59 research outputs found
Chiral fermion operators on the lattice
We only require generalized chiral symmetry and -hermiticity, which
leads to a large class of Dirac operators describing massless fermions on the
lattice, and use this framework to give an overview of developments in this
field. Spectral representations turn out to be a powerful tool for obtaining
detailed properties of the operators and a general construction of them. A
basic unitary operator is seen to play a central r\^ole in this context. We
discuss a number of special cases of the operators and elaborate on various
aspects of index relations. We also show that our weaker conditions lead still
properly to Weyl fermions and to chiral gauge theories.Comment: 25 pages, invited review article for Int. J. Mod. Phys.
Dirac operator normality and chiral fermions
Normality of the Dirac operator is shown to be necessary for chiral
properties.
From the global chiral Ward identity, which in the continuum limit gives the
index theorem, a sum rule results which constrains the spectrum. The
Ginsparg-Wilson relation is to be restricted to its simple form and is a member
of a set of spectral constraints. A family of alternative chiral
transformations is introduced. The one of L\"uscher is a special case which
transports only the anomaly term to the measure. An alternative transformation
would also be needed to correct Fujikawa's path-integral approach. From a
general function of the hermitean Wilson-Dirac operator the one of Neuberger
follows.Comment: 12 pages, LaTeX, cjp.sty included, talk at CHIRAL '99, Taipei, Sep.
13-18, 199
Formulation of chiral gauge theories
We present a formulation of chiral gauge theories, which admits more general
spectra of Dirac operators and reveals considerably more possibilities for the
structure of the chiral projections. Our two forms of correlation functions
both also apply in the presence of zero modes and for any value of the index.
The decomposition of the total set of pairs of bases into equivalence classes
is carefully analyzed. Transformation properties are derived.Comment: 3 pages, Lattice2004(chiral
Form and index of Ginsparg-Wilson fermions
We clarify the questions rised by a recent example of a lattice Dirac
operator found by Chiu. We show that this operator belongs to a class based on
the Cayley transformation and that this class on the finite lattice generally
does not admit a nonvanishing index, while in the continuum limit, due to
operator properties in Hilbert space, this defect is no longer there. Analogous
observations are made for the chiral anomaly. We also elaborate on various
aspects of the underlying sum rule for the index.Comment: 10 pages; v2: equation corrected, conclusions unchange
General chiral gauge theories
Only requiring that Dirac operators decribing massless fermions on the
lattice decompose into Weyl operators we arrive at a large class of them. After
deriving general relations from spectral representations we study correlation
functions of Weyl fermions for any value of the index, stressing the related
conditions for basis transformations and getting the precise behaviors under
gauge and CP transformations. Using the detailed structure of the chiral
projections we also obtain a form of the correlation functions with a
determinant in the general case.Comment: 3 pages, Lattice2003(chiral
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