Today a new paper by Dreiner, Haber, and Martin has me really excited: “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry,” arXiv:0812.1594. It appears to be a comprehensive (246 page) guide to working with two-component Weyl fermions.
Most quantum field theory texts of the 90s (i.e. Peskin and Schroeder) used four-component Dirac spinors. The primary motivation for this is that Dirac spinors are the irreducible representations of massive fermions. Weyl spinors, however, are mathematically the fundamental representations of the universal cover of the Lorentz group. (See our previous post on spinors.) The fermionic generators of supersymmetry, for example, are Weyl spinors.
More practically, two-component Weyl spinors are easier to work with than four-component Dirac spinors… by a factor of two—see how I did some tricky math, there? Instead of working with -matrices in some representation, one can work with -matrices (Pauli matrices) which have a single standard representation. The “gamma gymnastics” of calculations thus becomes simpler.
Most importantly, each Weyl fermion corresponds to a chiral state. This is arguably (i.e. for everyone but experimentalists) the most sensible representation because the Standard Model is a chiral theory. (We’ve got a previous post on this point, too.) Thus working with the two-component fermions give a better handle for the actual physics at hand; i.e. less confusion about chirality versus helicity, no more need for chiral projectors in Feynman rules, etc.
Of course, this all comes at a cost. One has to relearn the spinor formalism. Feynman rules have to be rewritten with different conventions for arrows and mass terms. (Recall that the mass terms are what mix the left- and right-chiral Weyl spinors into a propagating Dirac spinor.) Fortunately, the guide has ample worked examples to help get students up-and-running.
I’ve often heard more senior physicists bemoan the fact that we teach QFT using four-component spinors just because that’s what everyone else uses. They quote the absence of a thorough and unambiguous treatment of QFT using two-component spinors to once-and-for-all encourage the community to make a phase transition to a new vacuum. Well, I’m not sure if this will be the treatment that leads the revolution, but I know that I’ll certainly keep it handy.
The guide comes with a webpage that contains an alternate version using the GR/East-coast/mostly-plus metric. (Because if particle physicists are finaly breaking free of Dirac spinors, we might as well get rid of our metric as well…?) Stephen Martin has also updated his famous SUSY Primer to utilize Weyl spinors and the ‘subversive’ metric, along with a few content updates.
Boy, the department printer is sure going to get a work out this afternoon.