41 research outputs found
Making Sense of Singular Gauge Transformations in 1+1 and 2+1 Fermion Models
We study the problem of decoupling fermion fields in 1+1 and 2+1 dimensions,
in interaction with a gauge field, by performing local transformations of the
fermions in the functional integral. This could always be done if singular
(large) gauge transformations were allowed, since any gauge field configuration
may be represented as a singular pure gauge field. However, the effect of a
singular gauge transformation of the fermions is equivalent to the one of a
regular transformation with a non-trivial action on the spinorial indices. For
example, in the two dimensional case, singular gauge transformations lead
naturally to chiral transformations, and hence to the usual decoupling
mechanism based on Fujikawa Jacobians. In 2+1 dimensions, using the same
procedure, different transformations emerge, which also give rise to Fujikawa
Jacobians. We apply this idea to obtain the v.e.v of the fermionic current in a
background field, in terms of the Jacobian for an infinitesimal decoupling
transformation, finding the parity violating result.Comment: 14 pages, Late
Casimir effect in 2+1 dimensional noncommutative theories
We study the Dirichlet Casimir effect for a complex scalar field on two
noncommutative spatial coordinates plus a commutative time. To that end, we
introduce Dirichlet-like boundary conditions on a curve contained in the
spatial plane, in such a way that the correct commutative limit can be reached.
We evaluate the resulting Casimir energy for two different curves: (a) Two
parallel lines separated by a distance , and (b) a circle of radius . In
the first case, the resulting Casimir energy agrees exactly with the one
corresponding to the commutative case, regardless of the values of and of
the noncommutativity scale , while for the latter the commutative
behaviour is only recovered when . Outside of that regime,
the dependence of the energy with is substantially changed due to
noncommutative corrections, becoming regular for .Comment: 12 pages, 3 figure
Dynamical Domain Wall Defects in 2+1 Dimensions
We study some dynamical properties of a Dirac field in 2+1 dimensions with
spacetime dependent domain wall defects. We show that the Callan and Harvey
mechanism applies even to the case of defects of arbitrary shape, and in a
general state of motion. The resulting chiral zero modes are localized on the
worldsheet of the defect, an embedded curved two dimensional manifold. The
dynamics of these zero modes is governed by the corresponding induced metric
and spin connection. Using known results about determinants and anomalies for
fermions on surfaces embedded in higher dimensional spacetimes, we show that
the chiral anomaly for this two dimensional theory is responsible for the
generation of a current along the defect. We derive the general expression for
such a current in terms of the geometry of the defect, and show that it may be
interpreted as due to an "inertial" electric field, which can be expressed
entirely in terms of the spacetime curvature of the defects. We discuss the
application of this framework to fermionic systems with defects in condensed
matter.Comment: 12 pages, Late
Chiral zero modes in non local domain walls
We study a generalization of the Callan-Harvey mechanism to the case of a non
local mass. Using a 2+1 model as a concrete example, we show that both the
existence and properties of localized zero modes can also be consistently
studied when the mass is non local. After dealing with some general properties
of the resulting integral equations, we show how non local masses naturally
arise when radiative corrections are included. We do that for a 2+1 dimensional
example, and also evaluate the zero mode of the resulting non local Dirac
operator.Comment: 20 pages, LaTeX, 4 figures; typos and content of sections 2 and 3
correcte
Neumann Casimir effect: a singular boundary-interaction approach
Dirichlet boundary conditions on a surface can be imposed on a scalar field,
by coupling it quadratically to a -like potential, the strength of
which tends to infinity. Neumann conditions, on the other hand, require the
introduction of an even more singular term, which renders the reflection and
transmission coefficients ill-defined because of UV divergences. We present a
possible procedure to tame those divergences, by introducing a minimum length
scale, related to the non-zero `width' of a {\em nonlocal} term. We then use
this setup to reach (either exact or imperfect) Neumann conditions, by taking
the appropriate limits. After defining meaningful reflection coefficients, we
calculate the Casimir energies for flat parallel mirrors, presenting also the
extension of the procedure to the case of arbitrary surfaces. Finally, we
discuss briefly how to generalize the worldline approach to the nonlocal case,
what is potentially useful in order to compute Casimir energies in theories
containing nonlocal potentials; in particular, those which we use to reproduce
Neumann boundary conditions.Comment: New title and reference adde
Tunneling between fermionic vacua and the overlap formalism
The probability amplitude for tunneling between the Dirac vacua corresponding
to different signs of a parity breaking fermionic mass in dimensions
is studied, making contact with the continuum overlap formulation for chiral
determinants. It is shown that the transition probability in the limit when corresponds, via the overlap formalism, to the squared modulus of a
chiral determinant in two Euclidean dimensions. The transition probabilities
corresponding to two particular examples: fermions on a torus with twisted
boundary conditions, and fermions on a disk in the presence of an external
constant magnetic field are evaluated.Comment: Reference added. 12 pages, LateX, no figure
Casimir effect with nonlocal boundary interactions
We derive a general expression for the Casimir energy corresponding to two
flat parallel mirrors in d+1 dimensions, described by nonlocal interaction
potentials. For a real scalar field, the interaction with the mirrors is
implemented by a term which is a quadratic form in the field, with a nonlocal
kernel. The resulting expression for the energy is a function of the parameters
that define the nonlocal kernel. We show that the general expression has the
correct limit in the zero width case, and also present the exact solution for a
particular case.Comment: 13 pages, LaTeX; minor misprints correcte
One-loop effects in a self-dual planar noncommutative theory
We study the UV properties, and derive the explicit form of the one-loop
effective action, for a noncommutative complex scalar field theory in 2+1
dimensions with a Grosse-Wulkenhaar term, at the self-dual point. We also
consider quantum effects around non-trivial minima of the classical action
which appear when the potential allows for the spontaneous breaking of the U(1)
symmetry. For those solutions, we show that the one-loop correction to the
vacuum energy is a function of a special combination of the amplitude of the
classical solution and the coupling constant.Comment: Version to appear in JHE
Regularized overlap and the chiral determinant
We study the relationship between the continuum overlap and its corresponding
chiral determinant, showing that the former amounts to an unregularised version
of the latter. We then construct a regularised continuum overlap, and consider
the chiral anomalies that follow therefrom. The relation between these
anomalies and the ones derived from the formal (i.e., unregularised) overlap is
elucidated.Comment: 14 pages, late
Finite temperature regularization
We present a non-perturbative regularization scheme for Quantum Field
Theories which amounts to an embedding of the originally unregularized theory
into a spacetime with an extra compactified dimensions of length L ~
Lambda^{-1} (with Lambda an ultraviolet cutoff), plus a doubling in the number
of fields, which satisfy different periodicity conditions and have opposite
Grassmann parity. The resulting regularized action may be interpreted, for the
fermionic case, as corresponding to a finite-temperature theory with a
supersymmetry, which is broken because of the boundary conditions. We test our
proposal in a perturbative calculation (the vacuum polarization graph for a
D-dimensional fermionic theory) and in a non-perturbative one (the chiral
anomaly).Comment: 17 pages, LaTeX fil