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I was wondering if there are any other important classical/quantum analogs, along the lines of these examples:

  • Schrödinger Equation $\leftrightarrow$ Hamilton-Jacobi Formalism
  • Path Integrals $\leftrightarrow$ Lagrangian Formalism
  • Heisenberg Picture $\leftrightarrow$ Hamiltonian Formalism
  • Commutators $\leftrightarrow$ Poisson brackets

Unfortunately, there don't seem to be any easily searchable lists on Google. To clarify, by correspond I mean equivalent in the classical limit, as $\hbar \rightarrow 0$. For example:

  • As $\hbar \rightarrow 0$, the S.E. reduces to the H-J equation and a continuity equation.
  • As $\hbar \rightarrow 0$, all paths besides the path of least action cancel out.
  • As $\hbar \rightarrow 0$, the commutator reduces to the Poisson bracket as detailed here http://www.stat.ucla.edu/~ywu/Commutator.pdf, which also shows the correspondence between the Hamiltonian Formalism and the Heisenberg picture.
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    $\begingroup$ The issue here is that you are trying to paint this a lot neater than things actually are. Your arrows here don't really mean the same thing in each cases, beyond some vague, superficial notion of 'corresponding'. Quantization and taking the classical limit are not, in general, straightforward processes. This is why we can't simply draw up a list $\endgroup$
    – By Symmetry
    Commented Mar 28, 2022 at 16:59
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    $\begingroup$ I have edited my post to specify precisely what I mean by correspond, and to show that $\leftrightarrow$ means the exact same thing in each case. $\endgroup$
    – SSD
    Commented Mar 28, 2022 at 17:40
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    $\begingroup$ I think there are too many connections between quantum mechanics and classical mechanics to list. Some important ones that come to mind that aren't on your list are the WKB approximation and how it connects with old quantum theory, coherent states, and the large number-of-particles limit. $\endgroup$
    – Andrew
    Commented Mar 28, 2022 at 19:43
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    $\begingroup$ Also I don't know much about this, but there is a way to formulate classical mechanics in a Hilbert space formalism. en.wikipedia.org/wiki/… $\endgroup$
    – Andrew
    Commented Mar 28, 2022 at 19:43
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    $\begingroup$ Also, decoherence $\endgroup$
    – Andrew
    Commented Mar 28, 2022 at 19:44

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