Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

146 submissions , 123 unreviewed
3,953 questions , 1,403 unanswered
4,889 answers , 20,762 comments
1,470 users with positive rep
507 active unimported users
More ...

Classification of constraints in Hamiltonian formalism

+ 3 like - 0 dislike
118 views

I am reading this paper, where the claim is from hamiltonian formalism one can show that non-local theories of gravity are ghost-free. Before going to complication, I would like to see a clear definition (and mathematical formula of how to obtain the-) of primary constraint, secondary constraint, first class constraint and second class constraint. It would be appreciated to even point a toy example. 

Moreover, I wonder how one uses these constraint(s) to count the number of degrees of freedom? 

Also I wonder if one can uses such formalism and show that the given gravitational theory is bounded from below? 

asked Jul 21, 2016 in Theoretical Physics by Wiliam (65 points) [ revision history ]
recategorized Jul 22, 2016 by dimension10

1 Answer

+ 4 like - 0 dislike

In Hamiltonian formalism, the phase space is a symplectic manifold i.e. a manifold $M$ equipped with a 2-form $\omega$ which is non-degenerate and closed. In a constrained system, the dynamical variables are constrained to lie on a submanifold $N$ of $M$. If the restriction of $\omega$ to $N$ is 0, then $N$ is called a coisotropic submanifold and the constraint is called first class. If the restriction of $\omega$ to $N$ is non-degenerate, then $N$ is called a symplectic submanifold and the constraint is called second class. A general $N$ can always be locall written as a product $N_1 \times N_2$ with $N_1$ coisotropic (first class) and $N_2$ symplectic (second class).

Same thing in a slightly different language: the symplectic structure defines a Poisson bracket $\{,\}$ on the algebra of functions on $M$. The constraints are the equations $f_i=0$ defining $N$. The constraints are called first class if $\{f_i,f_j\}=0$ up to the constraints, for all $i$, $j$. The constraints are called second class if the matrix $\{f_i,f_j\}$ is non-degenerate. In general, up to a change of variables, one can write the matrix of Poisson brackets $\{f_i,f_j\}$ in a block diagonal form: the first block being zero and the second block being non-degenerate. Constraints corresponding to the first block are called first class, the ones corresponding to the second are called second class.

The distinction between primary and secondary constraints appears in the context of the Hamiltonian description of a system given in  Lagrangian form. Primary constraints are the constraints which are there if the momenta are not independent (if the matrix of second derivatives of the Lagrangian with respect to the velocities is not invertible). Secondary constraints are extra constraints which are imposed by consistency of the primary constraints with the equations of motion.

If by number of degrees of freedom, one means the dimension of the symplectic manifold where the Hamiltonian dynamics is well-defined without constraints, it is $dim M - 2 n_1-n_1$ where $n_1$ is the number of first class constraints and $n_2$ is the number of second class constraints. Indeed, for a second class constraint, one simply has to restrict ourselves to the constrained submanifold but for a first class constraint, one has to quotient this submanifold by redundancies of the description ("gauge transformations"), which are as numerous as the first class constraints (coisotropic reduction).

The original reference on first/second class constraints is Dirac:

https://books.google.co.uk/books?id=GVwzb1rZW9kC&redir_esc=y

which is still worth reading. A more recent and complete reference is the book by Henneaux and Teitelbaum, "Quantization of gauge systems":

http://press.princeton.edu/titles/5156.html

(one can easily find pdf for these two references but I am not sure of their legal status).

answered Jul 21, 2016 by 40227 (4,690 points) [ revision history ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$y$\varnothing$icsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...