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How do you generalize the formula for matrices (or operators)

$$\int d^d x \, \exp \Big\{ - \frac{1}{2} x^i A_{ij} x^j \Big\} = \sqrt{\frac{(2 \pi)^d}{\det A}} = \sqrt{\det (2 \pi A^{-1})}$$

for tensors, i.e.

$$\int [d^d x]^2 \, \exp \Big\{ - \frac{1}{2} x^{ij} T_{ijkl} x^{kl} \Big\} \, ?$$

By $[d^d x ]^2$ I mean the integral over all variables $dx^{11}, dx^{12}, ..., dx^{21}, dx^{22}, ..., dx^{dd}$.

MBolin
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    I think you can consider $T_{ijkl}$ to be a matrix with rows labeled by the pairs $ij$ and columns by the pairs $kl$, and take its determinant. – G. Smith Aug 13 '19 at 16:56
  • Ok, that's what I thought at first, but for some reason it didn't convince me. I see no problem now – MBolin Aug 13 '19 at 19:20

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