Editing Feynman diagram (section)

==== Connected diagrams: ''linked-cluster theorem'' ====
Roughly speaking, a Feynman diagram is called ''connected'' if all vertices and propagator lines are linked by a sequence of vertices and propagators of the diagram itself. If one views it as an [[Graph (discrete mathematics)|undirected graph]] it is connected. The remarkable relevance of such diagrams in QFTs is due to the fact that they are sufficient to determine the [[Partition function (quantum field theory)|quantum partition function]] {{math|''Z''[''J'']}}. More precisely, connected Feynman diagrams determine

:<math>i W[J]\equiv \ln Z[J].</math>

To see this, one should recall that

:<math> Z[J]\propto\sum_k{D_k}</math>

with {{mvar|D<sub>k</sub>}} constructed from some (arbitrary) Feynman diagram that can be thought to consist of several connected components {{mvar|C<sub>i</sub>}}. If one encounters {{mvar|n<sub>i</sub>}} (identical) copies of a component {{mvar|C<sub>i</sub>}} within the Feynman diagram {{mvar|D<sub>k</sub>}} one has to include a ''symmetry factor'' {{mvar|''n<sub>i</sub>''!}}. However, in the end each contribution of a Feynman diagram {{mvar|D<sub>k</sub>}} to the partition function has the generic form

:<math>\prod_i \frac{C_{i}^{n_i} }{ n_i!} </math>

where {{mvar|i}} labels the (infinitely) many connected Feynman diagrams possible.

A scheme to successively create such contributions from the {{mvar|D<sub>k</sub>}} to {{math|''Z''[''J'']}} is obtained by

:<math>\left(\frac{1}{0!}+\frac{C_1}{1!}+\frac{C^2_1}{2!}+\cdots\right)\left(1+C_2+\frac{1}{2}C^2_2+\cdots\right)\cdots </math>

and therefore yields

:<math>Z[J]\propto\prod_i{\sum^\infty_{n_i=0}{\frac{C_i^{n_i}}{n_i!}}}=\exp{\sum_i{C_i}}\propto \exp{W[J]}\,.</math>

To establish the ''normalization'' {{math|''Z''<sub>0</sub> {{=}} exp ''W''[0] {{=}} 1}} one simply calculates all connected ''vacuum diagrams'', i.e., the diagrams without any ''sources'' {{mvar|J}} (sometimes referred to as ''external legs'' of a Feynman diagram).

The linked-cluster theorem was first proved to order four by [[Keith Brueckner]] in 1955, and for infinite orders by [[Jeffrey Goldstone]] in 1957.<ref>{{Cite book |last1=Fetter |first1=Alexander L. |url=https://books.google.com/books?id=0wekf1s83b0C |title=Quantum Theory of Many-particle Systems |last2=Walecka |first2=John Dirk |date=2003-06-20 |publisher=Courier Corporation |isbn=978-0-486-42827-7 |language=en}}</ref>