Lévy-Leblond’s Taxonomy of Non-Relativistic Spinors

• Lévy-Leblond’s Taxonomy is a systematic classification framework for non-relativistic spinors using first-order differential equations that act as square roots of the Schrödinger equation.
• It distinguishes spinor types by employing real, complex, and quaternionic division-algebra structures and by enforcing chirality and reality constraints.
• The framework extends to include potential-dependent interactions, leading to operator realizations of the osp(1|2) superalgebra and superconformal structures.

Lévy-Leblond’s taxonomy is a systematic classification of non-relativistic spinors, building in direct analogy to the taxonomy of relativistic spinors that distinguishes Dirac, Weyl, Majorana, Majorana-Weyl, and Quaternionic types. Based on the formalism of Lévy-Leblond differential equations (LLEs)—first-order operator equations that serve as "square roots" of the Schrödinger equation in $(1+d)$ space-time dimensions—the taxonomy partitions LLE spinors by real, complex, and quaternionic division-algebra structures and by the imposition of reality and/or chirality constraints. This classification determines, for each spinor dimension $n=2^k$, the possible Galilean-invariant non-relativistic first-order equations and their algebraic properties, including a precise map to the allowable number of spatial dimensions. The scheme is extended to encompass interactions via the systematic inclusion of prepotential-dependent terms, and for certain conformal potentials, gives rise to operator realizations of the $\osp(1|2)$ superalgebra (Miranda et al., 2024).

1. Lévy-Leblond Equations in $(1+d)$ Dimensions

Let $\Psi(t,\mathbf{x})$ be an $n$-component spinor, with natural units $\hbar=1$ and $m=1/2$. The free LLE in $(1+d)$ dimensions is

$$Q\otimes \mathbb{I}_r\,\Psi(t,\mathbf{x}) = \sum_{i=1}^d \Gamma^i\,\partial_{x^i}\,\Psi(t,\mathbf{x}),$$

where $n=2^k$0 is a $n=2^k$1 "time-square-root" block satisfying $n=2^k$2, and $n=2^k$3 are real $n=2^k$4 gamma-matrices built from the $n=2^k$5 matrices

$n=2^k$6

generating Clifford algebras $n=2^k$7 through tensor products—an approach labeled the "alphabetic presentation." The consistency condition $n=2^k$8 guarantees that all solutions of the LLE also solve the Schrödinger equation $n=2^k$9 with $\osp(1|2)$0. In this presentation, spatial $\osp(1|2)$1 are realized by selecting distinct "words" in the four-letter alphabet, with the new $\osp(1|2)$2 matrix functioning analogously to $\osp(1|2)$3 in the relativistic Dirac framework.

2. Division-Algebra Structures and Constraints

The division-algebraic structure of an LLE spinor is determined by the set of matrices commuting with all $\osp(1|2)$4, per Schur’s lemma:

• Real (Majorana): Only real scalars $\osp(1|2)$5 commute.
• Complex (Dirac): Matrices $\osp(1|2)$6 with $\osp(1|2)$7.
• Quaternionic: Linear combinations $\osp(1|2)$8 with $\osp(1|2)$9.

Chirality (Weyl) is realized if every $(1+d)$0 is block-antidiagonal; in the alphabetic formalism, each letter must start with $(1+d)$1 or $(1+d)$2. The imposition of both Majorana (reality) and Weyl (chirality) yields the Majorana–Weyl class.

3. Classification Table: Types, Dimensions, and Structure

For spinor dimensions corresponding to $(1+d)$3 ($(1+d)$4), Lévy-Leblond’s taxonomy yields the following inequivalent classes. The classification is parallel to the relativistic case, with allowable $(1+d)$5 set by the structure of the Clifford algebra built from alphabetic words. Table: classes, dimensions, and spinor types up to $(1+d)$6:

$(1+d)$7	Type	$(1+d)$8	Real dim. of spinor
$(1+d)$9	M	$\Psi(t,\mathbf{x})$0	$\Psi(t,\mathbf{x})$1
$\Psi(t,\mathbf{x})$2	M	$\Psi(t,\mathbf{x})$3	$\Psi(t,\mathbf{x})$4
$\Psi(t,\mathbf{x})$5	MW	$\Psi(t,\mathbf{x})$6	$\Psi(t,\mathbf{x})$7
$\Psi(t,\mathbf{x})$8	M	$\Psi(t,\mathbf{x})$9	$n$0
$n$1	MW	$n$2	$n$3
$n$4	D	$n$5	$n$6
$n$7	M	$n$8	$n$9
$\hbar=1$0	MW	$\hbar=1$1	$\hbar=1$2
$\hbar=1$3	D	$\hbar=1$4	$\hbar=1$5
$\hbar=1$6	W	$\hbar=1$7	$\hbar=1$8
$\hbar=1$9	H	$m=1/2$0	$m=1/2$1

Type legend: M = Majorana; W = Weyl (complex); D = Dirac (complex); MW = Majorana–Weyl; H = Quaternionic. The "real dimension" column specifies the real degrees of freedom.

4. Inclusion of Potential Terms

Potential terms are incorporated in the LLE framework by augmenting the spatial differential operators with a "pre-potential" $m=1/2$2. In $m=1/2$3 dimensions, the formalism requires advancing to $m=1/2$4 matrices. The generalized LLE becomes: $m=1/2$5 This leads, at the component level, to a pair of coupled Schrödinger equations,

$m=1/2$6

with effective potentials $m=1/2$7 (Miranda et al., 2024).

5. Conformal Potential and $m=1/2$8 Superalgebra Realization

Specializing to a conformal prepotential $m=1/2$9, the LLE operator takes the form

$(1+d)$0

Here, $(1+d)$1 squares to a generalized conformal Hamiltonian, and, together with operators $(1+d)$2, $(1+d)$3, and the second supercharge $(1+d)$4 (constructed from the Clifford basis plus auxiliary diagonals $(1+d)$5, $(1+d)$6), closes the $(1+d)$7 superalgebra. Notably, the system of (anti)commutators,

$(1+d)$8

along with $(1+d)$9 and related relations, constitutes an explicit differential-operator realization of $$Q\otimes \mathbb{I}_r\,\Psi(t,\mathbf{x}) = \sum_{i=1}^d \Gamma^i\,\partial_{x^i}\,\Psi(t,\mathbf{x}),$$0 in terms of first-order (in $$Q\otimes \mathbb{I}_r\,\Psi(t,\mathbf{x}) = \sum_{i=1}^d \Gamma^i\,\partial_{x^i}\,\Psi(t,\mathbf{x}),$$1) operators, signifying a novel superconformal structure for non-relativistic spinors (Miranda et al., 2024).

6. Synthesis and Structural Significance

Lévy-Leblond’s taxonomy fully extends the Atiyah–Bott–Shapiro framework known from relativistic spinor theory to the Galilean-invariant, non-relativistic context. The construction of free LLEs as irreducible solutions of

$$Q\otimes \mathbb{I}_r\,\Psi(t,\mathbf{x}) = \sum_{i=1}^d \Gamma^i\,\partial_{x^i}\,\Psi(t,\mathbf{x}),$$2

with $$Q\otimes \mathbb{I}_r\,\Psi(t,\mathbf{x}) = \sum_{i=1}^d \Gamma^i\,\partial_{x^i}\,\Psi(t,\mathbf{x}),$$3 and spatial $$Q\otimes \mathbb{I}_r\,\Psi(t,\mathbf{x}) = \sum_{i=1}^d \Gamma^i\,\partial_{x^i}\,\Psi(t,\mathbf{x}),$$4 from the tensor-alphabet, enforces a finite list of free-equation classes for each spinor size $$Q\otimes \mathbb{I}_r\,\Psi(t,\mathbf{x}) = \sum_{i=1}^d \Gamma^i\,\partial_{x^i}\,\Psi(t,\mathbf{x}),$$5. The inclusion of potential terms, and notably the conformal (inverse-square) case, reveals supersymmetric and superconformal algebraic structures, anchoring the classification in broader mathematical physics. This taxonomy, thus, rigorously charts the space of Galilean-invariant, first-order, non-relativistic spinor equations and their algebraic manifestations (Miranda et al., 2024).