Quantum Mechanics as a Relativistic Statistical Theory: A Mass-Shell Factorisation Approach
Abstract
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 2×2 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 ℏ, supplies the scale of physical spin-1/2. 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.
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BibTeX
@misc{everitt2026quantum,
title = {{Quantum Mechanics as a Relativistic Statistical Theory: A Mass-Shell Factorisation Approach}},
author = {Mark Everitt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.29353},
eprint = {2608.29353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29353},
}
RIS
TY - UNPB
TI - Quantum Mechanics as a Relativistic Statistical Theory: A Mass-Shell Factorisation Approach
AU - Mark Everitt
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2608.29353
UR - https://arxiv.org/abs/2608.29353
AB - 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 2×2 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 ℏ, supplies the scale of physical spin-1/2. 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.
ER -