Unified Effective Field Theory

• Unified Effective Field Theory is a comprehensive framework that unifies multiple symmetry domains and matter sectors to systematically construct effective field actions.
• It employs geometric decomposition and derivative power-counting to classify invariants and unify analyses in gravitating continua, hydrodynamics, and cosmology.
• The approach connects diverse physical systems, including open quantum systems and nonlinear optics, enabling consistent treatment of dissipative and non-equilibrium phenomena.

Unified Effective Field Theory (UEFT) refers to a collection of frameworks and methodologies that systematize the construction, analysis, and classification of effective field theories by unifying multiple symmetry domains, matter sectors, or regimes (relativistic/non-relativistic, local/nonlocal, equilibrium/out-of-equilibrium) into a single formalism. Such approaches exploit internal and spacetime symmetries, geometric structures, and the mapping between different physical systems to produce actions, building blocks, and predictions valid across diverse phenomena. This unification also reveals deep connections between the structure of low-energy physics and underlying geometric or algebraic properties, extending the reach and predictive power of EFTs. The concept finds realization in gravitating continua (solids, fluids, aether), hydrodynamics with and without boost symmetry, inflationary cosmology, non-equilibrium quantum systems, nonlinear optics, and the geometry of field amplitudes.

1. Symmetry Classification and Unified Actions for Continua

In the context of gravitating continua, UEFT organizes solids, fluids, and aether by their distinct internal symmetry groups. In the unitary (comoving) gauge, the fields labeling the material coordinates are fixed ($\phi^i(x)=x^i$), and the only dynamical fields are components of the spacetime metric, recast via

$$g_{00} = -U^2, \qquad g_{0i} = U^2 U_i, \qquad g_{ij} = h_{ij} - U^2 U_i U_j$$

where $h_{ij}$ is the induced ortho-spatial metric, and $U$, $U_i$ parametrize the local fluid velocity.

The symmetry breaking structure is as follows:

• Solid: Invariant under diagonal $\mathrm{ISO}_{\mathrm{diag}}(3)$ (constant translations and rotations acting identically in coordinate and material spaces), with symmetry breaking pattern $$\mathrm{ISO}(3) \times \mathrm{ISO}(3) \to \mathrm{ISO}_{\mathrm{diag}}(3)$$.
• Fluid: Invariant under volume-preserving diffeomorphisms, $\phi^i \to \phi'{}^i(\phi)$ with $\det \partial \phi'/\partial \phi = 1$.
• Aether: Invariant under general internal diffeomorphisms, $\phi^i \to \phi'{}^i(\phi)$, with only the four-velocity $u^\mu$ as an invariant; a homogeneous, isotropic aether acts as a cosmological constant.

The unified effective action in unitary gauge is constructed as

$$S[g] = \int d^4x \sqrt{-g} \; \mathcal{L}(g_{\mu\nu}, u^\mu, \nabla_\mu u_\nu, \dots)$$

with kinematic tensors—expansion, shear, vorticity, and acceleration—built from $u^\mu$ and $h_{ij}$. The symmetries dictate which invariants (e.g., $\det h_{ij}$, shear terms, higher derivative contractions) appear at each order in the derivative expansion.

2. Geometric Decomposition and Power-Counting

UEFT for continua utilizes a thread-based spacetime decomposition—contrasting with ADM foliation—where derivatives are split along the material flow ("thread"), defined by the timelike congruence $u^\mu$. The orthonormal derivative basis is

$$D_0 \equiv u^\mu \partial_\mu = U^{-1} \partial_0, \qquad D_i \equiv h^\mu_i \partial_\mu = \partial_i + U_i \partial_0$$

with Lie brackets encoding kinematic quantities: $$[D_0, D_i] = a_i D_0, \quad [D_i, D_j] = 2 \omega_{ij} D_0$$ This decomposition preserves the symmetries of the continuum class (e.g., thread-preserving diffs) and, unlike ADM, allows systematic identification of symmetry-invariants even at higher order.

Low-energy expansion employs power-counting in derivatives: zeroth order in $D$ gives invariants built solely from $h_{ij}$ (e.g., determinant, elementary symmetric polynomials), while higher orders yield contracted combinations of expansion, shear, vorticity, acceleration, and three-curvature.

Classification by continuum type proceeds as follows (where $F$ is a function of the basic invariants; $b$'s are coefficients of higher-derivative corrections):

Type	Condition on $F$ and $b_{\cdots}$
Solid	none
Fluid	$\partial^p_{e_2}\partial^q_{e_3}F \to 0$ for $p+q \ge 1$
Aether	$\partial_{h,e_2,e_3}F \to 0$ (i.e., $F=\text{const}$); only higher-deriv ops

3. Unified Treatment of Hydrodynamics and Dissipative Systems

A universal Schwinger–Keldysh effective action formalism encapsulates Lorentzian, Galilean, and Lifshitz hydrodynamics within a single covariant, frame-flexible UEFT framework (Armas et al., 2020). The master action is constructed with doubled fields and incorporates thermal, chemical, and velocity sources as well as dissipative contributions, yielding (at first derivative order) a total of 29 transport coefficients, classified into hydrostatic, nonhydrostatic-nondissipative, and dissipative sectors. The formalism provides a stable, frame-covariant map between all constitutive quantities and guarantees consistency across all symmetry limits, with proper handling of frame transformations and response to out-of-equilibrium perturbations. The same framework delivers explicit non-Galilean dispersion relations and fluctuation spectra.

4. Unified EFTs in Cosmology and Out-of-Equilibrium Dynamics

In inflationary cosmology, the UEFT is instantiated through a general single-clock action in ADM form that encompasses Horndeski and GLPV operators—characterized by functions $M_2^4(t)$, $\bar M_1^3(t)$, $\bar M_2^2(t)$, $H(t)$, and $\dot H(t)$—yielding scalar and tensor quadratic actions with time-dependent sound speeds and normalization functions (Motohashi et al., 2017). The framework leverages the Generalized Slow-Roll (GSR) formalism, in which both scalar and tensor power spectra are governed by source functions associated to sound-horizon integrals,

$$\ln\Delta_X^2(k) \approx\ln[f_X^{-2}] -2\int_{0}^{\infty}\frac{dx}{x}\,W'(x)\,S_X(\ln x)$$

with higher-order corrections tractably parameterized by slow-roll hierarchies. Observables derived in unitary (gauge-fixed) or comoving slices are systematically related, and existing principal-component constraints on spectral source functions are directly translated into percent-level constraints on EFT coefficients. This formalism unites all single-clock inflationary models admitting second-order perturbations into a universal language.

For open quantum systems, the non-equilibrium UEFT is constructed by integrating out heavy bath degrees of freedom, yielding an influence action with Hermitian (unitary) and dissipative parts, and leading to a unified description—via path integrals, Langevin equations, Lindblad master equations, and kinetic/Boltzmann equations—of thermalization, noise, and renormalization (Boyanovsky, 2015). All approaches recover the same effective dynamics, fluctuation–dissipation relations, and rates, both at $T=0$ (renormalization of local unitary EFT) and $T\neq0$ (thermalization driven by anomalous thresholds in the bath spectral density).

5. Geometric Unification of Amplitude Building Blocks

UEFT methodology can also be framed geometrically at the level of the off-shell building blocks of perturbation theory. For scalar EFTs, tree-level amplitudes remain invariant under arbitrary (derivative and nonderivative) field redefinitions, with on-shell covariance realized as diffeomorphism invariance on the target or field-configuration manifolds (Cohen et al., 24 Sep 2025). The covariant building blocks are:

• Target manifold (field-space) geometry: For $N$ scalars with action $$S[\phi] = \int d^d x [\frac{1}{2}g_{ij}(\phi)\partial_\mu\phi^i\partial^\mu\phi^j - U(\phi)]$$, nonderivative redefinitions act as coordinate changes, with Levi-Civita connection and curvature entering recursively in 3- and 4-point covariant vertices.
• Field configuration (functional geometry): Derivative redefinitions act as diffeomorphisms in the infinite-dimensional configuration space, with normal-coordinate expansions and connections yielding recursively covariant n-point vertices organized analogously to FS geometry.

Amplitudes can be constructed universally as functions $F(\Delta, \mathcal V)$, with $\Delta$ the propagator and $\mathcal V$ covariant vertices, ensuring on-shell covariance and field-redefinition invariance at every stage. Functional geometry subsumes field-space geometry, and the geometric expansion isomorphically lifts Feynman diagrams to the infinite-dimensional context.

6. UEFT in Nonlinear and Quantum Optics

In nonlinear and quantum optics, a single covariant action coupled to vector polarization modes $P_i^a$ unifies few-photon quantum phenomena with strong-field nonlinear optics, valid up to a frequency cutoff $\Lambda$ set by material scales (Liu et al., 6 Nov 2025). The action

$$S_U = \int d^4x\left[-\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu} + \tfrac{1}{2}h^{\mu\nu}\partial_\mu P^a_i \partial_\nu P^a_i + \mathcal L_{\text{int}}\right]$$

combines electromagnetic, kinetic, isotropic potential, and topological (axion-like) magnetoelectric terms. Expansion of the potential $\mathcal L_{\text{int}}$ yields the hierarchy of susceptibilities $\chi^{(1,2,3)}$, with $\chi^{(3)}(\omega)$ determined quantitatively at one loop. Gauge-invariant Schwinger–Keldysh/BRST quantization guarantees physical consistency in dissipative media. Real-time 1D dynamics are solved with matrix-product operator techniques, yielding percent-level agreement with measured nonlinear effects in, e.g., GaAs microcavities, epsilon-near-zero ITO films, and superconducting circuits. The action, when combined with appropriate gauge and bath couplings, provides a unified description across quantum and classical regimes, with all observables anchored by a single $\chi^{(3)}$ parameter.

7. Outlook and Significance

Unified Effective Field Theory elevates the power of EFT frameworks by systematizing the inclusion of multiple internal or spacetime symmetries, dissipative and nonequilibrium effects, and geometric structures. The approach clarifies the relationships between seemingly disparate physical systems, rigorously organizes invariants and power counting, and supports model-independent constraints derived from data. In each domain—gravitating matter, hydrodynamics, cosmology, open quantum systems, nonlinear optics, and amplitude geometry—the UEFT methodology achieves both internal completeness (all invariants systematically constructed and classified) and external unification (mapping and connecting systems across different physical regimes). A plausible implication is that future developments along these lines will increasingly leverage geometric and algebraic methods to further distill universal structures from phenomenologically rich effective theories.