Canonical Quantization - Issues and Limitations

Issues and Limitations

Dirac's book details his popular rule of supplanting Poisson brackets by commutators:

This rule is not as simple or well-defined as it appears. It is ambiguous when products of classical observables are involved which correspond to noncommuting products of the analog operators, and fails in polynomials of sufficiently high order.

For example, the reader is encouraged to check the following pair of equalities invented by Groenewold, assuming only the commutation relation :

The right-hand-side "anomaly" term −3ħ2 is not predicted by application of the above naive quantization rule. In order to make this procedure more rigorous, one might hope to take an axiomatic approach to the problem. If Q represents the quantization map that acts on functions f in classical phase space, then the following properties are usually considered desirable:

1. and (elementary position/momentum operators)
2. is a linear map
3. (Poisson bracket)
4. (von Neumann rule)

However, not only are these four properties mutually inconsistent, any three of them is also inconsistent! As it turns out, the only pairs of these properties that lead to self-consistent, nontrivial solutions are 2+3 and possibly 1+3 or 1+4. Accepting properties 1+2 along with a weaker condition that 3 be true only asymptotically in the limit ħ→0 (see Moyal bracket) is deformation quantization, and some extraneous information must be provided, as in the standard theories utilized in most of physics. Accepting properties 1+2+3 but restricting the space of quantizable observables to exclude terms such as the cubic ones in the above example amounts to geometric quantization.