Abstract
Starting from the continuum Dirac operator, I construct a renormalisation group blocking which transforms the continuum action into a lattice action, and I specifically consider the Wilson and overlap formalisms. For Wilson fermions the inverse blocking is non-local and thus invalid. However, I proceed to demonstrate that it is possible to construct a valid, local, blocking which, though dependent on the lattice spacing, generates the lattice overlap fermion action from the continuum action. Using this renormalisation group blocking for overlap fermions, I re-derive the Ginsparg–Wilson equations and the lattice chiral symmetry, and show that the standard Ginsparg–Wilson relation is not the most general way of expressing chiral symmetry on the lattice, nor, for overlap fermions, the most natural. I suggest how this reformulation of the Ginsparg–Wilson relation combined with the renormalisation group formulation of overlap fermions could allow the construction of a CP -invariant lattice chiral gauge theory.
Highlights
Chiral symmetry is one of the most important properties of the massless continuum QCD Lagrangian
Afterwards, based on a construction derived from the renormalisation group, Ginsparg and Wilson described a way in which chiral symmetry could be maintained on the lattice, namely that the right-hand side of equation (2) could be modified to give a term which is both local and which vanishes in the continuum limit [2]
I differ from almost all previous authors considering renormalisation group blockings within the context of lattice QCD because I do not assume, as they did, that the blocking matrices commute with γ5, and because, whereas they blocked from a continuum theory to a lattice or a lattice to another lattice with a different lattice spacing, I will block from a continuum theory with one action to another continuum theory but with a different action
Summary
Chiral symmetry is one of the most important properties of the massless continuum QCD Lagrangian. Troubling at more of a theoretical than practical level, the Ginsparg-Wilson equation and the fixed point fermions were derived from renormalisation group considerations; while overlap and domain wall fermions were derived by an entirely different approach. That they satisfy (or approximately satisfy) the Ginsparg-Wilson relation hints that there could be some relationship between these operators and the renormalisation group. A preliminary outline of sections 2-6 was presented in reference [33]
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