Galilean Fields, Scales, and Low-Energy Degrees of Freedom

Galilean symmetry fixes the kinematics of a nonrelativistic field: its particle-number charge carries a mass, boosts shift momentum by $m\mathbf v$ , and a free mode disperses as $\mathbf p^2/(2m)$ . It does not fix the interactions. Those are organized by scale separation, the retained degrees of freedom, and matching to microscopic observables.

Required background. Lie Groups, Lie Algebras, and Exponential and Adjoint Maps supplies generator algebra; The Action Principle and Field Equations supplies variational dynamics; One-Particle States: Mass, Spin, and Relativistic Normalization supplies the state-normalization contrast with relativistic fields.

Helpful background. What Is a Symmetry of a QFT? clarifies symmetry action, and Modes, Virtualities, and EFT Scale Separation supplies the general separation-of-scales method.

Galilean fields and the mass charge

Consider a complex field $\psi_a(t,\mathbf x)$ for species $a$ in $d$ spatial dimensions. We normalize particle number by

N_a=\int \mathrm d^d x\,\psi_a^\dagger\psi_a,

so $\psi_a$ has engineering dimension $d/2$ in momentum units. For an active boost by velocity $\mathbf v$ , choose

\psi_a(t,\mathbf x)\longmapsto e^{i m_a(\mathbf v\cdot\mathbf x-\frac12\mathbf v^2t)} \psi_a(t,\mathbf x-\mathbf v t).

The phase is not optional. Substituting it into

S_0=\int \mathrm dt\,\mathrm d^d x\, \psi_a^\dagger \left(i\partial_t+\frac{\boldsymbol\nabla^2}{2m_a}\right)\psi_a

changes the Lagrangian only by the coordinate relabeling. A plane wave of momentum $\mathbf p$ is sent to one of momentum $\mathbf p+m_a\mathbf v$ and energy $E+\mathbf v\cdot\mathbf p+\tfrac12m_a\mathbf v^2$ .

At the algebraic level, spatial translations $P_i$ and boosts $K_i$ obey

[K_i,P_j]=i\delta_{ij}M, \qquad M=\sum_a m_aN_a.

Thus mass is the Bargmann central charge. Superpositions of sectors with different total mass are not ordinary irreducible Galilean representations. A chemical potential couples to $N$ , not directly to $M$ ; the two are proportional only for a single species with fixed mass.

This central extension is the invariant checkpoint for boost conventions. Reversing active and passive boosts reverses several intermediate signs, but the momentum shift, dispersion, and central commutator must agree Bargmann 1954, pp. 1–20.

Scales and low-energy degrees of freedom

The quadratic dispersion implies dynamical exponent $z=2$ : under

\mathbf x\mapsto\lambda\mathbf x, \qquad t\mapsto\lambda^2t, \qquad \psi\mapsto\lambda^{-d/2}\psi,

the free action is invariant. This is kinematic power counting, not yet a claim of full scale invariance. A mass, density, scattering length, lattice spacing, temperature, or anomaly may introduce a physical scale.

A useful low-energy specification distinguishes at least four scales:

a microscopic inverse range $\Lambda_{\rm micro}$ or band-separation scale;
a regulator cutoff $\Lambda$ , chosen below degrees of freedom that have been removed;
characteristic external momenta such as $k$ , $k_F$ , or the inverse healing length; and
state scales such as $T$ and $\mu$ .

Predictivity requires $Q/\Lambda_b\ll1$ , where $Q$ is the largest retained external scale and $\Lambda_b$ is the first omitted physical scale. The numerical cutoff $\Lambda$ may be varied inside a useful window; it is not itself the breakdown scale. Couplings run with $\Lambda$ so that matched observables do not.

For a dilute one-component gas, the minimal field content can be one bosonic or fermionic field. Near a lattice band minimum, the slowly varying envelope field is the correct degree of freedom; away from that minimum, additional valleys or bands may be required. Near a narrow resonance, a molecular field can become low energy. Near a broken-symmetry state, phase and density fluctuations may be more efficient than the microscopic particle field. “Low energy” therefore describes modes relative to a specified state, not a permanent label attached to an operator.

A lattice-to-continuum checkpoint

For nearest-neighbor hopping on a hypercubic lattice,

H_0=-t\sum_{\langle ij\rangle} (c_i^\dagger c_j+\text{h.c.}),

the dispersion measured from its minimum is

\varepsilon(\mathbf p) =2t\sum_{i=1}^d[1-\cos(p_i a)] =ta^2\mathbf p^2- \frac{ta^4}{12}\sum_i p_i^4+O(a^6p^6).

Matching the quadratic term gives

m^*=\frac{1}{2ta^2}.

The $p_i^4$ correction remembers the lattice: it breaks continuous rotations to the point group and estimates the first kinematic error. The continuum field alone is reliable only while $\lvert\mathbf p\rvert a\ll1$ and other minima or bands are separated. This worked reduction is a first application of Galilean field content, but exact Galilean symmetry is emergent rather than microscopic.

Symmetry does and does not determine

Particle-number symmetry permits interactions with equal numbers of $\psi$ and $\psi^\dagger$ . Translation, rotation, parity, time reversal, statistics, and internal symmetries further restrict them. For a short-range single-species boson, a representative action is

S=\int \mathrm dt\,\mathrm d^d x \left[ \psi^\dagger\left(i\partial_t+\frac{\nabla^2}{2m}\right)\psi -\frac{C_0}{2}\psi^\dagger\psi^\dagger\psi\psi +\frac{C_2}{2}\,\mathcal O_2+\cdots \right].

Galilean invariance constrains derivatives to relative momenta in two-body operators; it does not determine $C_0,C_2,\ldots$ . These coefficients must be matched. For identical spinless fermions the local $s$ -wave term vanishes by antisymmetry, so the leading interaction contains derivatives.

The strongest conclusion available from symmetry and scales alone is therefore conditional: the field content and operator forms are fixed within a declared low-energy sector, while their coefficients and the accuracy of the truncation come from matching and power counting Braaten and Hammer 2006, §§ 2–3.

Common pitfalls

Treating mass as merely a parameter in the dispersion. In a Galilean quantum theory it is also the central charge. Mixing mass sectors without stating the enlarged field content loses the representation structure.

Calling the cutoff the physical range. A regulator is adjustable; the range or band gap is physical. Cutoff independence after refitting couplings is a test, not a reason to identify the two.

Assuming the continuum inherits every symmetry exactly. A lattice can yield an approximately quadratic band while leaving rotational corrections at $O(a^2p^4)$ . Exact and emergent symmetries must be distinguished.

Exercises

Verify the boost phase

Insert the boosted field into the free Schrödinger equation and show that the transformed plane wave has the claimed momentum and energy.

Solution

For $\psi=e^{i(\mathbf p\cdot\mathbf x-Et)}$ , the transformed field is

\psi_{\mathbf v} =e^{i[(\mathbf p+m\mathbf v)\cdot\mathbf x -(E+\mathbf v\cdot\mathbf p+\frac12m\mathbf v^2)t]}.

If $E=\mathbf p^2/(2m)$ , then the new energy is $(\mathbf p+m\mathbf v)^2/(2m)$ . The dispersion is therefore boost covariant.

Estimate the continuum error

For momentum along one lattice axis, find the fractional correction to the quadratic dispersion through $O((pa)^2)$ .

Solution

Along one axis,

\varepsilon(p)=ta^2p^2\left[1-\frac{(pa)^2}{12}+O((pa)^4)\right].

The leading fractional error is $-(pa)^2/12$ . A desired one-percent kinematic accuracy requires $pa\lesssim\sqrt{0.12}\simeq0.35$ , before considering interaction or higher-band errors.

Continue

Second-Quantized Bosons and Fermions constructs the field algebra. Schrödinger Symmetry, Scale Invariance, and Anomalies asks when the $z=2$ scaling extends beyond free kinematics. From Microscopic Hamiltonians to Continuum Fields turns the scale inventory into a matching calculation.

References

Bargmann, Valentine. “On Unitary Ray Representations of Continuous Groups.” Annals of Mathematics 59 (1954): 1–46. DOI.
Braaten, Eric, and Hans-Werner Hammer. “Universality in Few-Body Systems with Large Scattering Length.” Physics Reports 428 (2006): 259–390. DOI. Open PDF.