Primary constraints for Hamiltonian field theories

In summary, the conversation discusses the determination of primary constraints in the construction of a generalised Hamiltonian for a field theory, based on Dirac's definition. The question is whether a momentum variable that depends only on spatial derivatives of the generalised coordinates is a primary constraint. There are conflicting thoughts on this, with some sources stating that a primary constraint occurs when the momentum is not invertible for the corresponding velocity, while Dirac defines it as a function of the momenta that reduces the number of independent phase space variables. The conversation also touches on the example of a Lagrangian system with no constraints, and the confusion that can arise in determining primary constraints in certain theories.
  • #1
Bobhawke
144
0
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 [tex] \chi (q's, p's)=0 [/tex] 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.
 
Physics news on Phys.org
  • #2
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 [itex] \chi(q,p)\simeq 0[/itex].
 
  • #3
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.
 
  • #4
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.
 
  • #5
As a simple example, consider a theory where the conjugate momentum is defined by [tex] \pi_i = \partial_i \phi [/tex] where [tex] \phi [/tex] is the field variable, and [tex] \partial_i [/tex] is a spatial derivative only. By your criteria, this is a primary constraint because it does not depend on the time derivative of [tex] \phi [/tex] 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; [tex] \pi_i - \partial_i \phi = 0 [/tex] does not do that. What it does do is fix the spatial derivative of [tex] \phi [/tex] as a function of [tex] \pi_i [/tex] but the phase space variables are all still independent of one another. Thus it is not a primary constraint.
 
Last edited:
  • #6
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).
 
  • #7
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 [tex] \chi(q,p)=0 [/tex] It is a function of the form [tex] \psi(q,p,\partial q) =0 [/tex] 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.
 
Last edited:
  • #8
For the life of me I can't find a proper Lagrangian without constraints which would have πi−∂iϕ=0 coming from the momenta's definition.
 
  • #9
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.
 

1. What are primary constraints for Hamiltonian field theories?

Primary constraints are conditions that must be satisfied in order for the Hamiltonian formulation of a field theory to be valid. They arise from the requirement that the Poisson brackets of the constraints with the Hamiltonian must be zero.

2. How are primary constraints different from secondary constraints?

Primary constraints are the initial set of constraints that arise from the Hamiltonian formulation, while secondary constraints are constraints that arise from the consistency conditions of the primary constraints. Secondary constraints are derived from primary constraints and can be used to further restrict the system.

3. What is the role of primary constraints in Hamiltonian field theories?

Primary constraints play a crucial role in Hamiltonian field theories as they help to determine the number of degrees of freedom of the system. They also help to simplify the equations of motion by eliminating any unphysical degrees of freedom.

4. How are primary constraints treated in the Hamiltonian formulation?

In the Hamiltonian formulation, primary constraints are incorporated into the Hamiltonian equations through the use of Lagrange multipliers. These multipliers are added to the Hamiltonian to enforce the primary constraints and ensure that they are satisfied.

5. Can primary constraints be eliminated from the Hamiltonian formulation?

In some cases, it is possible to eliminate primary constraints from the Hamiltonian formulation by using the equations of motion. However, this is not always possible and in some cases, the primary constraints must be retained in order to maintain the consistency of the formulation.

Similar threads

Replies
4
Views
1K
  • Classical Physics
Replies
1
Views
1K
  • Classical Physics
Replies
33
Views
1K
  • Classical Physics
Replies
1
Views
622
Replies
6
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
375
  • Atomic and Condensed Matter
Replies
1
Views
515
  • Classical Physics
Replies
1
Views
568
  • Classical Physics
Replies
1
Views
942
Replies
2
Views
3K
Back
Top