Schwinger–Keldysh Effective Actions for Fluids

A hydrodynamic Schwinger–Keldysh effective action is a doubled low-energy generating functional whose saddle gives constitutive evolution and whose imaginary terms generate fluctuations. Locality and symmetry organize the derivative expansion, while SK normalization, reality, causal response, and positivity constrain every term. Thermal KMS is an additional state-dependent symmetry, not a consequence of doubling alone.

Required background. Hydrodynamic Effective-Theory Architecture supplies the EFT logic. Hydrodynamic Fluctuations and Noise fixes the diffusion kernel to reproduce.

Helpful background. Unitarity, Normalization, and Largest-Time Identities derives the microscopic contour constraints.

The SK constraints in the r/a basis

For two contour copies, define $\phi_r=(\phi_1+\phi_2)/2$ and $\phi_a=\phi_1-\phi_2$. With weight $e^{iS_{\mathrm{eff}}}$, the effective action must obey

$$S_{\mathrm{eff}}[\phi_r,\phi_a=0]=0,$$ $$S_{\mathrm{eff}}[\phi_r,\phi_a]^* = -S_{\mathrm{eff}}[\phi_r,-\phi_a],$$ $$\operatorname{Im}S_{\mathrm{eff}}\ge0.$$

The first is $Z[J,J]=1$. The second is SK reality. The third makes the noise weight convergent and its quadratic covariance nonnegative. Causal or largest-time identities further forbid a response before the latest source insertion. Crossley, Glorioso, and Liu formulate these constraints for nonlinear dissipative fluids and show how the source gauge symmetries implement conservation Crossley, Glorioso, and Liu 2017, §§2–4, pp. 5–29, Open PDF.

The fundamental fluid construction uses Stueckelberg maps and gauge-invariant combinations on both contour legs. A reduced density action is sufficient to display the mechanism, but it is not a substitute for those symmetries in a full charged fluid.

Quadratic diffusion action

Let $n_r$ be a conserved density and $\varphi_a$ its response field. The leading classical action is

$$S_{\mathrm{diff}} = \int\!\mathrm dt\,\mathrm d^{d_s}x \left[ \varphi_a(\partial_t n_r-D\nabla^2n_r) +iT\sigma(\boldsymbol\nabla\varphi_a)^2 \right],$$

where $D=\sigma/\chi$. Variation with respect to $\varphi_a$ at the classical saddle gives diffusion. The imaginary term is positive for $T\sigma\ge0$ and vanishes at $\varphi_a=0$.

Completing a Gaussian Hubbard–Stratonovich transformation rewrites

$$e^{-\int T\sigma(\nabla\varphi_a)^2}$$

as an average over a current noise $\boldsymbol\xi$ with covariance $2T\sigma$. Integrating $\varphi_a$ then imposes

$$\partial_t n_r-D\nabla^2n_r =-\boldsymbol\nabla\!\cdot\boldsymbol\xi.$$

Thus dissipation and noise are two parts of one quadratic kernel.

In Fourier space the retarded inverse operator is

$$K_R(\omega,k)=-i\omega+Dk^2.$$

With the site convention $G_R=-i\theta\langle[n,n]\rangle$ and a Hamiltonian source $H\to H-\mu n$,

$$G^R_{nn}(\omega,k) = -\chi\frac{Dk^2}{-i\omega+Dk^2},$$

while the symmetric correlator is

$$G^S_{nn}(\omega,k) = \frac{2T\chi Dk^2}{\omega^2+(Dk^2)^2}.$$

Their pole, static limit, and KMS low-frequency relation are independent checks of signs and normalization.

Sources, gauge redundancy, and matching

For a conserved $U(1)$ charge, introduce contour phases $\phi_s$ and sources $A_{s\mu}$ only through

$$B_{s\mu}=A_{s\mu}+\partial_\mu\phi_s, \qquad s=1,2.$$

Independent gauge transformations on the two legs leave $B_{s\mu}$ invariant and generate the Ward identity. The local chemical potential is built from $B_{r0}$ in the thermal frame; $B_a$ generates current insertions. A full action must match its coefficients to the same equation of state, hydrodynamic frame, and Kubo normalization as the constitutive tensors.

Field redefinitions can move terms between the real response sector and imaginary noise sector. The path-integral Jacobian and any ghost action must move with them. Omitting those induced terms is the stochastic analogue of an incomplete hydrodynamic-frame transformation.

Nonlinear and quantum qualifications

Higher powers of $\phi_a$ encode non-Gaussian noise; nonlinear $r$ dependence makes the noise multiplicative. Loops require a cutoff and renormalize thermodynamics and transport. The classical $a$-field power counting assumes $\lvert\omega\rvert\ll T$; at quantum frequencies the KMS kernel is nonlocal in time and the simple $iT\sigma$ coefficient is insufficient.

SK consistency alone allows many nonequilibrium actions. Imposing dynamical KMS relates their dissipative and noise vertices and yields nonlinear Onsager-type constraints. Glorioso, Crossley, and Liu derive this classical-limit thermal symmetry and its relation to entropy currents Glorioso, Crossley, and Liu 2017, §§2–5, Open PDF.

The complete checklist is the fluctuating-hydrodynamics and SK consistency reference.

The fluctuation/SK branch is the effective-action completion developed here. Inspect its separation from the superfluid, anomaly, flux, integrable, spin, and critical branches: the $r/a$ doubling organizes response and noise but does not by itself supply those additional slow fields.

The Schwinger–Keldysh branch packages constitutive response and fluctuations into an $r/a$ action constrained by normalization, reality, and nonnegative noise. Equilibrium dynamical KMS is an additional symmetry, not a consequence of contour doubling alone. The diagram is a qualitative taxonomy and does not imply that its seven extensions can be superposed without rematching fields, symmetries, and regulators.

In text: the $a$ field enforces the diffusion equation and generates retarded response, the imaginary $a^2$ term generates noise, and the KMS relation fixes their equilibrium normalization. Goldstone phases, flux currents, rapidity occupations, spin density, and critical modes require separate additions to this action.

Exercise

Check SK reality and positivity of $S_{\mathrm{diff}}$, and explain why a real term proportional to $(\nabla\varphi_a)^2$ is forbidden.

Solution

The term linear in $\varphi_a$ is real and changes sign under $\varphi_a\to-\varphi_a$. The quadratic term is purely imaginary and is unchanged. Hence

$$S^*[n_r,\varphi_a]=-S[n_r,-\varphi_a].$$

Its imaginary part is

$$\operatorname{Im}S=T\sigma\int(\nabla\varphi_a)^2\ge0.$$

A real $a^2$ term would be unchanged by $a\to-a$ but would have to change sign under the reality relation, forcing its coefficient to vanish. A purely imaginary coefficient is allowed and represents noise.

Common pitfalls

Equating $S[a=0]=0$ with KMS. The first follows from contour normalization. KMS requires a thermal state and a thermal transformation.

Writing an action without source gauge invariance. Conservation must follow as a Ward identity, not only after setting sources to zero.

Ignoring the imaginary sector. A dissipative retarded term without its equilibrium noise partner violates fluctuation–dissipation.

Where this leads

Dynamical KMS and Topological Symmetries separates the thermal and normalization symmetries. Long-Time Tails and Fluctuation Renormalization uses the nonlinear vertices generated by this EFT.

References

• Crossley, Michael, Paolo Glorioso, and Hong Liu. 2017. “Effective Field Theory of Dissipative Fluids.” Journal of High Energy Physics 2017 (9): 095. DOI. Open PDF.

• Glorioso, Paolo, Michael Crossley, and Hong Liu. 2017. “Effective Field Theory of Dissipative Fluids (II): Classical Limit, Dynamical KMS Symmetry and Entropy Current.” Journal of High Energy Physics 2017 (9): 096. DOI. Open PDF.