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arXiv:0807.3508

On a New Form of Quantum Mechanics

Natalia N. Gorobey, Alexander S. Lukyanenko

Published Jul 22, 2008• 0 citations
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Abstract

We propose a new form of nonrelativistic quantum mechanics which is based on a quantum version of the action principle.
DOI
10.48550/arxiv.0807.3508

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For the largest class of physical systems having a classical analog, a new rigorous, but not probabilistic, Lagrangian version of nonrelativistic quantum mechanics is given, in terms of a notion of regularized action function. As a consequence of the study of the symmetries of this action, an associated Nœther theorem is obtained. All the quantum symmetries resulting from the canonical quantization procedure follow in this way, as well as a number of symmetries which are new even for the case of the simplest systems. The method is based on the study of a corresponding Lie algebra and an analytical continuation in the time parameter of the probabilistic construction given in paper I of this work. Generically, the associated quantum first integrals are time dependent and the probabilistic model provides a natural interpretation of the new symmetries. Various examples illustrate the physical relevance of our results.