Dirac's constraint dynamics

Tags:
1. Oct 27, 2015

HeavyWater

I am failing to understand Dirac's constraint dynamics. Can anyone suggest a useful resource?
Thanks

2. Oct 28, 2015

strangerep

Presumably you are already learning it from somewhere? If so, where? (It's a bit hard to help if you don't say where/what you're failing to understand...)

I originally learned this stuff from the Dirac bracket Wikipedia page, combined with Dirac's original Yeshiva lecture notes on QM.

Last edited: Oct 28, 2015
3. Oct 28, 2015

HeavyWater

An obvious question strangerep. Let me pull together the references for you later today.

4. Oct 28, 2015

Heinera

Last edited: Oct 28, 2015
5. Oct 28, 2015

strangerep

6. Oct 29, 2015

Heinera

Yes, now it does when I check it from a different browser.

Last time I clicked the link, it pointed to this book by Gordon Baym: https://books.google.no/books?id=1125sVZ2_GcC&redir_esc=y.

Weird. Maybe I have some plug-in in the other browser that substitutes links.

Edit: It seems that Physicsforums is using VigLink:

This is a script that will sometimes redirect links to another vendor, in my case from Amazon to a seller called alibris.com (and also a different product).

Last edited: Oct 29, 2015
7. Oct 29, 2015

Elemental

HeavyWater,

I recall the subject was hardly touched in graduate school, but Dirac and Bergmann did a great deal of work on constraint theory in the 1950s to develop a canonical description of general relativity. If that's your focus you can check out Dirac's paper at http://rspa.royalsocietypublishing.org/content/246/1246/333 (free PDF online).

Constraint theory has many other applications, and it is not too difficult to understand if you're familiar with canonical formalism. One usually starts with a Lagrangian or the corresponding Hamiltonian, identifies any dependencies between the coordinates, velocities, or momenta (whether arising naturally within the dynamics or imposed by fiat) and calls them primary constraints. Then one identifies whether the requirement that these constraints be maintained over times lead to additional constraints; if so, one calls them secondary constraints. Write all primary or secondary constraints in the form of equations equal to zero, multiplied by a factor (to be determined later). Add these functions multiplied by their undetermined multipliers to the Hamiltonian (which, since they're equal to zero, won't change its value). This is the extended Hamiltonian.

Now comes the trick. You can write Hamilton's equations of motion, like dA/dt = [A,H], using the extended Hamiltonian, where the brackets mean Poisson brackets for classical physics, commutator for the quantum physics version. Every term in the extended Hamiltonian generates a bracket term in the equation of motion, including the constraint terms and their coeeficients. However, the fact that those extra brackets contain a function (i.e., a constraint) which is equal to zero doesn't mean the entire bracket can just be eliminated (otherwise, why did you add it to the Hamiltonian in the first place?). One computes the brackets anyway, thus generating new terms in the equations of motion, and only imposes the constraint (that the constraint relation equals zero) after all brackets are computed. One says that the constraints are only weakly equal to zero (usually denoted with an approximately equal sign). In quantum physics, constraints are imposed only on the states, not the operators.

But before proceeding to the equations of motion, a simplification process is performed. Calculate the Poisson bracket of every known constraint with every other known constraint, and also with the original Hamiltonian (the version which does not have constraints added). Each constraint whose bracket vanishes weakly with all other constraints and the unconstrained Hamiltonian is relabeled a first class constraint. If this process generates new constraints, they are relabeled second class constraints. If a contradiction is found, the original Lagrangian or Hamiltonian is invalid. If there are second class constraints, one then tries to form linear combinations of them which will weakly Poisson commute with each other and the Hamiltonian; if they do, they are relabeled first class constraints. Any remaining second class constraints are in pairs and represent unphysical degrees of freedom; therefore, their coordinates are eliminated from the description of the system, and their corresponding constraints are eliminated from the extended Hamiltonian. The result is the Dirac Hamiltonian. All equations of motion can then be written in the form of brackets with the Dirac Hamiltonian. The Dirac Hamiltonian will contain first class constraints, multiplied by an undetermined coefficient. This undetermined coefficient is a gauge freedom and will appear in the equations of motion. The gauge may be fixed in various ways (whole new subject).

I apologize if this is too long, but it is interesting and neglected. For instance, one can treat time as an operator in quantum mechanics using constraints, which has applications in relativity. The result is a Hamiltonian which vanishes (the problem of time) but nevertheless generates motion (but that's another whole new subject).

Elemental

Last edited: Oct 29, 2015
8. Oct 29, 2015

strangerep

Heh, it's not "too long" at all. I also find it interesting, and I agree it's "neglected". The details involved in quantizing non-trivial classical systems can be quite subtle.

9. Oct 30, 2015

A. Neumaier

10. Nov 1, 2015

HeavyWater

Thanks to all of you. You went way above and beyond. Now I need to spend some time with your suggested readings. I can tell you, I have looked at Diracs papers and book...and looked...and looked. He is way beyond my level of understanding.

Thanks to each of you.

11. Nov 1, 2015

stevendaryl

Staff Emeritus
Ah. I had noticed the effect before, but didn't know the cause.