dorsal/arxiv
View SchemaObservables and States p-Mechanics
| Authors | Alastair Brodlie, Vladimir V. Kisil |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0304023 |
| URL | https://arxiv.org/abs/quant-ph/0304023 |
| Journal | Advances in Mathematics Research, V, Nova Sci., 2003, pp. 101-136. |
Abstract
This is an up-to-date survey of the p-mechanical construction (see funct-an/9405002, quant-ph/9610016, math-ph/0007030, quant-ph/0212101, quant-ph/0303142), which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics. Observables in p-mechanics are defined to be convolution operators on the Heisenberg group H^n. Under irreducible representations of H^n the p-observables generate corresponding observables in classical and quantum mechanics. p-States are defined as positive linear functionals on p-observables. It is shown that both states and observables can be realised as certain sets of functions/distributions on the Heisenberg group. The dynamical equations for both p-observables and p-states are provided. The construction is illustrated by the forced and unforced harmonic oscillators. Connections with the contextual interpretation of quantum mechanics are discussed. Keywords: Classical mechanics, quantum mechanics, Moyal brackets, Poisson brackets, commutator, Heisenberg group, orbit method, deformation quantisation, symplectic group, representation theory, metaplectic representation, Berezin quantisation, Weyl quantisation, Segal--Bargmann--Fock space, coherent states, wavelet transform, Liouville equation, contextual interpretation, interaction picture, forced harmonic oscillator.
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"abstract": "This is an up-to-date survey of the p-mechanical construction (see\nfunct-an/9405002, quant-ph/9610016, math-ph/0007030, quant-ph/0212101,\nquant-ph/0303142), which is a consistent physical theory suitable for a\nsimultaneous description of classical and quantum mechanics. Observables in\np-mechanics are defined to be convolution operators on the Heisenberg group\nH^n. Under irreducible representations of H^n the p-observables generate\ncorresponding observables in classical and quantum mechanics. p-States are\ndefined as positive linear functionals on p-observables. It is shown that both\nstates and observables can be realised as certain sets of\nfunctions/distributions on the Heisenberg group. The dynamical equations for\nboth p-observables and p-states are provided. The construction is illustrated\nby the forced and unforced harmonic oscillators. Connections with the\ncontextual interpretation of quantum mechanics are discussed. Keywords:\nClassical mechanics, quantum mechanics, Moyal brackets, Poisson brackets,\ncommutator, Heisenberg group, orbit method, deformation quantisation,\nsymplectic group, representation theory, metaplectic representation, Berezin\nquantisation, Weyl quantisation, Segal--Bargmann--Fock space, coherent states,\nwavelet transform, Liouville equation, contextual interpretation, interaction\npicture, forced harmonic oscillator.",
"arxiv_id": "quant-ph/0304023",
"authors": [
"Alastair Brodlie",
"Vladimir V. Kisil"
],
"categories": [
"quant-ph"
],
"journal_ref": "Advances in Mathematics Research, V, Nova Sci., 2003, pp. 101-136.",
"title": "Observables and States p-Mechanics",
"url": "https://arxiv.org/abs/quant-ph/0304023"
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