Here we argue that spinor structure arises naturally if relativistic statistical mechanics is formulated directly on phase spacetime. Requiring a first-order phase-spacetime description that retains both mass-shell branches leads to a Clifford factorisation of the relativistic constraint and hence to a $4\times4$ spinor-matrix distribution function. We show that deformation quantisation leads to a phase-space formulation of spin quantum mechanics. We argue that projection onto positive- and negative-energy sectors recovers the standard relativistic classical transport equations in the appropriate scalar limits, while the corresponding left- and right- stargenvalue equations reproduce the constraint structure of the Dirac-Wigner formulation. The result is a phase-space route from relativistic statistical mechanics to spinor quantum mechanics, in which spin algebra emerges as the internal structure required by any relativistic statistical theory containing both mass-shell branches and the dimensions of angular momentum from quantum non-locality.Here we argue that spinor structure arises naturally if relativistic statistical mechanics is formulated directly on phase spacetime. Requiring a first-order phase-spacetime description that retains both mass-shell branches leads to a Clifford factorisation of the relativistic constraint and hence to a $4\times4$ spinor-matrix distribution function. We show that deformation quantisation leads to a phase-space formulation of spin quantum mechanics. We argue that projection onto positive- and negative-energy sectors recovers the standard relativistic classical transport equations in the appropriate scalar limits, while the corresponding left- and right- stargenvalue equations reproduce the constraint structure of the Dirac-Wigner formulation. The result is a phase-space route from relativistic statistical mechanics to spinor quantum mechanics, in which spin algebra emerges as the internal structure required by any relativistic statistical theory containing both mass-shell branches and the dimensions of angular momentum from quantum non-locality.

arXiv:2505.03551May 6, 20250 citations

Algebraic and Quantum Mechanical Approach to Spinors

Tahir Manzoor, S. N. Hasan

Springer proceedings in mathematics & statisticsNov 15, 20230 citations

Algebra of the spinor invariants and the relativistic hydrogen atom

A. A. Eremko, Larissa S. Brizhik, В. М. Локтев

It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.

arXiv:2211.01857Annals of PhysicsMar 2, 20236 citations

Quantum Mechanics as a Relativistic Statistical Theory: A Mass-Shell Factorisation Approach

Mark Everitt

Spinor structure is usually introduced as part of relativistic quantum theory. We show that its underlying algebra can instead emerge without quantisation from a covariant statistical description of a massive relativistic particle subject to an explicit finite-dimensional linear factorisation of the complete massive mass-shell constraint motivated by requiring the theory retain both sheets of the massive mass shell. A finite-dimensional factorisation of the complete mass-shell constraint, linear in all four momentum components, forces a Clifford algebra. In (3+1) dimensions its minimal complex representation is four-dimensional, while each on-shell factor selects a rank-two sector; in this representation statistical completeness requires an arbitrary $2\times2$ matrix distribution within that sector before quantisation. Projecting the Weyl-ordered matrix Liouvillian gives relativistic transport while preserving all internal populations and coherences. For universal deformations on flat canonical phase space satisfying the stated covariance, matrix-neutrality, homogeneity and associativity assumptions, the resulting product is the matrix Weyl--Moyal product. Its parameter has dimensions of action and, when identified with $\hbar$ , supplies the scale of physical spin- $\tfrac12$ . Requiring all-order constraint preservation gives a vanishing star commutator, while preservation of the selected star-spectral sector yields the two-sided Dirac--Wigner equations. Our central conclusion is therefore that relativistic mass-shell factorisation supplies spinor structure, while completion of the constraint-preservation hierarchy within the stated deformation class leads to the Weyl--Moyal phase-space formulation of quantum mechanics.Spinor structure is usually introduced as part of relativistic quantum theory. We show that its underlying algebra can instead emerge without quantisation from a covariant statistical description of a massive relativistic particle subject to an explicit finite-dimensional linear factorisation of the complete massive mass-shell constraint motivated by requiring the theory retain both sheets of the massive mass shell. A finite-dimensional factorisation of the complete mass-shell constraint, linear in all four momentum components, forces a Clifford algebra. In (3+1) dimensions its minimal complex representation is four-dimensional, while each on-shell factor selects a rank-two sector; in this representation statistical completeness requires an arbitrary $2\times2$ matrix distribution within that sector before quantisation. Projecting the Weyl-ordered matrix Liouvillian gives relativistic transport while preserving all internal populations and coherences. For universal deformations on flat canonical phase space satisfying the stated covariance, matrix-neutrality, homogeneity and associativity assumptions, the resulting product is the matrix Weyl--Moyal product. Its parameter has dimensions of action and, when identified with $\hbar$, supplies the scale of physical spin-$\tfrac12$. Requiring all-order constraint preservation gives a vanishing star commutator, while preservation of the selected star-spectral sector yields the two-sided Dirac--Wigner equations. Our central conclusion is therefore that relativistic mass-shell factorisation supplies spinor structure, while completion of the constraint-preservation hierarchy within the stated deformation class leads to the Weyl--Moyal phase-space formulation of quantum mechanics.

quant-pharXiv:2608.29353Sep 1, 20260 citations

Aspects of relativistic quantum mechanics on phase space

S. Twareque Ali

Springer eBooksApr 6, 20082 citations

Relativistic Dynamical Groups in Quantum Theory and Some Possible Applications

P. Román

Springer eBooksJan 1, 19801 citation

4\times4

spinor-matrix distribution function. We show that deformation quantisation leads to a phase-space formulation of spin quantum mechanics. We argue that projection onto positive- and negative-energy sectors recovers the standard relativistic classical transport equations in the appropriate scalar limits, while the corresponding left- and right- stargenvalue equations reproduce the constraint structure of the Dirac-Wigner formulation. The result is a phase-space route from relativistic statistical mechanics to spinor quantum mechanics, in which spin algebra emerges as the internal structure required by any relativistic statistical theory containing both mass-shell branches and the dimensions of angular momentum from quantum non-locality.

2\times2

matrix distribution within that sector before quantisation. Projecting the Weyl-ordered matrix Liouvillian gives relativistic transport while preserving all internal populations and coherences. For universal deformations on flat canonical phase space satisfying the stated covariance, matrix-neutrality, homogeneity and associativity assumptions, the resulting product is the matrix Weyl--Moyal product. Its parameter has dimensions of action and, when identified with

\hbar

, supplies the scale of physical spin-

\tfrac12

. Requiring all-order constraint preservation gives a vanishing star commutator, while preservation of the selected star-spectral sector yields the two-sided Dirac--Wigner equations. Our central conclusion is therefore that relativistic mass-shell factorisation supplies spinor structure, while completion of the constraint-preservation hierarchy within the stated deformation class leads to the Weyl--Moyal phase-space formulation of quantum mechanics.