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Can someone give a simple physical example of first class and second class constraints? I mean, if you were giving a classical mechanics lecture for undergraduates, how would you explain this concept to them and which simple example would you use?

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1) Well, constrained Hamiltonian dynamics is a huge subject. Quoting Wikipedia:

A dynamical quantity is a 1st class constraint is if its Poisson bracket with all the other constraints vanishes on the constraint surface (the surface implicitly defined by the simultaneous vanishing of all the constraints). A 2nd class constraint is one that is not first class.

I assume the reader already has seen these definitions and is more interested in knowing exactly why these concepts are useful?

If time permits, one could go through some simple physical relevant cases, such as e.g. the Hamiltonian formulation of E&M where Gauss's law is a 1st class constraint.

2) However, if time is limited, then it might be more instructive to consider the following slightly artificial model to get the basic idea without a lot of computation.

2a) 2nd class constraints. Take an arbitrary Hamiltonian theory $T$ with a phase space $(M,\omega)$ (say without any constraints for simplicity) and a Hamiltonian $H$. Now artificially introduce two extra variables $q$ and $p$ with Poisson bracket

$$\tag{1} \{q,p\}_{PB}~=~1,$$

and impose the 2nd class constraints

$$\tag{2} q~\approx~ 0~ \approx~ p.$$

In other words, the new extended phase space is $M\times \mathbb{R}^2 $. Now, for all practical purposes, the new extended theory $\tilde{T}$ is effectively the same theory as the original theory $T$. But when we quantize (1), we get

$$\tag{3} [\hat{q},\hat{p}]~=~i\hbar{\bf 1}.$$

However the CCR (3) is inconsistent with the 2nd class constraints (2). The resolution is to use the Dirac bracket instead $$\tag{4} \{q,p\}_{DB}~=~0.$$ When we quantize (4), we get $$\tag{5} [\hat{q},\hat{p}]~=~0$$ as we should.

2b) 1st class constraints. Consider the same situation with the 2nd class constraints (2) replaced with a single 1st class constraint

$$\tag{6} q~\approx~ 0.$$

Now the shift

$$\tag{7} p~\longrightarrow~ p+c$$

is a gauge-symmetry generated by the 1st class constraint (6). We should gauge-fix this gauge-symmetry (7). We can e.g. chose the gauge-fixing condition

$$ \tag{8} p~\approx~p_0, $$

where $p_0$ is a constant. Again we effectively returned to the original theory $T$.

3) In more general situations, the phase space will not factorize in a simple way as in Section 2. They could be imbedded in a complicated way into the extended phase space. Instead we would have to develop a more geometric theory for how to treat 1st and 2nd constraints and the Dirac bracket. In the case of gauge symmetry, we should show that the theory does not depend on the choice of gauge-fixing conditions.

References:

  1. P.A.M. Dirac, Lectures on QM, (1964).

  2. A. Das, Lectures on QFT, (2008); chapter 10.

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  • $\begingroup$ Actually, I have learned the formalism after asking the question by going through Dirac's book. Another good reference that helped me a lot was "Lecture Notes on Quantum Field Theory" by Ashok Das. There is a chapter there explaining the formalism and giving simple examples. It is very helpful. Thanks for your answer by the way. $\endgroup$ – sahin Mar 11 '15 at 10:39

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