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American Journal of Physics

Addendum to “Note on the Inequivalence of Classical and Quantum Hamiltonians”

Frederick James Kennedy, Edward H. Kerner

Published Mar 1, 1966• 4 citations
Publisher
DOI
10.1119/1.1972904

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Note on the Inequivalence of Classical and Quantum Hamiltonians

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By working directly from Newtonian equations of motion and representing acceleration ẍ as [(x,H), H], families of Hamiltonians may be constructed which are functions of the energy and which, while describing the same classical motions, are not connected by canonical transformations. One dimensional examples are given, and the formal rules of quantization executed when H is (energy)12 to show that the quantum motion does not come out right. The reason is that the commutation rule (x,p) = iħ does not have universal standing; quantum theory in employing this rule is apparently requiring not merely that H be a generator of motion, but also that it be a very particular one, viz., the energy when the system is conservative.

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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.

arXiv:gr-qc/9904042•Apr 19, 1999•1 citation
By working directly from Newtonian equations of motion and representing acceleration ẍ as [(x,H), H], families of Hamiltonians may be constructed which are functions of the energy and which, while describing the same classical motions, are not connected by canonical transformations. One dimensional examples are given, and the formal rules of quantization executed when H is (energy)12 to show that the quantum motion does not come out right. The reason is that the commutation rule (x,p) = iħ does not have universal standing; quantum theory in employing this rule is apparently requiring not merely that H be a generator of motion, but also that it be a very particular one, viz., the energy when the system is conservative.
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.