# Resources on constrained Hamiltonian field theory

I apologise in advance if the title is not clear, but I don't know the exact name of the subject I'm looking for.

I started reading Dirac's "Lectures on Quantum Mechanics", and it sparked my interest on the formalism of the generalised Hamiltonian, when the equations for the momenta lead to constraint equations of the form $\phi(p,q) = 0$. It did a very good job of explaining this "constraint" formalism and the Dirac bracket.

However, I am having trouble when it goes on to the generalisation in field theory. Indeed, all the derivatives are changed by functional derivatives, and I am not very comfortable with those yet. So I would like to ask if there exists other books, which may go into more detail about this "generalised hamiltonian formalism", especially in the field theory part. I wasn't able to find any resources mainly due to the fact that I don't know the name of this formalism.

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I can suggest some reference that I found. I run into this topic when I try to understand light-cone quantization.

1, Some old books: K. Sundermeyer, Constrained Dynamics (Springer, Heidelberg, 1982). It stared with basic definitions, regular and singular system. Then move onto more advanced topics, like Yang-mills. Easy to follow.

A. Hanson, T. Regge, and C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale dei Lincei, Rome, 1976). I haven't read this one, the book strike me like as a survey. The author try to make connection with applications.

I also found a relatively new book, Classical and Quantum Dynamics of Constrained Hamiltonian Systems by Heinz J. Rothe and Klaus D. Rothe. It published 2010, so it's easy to read. Also, they put a lot of examples in the book. 2, Papers: If you read the first book, the author would mention these papers. P. A. M. Dirac, Can. J. Math. 2, 129 (1950)

P. Bergmann, Helv. Phys. Acta Suppl. 4, 79 (1956)

J. Anderson and P. Bergmann, Phys. Rev. 83, 1018 (1951)

Personally, I recommend the first book and the third book. If you have some foundation and ask for the treatment of a specific topic, go to second one.

• Hi Haowu, please take a moment to read the resource-recommendation policy. Currently your answer doesn't satisfy the requirements, which may cause it to be deleted. – Kyle Kanos Feb 27 at 1:47

I have some vague recollections of Dirac's book as discussing some "slightly weaker" or "more relaxed" version of quantum mechanics involving constraints. But I have a hard time recalling the definition of constraints in that book, or the algorithm to generate them. So I am uncertain if the algorithm used today to discover constraints in a given system is "identical" to Dirac's system.

• Hamilton's Formalism for Systems with Constraints by Andreas W. Wipf gives a summary of the formalism, using Chern-Simons theory and Yang-Mills theory (both field theories) as in-depth examples.
• Lectures on Constrained Systems by Ghanashyam Date provides a good summary of the classical description of constrained systems. Section 3 provides a succinct description of the algorithm where we discover constraints in a given classical system described by the Lagrangian $$L$$. It motivates the entire discussion with analysing electromagnetism in section 2.
• Quantization of Gauge Systems by Henneaux and Teitelboim. This book is considered a Bible on the subject of constraints in gauge systems. Specifically the first 5 chapters discusses classical aspects to Hamiltonian analysis.

(The first 5 chapters develop the formalism of constraints, it's entirely classical (as I recollect it), and completely general (in the sense that it works for either fields or mechanical systems).)

It's a dense book, where each sentence needs to be unpacked. Definitions are made in passing without fanfare (e.g., the "first class Hamiltonian" $$H'$$ is defined in Eq (1.27) but the term appears at the end of the next section, the last paragraph before section 1.2, without much fanfare or announcement; on the other hand, the term "first class Hamiltonian" never is used again).

Occasionally the authors were, to me, frustratingly cryptic. I found parts of chapter 3 hard to read without also consulting Henneaux's "Lectures on the antifield-BRST formalism for gauge theories" Nuclear Physics B - Proceedings Supplements Volume 18, Issue 1, December 1990, Pages 47-105

For simplicity, most of the examples in the book focuses on mechanical systems, since it's easier to do calculations with finitely many degrees of freedom than with field quantities.