Editing Feynman diagram (section)

==== Symmetry factors ====
The number of ways to form a given Feynman diagram by joining half-lines is large, and by Wick's theorem, each way of pairing up the half-lines contributes equally. Often, this completely cancels the factorials in the denominator of each term, but the cancellation is sometimes incomplete.

The uncancelled denominator is called the ''symmetry factor'' of the diagram. The contribution of each diagram to the correlation function must be divided by its symmetry factor.

For example, consider the Feynman diagram formed from two external lines joined to one {{mvar|X}}, and the remaining two half-lines in the {{mvar|X}} joined to each other. There are 4&nbsp;×&nbsp;3 ways to join the external half-lines to the {{mvar|X}}, and then there is only one way to join the two remaining lines to each other. The {{mvar|X}} comes divided by {{nowrap|4! {{=}} 4 × 3 × 2}}, but the number of ways to link up the {{mvar|X}} half lines to make the diagram is only 4&nbsp;×&nbsp;3, so the contribution of this diagram is divided by two.

For another example, consider the diagram formed by joining all the half-lines of one {{mvar|X}} to all the half-lines of another {{mvar|X}}. This diagram is called a ''vacuum bubble'', because it does not link up to any external lines. There are 4! ways to form this diagram, but the denominator includes a 2! (from the expansion of the exponential, there are two {{mvar|X}}s) and two factors of 4!. The contribution is multiplied by {{sfrac|4!|2 × 4! × 4!}} =&nbsp;{{sfrac|1|48}}.

Another example is the Feynman diagram formed from two {{mvar|X}}s where each {{mvar|X}} links up to two external lines, and the remaining two half-lines of each {{mvar|X}} are joined to each other. The number of ways to link an {{mvar|X}} to two external lines is 4&nbsp;×&nbsp;3, and either {{mvar|X}} could link up to either pair, giving an additional factor of 2. The remaining two half-lines in the two {{mvar|X}}s can be linked to each other in two ways, so that the total number of ways to form the diagram is {{nowrap|4 × 3 × 4 × 3 × 2 × 2}}, while the denominator is {{nowrap|4! × 4! × 2!}}. The total symmetry factor is 2, and the contribution of this diagram is divided by 2.

The symmetry factor theorem gives the symmetry factor for a general diagram: the contribution of each Feynman diagram must be divided by the order of its group of automorphisms, the number of symmetries that it has.

An [[automorphism]] of a Feynman graph is a permutation {{mvar|M}} of the lines and a permutation {{mvar|N}} of the vertices with the following properties:

# If a line {{mvar|l}} goes from vertex {{mvar|v}} to vertex {{mvar|v′}}, then {{math|''M''(''l'')}} goes from {{math|''N''(''v'')}} to {{math|''N''(''v′'')}}. If the line is undirected, as it is for a real scalar field, then {{math|''M''(''l'')}} can go from {{math|''N''(''v′'')}} to {{math|''N''(''v'')}} too.
# If a line {{mvar|l}} ends on an external line, {{math|''M''(''l'')}} ends on the same external line.
# If there are different types of lines, {{math|''M''(''l'')}} should preserve the type.

This theorem has an interpretation in terms of particle-paths: when identical particles are present, the integral over all intermediate particles must not double-count states that differ only by interchanging identical particles.

Proof: To prove this theorem, label all the internal and external lines of a diagram with a unique name. Then form the diagram by linking a half-line to a name and then to the other half line.

Now count the number of ways to form the named diagram. Each permutation of the {{mvar|X}}s gives a different pattern of linking names to half-lines, and this is a factor of {{math|''n''!}}. Each permutation of the half-lines in a single {{mvar|X}} gives a factor of 4!. So a named diagram can be formed in exactly as many ways as the denominator of the Feynman expansion.

But the number of unnamed diagrams is smaller than the number of named diagram by the order of the automorphism group of the graph.

{{anchor|Linked-cluster theorem}}