Primary constraints for Hamiltonian field theories

[Bobhawke]

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 $$\chi (q's, p's)=0$$ 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.

 

[dextercioby]

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 $\chi(q,p)\simeq 0$.

 

[Bobhawke]

The crux of the primary constraint is that the 2n phase space variables are not all independent because there are one or more relations between them, given by the definition of the momenta. What I'm saying is that the definition of the momenta in this case, though independent of the time derivatives of the generalised coordinates, so that the Legendre transform is not invertible, are dependent on the spatial derivatives of the generalised coordinates. Therefore the definition of the momentum may be viewed as determining the spatial derivatives of the generalised coordinates in terms of independent phase space variables. In other words, there would still be no redundancy in the phase space variables, and therefore this should not be considered a primary constraint.

 

[dextercioby]

I don't get what you claim. Actually, the theory is independent of whether there are spatial derivatives of the variables or not. Think about the free relativistic particle with or without einbein. It's a 1D problem.

 

[Bobhawke]

As a simple example, consider a theory where the conjugate momentum is defined by $$\pi_i = \partial_i \phi$$ where $$\phi$$ is the field variable, and $$\partial_i$$ is a spatial derivative only. By your criteria, this is a primary constraint because it does not depend on the time derivative of $$\phi$$ But it can't be; it is not a relation between phase space variables. Primary constraints should define a constraint surface in the phase space; $$\pi_i - \partial_i \phi = 0$$ does not do that. What it does do is fix the spatial derivative of $$\phi$$ as a function of $$\pi_i$$ but the phase space variables are all still independent of one another. Thus it is not a primary constraint.

 

[dextercioby]

I'm sure you have read other sources than mine. By your criterion, the Dirac field has no constraints, yet it does. The dynamical system has primary constraints each time the Hessian matrix is degenerate (thing which would automatically prevent passing from the tangent bundle to the cotangent one by performing the Legendre transformation).

 

[Bobhawke]

I think you're greatly misunderstanding what I'm saying.

By my criterion, which is in fact the criterion given by Dirac in "Lectures on quantum mechanics", the Dirac field does have primary constraints. Dirac's definition of a primary constraint is a relation given by the definition of the momenta that reduces the number of independent degrees of freedom in the phase space.

Your definition of a primary constraint is when you have a velocity variable which is not invertible for the corresponding momentum. Now, in most cases this definition and Dirac's agree, but I am telling you a situation in which they do not.

The situation is when you have a momentum variable in a field theory which does not depend on any velocities, but does depend on spatial derivatives of the generalised coordinates. By your definition, this is a primary constraint. By Dirac's definition it is not, because it is not an equation which reduces the number of independent degrees of freedom in the phase space. This is because it is not a function of the form $$\chi(q,p)=0$$ It is a function of the form $$\psi(q,p,\partial q) =0$$ where the derivative is a spatial derivative only. That is, it is an equation which determines the spatial derivatives of q in terms of the other variables, and therefore does not reduce the number of independent degrees of freedom in the phase space.

I think that's about as clearly as I can explain it.

 

[dextercioby]

For the life of me I can't find a proper Lagrangian without constraints which would have π_i−∂_iϕ=0 coming from the momenta's definition.

 

[Bobhawke]

Ah yes, I think this may be the source of the confusion. This question arose in the context of the theory described in section 2 of the following paper: http://arxiv.org/abs/1309.1660. It is a very little known theory that is a "pure gauge" theory that is equivalent to the first order formalism for general relativity. Constraints of the type I have been discussing occur in this theory.