# Hamiltonian formulation of *classical* field theory

1. Mar 22, 2010

### DrFaustus

Hi,

I was looking for a book that would explain classical field theory in a Hamiltonian setting. What I mean by this is that there be no *actions* around, no *Lagrangians* and *Legendre transforms* to define the Hamiltonian and so on. What I'm looking fo is an exposition of (classical) field theory that starts with the Hamiltonian and the Poisson bracket, from which one can obviously derive the equations of motion and everything else. In this setting I can then *quantize* the theory in the usual way. And if the exposition is mathematically (semi-)rigorous, even better!

Can anyone suggest a reference? Review paper, book or similar?

(Mods, sorry if the post is not in the appropriate forum. It seemed the most appropriate to me as I'm interested in the quantization of the theory.)

2. Mar 22, 2010

### muppet

I know Goldstein has a section on classical field theory, but I don't know whether or not it develops it in the way you suggest; I'd guess not however.

Can I ask why you're so averse to obtaining it via the Legendre transformation of the Lagrangian?

3. Mar 22, 2010

### DrFaustus

muppet -> Because of "intellectual elegance". I need to write up something about canonical (Hamiltonian) quantization and it's conceptually much more elegant if I could avoid talking about Lagrangians and actions. Much like in ordinary QM where the Lagrangian doesn't enter till you want to talk about path integrals. Not to mention that the Legendre transform is not really well defined if the Lagrangian contains interaction terms with derivatives. I could put what I know together and a physicists would be happy about it, but mathematical rigour is a bit of an issue, so I wanted some reference on the subject. Thanks for the suggetions tho', I'll check it out.

4. Mar 22, 2010

### muppet

I see. The trouble is that it's defined as the Legendre transformation of the Lagrangian. In the setting of classical mechanics, the Legendre transformation only works out to be
$$T-V \rightarrow T+V$$
when the kinetic energy is a quadratic form in the velocity. If the hamiltonian were defined as the energy operator, then the Schroedinger equation $$H\Psi=E\Psi$$ would be trivial; it's because the energy operator can be defined as the generator of infinitesmal time translations, and the hamiltonian as the legendre transformation of the lagrangian, that it isn't.

Also, if you want to be careful about how you do things, I'd suggest having a look at chapter 7 of vol.1 of Weinberg's Quantum theory of fields; he's careful to emphasise some subtleties a lot of authors suppress for expedience.

5. Mar 22, 2010

### samalkhaiat

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Phase Space, Rep. Math. Phys. 41 (1998) 49-90, hep-th/9709229.
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regards

sam

Last edited: Mar 22, 2010
6. Mar 22, 2010

### muppet

Well, now I feel inadequate :tongue:

7. Mar 23, 2010

### DrFaustus

samalkhaiat -> Wow, I'm aboslutely impressed. You've given me more references than I could hope for! Thanks A LOT!!!

8. Mar 28, 2010

### Fredrik

Staff Emeritus
Last edited by a moderator: Apr 24, 2017