# Use of Dirac Bracket

1. Aug 19, 2011

### aries0152

I have searched in web and go through some papers. But the use of Dirac Bracket in constraint still unclear to me. It would be better if I have some examples.

2. Aug 19, 2011

### rkrsnan

What do you mean by constraint? Plz elaborate

Wikipedia article is reasonably good.

3. Aug 19, 2011

### dextercioby

The Dirac bracket provides the means to develop the Hamiltonian formalism for any Lagrangian system, for it generalizes the symplectic structure on the phase space. It can therefore serve as a tool in quantization of constrained systems.

There are a few books dealing with this subject that I know of, but I only reccomend Dirac's 1964 lectures and Henneaux & Teitelboim's <Quantization of Gauge Systems>. The latter contains an extended treatment of the BRST symmetry which is an alternative to the Dirac bracket for 1st class systems.

4. Aug 19, 2011

### inempty

You could see Weinberg I, chapter 7,8 to find examples. In simple words, the idea to quantize constrained systems is to do canonical transformations, making constraints the forms that just fix the values of several couples of conjugate variables, and to remove these couples. This method is equal to using Dirac brackets, of which details can be found in Weinberg I, and generally we'll use Dirac brackets. To addition, since poisson brackets could not be changed because of canonical transformations, 1st class constraints usually mean a lack of constraints to constrain both a variable and its conjugation, so we should add certain gauges as additional constraints.

5. Aug 20, 2011

### Simon_Tyler

A good paper to start on this topic is http://arxiv.org/abs/quant-ph/9606031
Then move onto the couple of text books on Dirac Quantization.

Here's the references used for a half-complete set of notes I wrote up on Dirac's canonical quantization.

[1] P. A. M. Dirac, “Lectures on Quantum Mechanics,” Belfer Graduate School of Science Monograph Series (1964)
[2] H. J. Matschull, “Dirac’s canonical quantization programme,” arXiv:quant-ph/9606031.
[3] K. Sundermeyer, “Constrained Dynamics With Applications To Yang-Mills Theory, General Relativity, Classical
Spin, Dual String Model,” Lect. Notes Phys. 169 (1982) 1.
[4] M. Henneaux and C. Teitelboim, “Quantization of gauge systems,” Princeton, USA: Univ. Pr. (1992) 520 p
[5] A. Hanson, T. Regge and C. Teitelboim, “Constrained Hamiltonian systems,” Accademia nazionale dei lincei
(1976)

Dirac's lecture notes [1] are a good read.