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arXiv:hep-th/0212321 · Journal of Physics A Mathematical and General

Lagrangian versus quantization

Ciprian Acatrinei

Published Jan 9, 2004• 6 citations
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Abstract

We discuss examples of systems which can be quantized consistently, although they do not admit a Lagrangian description.
DOI
10.1088/0305-4470/37/4/010

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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.