Solve Gaussian Integrals: QFT Explained & Math Book for Physicists

[RedX]

This is probably an easy question, but my math is not good enough to answer it.

For Gaussian integrals:

$$\frac{\int \Pi_i [dx_i] x_k x_l e^{-\frac{x_i A_{ij} x_j}{2}}} {\int \Pi_i [dx_i] e^{-\frac{x_i A_{ij} x_j}{2}}}=A^{-1}_{kl}$$

As far as I understand it, in QFT, A_ij is a local operator. So A_ij might be at most block diagonal, with very tiny sub-blocks corresponding to derivative terms (to know the derivative you only need to know the field an infinitismal distance away). Hence A^-1_ij should be block diagonal too, with tiny sub-blocks of the same dimension.

So how is it that the propagator $$G(x-y)=-iA^{-1}_{xy}$$ doesn't seem to vanish for (x-y) not infinitismal?

Also does anyone know of a good math book written for improving the math of physics students?

 

[weejee]

Hence A^-1_ij should be block diagonal too, with tiny sub-blocks of the same dimension.

I don't think this is true.

 

[RedX]

I don't think this is true.

Yeah. The matrix A was never block diagonal to begin with: I don't know what I was thinking there. Just because it has only nearest neighbors on the diagonals doesn't mean it's block diagonal. So I think in general the inverse is global.

 

[homology]

Hassani has a text on mathematics for physicists that quite complete and fairly modern and well written.