The Consequences of Non-Normality

TL;DR

This paper explores the effects of non-normality in Wilson-type lattice Dirac operators, proposing optimal approximation methods and examining their implications for quantum mechanics concepts and computational techniques.

Contribution

It introduces the truncated singular value expansion as the optimal approximation to the inverse operator and links it to the eigenmode expansion via gamma_5-hermiticity.

Findings

Truncated singular value expansion is optimal for approximating D^{-1}.

The expansion is equivalent to gamma_5 times the eigenmode expansion.

Non-normality impacts the definition of observables and computational methods.

Abstract

The non-normality of Wilson-type lattice Dirac operators has important consequences - the application of the usual concepts from the textbook (hermitian) quantum mechanics should be reconsidered. This includes an appropriate definition of observables and the refinement of computational tools. We show that the truncated singular value expansion is the optimal approximation to the inverse operator D^{-1} and we prove that due to the gamma_5-hermiticity it is equivalent to gamma_5 times the truncated eigenmode expansion of the hermitian Wilson-Dirac operator.