Field-Theory of Ideal Hydrodynamics

• The paper establishes that ideal hydrodynamics emerges as an EFT from the spontaneous breaking of spacetime symmetries, identifying hydrodynamic variables as Goldstone fields.
• It provides a precise mapping of field variables to thermodynamic quantities via invariants like b and y, ensuring the conservation of energy, momentum, and charge.
• The approach organizes higher-order corrections through a controlled derivative expansion, enabling systematic inclusion of non-dissipative terms and quantum anomalies.

A field-theoretic framework for ideal hydrodynamics is a formulation in which the irreversible, dissipationless dynamics of fluids are described in terms of field variables subject to action principles and symmetry constraints. This approach systematically reinterprets classical hydrodynamics as the low-energy effective field theory (EFT) of a system that realizes certain spacetime and internal symmetries, clarifies the identification of hydrodynamic variables in terms of Goldstone fields, and organizes corrections in a controlled derivative expansion. It accommodates the inclusion of conserved charges, provides a precise map between field-theoretic and thermodynamical quantities, and forms a natural setting for incorporating quantum anomalies.

1. Goldstone Field Content and Symmetry Structure

The foundational insight is that ideal hydrodynamics emerges as the EFT associated with the spontaneous breaking of spacetime symmetries by the existence of a local fluid rest frame. In the field-theoretic construction, the basic dynamical variables are:

• A set of scalar fields $\phi^I(x)$, $I=1,\ldots,d$, which label the comoving Lagrangian coordinates of fluid elements.
• A phase field $\psi(x)$ when a global U(1) charge (e.g., particle number) is present.

These fields realize a set of symmetries:

• Internal shifts and rotations: $\phi^I \mapsto \phi^I + a^I,\ R^I_J \phi^J$.
• Volume-preserving diffeomorphisms (with some implementations imposing area/volume relabeling symmetry).
• U(1) global symmetry: $\psi \mapsto \psi + c$.
• Chemical shift symmetry: $\psi \mapsto \psi + f(\phi^I)$, ensuring the charge is comoving.

The chemical shift symmetry constrains the allowed terms in the action so that the charge and energy currents are locked—i.e., there are no independent charge diffusion modes in the ideal limit (Dubovsky et al., 2011).

2. Construction of the Effective Action and Leading-Order Hydrodynamics

The action is built as a function of the lowest-derivative invariants constructed from these fields. The leading-order (ideal) hydrodynamic action in $d+1$ spacetime dimensions is: $S = \int d^{d+1}x\ F(b, y)$ Here,

• $$b = \sqrt{\det B^{IJ}},\quad B^{IJ} = \partial_\mu \phi^I \partial^\mu \phi^J$$ is interpreted as the (local) entropy density.
• $y = u^\mu \partial_\mu \psi$, with $u^\mu = J^\mu/ b$ the fluid four-velocity and $$J^\mu = \epsilon^{\mu \alpha_1\ldots\alpha_d} \partial_{\alpha_1} \phi^1 \ldots \partial_{\alpha_d} \phi^d$$. $y$ plays the role of a chemical potential (up to normalization).

These invariants are dictated by the symmetries: $b$ is invariant under internal and spacetime symmetries, while $y$ is invariant under chemical shift. This structure ensures the lowest-derivative (ideal) Lagrangian is entirely fixed up to the function $F$, which encodes the equation of state.

3. Mapping to Thermodynamic Quantities and Conservation Laws

Varying the action with respect to the metric yields a stress-energy tensor that takes the perfect-fluid form: $T^{\mu\nu} = (\rho + p) u^\mu u^\nu - p g^{\mu\nu}$ The identifications are: $$\begin{align*} p &= F - b F_b \ \rho &= y F_y - F \ n &= F_y \ j^\mu &= F_y u^\mu \end{align*}$$ where $F_b = \partial F/\partial b$ and $F_y = \partial F/\partial y$.

The entropy current is $J^\mu = b u^\mu$ and is identically conserved off-shell due to the symmetry structure. The Noether current for the U(1) symmetry yields the charge current, which—because of the chemical shift symmetry—is automatically comoving with the fluid velocity (Dubovsky et al., 2011). This enforces the absence of static charge diffusion in the ideal regime.

4. Derivative Expansion and Higher-Order Corrections

The EFT framework organizes corrections via a systematic derivative expansion. At leading order (one derivative per field), the theory exactly reproduces classical ideal hydrodynamics and its thermodynamic structure. Higher-order terms, compatible with the symmetries, encapsulate controlled corrections. For example, a second-order, non-dissipative correction is: $$\Delta S = \alpha \int d^{d+1}x\, (u^\mu \partial_\mu b)^2$$ Such terms produce explicit corrections to hydrodynamic observables. For example, the usual sound-wave dispersion relation $\omega^2 = c_s^2 k^2$ receives a $k^4$ correction: $\omega^2 = c_s^2 k^2 + \delta c\, k^4$ with $\delta c \propto \alpha$ (involving thermodynamic derivatives through $F_b, F_{bb}$) (Dubovsky et al., 2011). At first order, all non-dissipative corrections can be removed by field redefinitions, confirming that transport coefficients like viscosity are strictly dissipative at leading order.

5. Conserved Charges, Chemical Shift Symmetry, and Alignment of Currents

The chemical shift symmetry ensures that the charge current $j^\mu$ is strictly aligned with the entropy current $J^\mu$—i.e., $j^\mu \propto J^\mu$, or equivalently, charge is advected with the fluid without independent transport. This symmetry is responsible for the absence of additional propagating modes (relative to the basic sound wave) in the normal phase. Only in superfluids—where chemical shift is relaxed to a global shift symmetry—can independent relative motion between charge and energy occur (Dubovsky et al., 2011).

6. Anomalies and Extension to Anomalous Hydrodynamics

The EFT approach, by virtue of working at the action level and encoding all symmetries from the outset, is a natural framework for incorporating quantum anomalies. In anomalous fluids, additional terms such as Wess–Zumino terms can be systematically included in the action, leading to modified constitutive relations and new universal constraints (such as unitarity bounds and positivity) that are obscure in conventional hydrodynamic treatments.

7. Summary and Significance

The field-theoretic framework for ideal hydrodynamics provides:

• A fully symmetry-based construction of dissipationless fluid dynamics, establishing hydrodynamic variables as Goldstone fields.
• A precise map from field variables to thermodynamical quantities and conservation laws.
• A systematic, symmetry-constrained derivative expansion allowing controlled inclusion of subleading, possibly non-dissipative corrections.
• A clear method for handling fluids with conserved charges, ensuring physical alignment of charge and entropy transport in the ideal regime.
• A direct route to consistent inclusion of quantum anomalies within hydrodynamics via action-based methods.
• A clear conceptual foundation for extending hydrodynamics beyond ideal fluids—toward including dissipation, anomalies, and fluctuating effects—while guaranteeing all underlying symmetries are respected.

In this construction, ideal hydrodynamics is not merely a phenomenological macroscopic model but emerges as a specific, symmetry-protected effective field theory universally governing the long-wavelength limit of non-dissipative many-body systems (Dubovsky et al., 2011).