RedX
Feb26-11, 09:07 PM
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, Aij is a local operator. So Aij 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-1ij 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?
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, Aij is a local operator. So Aij 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-1ij 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?