I am currently trying to carry out the construction of the generalised Hamiltoni

Question

I am currently trying to carry out the construction of the generalised Hamiltonian, constraints and constraint algebra, etc for a particular field theory following the procedure in Dirac's "Lectures on quantum mechanics". My question is the following: I have momentum variables that depend on the spatial derivatives of the generalised coordinates, but not on the time derivatives of the generalised coordinates. Is this a primary constraint or not? I have conflicting thoughts on this. On the one hand, there are texts that say a primary constraint occurs when the definition of a momentum variable is not invertible for the corresponding velocity. By this criteria, I do have a primary constraint because the momentum does not depend on the time derivative of the generalised coordinates. On the other hand, Dirac for example says that a primary constraint is a function of the form [Math Processing Error] that comes from the definition of the momenta. This is not the case for me, since I have a function that also depends on the spatial derivatives of the q's. By this criteria, I don't have a primary constraint. Any help much appreciated.

Explanation / Answer

If you can't revert the Legendre transformation, you have a primary constraint. For Lagrangian systems linear in derivatives (such as the Dirac field), there's no 'time derivative' in the momentum definition, so that such systems are automatically degenerate. The primary constraints are ?(q,p)?0.

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