Schwinger-Keldysh Coset Construction

• Schwinger-Keldysh coset construction is a framework for deriving low-energy effective actions for non-equilibrium systems by combining symmetry breaking with the closed-time-path formalism.
• It employs Maurer–Cartan one-forms to systematically build covariant invariants that ensure unitarity, capture quantum and thermal fluctuations, and respect fluctuation-dissipation relations.
• The construction unifies effective field theories for fluids, solids, and spin systems via systematic low-energy expansions, inverse Higgs constraints, and dynamical KMS symmetry.

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 (Landry, 2019, Akyuz et al., 2023).

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: $H$0 with $H$1 denoting the SK legs (Landry, 2019, Akyuz et al., 2023).

2. Coset Parametrization and Maurer–Cartan One-Forms

The SK coset construction introduces worldvolume coordinates $H$2 and defines two coset elements:

$H$3

The Maurer–Cartan one-form for each leg is:

$H$4

Here,

• $H$5 is the vierbein;
• $H$6 is the covariant derivative of the broken Goldstones;
• $H$7 is the connection for the unbroken $H$8.

Transformations under $H$9 generate field redefinitions with residual local $G_1 \times G_2$0 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: $G_1 \times G_2$1, $G_1 \times G_2$2, $G_1 \times G_2$3
• Mixed terms (crucial for dissipation): $G_1 \times G_2$4, $G_1 \times G_2$5

The metrics $G_1 \times G_2$6 (one per leg) enable covariant contractions, and $G_1 \times G_2$7 captures the mean geometry. Inverse Higgs (IH) constraints eliminate redundant Goldstone fields according to the algebraic structure:

• $G_1 \times G_2$8
• Thermal IH: $G_1 \times G_2$9
• Unbroken IH: $G$0

The SK effective action for sources $G$1 is constructed such that it vanishes when $G$2 and respects unitarity:

$G$3

At leading order, the action factorizes, and subleading $G$4 terms introduce mixing and dissipation:

$G$5

Conservative terms derive from $G$6, while dissipative and fluctuation (noise) terms structure the imaginary part, guaranteeing $G$7 (Landry, 2019, Akyuz et al., 2023).

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 ($G$8, $G$9), DKMS acts to leading order as

$H$0

guaranteeing $H$1 for the action. For internal symmetry cosets:

• DKMS transformations for Goldstone and matter fields are specified up to $H$2.
• For antiferromagnets and ferromagnets, discrete symmetry structure requires variant DKMS implementations, distinguished by the behavior under spin-space rotations and time reversal (Akyuz et al., 2023).

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

Finite-Temperature Fluids:

The construction for neutral relativistic fluids involves

$H$3

with boosts $H$4 broken and spatial rotations $H$5, translations $H$6 unbroken. Inverse Higgs constraints express $H$7 and $H$8 in terms of derivatives of $H$9, leading to SK-invariant actions for hydrodynamics that reproduce $G_1 \times G_2$0 upon variation (Landry, 2019).

Solids, Supersolids, and Liquid Crystals:

• Solids: Additional internal $G_1 \times G_2$1 break ISO(3) $G_1 \times G_2$2 diagonal $G_1 \times G_2$3. The action depends on invariants $G_1 \times G_2$4, $G_1 \times G_2$5, $G_1 \times G_2$6.
• Supersolids: Internal $G_1 \times G_2$7 with broken Lorentz $G_1 \times G_2$8, leading to invariants $G_1 \times G_2$9.
• 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 $H_1 \times H_2$0, with adjoint matter fields for spin density. The quadratic action generates the standard diffusive pole in $H_1 \times H_2$1-$H_1 \times H_2$2 and $H_1 \times H_2$3-$H_1 \times H_2$4 correlators, matching the KMS requirement and yielding correct dynamical spin responses (Akyuz et al., 2023).

6. Power Counting and Physical Scaling

The SK coset construction admits a systematic low-energy expansion:

• Momenta $H_1 \times H_2$5 (with $H_1 \times H_2$6 set by $H_1 \times H_2$7 or a symmetry-breaking scale),
• Frequencies $H_1 \times H_2$8,
• Possible expansion in $H_1 \times H_2$9.

Field components and derivatives are assigned scaling weights, enabling homogeneous power counting in both derivative and $\rho \propto e^{-H/T}$0 expansions. This facilitates a rigorous organization of the EFT and identification of physically relevant terms at each order (Akyuz et al., 2023).

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 (Landry, 2019, Akyuz et al., 2023).

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.