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Acta Physica Sinica (Overseas Edition)

A proposal of quantization of systems without standard lagrangians

Dehai Bao, Zhongyuan Zhu

Published Jun 1, 1996• 2 citations
Publisher

Abstract

For many classical systems which disobey Helmholtz conditions and thus possess no standard Lagrangian formulation on velocity phase space TQ , ordinary quantization procedures cannot be directly applied. By shifting to an enlarged space TTQ and viewing the systems as typically constrained ones, we propose a quantization method which seems to be practicable as illustrated by the examples.

DOI
10.1088/1004-423x/5/6/001

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For many classical systems which disobey Helmholtz conditions and thus possess no standard Lagrangian formulation on velocity phase space TQ , ordinary quantization procedures cannot be directly applied. By shifting to an enlarged space TTQ and viewing the systems as typically constrained ones, we propose a quantization method which seems to be practicable as illustrated by the examples.
If higher order wave equations are reduced to first order by supplementary conditions, then quadratic Hamiltonians of the most general type will emerge. Even then, for several reasons elaborated below, canonical quantization can still not be applied consistently. The benefit of clinging to a Lagrangian description of non-localized phenomena is then questionable. Non canonical quantization procedures should be considered. Some initial attempts at Lie-admissible quantization are examined critically here, and it is concluded that further recourse to experiment seems to be necessary.
Some one-dimensional dissipative systems can be found to admit canonical-quantization procedure, contrary to opinions expressed occassionally in the literature. Also, a natural way of quantizing an equation $\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{q}+\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}(q)=0$ for a function V(q)V(q)V(q) is found to use a nonstandard commutation relation $[q,\ensuremath{\pi}]=i\ensuremath{\hbar}\ensuremath{\pi}$ together with Hamiltonian $H=\ensuremath{\pi}+V(q)$. Here, $\ensuremath{\pi}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}$ stands for the velocity operator rather than the momentum operator. Finally, a quantization problem of a Newtonian geodesic equation of motion in nonflat space is discussed.
Here we use the extended phase space formulation of quantum statistical mechanics proposed in an earlier work to define an extended lagrangian for Wigner's functions (WFs). The extended action defined by this lagrangian is a function of ordinary phase space variables. The reality condition of WFs is employed to quantize the extended action. The energy quantization is obtained as a direct consequence of the quantized action. The technique is applied to find the energy states of harmonic oscillator, particle in the box, and hydrogen atom as the illustrative examples.