Editing Feynman diagram (section)

=== Scalar propagator ===
Each mode is independently Gaussian distributed. The expectation of field modes is easy to calculate:

:<math> \left\langle \phi_k \phi_{k'}\right\rangle = 0 \,</math>

for {{math|''k'' ≠ ''k''′}}, since then the two Gaussian random variables are independent and both have zero mean.

:<math> \left\langle\phi_k \phi_k \right\rangle = \frac{V}{k^2} </math>

in finite volume {{mvar|V}}, when the two {{mvar|k}}-values coincide, since this is the variance of the Gaussian. In the infinite volume limit,

:<math> \left\langle\phi(k) \phi(k')\right\rangle = \delta(k-k') \frac{1}{k^2} </math>

Strictly speaking, this is an approximation: the lattice propagator is:

:<math>\left\langle\phi(k) \phi(k')\right\rangle = \delta(k-k') \frac{1}{2\big(d - \cos(k_1) + \cos(k_2) \cdots + \cos(k_d)\big) }</math>

But near {{math|''k'' {{=}} 0}}, for field fluctuations long compared to the lattice spacing, the two forms coincide.

The delta functions contain factors of 2{{pi}}, so that they cancel out the 2{{pi}} factors in the measure for {{mvar|k}} integrals.

:<math>\delta(k) = (2\pi)^d \delta_D(k_1)\delta_D(k_2) \cdots \delta_D(k_d) \,</math>

where {{math|''δ<sub>D</sub>''(''k'')}} is the ordinary one-dimensional Dirac delta function. This convention for delta-functions is not universal—some authors keep the factors of 2{{pi}} in the delta functions (and in the {{mvar|k}}-integration) explicit.