Densities, Currents, and Nonrelativistic Ward Identities

Nonrelativistic densities and currents are defined most reliably by coupling the theory to background sources and differentiating the generating functional. Local source invariance gives Ward identities, while second source derivatives generate contact or diamagnetic terms. These terms and the order of the $\omega\to0$ and $\mathbf q\to0$ limits are part of the observable, not optional corrections.

Required background. Galilean Fields, Scales, and Low-Energy Degrees of Freedom fixes mass and boosts; Coherent-State Path Integrals for Many-Body Systems fixes the source-coupled functional; Quantum Currents, Improvements, and Conservation and Localized Transformations and Ward–Takahashi Identities supply the general current construction.

Helpful background. Spurions, Local Counterterms, and Symmetry Response explains local source terms, and Sources, Linear Response, and Kubo Formulae supplies the response interface.

Number sources and the continuity equation

For a field of unit particle-number charge, introduce $D_\mu=\partial_\mu-iA_\mu$ and

$$S[A]=\int\mathrm dt\,\mathrm d^d x \left[ i\psi^\dagger D_t\psi -\frac{(D_i\psi)^\dagger D_i\psi}{2m} -\mathcal V(\psi^\dagger\psi) \right].$$

Under $\psi\mapsto e^{i\alpha}\psi$ and $A_\mu\mapsto A_\mu+\partial_\mu\alpha$, the action is invariant if the regulator preserves number symmetry. At zero source,

$$n=\left.\frac{\delta S}{\delta A_0}\right|_{A=0} =\psi^\dagger\psi, \qquad j_i=\left.\frac{\delta S}{\delta A_i}\right|_{A=0} =\frac{1}{2mi}\psi^\dagger\overleftrightarrow{\partial_i}\psi.$$

Localized invariance gives

$$\partial_t n+\partial_i j_i=0.$$

Inside time-ordered correlators, derivatives also act on ordering step functions and produce equal-time commutators. Those contact terms are the local Ward identity’s right-hand side.

Momentum, mass, energy, and stress

Translation sources define momentum density $g_i$, energy density $\mathcal H$, energy current, and stress $\Pi_{ij}$. For one Galilean species without background magnetic field,

$$g_i=mj_i.$$

This is a consequence of boost symmetry. It can fail in a lattice, in spin–orbit-coupled systems, with several species of unequal charge-to-mass ratio, or when external fields exchange momentum with matter.

Stress tensors admit improvement terms that change local expressions by total derivatives while preserving integrated charges under suitable boundary conditions. Therefore a quoted local stress operator must state the source convention. Coupling to spatial geometry or a nonrelativistic Newton–Cartan background gives a systematic definition Son and Wingate 2006, §§ 2–3.

The vertex Ward identity

Let $G(p)$ be the exact one-particle propagator and $\Gamma^\mu(p+q,p)$ the amputated number-current vertex in the source convention above. With four-momentum contraction defined consistently with the Fourier transform, gauge invariance gives

$$q_\mu\Gamma^\mu(p+q,p) =G^{-1}(p+q)-G^{-1}(p).$$

At $q\to0$, this relates the vertex to derivatives of the inverse propagator. It is an exact identity, not permission to use a bare vertex with an arbitrary approximate self-energy. If $G^{-1}=G_0^{-1}-\Sigma$, then a nontrivial momentum- or frequency-dependent $\Sigma$ generally requires vertex corrections.

The identity can be checked independently by inserting the current into a two-point function and differentiating the equation of motion. Agreement tests both source signs and contact terms.

Diamagnetic terms and response limits

Expanding the kinetic term to second order in $\mathbf A$ gives

$$S[A]\supset \int\mathrm dt\,\mathrm d^d x \left(A_i j_i-\frac{n}{2m}\mathbf A^2\right)$$

up to the action/source sign convention. Consequently the electromagnetic or number-current response contains a paramagnetic correlator plus a local diamagnetic kernel proportional to $n/m$. Dropping it violates the longitudinal Ward identity and the $f$-sum rule.

Static and dynamic limits ask different questions:

$$\chi_{\rm stat}=\lim_{\mathbf q\to0}\lim_{\omega\to0}\chi(\omega,\mathbf q), \qquad \chi_{\rm dyn}=\lim_{\omega\to0}\lim_{\mathbf q\to0}\chi(\omega,\mathbf q).$$

The first allows density rearrangement over long wavelengths; the second probes a spatially uniform system before relaxation. Conservation laws often force them to differ. A response coefficient without its order of limits is incomplete.

A free-field check

For $G_0^{-1}(p)=p_0-\mathbf p^2/(2m)+\mu$, the bare vertex can be chosen

$$\Gamma^0_0=1, \qquad \boldsymbol\Gamma_0=\frac{2\mathbf p+\mathbf q}{2m}.$$

Then

$$q_0\Gamma^0_0-\mathbf q\cdot\boldsymbol\Gamma_0 =q_0-\frac{(\mathbf p+\mathbf q)^2-\mathbf p^2}{2m} =G_0^{-1}(p+q)-G_0^{-1}(p).$$

This fixes the relative sign between temporal and spatial contraction. An interacting approximation must reproduce the same identity with its dressed $G$ and compatible $\Gamma$.

Common pitfalls

Defining a current only by intuition. Source variation fixes normalization and all local contact terms. Different improvement conventions can share the same integrated charge.

Using $\mathbf g=m\mathbf j$ on a lattice. The identity follows from continuous Galilean boosts, which a lattice does not possess microscopically.

Interchanging response limits. Compressibility, conductivity, and screening can use different orders. Write the order beside the result.

Exercises

Derive the number current

Vary the source-coupled kinetic term with respect to $A_i$ at $A=0$.

Solution

Expanding $(D_i\psi)^\dagger D_i\psi$ to first order gives

$$(\partial_i\psi^*)(\partial_i\psi) +iA_i\bigl[\psi^*\partial_i\psi-(\partial_i\psi^*)\psi\bigr]+O(A^2).$$

Including the overall $-1/(2m)$ yields $j_i=(2mi)^{-1}\psi^*\overleftrightarrow{\partial_i}\psi$ with the stated source convention.

Test an approximate vertex

Suppose $\Sigma(p)=\lambda p_0$. Can the bare density vertex $\Gamma^0=1$ satisfy the $\mathbf q=0$ Ward identity?

Solution

Now $G^{-1}=(1-\lambda)p_0-\varepsilon_{\mathbf p}+\mu$, so for $q=(q_0,\mathbf0)$ the right side is $(1-\lambda)q_0$. The bare vertex gives $q_0$. A compatible vertex requires $\Gamma^0=1-\lambda$ in this simplified limit; using the bare vertex violates the identity when $\lambda\ne0$.

Continue

Current Vertices and Ward-Consistent Response applies the identity to dressed propagators. Spectral Moments and Many-Body Sum Rules derives integrated constraints from these commutators. Finite-Density Goldstone Counting uses charge densities to count collective modes.

References

• Son, Dam T., and Michael Wingate. “General Coordinate Invariance and Conformal Invariance in Nonrelativistic Physics: Unitary Fermi Gas.” Annals of Physics 321 (2006): 197–224. DOI. Open PDF.

Further reading

• Baym, Gordon, and Leo P. Kadanoff. “Conservation Laws and Correlation Functions.” Physical Review 124 (1961): 287–299. DOI.
• Watanabe, Haruki, and Hitoshi Murayama. “Effective Lagrangian for Nonrelativistic Systems.” Physical Review X 4 (2014): 031057. DOI. Open PDF.