# Hamiltonian of weak gravitational field

1. Jun 19, 2015

### Andre' Quanta

In weak field regime i know that it is possible to quantize the gravitational field obtaining a quantum theory of free particles, called gravitons, which is very similar to the one for the electtromagnetic field.
Do you know some book in wiich i can study this theory?
In anycase what is the expression for the Hamiltonian of this theory in terms of creations and destructions operators?

2. Jun 19, 2015

### fzero

I'd suggest starting with Donoghue's lecture notes http://arxiv.org/abs/gr-qc/9512024. The first part of DeWitt's http://arxiv.org/abs/0711.2445, as well as the original papers by Feynman and DeWitt might be useful at some point too, but I think that the effective field theory point of view is important and those older references predate that whole perspective, so might not be the best pedagogy anymore.

3. Jun 19, 2015

### Andre' Quanta

I was interested in the Hamiltonian operator for the theory, is it the same as the one of the free electromagnetic field?

4. Jun 19, 2015

### fzero

The field operator in linearized gravity has spin 2 so it cannot be the same. I don't know off hand any place to find an explict expression for the Hamiltonian. It's best if you learn what the Lagrangian is, then you can see if it's easy to make a Legendre transformation to write a Hamiltonian. The Lagrangian is more useful for covariant QFT, though you might have some particular reason to study the Hamiltonian.

5. Jun 20, 2015

### Andre' Quanta

The weak gravitational field, the usual h mu nu, satisfies the simple D' Alemebert equation, so the lagrangian is the one of Klein-Gordon replacing the scalar field with the tensorial one in our theory: this will be valid also for the Hamiltonian so i would say the Hamiltonian that i am looking for is simply the same as the one used for the free Klein-Gordon field and the free electromagnetic field, am i wrong?
ps: there would be difference in some costant for the dimensionality, but we can neglect it, i am only looking for a formal expression

6. Jun 20, 2015

### fzero

Maybe you want to use the Wheeler-DeWitt and linearize. Otherwise, you can start with the linearized Lagrangian and you'll have to apply constraints for the gauge conditions etc.