Non-Relativistic Scaling Regime Explained

Updated 22 October 2025

Non-Relativistic Scaling Regime is defined as the limit where velocities, energies, and momentum transfers are much smaller than the relativistic scale, enabling systematic expansions in small parameters like v/c.
It employs methodologies such as operator expansions, block diagonalization, and Lie algebra contractions to derive effective non-relativistic models applicable to quantum field theory, fluid dynamics, and gravitational systems.
This regime facilitates controlled approximations and precise predictions in diverse domains—from quantum many-body dynamics to plasma physics—by systematically incorporating subleading corrections and ensuring unitarity in scattering processes.

The non-relativistic scaling regime refers to the physical, mathematical, and theoretical context in which characteristic velocities (or, more generally, energy and momentum transfers) are much smaller than the fundamental relativistic scale set by the speed of light. In this limit, relativistic corrections become subleading and the dynamics, symmetries, and observables of the system are governed by expansions in a small parameter (typically $v/c$ or $1/c$). This regime controls a vast landscape of phenomena, ranging from effective field theory reductions of quantum field theories, finite-temperature particle production, electron acceleration, and quantum fluid dynamics to the structure of spacetime itself, and sets the framework for model-building in condensed matter, nuclear, plasma, and astrophysical systems.

1. Reduction of Relativistic Theories: Methodologies and Key Structures

Fundamental relativistic theories often give rise to non-relativistic effective descriptions under appropriate scaling limits. The transition is usually realized by taking $c\to\infty$ (equivalent to $v/c\ll 1$ ), accompanied where needed by a double-scaling of couplings to ensure non-trivial limits:

Operator Expansions: In thermal field theory, such as for heavy right-handed neutrino production, operator product expansions (OPE) in powers of $(\pi T/M)^2$ , with $M$ the heavy scale and $\pi T$ the thermal scale, systematically extract leading and subleading effects. For the propagator, one writes:

$\Bigl[ \frac{1}{(K-P)^2} \Bigr]_P = \frac{1}{K^2} + \frac{2 K\cdot P}{K^4} + \frac{4 (K\cdot P)^2}{K^6} + \dots$

Block Diagonalization and Adiabatic Perturbation: The Dirac equation decouples into electronic and positronic branches. Space-adiabatic perturbation theory, aided by magnetic pseudodifferential calculus, provides systematic approximation of the positive-energy subspace leading to the Pauli Hamiltonian:

$H_{\rm Pauli} = \frac{1}{2m}(-i\hbar \nabla_x - A(x))^2 + V(x) - \frac{e\hbar}{2m}\sigma \cdot B(x)$

Corrections emerge as explicit orders in $1/c$ or $v/c$ (Fürst et al., 2012).

Double Scaling and Mode-splitting: Integrable quantum field theories with fermionic excitations (e.g., Gross–Neveu or supersymmetric models) require $c\to\infty$ while rescaling couplings (e.g., $gc =$ const) to maintain finite-energy excitations. Mode-splitting cancels the divergent rest-energy, funneling all such models to a small set of Galilean-invariant, integrable non-relativistic models (Bastianello et al., 2017).
Lie Algebra Contraction: The systematic expansion and contraction (e.g., Inönü–Wigner procedure) of the relativistic Poincaré algebra, sometimes using multiple copies, produces non-relativistic symmetry algebras (Galilei, Newton–Cartan, or extended Newtonian) and associated action principles, including multimetric limits yielding Newtonian gravity with constant background density (Ekiz et al., 2022).

2. Non-Relativistic Scaling and Effective Theories Across Physical Domains

The regime underpins classical limits, effective descriptions, and anomalies:

Thermal Field Theory and Early Universe Applications: In heavy right-handed neutrino thermal production and leptogenesis, the non-relativistic regime ( $\pi T \ll M$ ) enables expansion of rate equations, with momentum-independent leading rates and subleading momentum-dependent corrections. Radiative and relativistic corrections adjust the population and washout rates (by $\sim$ 1–20% for lepton number), essential for precision in baryogenesis scenarios (Laine et al., 2011, Bodeker et al., 2013).
Non-Relativistic Fluid Dynamics: The passage from relativistic to non-relativistic (Navier–Stokes) hydrodynamics involves power-series expansion in $1/c$. Post-Newtonian corrections are neglected at leading order, but remnants of relativistic causality remain—transport coefficients at first order are bounded and the celebrated shear viscosity to entropy density ratio ( $\eta/s$ ) is now both upper and lower bounded:

$\frac{\rho \kappa^{(0)}}{s(s + n (\partial_\epsilon \mu/\partial_\epsilon T))} > \eta^{(0)}/s$

Moreover, the Fourier law for heat conduction acquires gradient corrections that persist in the non-relativistic limit for causal, hyperbolic-hydrodynamics models (BDNK), while strictly parabolic behavior emerges in classical (first-order) limits (R et al., 2023).

Scaling Anomalies & Symmetry Breaking: For non-relativistic CFTs (e.g., $z=2$ Lifshitz or Schrödinger fixed points), scale anomalies are classified via cohomological methods. With Galilean boost and/or foliation (Frobenius) structure, $A$ - and $B$ -type anomalies manifest in $2+1$ dimensions and reveal intricate connections to causality and topological invariants, such as the Euler density after null reduction from higher dimensions (Arav et al., 2016).

3. Analytic and Numerical Exploration of Non-Relativistic Scaling in Many-Body and Plasma Systems

Electron Acceleration in Surface Plasma Waves: For $a_{sw} = e E_{sw}/(mc\omega) \ll 1$ , classical (non-relativistic) motion dominates and the electron quiver energy sets the scaling:

$U_{osc} = \frac{e^2 E_{sw}^2}{4 m \omega^2}$

Phase of injection and wave inhomogeneity control kinetic energy gain; ponderomotive scaling breaks down near strong inhomogeneity or short SPW lifetimes (Riconda et al., 2015).

Domain Wall and Defect Network Evolution: In cosmological scenarios, non-relativistic regimes emerge for networks subject to high Hubble expansion (damping) rates. The velocity-dependent one-scale (VOS) model describes the evolution, predicting scaling regimes—conformal stretching ( $L \propto a$ ) and non-relativistic Kibble scaling ( $L \propto t$ with $v =$ const)—well corroborated by $4096^3$ lattice simulations (Martins et al., 2016).
Plasma Instabilities and Nonlinear Wave Interaction: In weak-field, non-magnetized pair plasmas ( $a = 2e|A_0|/m \ll 1$ $a = 2 e ∣ A_{0} ∣/ m ≪ 1$ ), nonlinear processes such as induced scattering and filamentation instability are analytically tractable:
- Induced scattering rate:
$\kappa \sim \sqrt{\frac{\pi \omega_p^2}{32 e} \frac{m a^2}{T}}$ - Filamentation instability growth rate:

$\Gamma_{max} \sim \frac{m}{8T} \frac{\omega_p^2}{\omega_0} a_0^2$

Simulations confirm theory and facilitate further work in the non-relativistic and weakly nonlinear plasma regimes (Ghosh et al., 2021).

4. Quantum Dynamics and the Limitations of Non-Relativistic Approximation

A widespread practice assumes non-relativistic quantum mechanics accurately describes dynamics whenever $v/c \ll 1$ . However, this expectation breaks down for long-time evolution:

Even minute relativistic corrections to the energy spectrum induce time-dependent phase differences. Given

$\Psi_{R}(t) = \sum_n A_n(0) e^{-i E_n t/\hbar} \phi_n, \quad \Psi_{NR}(t) = \sum_n a_n(0) e^{-i E_n^{NR} t/\hbar} \phi_n$

with $E_n - E_n^{NR} \ll E_n^{NR}$ , the phase accumulates and leads to large deviations in observable quantities (mean, variance, autocorrelation) on a timescale $T_{critical} \sim \hbar/\delta E_n$ —even in the deep non-relativistic regime.

For atomic Rydberg wave packets, this breakdown is experimentally accessible prior to spontaneous emission, necessitating full relativistic treatment for quantum revivals and interference (Lan et al., 2018).

5. Non-Relativistic Scaling in Spacetime Geometry and Gravity

Moving beyond matter, scaling limits organize the geometry of space and the structure of gravity:

Connections and Topology in Gravity: The non-relativistic scaling limit of General Relativity requires consistent reduction of the Lorentzian metric:

$g_{\alpha\beta} = -\frac{1}{\lambda} \tau_{\alpha} \tau_{\beta} + g^{(0)}_{\alpha\beta} + \lambda g^{(1)}_{\alpha\beta} + \mathcal{O}(\lambda^2), \;\; \lambda = 1/c^2$

However, standard Einstein equations admit a well-defined Newtonian (Newton–Cartan) limit only if spatial topology is Euclidean; otherwise, a modified "bi-connection" theory introduces a topological reference curvature so the Newtonian limit exists on arbitrary topology (Vigneron, 2022).

Gravity Duals and Holographic Perspective: In non-relativistic holography, metrics built to realize anisotropic scaling isometries ( $x \to \lambda^k x$ , $t \to \lambda t$ ) admit infinite-dimensional symmetry enhancements (e.g., spin- $k$ Galilean/CFT algebras) as their asymptotic symmetry groups, mirroring local conformal extensions in the dual field theory (Lü et al., 2023).
Multimetric Gravity and Contraction: By systematic contraction of multiple copies of the Einstein–Hilbert action (multimetric gravity), one obtains well-defined non-relativistic limits. The resulting theories reproduce Newtonian gravity (with optional constant background mass density) by constructing the Lagrangian as a regulated sum over expanded fields in a contraction parameter $\lambda$ (Ekiz et al., 2022).

6. Unitarity, Long-Range Interactions, and Scattering in the Non-Relativistic Regime

S-matrix unitarity imposes strict upper bounds on partial-wave cross-sections in quantum mechanics and quantum field theory, notably important when the scaling regime ( $k \to 0$ ) would tend to yield divergent cross-sections for long-range interactions.

In models with long-range potentials (e.g., dark matter coupled to light mediators), perturbative (e.g., Sommerfeld enhanced) calculations of cross-sections can violate these unitarity bounds. Resolution requires resummation of all two-particle-irreducible (2PI) diagrams, leading to a complex, non-local effective potential in the non-relativistic Schrödinger (or Bethe–Salpeter) equation, where the imaginary part reflects summed (squared) inelastic amplitudes:

$\operatorname{Im}[V_\ell(r, r')] = - \sum_j \nu_\ell^j{}^*(r) \nu_\ell^j(r')$

Unitarized cross-sections for elastic and each inelastic channel are then given by:

$x_\ell^{(\textrm{reg})} = \frac{x_\ell^{(\textrm{unreg})} + (1 - x_\ell^{(\textrm{unreg})}) [y_\ell^{(\textrm{unreg})}]^2}{[1 + y_\ell^{(\textrm{unreg})}]^2},\quad y_\ell^{j(\textrm{reg})} = \frac{y_\ell^{j(\textrm{unreg})}}{[1 + y_\ell^{(\textrm{unreg})}]^2}$

which ensures the unitarity bound $\sigma^{(\textrm{el})}_\ell + \sigma^{(\textrm{inel})}_\ell \leq \sigma_\ell^{(u)}$ is never violated.

This formalism is broadly applicable, for example, in non-relativistic dark matter freeze-out (joining relic abundance and unitarity mass bounds), indirect detection (regulating large annihilation rates), and self-interactions (enforcing bounds critical for small-scale structure modeling) (Flores et al., 3 May 2024).

7. Summary Table: Representative Themes in the Non-Relativistic Scaling Regime

Physical Domain	Key Scaling Principle	Characteristic Feature/Result
Field theory / particle physics	$v/c \ll 1$ , $(\pi T/M) \ll 1$	Leading and subleading OPE terms; radiative corrections; reduced rate equations (Laine et al., 2011, Bodeker et al., 2013)
Quantum many-body	Double scaling ( $c\to\infty$ , couplings)	Integrable models reduce to Lieb–Liniger or Yang–Gaudin (Bastianello et al., 2017)
Fluid dynamics	Series in $1/c$	Non-relativistic Navier–Stokes emerges; transport coefficient bounds; gradient corrections (R et al., 2023)
Defect/network evolution	$v\ll 1$ , expansion rate	Kibble scaling, VOS model, Hubble damping (Martins et al., 2016)
Quantum dynamics	$v/c \ll 1$ over long $t$	Relativistic vs non-relativistic phase drift—breakdown for $t\gg T_{crit}$ (Lan et al., 2018)
Gravitational theory	Contraction in $\lambda = 1/c^2$	Newton–Cartan limit; bi-connection for general topologies (Vigneron, 2022)
Holography	Anisotropic scaling exponents	Infinite-dimensional non-relativistic conformal algebra (Lü et al., 2023)
Scattering/unitarity	Small $k$ , long-range $V(r)$	Non-perturbative unitarization via 2PI resummation; regulated cross-sections (Flores et al., 3 May 2024)

The non-relativistic scaling regime serves as the organizing principle for extracting universal, effective dynamics and symmetry structures in systems where the fundamental relativistic scale is hierarchically large relative to observed energies, velocities, or gradients. Rigorous application of scaling methodology—often with resummations, operator expansions, or algebra contractions—is essential to controlling precision, identifying anomalies, guaranteeing unitarity, and connecting to experimental signatures or cosmological constraints. The regime remains central in both model building and foundational analysis across all scales of physical theory.