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(Qualifier: I know virtually nothing about quantum mechanics)

In classical physics, Newton's laws guarantee that any physically relevant quantity is a function of the position and momentum of the particles in a system studied. Given a function $a(x,\xi)$ of position and momentum, quantizing gives a psuedodifferential operator $a(X,D)$ which somehow generalizes the function $a$ to operate on the wave functions of particles. Does this idea resulting from Newton's laws generalize to quantum mechanics, i.e. do we expect any physically meaningful quantity in quantum mechanics to be represented by a pseudodifferential operator? Are there principles in quantum mechanics that describe this property?

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    $\begingroup$ This is appropriate for Physics StackExchange (and the question needs more clarification), or opening a book on QM looking for the buzzword "observables". $\endgroup$
    – Chris Gerig
    Commented Sep 22, 2020 at 19:33
  • $\begingroup$ Quantum mechanics is far from a well-defined mathematical theory, where such questions could be answered... $\endgroup$
    – Francois Ziegler
    Commented Sep 22, 2020 at 20:24
  • $\begingroup$ If I were particularly prone to point out irony, I might suppose that the OP used the word "quantize" without realizing its origin. But perhaps a more charitable interpretation of the question is this: Do all quantum observables (physically relevant quantities) arise by quantization? The answer may be both Yes and No, depending on how strict one wants to be with the terminology. $\endgroup$
    – Igor Khavkine
    Commented Sep 22, 2020 at 21:06
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    $\begingroup$ I’m voting to close this question because it belongs on physics.se $\endgroup$
    – user44191
    Commented Sep 22, 2020 at 21:10
  • $\begingroup$ @IgorKhavkine I admit I know much more about harmonic analysis than quantum mechanics (I'm coming at the theory more as intuition for understanding psuedodifferential operators) :P. My understanding is that quantization is when one generalizes classical measurements to measurements in quantum mechanics. Where might I find discussion on whether observables arise by quantization? $\endgroup$
    – Jacob Denson
    Commented Sep 22, 2020 at 21:25

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The spin degree of freedom is generally not represented by pseudodifferential operators (though such a representation can be constructed a posteriori, as pointed out by Francois Ziegler in comments). What matters ultimately are representation-independent statements, i.e., the operator algebras. I don't know whether any physically relevant algebra could in principle be represented using pseudodifferential operators, but spin is a ready example where such a representation at least does not initially arise from the underlying physics.

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    $\begingroup$ This I feel is really misleading. The representations of $\mathrm{SU}(2)$ describing spin arise by geometric quantization in holomorphic sections of a line bundle over the classical 2-sphere (Borel-Weil theory), and there the components of spin are represented by differential operators in a very non-“contrived” way. $\endgroup$
    – Francois Ziegler
    Commented Sep 23, 2020 at 0:26
  • $\begingroup$ @FrancoisZiegler - thank you for pointing out the construction. I'll edit and dispose of "contrived". What I was trying to express is that representations of the spin operators weren't originally arrived at by "quantizing" some classical coordinates using differential operators. The representation through differential operators is constructed a posteriori, and in practice seldom invoked in physics applications. $\endgroup$
    – Michael Engelhardt
    Commented Sep 23, 2020 at 0:58
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    $\begingroup$ Historically you are quite right. For the longest time the claim was made that spin had no “classical counterpart” of which it would be the “quantization”. Same for $\mathrm{SU}(n)$ with isospin, hypercharge and other baryonic quantities. The development of symplectic geometry has now put us well past that stage. $\endgroup$
    – Francois Ziegler
    Commented Sep 23, 2020 at 1:07

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