<P> where S is the sign of the permutation that reorders the sequence of ψ and ψ to put the ones that are paired up to make the delta - functions next to each other, with the ψ coming right before the ψ . Since a ψ, ψ pair is a commuting element of the Grassmann algebra, it doesn't matter what order the pairs are in . If more than one ψ, ψ pair have the same k, the integral is zero, and it is easy to check that the sum over pairings gives zero in this case (there are always an even number of them). This is the Grassmann analog of the higher Gaussian moments that completed the Bosonic Wick's theorem earlier . </P> <P> The rules for spin - 1 / 2 Dirac particles are as follows: The propagator is the inverse of the Dirac operator, the lines have arrows just as for a complex scalar field, and the diagram acquires an overall factor of − 1 for each closed Fermi loop . If there are an odd number of Fermi loops, the diagram changes sign . Historically, the − 1 rule was very difficult for Feynman to discover . He discovered it after a long process of trial and error, since he lacked a proper theory of Grassmann integration . </P> <P> The rule follows from the observation that the number of Fermi lines at a vertex is always even . Each term in the Lagrangian must always be Bosonic . A Fermi loop is counted by following Fermionic lines until one comes back to the starting point, then removing those lines from the diagram . Repeating this process eventually erases all the Fermionic lines: this is the Euler algorithm to 2 - color a graph, which works whenever each vertex has even degree . Note that the number of steps in the Euler algorithm is only equal to the number of independent Fermionic homology cycles in the common special case that all terms in the Lagrangian are exactly quadratic in the Fermi fields, so that each vertex has exactly two Fermionic lines . When there are four - Fermi interactions (like in the Fermi effective theory of the weak nuclear interactions) there are more k - integrals than Fermi loops . In this case, the counting rule should apply the Euler algorithm by pairing up the Fermi lines at each vertex into pairs that together form a bosonic factor of the term in the Lagrangian, and when entering a vertex by one line, the algorithm should always leave with the partner line . </P> <P> To clarify and prove the rule, consider a Feynman diagram formed from vertices, terms in the Lagrangian, with Fermion fields . The full term is Bosonic, it is a commuting element of the Grassmann algebra, so the order in which the vertices appear is not important . The Fermi lines are linked into loops, and when traversing the loop, one can reorder the vertex terms one after the other as one goes around without any sign cost . The exception is when you return to the starting point, and the final half - line must be joined with the unlinked first half - line . This requires one permutation to move the last ψ to go in front of the first ψ, and this gives the sign . </P>

For feynman diagrams match the particle with the type of line