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,

$$ Z[J] = \text{exp}\bigg(i\int \mathrm d^4x \;\mathrm d^4x'J(x')\Delta_F(x-x') J(x) \bigg) \qquad \qquad (1) $$

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
$$ Z[J] = \frac{1}{Z_0} \int [d \phi] \text{exp}\bigg(-i\int d^4x \bigg[ \frac{1}{2}\phi (\Box + m^2 - i\epsilon)\phi - \phi J\bigg]\bigg) $$

All I did was to introduce $\phi \rightarrow \phi + \phi_0 $ and demand that

$$ (\Box + m^2 - i\epsilon)\phi_0 = J(x) $$ and $\Delta_F(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