Galilean-Invariant NREFT Framework

• Galilean-Invariant NREFT is a framework that defines the non-relativistic limit of quantum field theory by retaining essential rest-energy phases and ensuring Galilean symmetry.
• It utilizes specific boost transformation laws, extra phase factors, and gravitational potential terms to maintain physical consistency in non-inertial frames.
• The approach bridges the gap between relativistic and non-relativistic quantum mechanics, offering practical insights for quantum systems in gravitational settings.

A Galilean-Invariant Non-Relativistic Effective Theory (NREFT) is a theoretical framework that rigorously describes the non-relativistic limit ($c \rightarrow \infty$) of relativistic quantum field theory, ensuring compatibility with Galilean symmetry, and maintaining subtle "residues" of relativistic structure such as rest-energy phases and equivalence to gravitational phenomena. The correct limiting procedure is crucial for both physical consistency and faithful reproduction of symmetry properties; extra phase factors, transformation laws, and potential terms emerge in the process and have deep implications for quantum mechanics and the principle of equivalence.

1. Galilean Transformation, Action, and the Rest-Energy Phase

The action for a free relativistic particle,

$$\mathcal{A} = - m c^2 \int \sqrt{1 - \frac{v^2}{c^2}}\, dt$$

is Lorentz invariant. Expanding in $1/c^2$ yields the non-relativistic kinetic term and an additional constant term $-m c^2 t$ (the "rest energy"), typically neglected in classical contexts. However, under a Galilean boost,

$x' = x - \xi(t),\quad t' = t$

(where $\xi(t) = Vt$ or a more general function for non-inertial frames), the non-relativistic action is not strictly invariant, but changes by a total time derivative: $$L'(x',\dot{x}') = L(x, \dot{x}) + \frac{d}{dt} f(x, t)$$ with

$$f(x, t) = -m x\, \dot{\xi}(t) + \frac{1}{2} m \int \dot{\xi}(t)^2 dt$$

While this boundary term leaves classical trajectories unchanged, it alters the quantum-mechanical phase of the wave function. Explicitly, the transformation for the non-relativistic (Schrödinger) wave function involves

$$\Psi(t, x) = \Psi'(t, x - \xi(t)) \exp \left[-\frac{i}{\hbar} f(x, t)\right]$$

and the phase $f(x, t)$ can be directly related to the rest-energy term arising from expanding Lorentz transformations to $O(1/c^2)$. Thus, the "irrelevant" $-mc^2 t$ piece is essential for the correct Galilean boost law in quantum mechanics, and the residual $e^{-i m c^2 t/\hbar}$ phase must be kept, even as $c \to \infty$.

Table: Galilean Transformation Law for the Wave Function

Frame Transformation	Extra Phase (Quantum)	Physical Origin
$x' = x - \xi(t)$	$-\frac{i}{\hbar} f(x, t)$	Expansion of $mc^2(t-t')$
Schrödinger $\psi \to \psi'$	$\exp(-\frac{i}{\hbar} f(x, t))$	Rest energy phase from relativity
$f(x, t)$	$$-m x\, \dot{\xi}(t) + \frac{1}{2} m \int \dot{\xi}(t)^2 dt$$	Relativistic expansion

The consequence is that "rest energy" is not discarded in the quantum mechanical limit, but leaves a measurable imprint on phase evolution and transformation properties.

2. Non-Inertial Frames, Gravitational Field, and the Principle of Equivalence

The non-relativistic limit of a covariant Klein-Gordon equation in a non-inertial (accelerated) frame, such as the Rindler metric,

$$ds^2 = \left(1 + \frac{g(t)\, x}{c^2}\right) dt^2 - dx^2$$

results, for $c \to \infty$, in a Schrödinger equation with a time-dependent gravitational potential: $$i\hbar\, \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + m g(t) x\, \psi$$ Thus, the gravitational field is naturally encoded as a linear potential term in the non-relativistic quantum theory. The transformation law for the wave function in an accelerated frame inherits an extra phase—

$m c^2 (t - t') \approx m V x' + \frac{m V^2 t'}{2}$

—matching the phase from gravitational time dilation. This demonstrates that the principle of equivalence between acceleration and gravity survives the non-relativistic limiting process as a physical phase in quantum mechanics.

3. Path Integrals and the Feynman Propagator in the Non-Relativistic Limit

The relativistic Feynman propagator, expressed in the proper time (Schwinger) representation as

$$\mathcal{A} = -m \int ds = -m \int_0^1 d\eta\, \sqrt{\frac{dx^\mu}{d\eta} \frac{dx_\mu}{d\eta}}$$

when expanded in the non-relativistic limit (using saddle-point methods), reduces to the standard non-relativistic kernel: $$K \simeq \left(\frac{1}{2m}\right) \left(\frac{m}{2\pi i t}\right)^{3/2} \exp \left[ -\frac{i m c^2 t}{\hbar} + \frac{i m |\mathbf{x}_2 - \mathbf{x}_1|^2}{2 \hbar t} \right]$$ The exponential factor $e^{-i m c^2 t / \hbar}$ again captures the rest-energy "memory" in the NR limit. In a weak gravitational field (Newtonian metric)

$$ds^2 = \left(1 + \frac{2\phi}{c^2}\right) c^2 dt^2 - d\mathbf{x}^2$$

the path integral yields

$$K(x_b, t_b; x_a, t_a) \propto \exp \left\{ \frac{i}{\hbar} \left[ \int_{t_a}^{t_b} \left( \frac{1}{2}m\dot{x}^2 - m\phi(x) \right) dt - m c^2 (t_b - t_a) \right] \right\}$$

The gravitational potential is encoded exactly as in the Schrödinger equation, while the rest-energy phase separates out.

Regarding composition (transitivity), the NR kernel satisfies

$$K(x_3, t_3; x_1, t_1) = \int dx_2\, K(x_3, t_3; x_2, t_2) K(x_2, t_2; x_1, t_1)$$

but the relativistic Feynman propagator is only "transitive" up to a derivative in the mass,

$$\int d^4x_2\, G_F(x_3; x_2) G_F(x_2; x_1) = i \frac{\partial}{\partial(m^2)} G_F(x_3; x_1)$$

This property, after suitable Fourier transformation, reduces to the standard NR behavior.

4. Physical Implications for Galilean-Invariant NREFT

The key results underscore that the peculiar features of NR quantum theory under Galilean boosts (extra phase), and in the transition to non-inertial or gravitational backgrounds (potential energy shifts and phase changes), are direct consequences of taking the large-$c$ limit of relativistic theory while retaining the rest-energy term. These structures are not arbitrary insertions but are dictated by a careful limiting process, which establishes physical consistency and preserves deep symmetries, such as the principle of equivalence.

The NREFT framework derived in this manner thus:

• Correctly reproduces transformation laws for the quantum wave function under the full Galilean group, including boosts with time-dependent velocities.
• Incorporates gravitational effects and coordinate accelerations into quantum dynamics via potential terms and nontrivial phases.
• Ensures self-consistency between path integral (propagator) and operator methods in the quantum theory.

5. Summary of Key Mathematical Structures

The systematic approach to the NR limit yields central formulas that govern the Galilean-invariant NREFT:

Structure	Formula/Formulation
Relativistic action	$$\displaystyle \mathcal{A} = - m c^2 \int \sqrt{1 - \frac{v^2}{c^2}}\, dt$$
Galilean transformation of $\Psi$	$$\displaystyle \Psi(t, x) = \Psi'(t, x - \xi(t)) \exp\left[ -\frac{i}{\hbar}f(x, t) \right]$$ with $f(x, t)$ as above
NR Schrödinger equation with gravity	$$\displaystyle i\hbar\, \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2\psi}{\partial x^2} + m g(t) x\, \psi$$
NR Feynman kernel	$$\displaystyle K \simeq \left(\frac{1}{2m}\right) \left(\frac{m}{2\pi i t}\right)^{3/2} \exp \left[ -\frac{i m c^2 t}{\hbar} + \frac{i m |\mathbf{x}_2 - \mathbf{x}_1|^2}{2\hbar t} \right]$$

6. Applications and Theoretical Significance

This Galilean-invariant NREFT formalism provides the foundation for:

• Consistent quantum mechanical descriptions of particles in non-inertial and gravitational backgrounds, preserving the principle of equivalence in the quantum regime.
• The correct inclusion of gravitational phases in interferometry and quantum experiments involving reference frame changes.
• Extending NREFT to composite and multi-particle systems, where phase factors arising from Galilean covariance are essential.

The explicit retention of rest-energy phases and correct transformation properties in the NR limit clarifies previously subtle points in both quantum mechanics and effective field theory, situating Galilean-invariant NREFT as a systematic and robust framework for non-relativistic quantum physics rooted in relativistic first principles.