Relation between Dirac's generalized Hamiltonian dynamics method and path integral method to deal with constraints

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?

(Thanks to Ron Maimon for phrasing suggestion)

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1 Answer

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

1. Dirac's approach "Quantize, then Constrain";
2. 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. 2960–2975 eprint).

Halliwell and Hartle's "Wave functions constructed from an invariant sum over histories satisfy constraints" (Phys. Rev. D 43 (1991) pp. 1170–1194 eprint ) generalize this result for parametric systems.

Barvinsky shows in "Solution of quantum Dirac constraints via path integral" (arXiv:hep-th/9711164) that the path integral directly solves the quantum constraints, for a generic first-class constrained system at the level of one-loop ("semiclassical") approximations.

Klauder's "Path integrals, and classical and quantum constraints" eprint is a pedagogical review of quantizing constrained systems.

You might want to look at Henneaux and Teiteilboim's Quantization of Gauge Systems, specifically Chapter 16.

Addendum

The path integral using the Faddeev-Poppov method is completely equivalent to the following: suppose we want to change coordinates from just position $q$ to the gauge orbit $\Lambda$ plus the physically meaningful position $\bar{q}$. Then the functional integral changes as

$$\begin{align} \int \exp(I[q])\mathcal{D}q &= \int \exp(I[\bar{q}])\Delta_{fp}\,\mathcal{D}\bar{q}\mathcal{D}\Lambda\\ &=\int\mathcal{D}\Lambda\int\exp(I[\bar{q}])\Delta_{fp}\,\mathcal{D}\bar{q} \end{align}$$

We have the $\int\mathcal{D}\Lambda$ be infinite but trivial (it's the volume of the gauge orbit). The $\Delta_{fp}$ is the Faddeev-Poppov determinant.

This approach is discussed in detail in Emil Mottola's "Functional Integration Over Geometries" arXiv:hep-th/9502109.

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Thank you. So then, is it redundant to study Dirac's method, provided that I'm also going to do path integrals later? Actually I'm studying the former method as some sort of pre-requisite for BRST symmetry. Is even BRST superseded by some other more powerful formalism (just as Dirac constraint method is by path integral)? I'd appreciate an answer to these since I want to know whether it would be really fruitful to study what I'm studying. Thanks again... – 1989189198 Jun 24 '12 at 7:39

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It depends what you want to do. IMHO, you should study both the Dirac approach and the path integral approach. The Dirac approach can lead to mathematically impossible constraints (e.g., the Wheeler-DeWitt equation in quantum gravity); that's probably why the path integral approach is taught. I'd strong suggest/urge/demand reading Henneaux and Teitelboim's Quantization of Gauge Systems after whatever you're reading currently (it's a book focusing on BRST quantization, but discusses quite thoroughly Dirac and Reduced Phase Space approaches). – Alex Nelson Jun 24 '12 at 15:02
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