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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.

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1 Answer 1

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When considering Hamiltonian formalism, it is important to distinguish between the following two frameworks:

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

  2. 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.

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So, using your notation, in the field theory case if I had a momentum given by $\pi = \partial \phi$, would that be a primary constraint? I am inclined to answer no, because it seems to me this simply determines the variable $\partial \phi$ in terms of $\pi$, and does not reduce the number of independent variables in the phase space. –  Steven Oct 6 '13 at 14:46
    
Moreover, if my example is indeed a primary constraint, then how does one go about imposing it? The primary constraints should define a constraint surface in the phase space, but $\pi - \partial \phi =0$ does not do that. –  Steven Oct 6 '13 at 16:42
    
@Steven: Yes e.g. the model ${\cal L}=\dot{\phi} \partial \phi$ leads to the primary constraint $\pi = \partial \phi$. What FT are you looking at? What is the Lagrangian density ${\cal L}$? –  Qmechanic Oct 6 '13 at 16:49
    
The theory that I'm actually dealing with can be found in section 2 of the following paper: arxiv.org/abs/1309.1660. I am working with the four dimensional version of this theory (things are qualitatively different for dimensions $\leq$ 3). It is a 'pure gauge' theory that is equivalent to the first order form of general relativity. The momentum conjugate to the $\phi$ and the spatial part of $\omega$ (spin connection) variables involve spatial derivatives of $\phi$ and $\omega$, but do not depend on the corresponding velocities. –  Steven Oct 6 '13 at 17:02
    
To save you some time, the momentum conjugate to $\phi^a$ for example is given by $\pi_a = 2 (D_j \phi)^b F_{kl}^{cd} \epsilon_{abcd} \epsilon^{jkl}$. The indices $i,j,k=1,2,3$ are "spacetime" indices that are restricted to run over the space part. $D$ is the covariant derivative for the gauge group, which is $ISO(4)$, F is the field strength tensor, and the indices $a,b,c,d...=1,2,3,4$ are in a particular representation of the four dimensional Euclidean group $ISO(4)$. –  Steven Oct 6 '13 at 17:10

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