 , c 2 , c 3 are independent functions of τ, µ. With (5.92), we find that with conductivity σ given by (5.98) 19 Note that ∂ui = ∂i log b. 20 With some specific dimensions, one may be able to construct first derivative terms using the tensor.

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Comparing with (5.96), we see that the thermal conductivity is determined from conductivity in the usual way and the c 2 term vanishes. In the conventional formulation, both of these relations follow from entropy current constraints.
The bulk viscosity ζ can be obtained by examining the other frame-independent quantity where one needs to use the zeroth derivative order equations of motion to obtain the right hand side.
One can also check that the reality condition in (2.59) does not appear to impose any additional constraints at these orders.

Generalized Onsager relations
Let us now consider the implications of the generalized Onsager relations (2.59) and (2.63). The nonlinear source action for (5.43) can be written as where we have used the decomposition (5.3) and Applying (2.63) to (5.100), we then find that, Note that all the relations above can be obtained from the Onsager relations at linearized level. So to first derivative order, nonlinear generalizations do not yield new relations.

Non-equilibrium fluctuation-dissipation relations
Now let us consider the relations between coefficients of I (1) and I (2) which follow from the local KMS conditions. We find the source action by following the procedure outlined

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For a neutral fluid, the corresponding expression is which is again non-negative from (5.80).

Full action to O(a 2 ) in physical spacetime
Let us now collect (5.72) and (5.73), and all the relations on the coefficients found in section 5.6 and section 5.7. We have up to order O(a 2 ) and to first derivative order T µν hydro = ( 0 +h )u µ u ν +(p 0 +h p )∆ µν +2u (µ q ν) −ησ µν ,Ĵ µ hydro = (n 0 +h n )u µ +j µ , (5.115) with (using (5.92) and (5.102)) At order O(a 2 ) we have at zeroth order in derivatives, where we have used the non-equilibrium fluctuation-dissipation relations (5.104)-(5.106). Notice that in (5.121), other than X µ a , ϕ a , the dynamical variables appear through standard hydrodynamical variables u µ , µ and τ . Also recall that u µ , µ are derived variables constructed from σ a (x), ϕ(x) as discussed in section 5.4. Below we will refer to u µ , µ and τ as hydro variables, and X µ a , ϕ a as noises. JHEP09(2017)095

Stochastic hydrodynamics
Approximating all the hydro variables by their background values, we obtain a Gaussian action for the noises X µ a and ϕ a . As in section 4.3, introducing the Legendre conjugates t µν and j µ for ∂ (µ X aν) and ∂ µ ϕ a respectively, the equations of motion for X µ a and ϕ a become where t µν and j µ can be interpreted as the noise contribution to the full stress tensor and current respectively, and satisfy Gaussian distributions. More explicitly, around equilibrium values, i.e. τ = ϕ = 0 and u µ = (1, 0), we find the path integrals for t µν and j µ have the form All coefficients in (5.123) should be understood as equilibrium values.
Beyond the quadratic approximation, as in the vector case again, there appears to be no benefit to introducing the Legendre conjugate for X µ a and ϕ a . The action (5.113) provides an interacting effective field theory among hydro variables and noises.

Entropy current
Now consider the O(a) action (5.114) in the ideal fluid limit, i.e.
which are respectivelyT µν hydro andĴ µ hydro at zeroth order in the derivative expansion. The ideal fluid action (5.126) has an "accidental" symmetry: it is invariant under δX aµ = λe τ u µ , δϕ a = λμ is a total derivative. To see this, note that and (5.129) follows, since from (5.89) we have The conserved Noether current S µ corresponding to (5.128) can be written as which is precisely the standard covariant form of the entropy current [73]. The entropy current has previously appeared as a Noether current in [32,33,57]. In fact this connection was central to developing the framework proposed in [32,33]. It can now be readily checked that (5.128) is no longer a symmetry either beyond the leading order in the derivative expansion inĨ (1) or ofĨ (2) 0 . We have also not been able to find a generalization of (5.128) under which the action is invariant beyondĨ (1) 0 . That (5.128) is present only forĨ (1) 0 is consistent with the physical expectation that a conserved entropy current is an accident at the ideal fluid level. With noises or dissipations, we do not expect a conserved entropy current.
It is natural to ask what happens to the entropy current beyond the ideal fluid level at O(a). The local KMS condition will ensure that it has a non-negative divergence from the following reasoning. As discussed in section 3.2, the partition function prescription of [13,14] arises as a subset of the local KMS condition at O(a). It has been shown by [15,16] that constraints from the partition function prescription are equivalent to equalitytype requirements from the non-negative divergence of the entropy current to all orders in derivatives. As seen in section 5.8 the inequality constraints from non-negative divergence of the entropy current follow in our context from the well-definedness of the integration measure. We have examined this to first derivative order. In [15,16] it has been argued these first order inequalities are the only inequality constraints coming from the entropy current to all derivative orders. Thus at O(a), the entropy current (suitably corrected at each derivative order) will have a non-negative divergence to all orders in derivatives. At O(a 2 ) level, where noises are included, we do not expect the divergence of the entropy current should be non-negative as noises are random fluctuations.

Two-point functions
Now let us consider (5.43) and (5.48) in the small amplitude expansion in the sources and dynamical fields. More explicitly, we write X µ s (σ) = δ µ a σ a + π µ (σ) + · · · , g sµν (x) = η µν + γ sµν (x), (5.133) and expand (5.43) and (5.48) to quadratic order in γ µν , A µ and π µ , ϕ, τ with dynamical and source fields considered to be of the same order. It is then straightforward, but a bit tedious, to integrate out the dynamical fields to obtain the generating functional for all JHEP09(2017)095 retarded and symmetric two-point functions among components of the stress tensor and current in the hydrodynamical regime. One can readily verify that with thermodynamical relations (5.89), the Onsager relations (5.102), and the local FDT relations (5.104)-(5.106), the full quadratic Green functions satisfy the FDT relations (2.39) and (2.41).
The explicit quadratic action and the final expressions are a bit long. Here we will first outline the general structure and then present the final expression of the generating functional for a neutral fluid.
We will take the spatial momentum k of external fields to be along the z direction, i.e. k z = q and k α = 0 with α denoting all the transverse spatial directions. Then the background metric and gauge fields can be separated into three sectors vector : where we have again suppressed 1, 2 subscripts. Again, below, r, a will be used to denote the symmetric and antisymmetric combinations of these variables. After integrating out the dynamical modes, the final generating functional should be diffeomorphism and gauge invariant, i.e. invariant under for arbitrary infinitesimal fields ξ µ and σ. Then to quadratic order in external fields, the final generating functional can be written as where W 1 is linear in the external fields, i.e. giving one-point functions with 0 , p 0 , n 0 all constants. Clearly W 1 is invariant under the linear part of (5.137).
Its variations under the quadratic part of (5.137) are canceled by the variations of the quadratic pieceW 2 under the linear part of (5.137). The other quadratic piece, W 2 , is invariant under the linear part of (5.137) by itself, and thus must be expressed in terms of the following (linear) gauge invariant combinations: γ αβ , Z α = qa α + ωb α , A α , Z = q 2 γ 00 + 2ωqγ 0z + ω 2 γ zz , γ, E z = ωA z + qA 0 , (5.140) where we have again suppressed r, a indices.

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Let us now give the explicit expressions forW 2 and W 2 for a neutral fluid. For the tensor sector, we haveW tensor 2 = 0 and where we have used the first equation of (5.104).
For the vector sector, we havẽ and where we have kept only the leading term in the numerators in the small ω and q expansion, and the momentum diffusion constant D takes its expected value: For the scalar sector, we havẽ and In the above expressions, is the sound velocity. Clearly the expressions exhibit the expected sound pole and attenuation constant. One can also check that the apparent singularity at ω = 0 and q = 0 in (5.145) and (5.146) cancel.

JHEP09(2017)095 6 Discussion
We conclude this paper by mentioning some future directions. Firstly, it would be interesting to explore the physical implications of the new constraints for hydrodynamical equations of motion from the generalized Onsager relations proposed in this paper. We already saw that these relations lead to nontrivial new constraints for the vector theory starting at the second derivative order for cubic terms. For a full charged fluid, these relations will also lead to new constraints at second derivative order. It would be of clear interest to work them out explicitly and to understand their physical implications. We also hinted in section 3.2 that local KMS condition may give rise to new inequality constraints at higher derivative orders. It would also be interesting to explore it further.
Secondly, the discussion of the bosonic action can be generalized in many different respects, to more than one conserved currents or non-Abelian global symmetries, parity and time reversal violations, inclusion of a magnetic field, anomalies, non-relativistic systems, superfluids, as well as anisotropic and inhomogeneous systems. Also important is to generalize it to situations with additional gapless modes, such as systems near a phase transition or with a Fermi surface.
Thirdly, it is clearly of importance to use our formalism to study effects of hydrodynamical fluctuations in various physical contexts, 22 in particular to non-equilibrium situations. Furthermore, it would be very interesting to understand physical implications of "ghost" fields.
Finally, the relation between supersymmetry and the KMS conditions should be understood better. Even for the theory of a single vector current, our understanding of the role of supersymmetry at both the classical statistical and quantum level can be much improved. At the classical statistical level, do the local KMS conditions combined with supersymmetry ensure all the KMS conditions at all loop levels? While it is tempting to conjecture the answer is the affirmative we do not yet have a full proof. At the quantum level, how should the deformed "supersymmetric" algebra [δ,δ] =¯ 2 tanh iβ 0 ∂ t 2 (6.1) be generalized to nonlinear level? Another important problem is to write down the fermonic part of the full charged fluid action. This is straightforward to do in a small amplitude expansion at quadratic, cubic, or higher orders, as in the theory of a single vector current, but the number of terms greatly proliferate and the analysis gets tedious. It is certainly more desirable to write down a full nonlinear fermonic action. This appears to require a supergravity theory at the classical statistical level due to the time diffeomorphism in the fluid spacetime, and a "quantum deformed" supergravity theory at the quantum level.

B Fluctuation-dissipation theorem at general orders
In this appendix, we first review and slightly extend the formulation of KMS conditions at general orders developed in [63], and then use the formalism to prove the relation (2.59).

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where i k label different operators, a i = 1, 2, and n 1,2 are the number of 1 and 2 indices respectively. In the above equations, integrations over the positions of φ's should be understood. Below, we will use a simplified notation to denote G a 1 i 1 a 2 i 2 ···anin as G αI , with Gᾱ I denoting the corresponding Greens function obtained from G αI by switching 1↔2. By definition, in coordinate space and in momentum space where we use x and k to collectively denote x 1 , x 2 · · · and k 1 , k 2 , · · · respectively. It is also convenient to introduce There is a parallel relation forG. Note that the response functions can be expressed as where a i = 1, 2 and n 2 counts the number of 2-index among a 1 , · · · a n−1 .

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withÕ Thus correlation functions ofÕ A with themselves are also all zero. In the → 0 limit discussed in section 1.7 and section 1.8, Note that for two-point functions (B.17) is the full condition, but this is not the case for n ≥ 3.

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Given that B µ = A µ + ∂ µ ϕ, the above equation implies that where tildes again act as in (C.6) and now I is the full bosonic action. From (C.3), we then conclude that the local KMS conditions lead to KMS conditions for the tree-level generating functional.

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Now let us consider the variation of Ξ under δg 1µν , which is tricky due to the logarithm. As discussed in the main text, both the action and the stress tensor are organized as expansions of a-variables, it is thus enough for us to work out the variation as an expansion of Ξ. For this purpose, let us first introduce Then expanding both sides ofâ −1 2 δâ 1 = e Ξ δ 1 = δe Ξ (E. 12) in Ξ, we find that Similarly, under a variation of g 2 we find that . (E.14) F Structure of stress tensor and current at order O(a 0 ) In this appendix, we prove that at leading order in a expansion, the stress tensor and current can be expressed in terms of velocity-type variables u µ , µ, τ to all derivative orders. The stress tensor at O(a 0 ) can be obtained by varying the action with respect to g 1µν and setting the a-type fields to zero. At this order, there is only one set of background fields and dynamical variables (see (5.56)). The r-subscripts can thus be dropped. From (5.53), we then find T µν (x) = µ δL δµ a − δL δE a u µ u ν + δL δχ a ∆ µν where we have used (5.10). Similarly, the current can be written aŝ We will now show that for the most general L invariant under (1.22)-(1.23) and (1.24), only velocity-type variables u µ , τ, µ and their derivatives will occur in (F.1)-(F.2).
For this purpose, let us consider a general tensor under spatial diffeomorphisms (1.22), invariant under (1.23) and (1.24), which are constructed out of r-variables. Below we will refer to such a quantity as a spatial tensor. From our discussion of covariant derivatives in section (5.1.2), a spatial tensor of any rank can be constructed by acting with D 0 , D i on the following basic objects: