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Physical Review A

Canonical quantization of some dissipative systems and nonuniqueness of Lagrangians

Susumu Ôkubo

Published Jun 1, 1981• 26 citations
Publisher

Abstract

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 $V(q)$ 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.

DOI
10.1103/physreva.23.2776

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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 q\ifmmode\else¨\textasciidieresis\fi+V\ifmmode˙\else˙\fi(q)=0\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{q}+\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}(q)=0q\ifmmode\else¨​\textasciidieresis\fi​+V\ifmmode˙\else˙\fi​(q)=0 for a function V(q)V(q)V(q) is found to use a nonstandard commutation relation [q,\ensuremathπ]=i\ensuremathℏ\ensuremathπ[q,\ensuremath{\pi}]=i\ensuremath{\hbar}\ensuremath{\pi}[q,\ensuremathπ]=i\ensuremathℏ\ensuremathπ together with Hamiltonian H=\ensuremathπ+V(q)H=\ensuremath{\pi}+V(q)H=\ensuremathπ+V(q). Here, \ensuremathπ=q\ifmmode˙\else˙\fi\ensuremath{\pi}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}\ensuremathπ=q\ifmmode˙\else˙\fi​ 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.
The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system. But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.
Noncanonical quantization schemes have been proposed in quantum physics long time ago. Recalling some of them I associate these investigations to the quantum mechanical im-plications of the canonically inequivalent Hamilton formula-tions of classical Newtonian dynamics. As the simplest illus-tration of such alternative Hamiltonian descriptions I consider one dimensional system of two particles interacting harmoni-cally. Description of the motion of their center of mass is stan-dard and guarantees the Galilean covariance but the internal motion is described as a noncanonical oscillator for which the Hamiltonian and the total energy are not related by a simple proportionality. Alternative Hamiltonians lead to alternative Poisson structures which may be considered as classical ana-logues of the noncanonical quantum commutation rules. Gen-Concepts of Physics, Vol. I (2004) 11 erated alternative quantizations are consistent with the same Heisenberg equations of motion, i.e., they satisfy the Wigner principle of quantization. For the one dimensional harmonic oscillator the simplest noncanonical algebraic description is given in terms of the Lie algebra so(p, q), p+ q = 3. In such a case the Hamiltonian is still a linear function while the energy is a quadratic polynomial of the main quantum number. This leads to nonstandard termodynamical properties obeyed by such a noncanonical oscillator. The 3-dimensional harmonic oscillator also has a noncanonical description given in terms of the simple Lie algebra, so(p, q), p+ q = 5 in the case under consideration. The most important physical consequence of such a description is the noncommutativity of the coordinate operators, interpreted to be an implication of the noncommu-tative geometry of the underlying spacetime.
One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.