Finite-dimensional Feynman Diagrams

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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 theintegral in polar coordinates. The general identity follows by change of variablefrom  x to .

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

Proposition 2 We use the fact that a symmetric matrix A isdiagonalizable: there exists an orthogonal matrix U(so Ut = U-1)such that UAU-1 is the diagonal matrix Bwhose onlynonzero entries are b11, ... , bddalong the diagonal. Then A = U-1BUand vtAv = vt U-1B U v =vtUtB U v =wtB w where w = Uv,using Ut = U-1 and (Uv)t =vtUt. Since Uis orthogonal   detU = 1   and the change of variable from v to wdoes notchange 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 ddimensions replaces awith Aas before and bwiththe vector b = (b1, ... , bd)

Proposition 4 This is proven exactly like Proposition 2.If we write this integral as Zbthen the integral of Proposition 2 is Z0and this proposition can be rewritten as 