Primary constraints for Hamiltonian field theories 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
$\chi(p,q)=0$
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.
 A: When considering Hamiltonian formalism, it is important to distinguish between the following two frameworks: 


*

*Point mechanics (PM). Variables$^1$: $q^i(t)$ and $p_j(t)$. The Hamiltonian $H$ depends on the following arguments: $$\tag{1} H(q(t);p(t);t).$$

*Field theory (FT) in $d+1$ spacetime dimensions. Variables$^1$: $\phi^{\alpha}(x,t)$ and $\pi_{\beta}(x,t)$. The Hamiltonian density ${\cal H}$ depends on the following arguments:
$${\cal H}\left(\phi(x,t), \partial \phi(x,t),\partial^2\phi(x,t),\ldots,\partial^N\phi(x,t) ;\right. $$ 
$$\tag{2}\left. \pi(x,t) , \partial \pi(x,t),\partial^2\pi(x,t),\ldots,\partial^N\pi(x,t)  ;x,t\right).$$
where $\partial$ denotes spatial (as opposed to temporal) derivative. Here $N$ is finite for a local FT, and $N\leq 1$ for a relativistic FT. 
PM is the $d=0$ case of FT; while FT can be viewed as PM if we treat spatial coordinates as a continuous index $i=(\alpha,x)$, cf. DeWitt's condensed notation.
FT always has infinitely many degrees of freedom (DOF), while PM could have finitely or infinitely many DOFs.
In the Legendre transformation/Dirac-Bergmann procedure for FT, the spatial derivatives (unlike the temporal derivatives) have no special status/role. Equivalently, the spatial derivatives are passive spectators. 
In FT the definition of primary constraints carries over from the PM case without modification. In particular, the presence of spatial derivatives doesn't alter the status of an equation as a constraint or not.
--
$^1$ Note that in the case of constraints, the variables (besides dynamical variables) include auxiliary variables as well.
