---
title: Schwinger-Keldysh Coset Construction
url: https://www.emergentmind.com/topics/schwinger-keldysh-coset-construction
type: topic
---

# Schwinger-Keldysh Coset Construction

The Schwinger-Keldysh coset construction is a framework for systematically deriving low-energy effective actions for systems with spontaneous symmetry breaking in a non-equilibrium, finite-temperature, or mixed-state context. By combining the methods of the closed-time-path (SK/CTP) formalism and coset constructions based on symmetry-breaking patterns, this approach enables a first-principles derivation of real-time effective field theories (EFTs) that incorporate both quantum and thermal fluctuations, dissipation, and the constraints of unitarity and dynamical Kubo-Martin-Schwinger (KMS) symmetry. It provides a unified language for non-equilibrium systems including fluids, solids, supersolids, and liquid crystals at finite temperature, and applies equally to systems with internal or spacetime symmetries [1912.12301][2306.17232].

## 1. Symmetry-Breaking Patterns and Doubling

In equilibrium, the coset construction organizes the Goldstone modes associated with the breaking of a global symmetry group $G$ to a subgroup $H$. For non-equilibrium and finite-temperature EFTs, the SK formalism doubles the field content to encode forward and backward time-evolution ("1" and "2" legs). The relevant symmetry structure is therefore $G_1 \times G_2$, with both $G$ symmetries acting on the two legs. The density matrix (pure or mixed state) determines the diagonal unbroken subgroup $H$:

- In a pure state, $G_1 \times G_2$ may be spontaneously broken to $H_1 \times H_2$.
- In a thermal (mixed) state, $\rho \propto e^{-H/T}$, only the diagonal subgroup $H_{\text{diag}} \subset G_1 \times G_2$ is preserved.

The low-energy sector thus includes a Goldstone multiplet for each generator (broken or unbroken), doubled for the SK contour:
- Broken generators: $\pi_s^\alpha(\phi) \leftrightarrow \tau_\alpha$
- Unbroken generators: $\epsilon_s^A(\phi) \leftrightarrow T_A$
with $s=1,2$ denoting the SK legs [1912.12301][2306.17232].

## 2. Coset Parametrization and Maurer–Cartan One-Forms

The SK coset construction introduces worldvolume coordinates $\phi^M$ and defines two coset elements:
$$
g_s(\phi) = e^{iX_s^\mu(\phi)\bar{P}_\mu} e^{i\pi_s^\alpha(\phi) \tau_\alpha} e^{i\epsilon_s^A(\phi) T_A}
$$
The Maurer–Cartan one-form for each leg is:
$$
\omega_s = g_s^{-1} d g_s = i E_{sM}^\mu d\phi^M \bar{P}_\mu + i \nabla_M\pi_s^\alpha d\phi^M \tau_\alpha + i \mathcal{A}_{sM}^A d\phi^M T_A
$$
Here,
- $E_{sM}^\mu$ is the vierbein;
- $\nabla_M\pi_s^\alpha$ is the covariant derivative of the broken Goldstones;
- $\mathcal{A}_{sM}^A$ is the connection for the unbroken $T_A$.

Transformations under $g \in G$ generate field redefinitions with residual local $H$ transformations. Finite-temperature systems exhibit additional emergent gauge redundancies (chemical shifts, worldvolume diffeomorphisms).

## 3. Schwinger-Keldysh-Invariant Building Blocks and Invariant Actions

Covariant invariants in the SK coset construction arise from the Maurer–Cartan components:
- On each leg: $\nabla_M\pi_s^\alpha$, $E_{sM}^\mu$, $\mathcal{A}_{sM}^A$
- Mixed terms (crucial for dissipation): $E_{1M}^\mu (E_2^{-1})^M{}_\nu$, $\mathcal{A}_{aM}^A = \mathcal{A}_{1M}^A - \mathcal{A}_{2M}^A$

The metrics $G_{sMN} = E_{sM}^\mu \eta_{\mu\nu} E_{sN}^\nu$ (one per leg) enable covariant contractions, and $G_{rMN} = \frac{1}{2}(G_{1MN}+G_{2MN})$ captures the mean geometry. Inverse Higgs (IH) constraints eliminate redundant Goldstone fields according to the algebraic structure:
- $[\bar{P}, \tau'] \supset \tau$
- Thermal IH: $[\tau, \bar{P}_0] \supset \bar{P}$
- Unbroken IH: $[T, \bar{P}] \supset \bar{P}'$

The SK effective action for sources $J_1, J_2$ is constructed such that it vanishes when $\psi_1 = \psi_2$ and respects unitarity:
$$
e^{W[J_1, J_2]} = \mathrm{Tr}[U(+\infty, -\infty; J_1)\, \rho\, U^\dagger(+\infty, -\infty; J_2)]
$$
At leading order, the action factorizes, and subleading $\mathcal{O}(\hbar)$ terms introduce mixing and dissipation:
$$
I = S_\text{inv}[\phi_1] - S_\text{inv}[\phi_2] - \int (\mathrm{mixing}) + i\,(\mathrm{noise})
$$
Conservative terms derive from $S_\text{inv}$, while dissipative and fluctuation (noise) terms structure the imaginary part, guaranteeing $ \operatorname{Im} I \geq 0 $ [1912.12301][2306.17232].

## 4. Imposing Dynamical KMS Symmetry

Thermal systems on the SK contour must satisfy the dynamical KMS (DKMS) symmetry, ensuring the correct fluctuation-dissipation and equilibrium relations. In the Keldysh (r/a) basis ($\psi_r = (\psi_1 + \psi_2)/2$, $\psi_a = \psi_1 - \psi_2$), DKMS acts to leading order as
\[
\psi_r(x) \rightarrow \Theta \psi_r(x), \quad \psi_a(x) \rightarrow \Theta \left[\psi_a(x) + i \beta_0 \partial_t \psi_r(x)\right], 
\]
guaranteeing $\delta_{KMS} I = 0$ for the action. For internal symmetry cosets:
- DKMS transformations for Goldstone and matter fields are specified up to $\mathcal{O}(E/T)$.
- For antiferromagnets and ferromagnets, discrete symmetry structure requires variant DKMS implementations, distinguished by the behavior under spin-space rotations and time reversal [2306.17232].

## 5. Worked Examples: Fluids, Solids, and Spin Systems

**Finite-Temperature Fluids:**  
The construction for neutral relativistic fluids involves
$$
g_s = e^{i X_s^\mu \bar{P}_\mu} e^{i \eta_s^i K_i} e^{i \theta_s^i J_i},
$$
with boosts $K_i$ broken and spatial rotations $J_i$, translations $P_\mu$ unbroken. Inverse Higgs constraints express $\eta_s^i$ and $\theta_s^i$ in terms of derivatives of $X_s^\mu$, leading to SK-invariant actions for hydrodynamics that reproduce $\partial_\nu T^{\mu\nu} = 0$ upon variation [1912.12301].

**Solids, Supersolids, and Liquid Crystals:**
- **Solids:** Additional internal $Q_i$ break ISO(3) $\rightarrow$ diagonal $P_i + Q_i$. The action depends on invariants $Y^{ij}$, $Z^i$, $G_{00}$.
- **Supersolids:** Internal $Q_\mu$ with broken Lorentz $J_{\mu\nu}$, leading to invariants $Y^{\mu\nu}$.
- **Liquid Crystals (nematic, smectic):** Subgroups of rotations and translations dictate the Goldstone field content and SK-mixing structure of the action.

**Non-Relativistic Spin Systems:**
- **Paramagnets, Antiferromagnets, Ferromagnets:** SK coset for $SO(3)_1 \times SO(3)_2 \to SO(3)_\text{diag}$, with adjoint matter fields for spin density. The quadratic action generates the standard diffusive pole in $r$-$r$ and $r$-$a$ correlators, matching the KMS requirement and yielding correct dynamical spin responses [2306.17232].

## 6. Power Counting and Physical Scaling

The SK coset construction admits a systematic low-energy expansion:
- Momenta $k \ll M$ (with $M$ set by $T$ or a symmetry-breaking scale),
- Frequencies $E \ll M$,
- Possible expansion in $\hbar$.

Field components and derivatives are assigned scaling weights, enabling homogeneous power counting in both derivative and $E/T$ expansions. This facilitates a rigorous organization of the EFT and identification of physically relevant terms at each order [2306.17232].

## 7. Significance and Extensions

The SK coset construction unifies non-equilibrium EFTs under symmetry and thermodynamic consistency principles, providing a consistent derivation of hydrodynamics and generalized Goldstone dynamics with dissipation and fluctuations. It reproduces known theories of finite-temperature fluids and superfluids, and generates new EFTs for solids, supersolids, and complex liquid crystalline phases. The formalism is extendable order-by-order to include higher-derivative dissipative and noise effects, and produces non-equilibrium correlators that obey unitarity and fluctuation-dissipation theorems by construction [1912.12301][2306.17232].

A plausible implication is that the SK coset construction provides a template extendable to other non-equilibrium many-body systems, allowing new insights into collective excitations, real-time response, and universal constraints rooted in symmetry and thermodynamic structure.

Source: https://www.emergentmind.com/topics/schwinger-keldysh-coset-construction