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We want to calculate

[tex]\hbar \ln \int D x_i \exp[\frac{1}{32 \pi^3} \int ds \int d^3 r x_i(-is,r) M_{ij}(s,r) x_j(is,r)][/tex]

The answer is

[tex]\hbar \int \frac{ds}{2\pi} \ln \det[M_{ij}\delta^3(r-r')][/tex]

I know that

[tex]\int d^3 x_i e^{\frac{1}{2}x_i B_{ij} x_j} = \sqrt{\frac{(2\pi)^n}{\det B_{ij}}}[/tex]

and that standard logarithmic properties will be used. Also the [tex]\delta^3(r-r')[/tex] means that a Fourier transform involving that delta function will be employed at some point.

Beyond that I'm at a complete loss as to how to continue. One question is why we don't need to employ a Fourier transform involving a [tex]\delta(s-s')][/tex]. Any help would be much appreciated.

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# Homework Help: Gaussian Path Integral

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