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I inverted equation 1.3 to get ## \phi_{\mathbf k}(t)=\int \frac{e^{-i \mathbf k \cdot \mathbf x}}{(2\pi)^{\frac 3 2}} \phi(\mathbf x,t) d^3 \mathbf x ##. Then I put it in I to get:The vacuum wave functional (1.9)(## \Psi[\phi]\propto \exp[- \frac 1 2 \int d^3 \mathbf k|\phi_{\mathbf k}|^2 \omega_k]##) contains the integral

## I \equiv \int d^3\mathbf k |\phi_{\mathbf k}|^2 \sqrt{k^2+m^2} ##,

where##\phi_{\mathbf k}## are defined in equation (1.3)(##\phi(\mathbf x,t)=\int \frac{d^3 \mathbf k}{(2\pi)^{\frac 3 2}} e^{i\mathbf k\cdot \mathbf x} \phi_{\mathbf k}(t)##). This integral can be expressed directly in terms of the function ##\phi(\mathbf x) ##,

## I=\int d^3\mathbf x d^3 \mathbf y \phi(\mathbf x) K(\mathbf x,\mathbf y) \phi(\mathbf y) ##.

Determine the required kernel ##K(\mathbf x,\mathbf y) ##.

## I=\int \int d^3 \mathbf x d^3 \mathbf y \phi(\mathbf x,t)\left[ \int \frac{d^3 \mathbf k}{(2\pi)^3} \sqrt{k^2+m^2} e^{i(\mathbf y-\mathbf x)\cdot \mathbf k} \right]\phi^*(\mathbf y,t)##.

So it seems that I found the kernel but there are two problems:

1) I'm sure that one of the fields should be a complex conjugate but in the form suggested by the exercise, no one of them is!

2) The kernel is a divergent integral, both its real and imaginary parts diverge. Just try wolframalpha.com to see this.

Can anybody help me on this?

Thanks