On the Schrödinger functional in QCD

In a series of publications (Nucl. Phys. B384 (1994) 168; B389 (1994) 247), Lüscher et al. have demonstrated the usefulness of the Schrödinger functional in pure SU(2) and SU(3) gauge theory. In this paper, it is shown how their formalism can be extended to include fermions. In the framework of Wilson's lattice QCD, we define the Schrödinger functional by making use of the transfer matrix formalism. Boundary conditions for the fermions arise naturally. We then take the naive continuum limit of the action and show that no lattice peculiarities are left over. The corresponding free Dirac operator has a unique self-adjoint extension with purely discrete spectrum and no zero-modes.