On the Schroedinger functional in QCD

TL;DR

This paper extends the Schr"odinger functional formalism from pure gauge theories to include fermions within Wilson's lattice QCD, ensuring a well-defined continuum limit and spectral properties.

Contribution

It introduces a method to incorporate fermions into the Schr"odinger functional framework in lattice QCD, maintaining proper continuum behavior.

Findings

Fermions can be naturally included in the Schr"odinger functional formalism.

The continuum limit of the fermionic action is well-defined with no lattice artifacts.

The free Dirac operator has a unique self-adjoint extension with discrete spectrum.

Abstract

In a series of publications [\ref{LNWW},\ref{Schroedinger}], L\"uscher et al. have demonstrated the usefulness of the Schr\"odinger 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 Schr\"odinger 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.