dorsal/arxiv
View SchemaRelationships Between Quantum and Classical Mechanics using the Representation Theory of the Heisenberg Group
| Authors | Alastair Brodlie |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0412015 |
| URL | https://arxiv.org/abs/quant-ph/0412015 |
Abstract
This thesis is concerned with the representation theory of the Heisenberg group and its applications to both classical and quantum mechanics. We continue the development of $p$-mechanics which is a consistent physical theory capable of describing both classical and quantum mechanics simultaneously. $p$-Mechanics starts from the observation that the one dimensional representations of the Heisenberg group play the same role in classical mechanics which the infinite dimensional representations play in quantum mechanics. In this thesis we introduce the idea of states to $p$-mechanics. $p$-Mechanical states come in two forms: elements of a Hilbert space and integration kernels. In developing $p$-mechanical states we show that quantum probability amplitudes can be obtained using solely functions/distributions on the Heisenberg group. This theory is applied to the examples of the forced, harmonic and coupled oscillators. In doing so we show that both the quantum and classical dynamics of these systems can be derived from the same source. Also using $p$-mechanics we simplify some of the current quantum mechanical calculations. We also analyse the role of both linear and non-linear canonical transformations in $p$-mechanics. We enhance a method derived by Moshinsky for studying the passage of canonical transformations from classical to quantum mechanics. The Kepler/Coulomb problem is also examined in the $p$-mechanical context. In analysing this problem we show some limitations of the current $p$-mechanical approach. We then use Klauder's coherent states to generate a Hilbert space which is particularly useful for the Kepler/Coulomb problem.
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"abstract": "This thesis is concerned with the representation theory of the Heisenberg\ngroup and its applications to both classical and quantum mechanics. We continue\nthe development of $p$-mechanics which is a consistent physical theory capable\nof describing both classical and quantum mechanics simultaneously.\n$p$-Mechanics starts from the observation that the one dimensional\nrepresentations of the Heisenberg group play the same role in classical\nmechanics which the infinite dimensional representations play in quantum\nmechanics.\n In this thesis we introduce the idea of states to $p$-mechanics.\n$p$-Mechanical states come in two forms: elements of a Hilbert space and\nintegration kernels. In developing $p$-mechanical states we show that quantum\nprobability amplitudes can be obtained using solely functions/distributions on\nthe Heisenberg group. This theory is applied to the examples of the forced,\nharmonic and coupled oscillators. In doing so we show that both the quantum and\nclassical dynamics of these systems can be derived from the same source. Also\nusing $p$-mechanics we simplify some of the current quantum mechanical\ncalculations.\n We also analyse the role of both linear and non-linear canonical\ntransformations in $p$-mechanics. We enhance a method derived by Moshinsky for\nstudying the passage of canonical transformations from classical to quantum\nmechanics. The Kepler/Coulomb problem is also examined in the $p$-mechanical\ncontext. In analysing this problem we show some limitations of the current\n$p$-mechanical approach. We then use Klauder\u0027s coherent states to generate a\nHilbert space which is particularly useful for the Kepler/Coulomb problem.",
"arxiv_id": "quant-ph/0412015",
"authors": [
"Alastair Brodlie"
],
"categories": [
"quant-ph"
],
"title": "Relationships Between Quantum and Classical Mechanics using the Representation Theory of the Heisenberg Group",
"url": "https://arxiv.org/abs/quant-ph/0412015"
},
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