9,342 research outputs found
Spectral asymptotics of Euclidean quantum gravity with diff-invariant boundary conditions
A general method is known to exist for studying Abelian and non-Abelian gauge
theories, as well as Euclidean quantum gravity, at one-loop level on manifolds
with boundary. In the latter case, boundary conditions on metric perturbations
h can be chosen to be completely invariant under infinitesimal diffeomorphisms,
to preserve the invariance group of the theory and BRST symmetry. In the de
Donder gauge, however, the resulting boundary-value problem for the Laplace
type operator acting on h is known to be self-adjoint but not strongly
elliptic. The latter is a technical condition ensuring that a unique smooth
solution of the boundary-value problem exists, which implies, in turn, that the
global heat-kernel asymptotics yielding one-loop divergences and one-loop
effective action actually exists. The present paper shows that, on the
Euclidean four-ball, only the scalar part of perturbative modes for quantum
gravity are affected by the lack of strong ellipticity. Further evidence for
lack of strong ellipticity, from an analytic point of view, is therefore
obtained. Interestingly, three sectors of the scalar-perturbation problem
remain elliptic, while lack of strong ellipticity is confined to the remaining
fourth sector. The integral representation of the resulting zeta-function
asymptotics is also obtained; this remains regular at the origin by virtue of a
spectral identity here obtained for the first time.Comment: 25 pages, Revtex-4. Misprints in Eqs. (5.11), (5.14), (5.16) have
been correcte
One-Loop Effective Action for Euclidean Maxwell Theory on Manifolds with Boundary
This paper studies the one-loop effective action for Euclidean Maxwell theory
about flat four-space bounded by one three-sphere, or two concentric
three-spheres. The analysis relies on Faddeev-Popov formalism and
-function regularization, and the Lorentz gauge-averaging term is used
with magnetic boundary conditions. The contributions of transverse,
longitudinal and normal modes of the electromagnetic potential, jointly with
ghost modes, are derived in detail. The most difficult part of the analysis
consists in the eigenvalue condition given by the determinant of a
or matrix for longitudinal and normal modes. It is shown that the
former splits into a sum of Dirichlet and Robin contributions, plus a simpler
term. This is the quantum cosmological case. In the latter case, however, when
magnetic boundary conditions are imposed on two bounding three-spheres, the
determinant is more involved. Nevertheless, it is evaluated explicitly as well.
The whole analysis provides the building block for studying the one-loop
effective action in covariant gauges, on manifolds with boundary. The final
result differs from the value obtained when only transverse modes are
quantized, or when noncovariant gauges are used.Comment: 25 pages, Revte
New Kernels in Quantum Gravity
Recent work in the literature has proposed the use of non-local boundary
conditions in Euclidean quantum gravity. The present paper studies first a more
general form of such a scheme for bosonic gauge theories, by adding to the
boundary operator for mixed boundary conditions of local nature a two-by-two
matrix of pseudo-differential operators with pseudo-homogeneous kernels. The
request of invariance of such boundary conditions under infinitesimal gauge
transformations leads to non-local boundary conditions on ghost fields. In
Euclidean quantum gravity, an alternative scheme is proposed, where non-local
boundary conditions and the request of their complete gauge invariance are
sufficient to lead to gauge-field and ghost operators of pseudo-differential
nature. The resulting boundary conditions have a Dirichlet and a
pseudo-differential sector, and are pure Dirichlet for the ghost. This approach
is eventually extended to Euclidean Maxwell theory.Comment: 19 pages, plain Tex. In this revised version, section 5 is new,
section 3 is longer, and the presentation has been improve
One-Loop Divergences in Simple Supergravity: Boundary Effects
This paper studies the semiclassical approximation of simple supergravity in
Riemannian four-manifolds with boundary, within the framework of
-function regularization. The massless nature of gravitinos, jointly
with the presence of a boundary and a local description in terms of potentials
for spin , force the background to be totally flat. First, nonlocal
boundary conditions of the spectral type are imposed on spin-
potentials, jointly with boundary conditions on metric perturbations which are
completely invariant under infinitesimal diffeomorphisms. The axial
gauge-averaging functional is used, which is then sufficient to ensure
self-adjointness. One thus finds that the contributions of ghost and gauge
modes vanish separately. Hence the contributions to the one-loop wave function
of the universe reduce to those values resulting from physical modes
only. Another set of mixed boundary conditions, motivated instead by local
supersymmetry and first proposed by Luckock, Moss and Poletti, is also
analyzed. In this case the contributions of gauge and ghost modes do not cancel
each other. Both sets of boundary conditions lead to a nonvanishing
value, and spectral boundary conditions are also studied when two concentric
three-sphere boundaries occur. These results seem to point out that simple
supergravity is not even one-loop finite in the presence of boundaries.Comment: 37 pages, Revtex. Equations (5.2), (5.3), (5.5), (5.7), (5.8) and
(5.13) have been amended, jointly with a few misprint
From Peierls brackets to a generalized Moyal bracket for type-I gauge theories
In the space-of-histories approach to gauge fields and their quantization,
the Maxwell, Yang--Mills and gravitational field are well known to share the
property of being type-I theories, i.e. Lie brackets of the vector fields which
leave the action functional invariant are linear combinations of such vector
fields, with coefficients of linear combination given by structure constants.
The corresponding gauge-field operator in the functional integral for the
in-out amplitude is an invertible second-order differential operator. For such
an operator, we consider advanced and retarded Green functions giving rise to a
Peierls bracket among group-invariant functionals. Our Peierls bracket is a
Poisson bracket on the space of all group-invariant functionals in two cases
only: either the gauge-fixing is arbitrary but the gauge fields lie on the
dynamical sub-space; or the gauge-fixing is a linear functional of gauge
fields, which are generic points of the space of histories. In both cases, the
resulting Peierls bracket is proved to be gauge-invariant by exploiting the
manifestly covariant formalism. Moreover, on quantization, a gauge-invariant
Moyal bracket is defined that reduces to i hbar times the Peierls bracket to
lowest order in hbar.Comment: 14 pages, Late
On the Zero-Point Energy of a Conducting Spherical Shell
The zero-point energy of a conducting spherical shell is evaluated by
imposing boundary conditions on the potential, and on the ghost fields. The
scheme requires that temporal and tangential components of perturbations of the
potential should vanish at the boundary, jointly with the gauge-averaging
functional, first chosen of the Lorenz type. Gauge invariance of such boundary
conditions is then obtained provided that the ghost fields vanish at the
boundary. Normal and longitudinal modes of the potential obey an entangled
system of eigenvalue equations, whose solution is a linear combination of
Bessel functions under the above assumptions, and with the help of the Feynman
choice for a dimensionless gauge parameter. Interestingly, ghost modes cancel
exactly the contribution to the Casimir energy resulting from transverse and
temporal modes of the potential, jointly with the decoupled normal mode of the
potential. Moreover, normal and longitudinal components of the potential for
the interior and the exterior problem give a result in complete agreement with
the one first found by Boyer, who studied instead boundary conditions involving
TE and TM modes of the electromagnetic field. The coupled eigenvalue equations
for perturbative modes of the potential are also analyzed in the axial gauge,
and for arbitrary values of the gauge parameter. The set of modes which
contribute to the Casimir energy is then drastically changed, and comparison
with the case of a flat boundary sheds some light on the key features of the
Casimir energy in non-covariant gauges.Comment: 29 pages, Revtex, revised version. In this last version, a new
section has been added, devoted to the zero-point energy of a conducting
spherical shell in the axial gauge. A second appendix has also been include
Lack of strong ellipticity in Euclidean quantum gravity
Recent work in Euclidean quantum gravity has studied boundary conditions
which are completely invariant under infinitesimal diffeomorphisms on metric
perturbations. On using the de Donder gauge-averaging functional, this scheme
leads to both normal and tangential derivatives in the boundary conditions. In
the present paper, it is proved that the corresponding boundary value problem
fails to be strongly elliptic. The result raises deep interpretative issues for
Euclidean quantum gravity on manifolds with boundary.Comment: 14 pages, Plain Tex, 33 KB, no figure
Field theoretic description of the abelian and non-abelian Josephson effect
We formulate the Josephson effect in a field theoretic language which affords
a straightforward generalization to the non-abelian case. Our formalism
interprets Josephson tunneling as the excitation of pseudo-Goldstone bosons. We
demonstrate the formalism through the consideration of a single junction
separating two regions with a purely non-abelian order parameter and a sandwich
of three regions where the central region is in a distinct phase. Applications
to various non-abelian symmetry breaking systems in particle and condensed
matter physics are given.Comment: 10 pages no figure
Path integral quantization of electrodynamics in dielectric media
In the present paper we study the Faddeev-Popov path integral quantization of
electrodynamics in an inhomogenious dielectric medium. We quantize all
polarizations of the photons and introduce the corresponding ghost fields.
Using the heat kernel technique, we express the heat kernel coefficients in
terms of the dielectricity and calculate the ultra violet
divergent terms in the effective action. No cancellation between ghosts and
"non-physical" degrees of freedom of the photon is observed.Comment: 10 pages, Latex, submitted to J.Phys.A, revised (a misprint in the
bibliography
Ellipticity Conditions for the Lax Operator of the KP Equations
The Lax pseudo-differential operator plays a key role in studying the general
set of KP equations, although it is normally treated in a formal way, without
worrying about a complete characterization of its mathematical properties. The
aim of the present paper is therefore to investigate the ellipticity condition.
For this purpose, after a careful evaluation of the kernel with the associated
symbol, the majorization ensuring ellipticity is studied in detail. This leads
to non-trivial restrictions on the admissible set of potentials in the Lax
operator. When their time evolution is also considered, the ellipticity
conditions turn out to involve derivatives of the logarithm of the
tau-function.Comment: 21 pages, plain Te
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