Spin 1/2: Grassmann Integrals
The field path-integral can be extended to the Fermi case, but only if the notion of integration is expanded. A Grassman integral of a free Fermi field is a high-dimensional determinant or Pfaffian which defines the new type of Gaussian integration appropriate for Fermi fields.
The two fundamental formulas of Grassmann integration are:
where M is an arbitrary matrix and are independent Grassmann variables for each index i, and
Where A is an antisymmetric matrix, is a collection of Grassmann variables, and the 1/2 is to prevent double-counting (since ). In matrix notation, where and are Grassman valued row vectors, and are Grassman valued column vectors, and M is a real valued matrix:
Where the last equality is a consequence of the translation invariance of the Grassman integral. The Grassman variables are external sources for, and differentiating with respect to pulls down factors of .
again, in a schematic matrix notation. The meaning of the formula above is that the derivative with respect to the appropriate component of and gives the matrix element of . This is exactly analogous to the Bosonic path integration formula for a Gaussian integral of a complex Bosonic field:
So that the propagator is the inverse of the matrix in the quadratic part of the action in both the Bose and Fermi case.
For real Grassmann fields, for Majorana fermions, the path integral is a Pfaffian times a source quadratic form, and the formulas give the square root of the determinant, just as they do for real Bosonic fields. The propagator is still the inverse of the quadratic part.
The free Dirac Lagrangian:
formally gives the equations of motion and the anticommutation relations of the Dirac field, just as the Klein Gordon Lagrangian in an ordinary path integral gives the equations of motion and commutation relations of the scalar field. By using the spatial fourier-transform of the Dirac field as a new basis for the Grassmann algebra, the quadratic part of the Dirac action becomes simple to invert:
The propagator is the inverse of the matrix M linking and, since different values of k do not mix together.
The analog of Wick's theorem matches psi and psi-bars in pairs:
where S is the sign of the permutation which reorders the sequence of psi-bars and psis to put the ones which are paired up to make the delta-functions next to each other, with the psi-bar coming right before the psi. Since a psi-psi-bar pair is a commuting element of the Grassman algebra, it doesn't matter what order the pairs are in. If more than one psi/psi-bar 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 Grassman analog of the higher Gaussian moments which completed the Bosonic Wick's theorem earlier.
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 Grassman integration.
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 loops 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 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 which 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.
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 Grassman 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 psi-bar to go in front of the first psi, and this gives the sign.
This rule is the only visible effect of the exclusion principle in internal lines. When there are external lines, the amplitudes are antisymmetric when two Fermi insertions for identical particles are interchanged. This is automatic in the source formalism, because the sources for Fermi fields are themselves Grassman valued.
Famous quotes containing the word spin:
“Words can have no single fixed meaning. Like wayward electrons, they can spin away from their initial orbit and enter a wider magnetic field. No one owns them or has a proprietary right to dictate how they will be used.”
—David Lehman (b. 1948)