Semiclassics: Quantum in form, Classical in spirit

Semiclassics is a wonderful regime where things look quantum but behave classically. What this precisely means I elaborate on in the document by bringing up a result from the paper by Dirac that would later inspire Feynmann. It can be found here.

The point of this post is to demonstrate that semiclassics doesn’t need any appeal to quantum dynamics, only quantum ontology. It is of course important though that regular quantum mechanics deforms into semiclassics which it does via \hbar \rightarrow 0. I demonstrate this deformation via the 2 classical formulations of quantum mechanics and then I present Dirac’s beautiful result where we quantize just the ontology, leaving the evolution classical. The demonstrations are independent so you can skip straight to Dirac if you want just the excitement.

In any case, the ultimate result is that we need only consider Hamilton’s principal function from Hamilton-Jacobi theory to understand what is happening semiclassically. The even more amazing thing is this shows that quantum mechanics is not much more than geometric optics of wavefunctions given the well-known correspondence between the principal function and optical path length.

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