# Finite-dimensional Feynman Diagrams

## 2. Facts from calculus and their d-dimensional analogues

The basic fact from calculus that powers the whole discussion is:

Proposition 1

The identity with  a = 1  is proved by the trick of calculating the square of the integral in polar coordinates. The general identity follows by change of variable from  x  to .

This fact generalizes to higher-dimensional integrals. Set   v = (v1, ..., vd) and   dv = (dv1 ... dvd), and let   A   be a symmetric   d by d   matrix.

Proposition 2

We use the fact that a symmetric matrix A is diagonalizable: there exists an orthogonal matrix U (so Ut = U-1) such that UAU-1 is the diagonal matrix B whose only nonzero entries are b11, ... , bdd along the diagonal. Then A = U-1BU and vtAv = vt U-1B U v = vtUtB U v = wtB w where w = Uv, using Ut = U-1 and (Uv)t = vtUt. Since U is orthogonal   detU = 1   and the change of variable from v to w does not change the integral:

Proposition 3

This follows from Proposition 1 by completion of the square in the exponent and a change of variables.

The generalization to d dimensions replaces a with A as before and b with the vector b = (b1, ... , bd)

Proposition 4

This is proven exactly like Proposition 2. If we write this integral as Zb then the integral of Proposition 2 is Z0 and this proposition can be rewritten as