# The use of different bases in QFT

1. May 9, 2013

### unchained1978

In ordinary QFT, everything is formulated in terms of a Fock basis so when we write $|\psi\rangle$ we mean that this is a product of single particle states covering every momentum mode. This leads to a Hamiltonian that's typically of the form $\hat H=\int \frac{d^{3}k}{(2\pi)^{3}} [\omega_{k}(\hat a^{\dagger}_{k}\hat a_{k}+\frac{1}{2}\delta^{3}(0))]$.
Is this Fock basis different from the field basis such that $\langle \phi |\psi\rangle=\Psi[\phi]$ where $\Psi[\phi]$ is the wavefunctional of the field? (I hope that makes sense the way I've asked it). The Hamiltonian seems to have a different form, in the case of a scalar field
$\hat H=\int d^{3}x (-\delta^{2}/\delta \phi^{2}+(\vec\nabla \phi)^{2}+m^{2}\phi^{2})$
I don't understand very well whether or not these Hamiltonians and quantum states are really describing the same thing, or what the different forms represent. If anyone could enlighten me I would be extremely grateful.

2. May 10, 2013

### dextercioby

I don't wish to speculate on an answer, but have you read Hatfield's QFT book ? AFAIK it contains the best presentation on the so-called Schrödinger approach to QFT.