Apparently nice duality between classical and quantum mechanics first noticed by Dirac. As a graduate student of mathematics I believe such a wonderful similarity in their mathematics have a deep root and of course is not accidental. But I'm not so expert in physics that I could find and explain this root.

I also added a comment below to somehow clarifying question.

  • $\begingroup$ Making quantisation (the procedure of going from a classical system to a quantum system) mathematically rigorous is actually a rich field in ongoing mathematical research. See e.g. ncatlab.org/nlab/show/deformation+quantization) $\endgroup$ May 20 '20 at 8:38
  • $\begingroup$ Good point from Nat and @Chiral about ambiguity of question. my main concern is the word "root" which appeared as "really" in the title. concerning "work" is related but is not main concern. a simple analogy: "Really" why both motion of planets and motion of comets in solar system could be formulated wonderfully in similar framework of Newtonian gravity? what is the root of this unified formulation? my expected answer: CONICS! "both orbits are special case of conics". then: conics are quadratic curves. Newtonian quadratic formula for gravity have all conics in itself! then... $\endgroup$
    – moshtaba
    May 20 '20 at 20:16
  • $\begingroup$ then ... make Newton formula repulsive to get hyperbolas! and both physics and math get something from this search of the "root" @Chiral. $\endgroup$
    – moshtaba
    May 20 '20 at 20:21

...why does promoting numerical variables to operators neatly work?

The question seems to be asking about quantization, a recipe for constructing a quantum model based on a given classical model. Why/when does it work? That depends on what you mean by "work." I'll illustrate this using examples from quantum field theory.

"Work" = good classical approximation?

Sometimes the motive is to construct a quantum model that has a given classical approximation. For example, the quantum theory of the electromagnetic field is well-approximated by the classical theory, under the right conditions, and "quantizing" the classical theory gives us the quantum theory.

Does this always work? No! In fact, it frequently doesn't work. A famous example is quantum chromodynamics (QCD). Although QCD is constructed using quantization, the classical model (lagrangian) from which we started is not a good approximation to the quantum model under any circumstances (as far as I know).

"Work" = preserves all symmetries?

Sometimes the motive for using quantization is completely different. Sometimes the goal is to construct a quantum model that has the same symmetries as a given classical model, even if the classical model isn't a good approximation to the quantum one.

Does this always work? No again! When it doesn't work, physicists call it an anomaly. "Anomaly" is an overloaded word, so sometimes physicists call it quantum symmetry breaking.

One of the most important examples of quantum symmetry breaking is the scale anomaly, and one of the most famous examples of this is Yang-Mills theory, which is QCD without fermions. The classical version of Yang-Mills theory (which is not a good approximation) has scale symmetry: it looks the same at all scales. But when we apply the quantization recipe, the scale symmetry is lost in a spectacular way: instead of a scale-invariant spectrum of massless particles (gluons), we get a spectrum of exclusively massive particles (glueballs). The non-zero masses do not respect scale symmetry. This phenomenon is not limited to Yang-Mills theory. In fact, the quantum breaking of scale symmetry is a central theme in all of quantum field theory. The keywords renormalization group will lead you to an endless supply of literature about this.

Another prominent example is the chiral anomaly, and there are many others. The closely related subject of 't Hooft anomalies is an active area of research today. It's a deep and beautiful subject (that's a euphemism for "I don't completely understand it yet"), one that a graduate student of mathematics might find very appealing.

"Work" = preserves Poincaré symmetry?

The fate of Poincaré symmetry is different. The full Poincaré group, including reflections, can be anomalous, but as far as I know the connected component that includes the identity operator is never anomalous. As far as I know, quantization always preserves that part of the Poincaré group, if it was present in the original classical model.$^\dagger$ In this restricted sense, quantization always seems to "work." At least, I think it always does... but I'm not sure, and apparently I'm not the only Physics SE user who isn't sure:

$^\dagger$ Caveat: the usual way to define a quantum field theory non-perturbatively is to treat spacetime as a discrete lattice, which clearly ruins Poincaré symmetry, but Poincaré symmetry is restored in the continuum limit, so this isn't an "anomaly".

References for related sub-topics

The subject of quantization is vast and includes many interesting sub-topics. These two books give a wealth of insight about the subject:

  • Henneaux and Teitelboim (1992), Quantization of Gauge Systems, Princeton University Press

  • DeWitt (2003), The Global Approach to Quantum Field Theory (2 volumes), Oxford University Press


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