## Finite-dimensional Feynman Diagrams

**Feature Column Archive**

### 4. Wick's Theorem

Calculating high-order derivatives of a function like can be very messy. A useful theorem reduces the calculationto combinatorics.

**Wick's theorem**

where the sum is taken over all pairings of *i*_{1}, ..., *i*_{m}

Wick's theorem is proved (a careful counting argument) in texts onQuantum Field Theory. The most detailed explanation is in S. S. Schweber,An Introduction to Relativistic Quantum Field Theory, Evanston, IL,Row, Peterson 1961.

Let us calculate a couple of examples.

To begin, it is useful to write with (the sums running from 1 to *d*)using the series expansion exp *x* = 1 + *x* +*x*^{2}/2 +*x*^{3}/3! ... .The typical termwill be . This term is a homogeneous polynomialin the *b*^{i} of degree 2*n*

Differentiating *k*times a homogeneous polynomialof degree 2*n*and evaluating at zero will give zero unless *k* = 2*n*. So the job is to analyzethe result of 2*n*differentiations on .

The differentiation carried out most frequently in these calculations is

where we use the symmetry of the matrix *A*^{-1},a direct consequence of the symmetry of *A*.

In what follows will be abbreviated as .

Note that (1,2) and (2,1) count as the same pairing.