Editing Feynman diagram (section)

==== Higher Gaussian moments — completing Wick's theorem ====
There is a subtle point left before Wick's theorem is proved—what if more than two of the <math>\phi</math>s have the same momentum? If it's an odd number, the integral is zero; negative values cancel with the positive values. But if the number is even, the integral is positive. The previous demonstration assumed that the <math>\phi</math>s would only match up in pairs.

But the theorem is correct even when arbitrarily many of the <math>\phi</math> are equal, and this is a notable property of Gaussian integration:
:<math> I = \int e^{-ax^2/2}dx = \sqrt\frac{2\pi}{a} </math>
:<math> \frac{\partial^n}{\partial a^n } I = \int \frac{x^{2n}}{2^n} e^{-ax^2/2}dx = \frac{1\cdot 3 \cdot 5 \ldots \cdot (2n-1) }{ 2 \cdot 2 \cdot 2 \ldots \;\;\;\;\;\cdot 2\;\;\;\;\;\;} \sqrt{2\pi}\, a^{-\frac{2n+1}{2}}</math>

Dividing by {{mvar|I}},

:<math> \left\langle x^{2n}\right\rangle=\frac{\displaystyle\int x^{2n} e^{-a x^2/2} }{\displaystyle \int e^{-a x^2/2} } = 1 \cdot 3 \cdot 5 \ldots \cdot (2n-1) \frac{1}{a^n} </math>
:<math> \left\langle x^2 \right\rangle = \frac{1}{a} </math>

If Wick's theorem were correct, the higher moments would be given by all possible pairings of a list of {{math|2''n''}} different {{mvar|x}}:

:<math> \left\langle x_1 x_2 x_3 \cdots x_{2n} \right\rangle</math>

where the {{mvar|x}} are all the same variable, the index is just to keep track of the number of ways to pair them. The first {{mvar|x}} can be paired with {{math|2''n'' − 1}} others, leaving {{math|2''n'' − 2}}. The next unpaired {{mvar|x}} can be paired with {{math|2''n'' − 3}} different {{mvar|x}} leaving {{math|2''n'' − 4}}, and so on. This means that Wick's theorem, uncorrected, says that the expectation value of {{math|''x''<sup>2''n''</sup>}} should be:

:<math> \left\langle x^{2n} \right\rangle = (2n-1)\cdot(2n-3)\ldots \cdot5 \cdot 3 \cdot 1 \left\langle x^2\right\rangle^n </math>

and this is in fact the correct answer. So Wick's theorem holds no matter how many of the momenta of the internal variables coincide.