Editing Feynman diagram (section)

==== Vacuum bubbles ====
An immediate consequence of the linked-cluster theorem is that all vacuum bubbles, diagrams without external lines, cancel when calculating correlation functions. A correlation function is given by a ratio of path-integrals:

:<math> \left\langle \phi_1(x_1) \cdots \phi_n(x_n)\right\rangle = \frac{\displaystyle\int e^{-S} \phi_1(x_1) \cdots\phi_n(x_n)\, D\phi }{\displaystyle \int e^{-S}\, D\phi}\,.</math>

The top is the sum over all Feynman diagrams, including disconnected diagrams that do not link up to external lines at all. In terms of the connected diagrams, the numerator includes the same contributions of vacuum bubbles as the denominator:

:<math> \int e^{-S}\phi_1(x_1)\cdots\phi_n(x_n)\, D\phi = \left(\sum E_i\right)\left( \exp\left(\sum_i C_i\right) \right)\,.</math>

Where the sum over {{mvar|E}} diagrams includes only those diagrams each of whose connected components end on at least one external line. The vacuum bubbles are the same whatever the external lines, and give an overall multiplicative factor. The denominator is the sum over all vacuum bubbles, and dividing gets rid of the second factor.

The vacuum bubbles then are only useful for determining {{mvar|Z}} itself, which from the definition of the path integral is equal to:

:<math> Z= \int e^{-S} D\phi = e^{-HT} = e^{-\rho V} </math>

where {{mvar|ρ}} is the energy density in the vacuum. Each vacuum bubble contains a factor of {{math|''δ''(''k'')}} zeroing the total {{mvar|k}} at each vertex, and when there are no external lines, this contains a factor of {{math|''δ''(0)}}, because the momentum conservation is over-enforced. In finite volume, this factor can be identified as the total volume of space time. Dividing by the volume, the remaining integral for the vacuum bubble has an interpretation: it is a contribution to the energy density of the vacuum.