I have just started learning QFT. I have just completed scalar fields, which I learnt in using Canonical Quantisation and Path integrals. I did calculation of Casimir force between two metal plates using just free scalar field theory (using the vacuum energy). However, I am not able to find a way to do this thing using Path integrals and propagators. The partition function for the case of free scalar field (i.e KG field) turns out to be,
Z[J]=exp(i∫d4xd4x′J(x′)ΔF(x−x′)J(x))(1)
which after setting the Z[J=0]=1. I wish to know, how to approach my problem from here.
PS : I have not learnt vector or spinor fields yet. Most of the references or notes that I checked either assumed a prior knowledge of that or did not say how to quantise scalar fields.
EDIT : This is the integral to begin with right
Z[J]=1Z0∫[dϕ]exp(−i∫d4x[12ϕ(◻+m2−iϵ)ϕ−ϕJ])
All I did was to introduce ϕ→ϕ+ϕ0 and demand that
(◻+m2−iϵ)ϕ0=J(x)
and
ΔF(x−x′) is the Green's function involved in solving this equation.
Then I obtain the equation (1).
This post imported from StackExchange Physics at 2014-05-04 11:12 (UCT), posted by SE-user user35952