Let $M=\mathbb{R}^{2n}$ be the phase space with standard Poisson bracket on smooth functions on $M$. Fix a classical hamiltonian $h$ (function on $M$) and function $f$ generating symmetry of $h$ i.e. $$ \{h,f\}_{PB}=0. $$ If we choose an ordering prescription we can quantize $f$ and $h$ and get operators $F$ and $H$, but a priori we can't expect that $$ [H,F]=0. $$ If we choose Weyl ordering, then equivalently we can say that Moyal bracket of $h$ and $f$ is not necessarily zero. In other words, Moyal bracket has Poisson bracket as leading term, but also has higher quantum corrections $\{h,f\}_{MB}=\{h,f\}_{PB}+O(\hbar^2)$ and vanishing of the leading term does not imply vanishing of higher terms.

Are there any examples (of any interest) in quantum mechanics when this happens? Even more interesting would be a Lie algebra of classical symmetries that fail to be symmetries after quantization.

Of course, there is the following general naive question. Suppose there is a finite dimensional Lie subalgebra $g=\operatorname{span}(f_1, \ldots, f_n)$ with $\{f_i,f_j\}=\sum_k C^k_{ij} f_k$, of the Poisson algebra and we generate new Lie algebra $\tilde{g}$ by the same elements, but replacing Poisson bracket with Moyal bracket, then what is a relation between $g$ and $\tilde{g}$?

  • $\begingroup$ Well, there is lots of work by Hietarinta, us, etc, striving for the opposite: namely "quantizing h and f by adding suitable $\hbar$ corrections such that the corrected h and f MB-commute, whereas the uncorrected don't! In fact, this is a criterion of desirable quantizations, preserving the symmetries, not forfeiting them! Going from a Lie algebra to another Lie algebra upon quantization is rich, but it might be feasible: reversing a Wigner-Inonu Lie algebra contraction. $\endgroup$ – Cosmas Zachos Sep 11 at 22:27
  • $\begingroup$ Hietarinta 1984. $\endgroup$ – Cosmas Zachos Sep 11 at 22:33
  • $\begingroup$ Thanks! Interesting set of examples. But is it known when such desirable quantization exists? In other words, if $g$ is a Lie algebra of symmetries (with Hamiltonian in the center) when can we extend basis of such algebra to some basis of series in $\hbar$ that maps PB to MB. Is it some condition on $g$? If that were true for any finite dimensional $g$ would not that violate Groenewold-Van Hove no-go theorem? $\endgroup$ – Alex Sep 12 at 7:28
  • $\begingroup$ I know of no general results. Our paper linked achieves that heuristically. Open problems are like that, and quantization problems are largely open, and as we emphasize, are driven by symmetry preservation. The G-vN theorem, qua theorem, is not violated by a finite set of PBs: it tells you that not all PBs are mappable to MBs. $\endgroup$ – Cosmas Zachos Sep 12 at 15:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.