# A Canonical quantisation of the EM field

Tags:
1. Aug 3, 2016

### Mr-R

I have just gone through chapter 14 on the QFT for the gifted amateur by Lancaster and Blundell. Quantising the electromagnetic field results in the Hamiltonian:
$$\hat{H}=\int d^3p \sum^{2}_{\lambda=1} E_p \hat{a}^\dagger_{p\lambda} \hat{a}_{p\lambda}$$
with $E_p=|p|$. In this post $p$ represents the momentum 3-vector.
My question is; how does the concept vacuum energy apply here? I think what is puzzling me is the fact that I see many authors arrive at this result:
$$\hat{H}=\sum_{p\lambda}\hbar \omega_p (\hat{a}^\dagger_{p\lambda}\hat{a}_{p\lambda}+\frac{1}{2})$$.
Also, the previous expression has an integration over $p$ as opposed to a sum.
Maybe I am comparing it to the wrong Hamiltonian, but I think that after applying normal ordering I get rid of the 1/2 term.

2. Aug 3, 2016

### vanhees71

The vacuum energy is subtracted in the first expression, and that's the correct one, because the 2nd is obviously divergent. In the usual formalism you call this description "normal ordering". The socalled "vacuum energy" is anyway unobservable within special relativity since there you can measure only energy differences and not the absolute energy of a system. This is a first (almost trivial) example for "renormalizing" an observable quantity like the energy of the electromagnetic field, whose associated operator is given by your first expression.

3. Aug 3, 2016

### Mr-R

Yes that makes sense. Now I remember going through that infinite energy problem. Cheers!