dorsal/arxiv
View SchemaCanonically Relativistic Quantum Mechanics: Representations of the Unitary Semidirect Heisenberg Group, U(1,3) *s H(1,3)
| Authors | Stephen G. Low |
|---|---|
| Categories | |
| ArXiv ID | physics/9703008 |
| URL | https://arxiv.org/abs/physics/9703008 |
| DOI | 10.1063/1.531968 |
| Journal | J.Math.Phys. 38 (1997) 2197-2209 |
Abstract
Born proposed a unification of special relativity and quantum mechanics that placed position, time, energy and momentum on equal footing through a reciprocity principle and extended the usual position-time and energy-momentum line elements to this space by combining them through a new fundamental constant. Requiring also invariance of the symplectic metric yields U(1,3) as the invariance group, the inhomogeneous counterpart of which is the canonically relativistic group CR(1,3) = U(1,3) *s H(1,3) where H(1,3) is the Heisenberg Group in 4 dimensions and "*s" is the semidirect product. This is the counterpart in this theory of the Poincare group and reduces in the appropriate limit to the expected special relativity and classical Hamiltonian mechanics transformation equations. This group has the Poincare group as a subgroup and is intrinsically quantum with the Position, Time, Energy and Momentum operators satisfying the Heisenberg algebra. The representations of the algebra are studied and Casimir invariants are computed. Like the Poincare group, it has a little group for a ("massive") rest frame and a null frame. The former is U(3) which clearly contains SU(3) and the latter is Os(2) which contains SU(2)*U(1).
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"abstract": "Born proposed a unification of special relativity and quantum mechanics that\nplaced position, time, energy and momentum on equal footing through a\nreciprocity principle and extended the usual position-time and energy-momentum\nline elements to this space by combining them through a new fundamental\nconstant. Requiring also invariance of the symplectic metric yields U(1,3) as\nthe invariance group, the inhomogeneous counterpart of which is the canonically\nrelativistic group CR(1,3) = U(1,3) *s H(1,3) where H(1,3) is the Heisenberg\nGroup in 4 dimensions and \"*s\" is the semidirect product. This is the\ncounterpart in this theory of the Poincare group and reduces in the appropriate\nlimit to the expected special relativity and classical Hamiltonian mechanics\ntransformation equations. This group has the Poincare group as a subgroup and\nis intrinsically quantum with the Position, Time, Energy and Momentum operators\nsatisfying the Heisenberg algebra. The representations of the algebra are\nstudied and Casimir invariants are computed. Like the Poincare group, it has a\nlittle group for a (\"massive\") rest frame and a null frame. The former is U(3)\nwhich clearly contains SU(3) and the latter is Os(2) which contains SU(2)*U(1).",
"arxiv_id": "physics/9703008",
"authors": [
"Stephen G. Low"
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"doi": "10.1063/1.531968",
"journal_ref": "J.Math.Phys. 38 (1997) 2197-2209",
"title": "Canonically Relativistic Quantum Mechanics: Representations of the Unitary Semidirect Heisenberg Group, U(1,3) *s H(1,3)",
"url": "https://arxiv.org/abs/physics/9703008"
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