Suppose you want to solve a linear partial differential equation of the form , which determines some quantity in terms of its source . Here x could stand for possibly many variables, and the differential operator can be pretty much anything. This is a very general type of problem, not even specific to physics. An example in physics could be the Klein-Gordon equation, or with some more bells and whistles the Maxwell equation, which determines the electric and magnetic fields.
Let us replace this problem with the following equivalent one. If we find a function such that:
with the initial condition , and assuming the regularity condition , then it is easy to see that the function satisfies the original equation we set out to solve.
Now, this new equation for looks kind of familiar, if we are willing to overlook a few details. If we wish, we can think about as a time dependent wave function, with the parameter s playing the role of time. The equation for could then be interpreted as a Schrödinger equation, with the original operator playing the role of the Hamiltonian. We are ignoring a few issues to do with convergence, analytic continuation, and the related fact that the Schrödinger equation is complex, and the one we are discussing is not. Never mind, these are subtleties which need to be considered at a later stage.
The point is that we can now use any technique we learned in quantum mechanics to solve the original equation – path integral, canonical quantization, you name it. We can talk about the states and the Hilbert space they form, Fourier transform to get another basis for that Hilbert space, even discuss “time” evolution (that is, the dependence of various states on the auxiliary parameter s). We can get the state by summing over all paths of a “particle” with an appropriate worldline action and boundary conditions. Depending on the problem, we may be interested in various (differential) operators acting on , and they of course do not commute, resulting in uncertainty relations. You get the picture.
This technique is sometimes called first quantization, or Schwinger proper time method, or heat Kernel expansion. Whatever you call it, it has a priori nothing to do with quantum mechanics, there are no probabilities, Planck constant or any wavefunctions in any real sense. At this point we may be discussing the financial markets, population dynamics of bacteria, or simply classical field theory.
In the second pass, we can apply this idea to linear fields, generating solutions to various linear differential equations. Some of those equations are Lorentz invariant (Klein-Gordon, Dirac, Maxwell equations), but they have nothing to do with quantum mechanics, despite the original way they were referred to as “relativistic wave equations”. Once we add spin to the game, we start having the fascinating structures of (worldline) fermions and supersymmetry (not to be confused with spacetime fermions and supersymmetry), and we are also in a good shape to make the leap from classical field theory to classical string theory. Maybe I’ll get to that sometime…