## Gaussian Integrals and Wick Contractions

**Theorem 1: **For nonzero real ,

**Theorem 2**: For a real symmetric matrix , a real vector and a real vector , we have

**Proof:**

Define to be the real orthogonal matrix which diagonalizes to its diagonal form . That is,

where . Further, let us choose to be a special orthogonal matrix, so that . Also, define . Now,

The argument of the exponential is

which equals

The first term is

Let (that is, ). So the second term is

So the argument of the exponential becomes

which can be written as

The integrand becomes

Hence the quantity to be evaluated is

The quantity in curly brackets involves definite Gaussian integrals, the term of which is

using the identity .

So the continued product is

whereas the exponential factor sticking outside is

This completes the proof.

**Theorem 3**:

**Proof**: If is the number of terms in , differentiate the identity proved in Theorem 2 above times with respect to each of .

As an example, differentiate the RHS of the identity in Theorem 2 with respect to . The derivative is

the prefactor of the exponential is

which becomes

using the fact that is symmetric. Repeating this exercise, we see that bringing down involves differentiating with respect to , and effectively introduces a matrix element of . It is obvious by induction that the sum must run over all possible permutations of matrix indices of . That is, the sum must run over every possible permutation of pairing the indices . This completes the “proof”.

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