4,588 research outputs found

### Quasi-Local Formulation of Non-Abelian Finite-Element Gauge Theory

Recently it was shown how to formulate the finite-element equations of motion
of a non-Abelian gauge theory, by gauging the free lattice difference
equations, and simultaneously determining the form of the gauge
transformations. In particular, the gauge-covariant field strength was
explicitly constructed, locally, in terms of a path ordered product of
exponentials (link operators). On the other hand, the Dirac and Yang-Mills
equations were nonlocal, involving sums over the entire prior lattice. Earlier,
Matsuyama had proposed a local Dirac equation constructed from just the
above-mentioned link operators. Here, we show how his scheme, which is closely
related to our earlier one, can be implemented for a non-Abelian gauge theory.
Although both Dirac and Yang-Mills equations are now local, the field strength
is not. The technique is illustrated with a direct calculation of the current
anomalies in two and four space-time dimensions. Unfortunately, unlike the
original finite-element proposal, this scheme is in general nonunitary.Comment: 19 pages, REVTeX, no figure

### Casimir Energies and Pressures for $\delta$-function Potentials

The Casimir energies and pressures for a massless scalar field associated
with $\delta$-function potentials in 1+1 and 3+1 dimensions are calculated. For
parallel plane surfaces, the results are finite, coincide with the pressures
associated with Dirichlet planes in the limit of strong coupling, and for weak
coupling do not possess a power-series expansion in 1+1 dimension. The relation
between Casimir energies and Casimir pressures is clarified,and the former are
shown to involve surface terms. The Casimir energy for a $\delta$-function
spherical shell in 3+1 dimensions has an expression that reduces to the
familiar result for a Dirichlet shell in the strong-coupling limit. However,
the Casimir energy for finite coupling possesses a logarithmic divergence first
appearing in third order in the weak-coupling expansion, which seems
unremovable. The corresponding energies and pressures for a derivative of a
$\delta$-function potential for the same spherical geometry generalizes the TM
contributions of electrodynamics. Cancellation of divergences can occur between
the TE ($\delta$-function) and TM (derivative of $\delta$-function) Casimir
energies. These results clarify recent discussions in the literature.Comment: 16 pages, 1 eps figure, uses REVTeX

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