Question

What is the relation between path integral methods for dealing with constraints (constrained Hamiltonian dynamics involving non-singular Lagrangian) and Dirac's method of dealing with such systems (which involves the Dirac bracket)?

And what is the advantage/relative significance of each method?

Answer 1

Well, when canonically quantizing a system with constraints, you have two methods:

Dirac's approach "Quantize, then Constrain";
Reduced Phase Space approach "Constrain, then Quantize".

Although these two approaches have analogs with path integral quantization, the Path integral approach sweeps a lot of problems under the rug when you pick a particular gauge (a la Fadeev-Poppov quantization).

That's why the path integral approach is usually taught in quantum field theory courses: it's a straightforward recipe with few subtleties. The Canonical approach requires a bit more work.

I am aware, in quantum gravity at least, that you can recover the Dirac quantized constraints from taking the functional derivative of the path integral with respect to the Lagrange multipliers, and demanding it vanish. So I suspect there is a way to recover the Dirac quantized version from the Path integral approach.

This is unique to General Relativity, due to the inclusion of time. A formal derivation may be found in Hartle and Hawking's "Wave Function of the Universe" (Physical Review D 28 12 (1983) pp. 2960