Non-Relativistic Symmetry-Group Formalism

• Non-relativistic symmetry-group formalism is a rigorous framework that defines the symmetry structure of Newtonian physics, emphasizing the Galilean and centrally extended Bargmann groups.
• It employs central extensions, Casimir invariants, and coadjoint orbits to systematically capture key physical properties such as mass conservation and the emergence of projective representations in quantum theory.
• Extended symmetries, including conformal and superconformal extensions, broaden the formalism to encompass noninertial dynamics and higher-order invariances, enriching its applications in mathematical physics.

Non-relativistic symmetry-group formalism provides the group-theoretical and geometric structure underlying nonrelativistic classical and quantum physics, most notably through the Galilean and its centrally extended (Bargmann) group, their representations, coadjoint orbits, and projective/unitary realization in quantum theory. The formalism extends to conformal, superconformal, and noninertial symmetries, linking physics with the orbit method, symplectic geometry, and Lie algebra cohomology. It contrasts in key respects with relativistic (Poincaré-invariant) frameworks, especially concerning the role of mass and the observable content of projective representations.

1. Galilean Group and Its Central Extension

The Galilean group acts as the automorphism group of nonrelativistic Newtonian space-time, generated by spatial rotations $J_i$, translations $P_i$, Galilean boosts %%%%2%%%%, time translations $H$, and augmented by a central charge $M$ (mass) in the quantum setting. The Bargmann group is the one-dimensional central extension of the Galilean group, essential for quantum-mechanical consistency.

The complete set of nonvanishing Lie algebra commutators (setting $\hbar=1$) is: $$\begin{aligned} [J_i, J_j] &= i\,\epsilon_{ijk}\,J_k, \ [J_i, P_j] &= i\,\epsilon_{ijk}\,P_k, \ [J_i, K_j] &= i\,\epsilon_{ijk}\,K_k, \ [K_i, P_j] &= i\,M\,\delta_{ij}, \ [K_i, H] &= i\,P_i, \ [M, \cdot] &= 0, \end{aligned}$$ with all other brackets vanishing (Kong et al., 2021). This central extension encodes the physical fact that nonrelativistic quantum mechanics requires projective (rather than linear) representations, the projective phase being proportional to mass (Torre, 2023).

2. Casimir Invariants and Mass as a Central Element

For the centrally extended algebra, there are two Casimir invariants: $C_1 = M, \quad C_2 = \mathbf{P}^2 - 2 M H.$ On an irreducible representation, $M \to m$ and $C_2 \to \varepsilon$, where $m$ (mass) and $\varepsilon$ (internal energy) classify the Bargmann irreducibles (Kong et al., 2021, Kong, 2022). The appearance of $M$ as a central element is the rigorous group-theoretical expression of inertial mass in Newtonian physics and underlies the superselection of mass sectors in quantum theory.

3. Coadjoint Orbits and the Symplectic-Geometric Perspective

The orbit method provides a natural realization of nonrelativistic symmetries via coadjoint orbits of the Bargmann group. The dual Lie algebra carries coordinates $(j_i, p_i, k_i, h, m)$ paired with the group generators, and the Kirillov–Kostant–Souriau symplectic form on the coadjoint orbit $O_{(m,\varepsilon,s)}$ is

$$\omega = dp_i \wedge dx^i + \frac{1}{2}\epsilon_{ijk} s^k d\hat n^i \wedge d\hat n^j,$$

with $(x^i, p_i)$ coordinates on Newtonian phase space and $s^k$ labeling spin (Kong et al., 2021). The group acts by

$$x^i \mapsto R^i{}_j x^j + v^i t + a^i, \quad p_i \mapsto R_i{}^j p_j + m v_i,$$

and the corresponding orbit carries a $G$-Hamiltonian symplectic structure.

4. Projective Representations and Quantum Realization

Quantum mechanics mandates projective representations of the Galilei group, which are equivalent to unitary representations of the Bargmann group. The generator action in the Schrödinger representation $(m > 0)$ on $L^2(\mathbb{R}^3)$ is

$$\begin{aligned} (P_i \psi)(x) &= -i \partial_i \psi(x), \ (H \psi)(x) &= -\tfrac{1}{2m}\Delta \psi(x), \ (K_i \psi)(x) &= m x_i \psi(x) - i t \partial_i \psi(x), \ (J_i \psi)(x) &= -i \epsilon_{ijk} x^j \partial^k \psi(x). \end{aligned}$$

These operators fulfill the centrally extended algebra, ensuring the projective phase introduced by Galilean boosts in the path integral formalism (Torre, 2023).

The explicit boost operator is

$$[U(v) \psi](x, t) = \exp\left[\tfrac{i}{\hbar}(m v \cdot x - \tfrac{1}{2} m |v|^2 t)\right] \psi(x - v t, t),$$

and the commutator $[K_i, P_j] = i \hbar m \delta_{ij}$ encapsulates the failure of the Galilei group to have linear representations on Hilbert space except via central extension (Torre, 2023).

5. Nonrelativistic Phase Space and Extended Symmetries

A broader symmetry structure emerges in the nonrelativistic phase-space formalism, where the phase space $P \cong \mathbb{R}^{2n+2}$ with symplectic form $$\omega = -dt \wedge d\varepsilon + \delta_{ij} dq^i \wedge dp^j$$ and degenerate metric $dt^2$ is preserved by the extended Jacobi group $HSp(2n) = Sp(2n) \ltimes H(n)$ (Low, 2021). Here, $H(n)$ is the (real) Weyl–Heisenberg group. The explicit action is

$$\begin{aligned} q'^j &= q^j + v^j t, \ p'^j &= p^j + f^j t, \ \varepsilon' &= \varepsilon + v \cdot p - f \cdot q + r t, \ t' &= t, \end{aligned}$$

with nonvanishing commutator $[V_i, F_j] = \delta_{ij} R$ in the Heisenberg ideal. This recovers the familiar Galilean sector for $v$ and $q$ while extending to include "force" generators and noninertial symmetries (Low et al., 2011).

The inhomogeneous Hamilton group $\mathrm{IHa}(n)$, with its central extension $\mathcal{Q}\mathrm{Ha}(n)$, unifies rotations, boosts, spatial and momentum translations, and time-energy transformations, providing the maximal local symmetry of Hamilton's equations in the nonrelativistic regime (Low et al., 2011).

6. Conformal, Superconformal, and Higher-Order Symmetry Extensions

The nonrelativistic symmetry group admits conformal and supersymmetric extensions. The $N$-Galilean Conformal Algebra (NGCA) generalizes the Galilei group to higher order, with generators for time translation, dilatations, special conformal transformations, spatial translations, and higher-order boosts. In $d$ dimensions, the algebra admits a central extension for $N$ odd (all $N$ for $d=2$), with nonvanishing commutators such as

$$[C_i^a, C_j^b] = i\,\delta^{ab}\,\delta_{i+j,N} (-1)^{\frac{N-i}{2}} \frac{i!j!}{N!} M,$$

yielding invariant higher-derivative free Lagrangians and a complete set of conserved Noether charges (Batlle et al., 2018, Andrzejewski et al., 2013).

Schrödinger (nonrelativistic conformal) symmetry arises geometrically as a contraction or null reduction from higher-dimensional Minkowski space, with mass as a central generator. The maximal symmetry is the centrally extended Schrödinger group, which in the case of spin-$1/2$ admits supersymmetric extension via the orthosymplectic algebra $\mathfrak{osp}(1|1)$, combining bosonic and fermionic charges and underpinning super-Schrödinger invariance (0807.0513).

7. Contrast with Relativistic (Poincaré) Symmetry

The Poincaré algebra in $3+1$ dimensions admits no nontrivial central extension; mass appears only as part of the quadratic Casimir $C_1^{P} = P_\mu P^\mu$ (Kong et al., 2021). The additional central mass generator in the nonrelativistic limit arises in the $c \to \infty$ contraction, providing a sharp demarcation between the algebraic structure of nonrelativistic and relativistic symmetry groups. The inability to fully accommodate interactions within the strict Galilean symmetry (with time translation as a generator) leads to the exclusion of $H$ from the fundamental nonrelativistic quantum symmetry in some formalizations, restoring freedom for interactions via the enveloping algebra (Kong, 2022).

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The nonrelativistic symmetry-group formalism is fundamental to the structure of classical and quantum Newtonian theory. Its formulation via central extensions, coadjoint orbits, projective representations, and higher symmetry generalizations continues to inform research in mathematical physics, representation theory, and geometric quantization (Kong et al., 2021, Torre, 2023, Low, 2021, Kong, 2022, Batlle et al., 2018, Low et al., 2011, Andrzejewski et al., 2013, 0807.0513).