Editing Feynman diagram (section)

=== Monte Carlo ===
The path integral defines a probabilistic algorithm to generate a Euclidean scalar field configuration. Randomly pick the real and imaginary parts of each Fourier mode at wavenumber {{mvar|k}} to be a Gaussian random variable with variance {{math|{{sfrac|1|''k''<sup>2</sup>}}}}. This generates a configuration {{math|''φ<sub>C</sub>''(''k'')}} at random, and the Fourier transform gives {{math|''φ<sub>C</sub>''(''x'')}}. For real scalar fields, the algorithm must generate only one of each pair {{math|''φ''(''k''), ''φ''(−''k'')}}, and make the second the complex conjugate of the first.

To find any correlation function, generate a field again and again by this procedure, and find the statistical average:

:<math> \left\langle \phi(x_1) \cdots \phi(x_n) \right\rangle = \lim_{|C|\rightarrow\infty}\frac{ \displaystyle\sum_C \phi_C(x_1) \cdots \phi_C(x_n) }{|C| } </math>

where {{math|{{abs|''C''}}}} is the number of configurations, and the sum is of the product of the field values on each configuration. The Euclidean correlation function is just the same as the correlation function in statistics or statistical mechanics. The quantum mechanical correlation functions are an analytic continuation of the Euclidean correlation functions.

For free fields with a quadratic action, the probability distribution is a high-dimensional Gaussian, and the statistical average is given by an explicit formula. But the [[Monte Carlo method]] also works well for bosonic interacting field theories where there is no closed form for the correlation functions.