I don't think there is one; non-relativistic spin-1/2 particles do not have any classical analogue that I'm aware of, as I said before. Note the word "particles", though; spin-1/2 fields can be written down classically, and you would indeed use Grassmann variables for a path integral involving spin-1/2 fields, as described, for example, in Zee's book Quantum Field Theory in a Nutshell. But doing this is inherently relativistic, in the sense that you are working with representations of the Poincare algebra; I'm not aware of any non-relativistic way to do it.which is the classical, non-relativistic Lagrangian, which results after quantization in the Pauli equation?
I don't have it. Do you know of any online reference?Look at Marcuse book
Very interesting! I was not aware of this result.This paper describes the general solutions of a wide variety of PDEs (including the wave equation) as path integrals.
Can you describe how this result is obtained? What I'm still having a problem with is that (as the paper you linked to shows for the particular example of the wave equation on ##\mathbb{E}^{(1, 3)}##) the wave equation is a spacetime equation--it relates the second derivative with respect to time to the second derivative(s) with respect to space. But the geometric optics Lagrangian is a Lagrangian in space; time does not appear. And, as I've said, its usage in geometric optics is to describe the spatial paths of light rays. So how can a path integral using this Lagrangian lead to a spacetime equation?Marcuse book gives what is this something
How can I do that if there's no time appearing anywhere?With the Lagrangian I have given to you, if you compute its Hamiltonian
Unless you mean ##x## to play the role of time?How can I do that if there's no time appearing anywhere?
What you look for is the path integral for a relativistic first-quantized particle. It can be found in some string theory textbooks, but also in the short paper attached here.Can the Klein-Gordon (and maybe the Dirac) equation be derived from the path integral quantization of a given classical (supposedly relativistic) Lagrangian of particles? If so, which Lagrangian?
Yes, that is the solution to the riddle. The Hamiltonian corresponds to the z coordinate (not time!), so the resulting equation is not the K-G (wave equation plus mass) but the Helmholtz equation (Laplace equation plus mass).Unless you mean ##x## to play the role of time?
Here's what I get for the Hamiltonian, if I rewrite your Lagrangian using ##t## instead of ##x## and ##q## instead of ##y## (with ##y'## becoming ##\dot{q}##), and using units where ##c = 1## to reduce clutter:that is the solution to the riddle
But you can't run this backwards; you can't derive the full time-dependent Schrodinger equation from the energy eigenvalue problem, and you can't derive the wave equation from the Helmholtz equation. You can only go in the other direction. By the logic you're giving here, the Lagrangian for a classical free particle should give us the Schrodinger energy eigenvalue problem, not the time-dependent Schrodinger Equation.the Helmholtz equation is just the wave equation once the time part of the wave equation is variable-separated (in the same way the Schrödinger equation becomes the energy eigenvalue problem).
Yes, and when I do that, I get (in the position representation) the operator ##\nabla^2 + n^2##. Which, as you have said, is the Helmholtz equation, not the K-G equation. There's no time derivative anywhere.Now square the Hamiltonian, and perform the usual substitutions (number to operator) for quantization.
Polyakov's Gauge Fields and Strings, chapter 9.11, seems to give an answer in (9.325): the Dirac equation can be obtained by quantizing a supersymmetric particle action. Really surprising (to me).What you look for is the path integral for a relativistic first-quantized particle. It can be found in some string theory textbooks, but also in the short paper attached here.
In principle all boson fields (Higgs, W, Z) can have classical states (coherent states) the same way the electromagnetic field does but I think they're just strongly suppressed by their masses. However the "classical field" of the pre-SSB Higgs field is DEFINITELY present and very measurable since it gives mass to fermions.Well, I need to find the equivalent of light rays (the classical Lagrangian which is quantized to result in classical electromagnetism) for KG and Dirac.