286 research outputs found
Surface Vacuum Energy in Cutoff Models: Pressure Anomaly and Distributional Gravitational Limit
Vacuum-energy calculations with ideal reflecting boundaries are plagued by
boundary divergences, which presumably correspond to real (but finite) physical
effects occurring near the boundary. Our working hypothesis is that the stress
tensor for idealized boundary conditions with some finite cutoff should be a
reasonable ad hoc model for the true situation. The theory will have a sensible
renormalized limit when the cutoff is taken away; this requires making sense of
the Einstein equation with a distributional source. Calculations with the
standard ultraviolet cutoff reveal an inconsistency between energy and pressure
similar to the one that arises in noncovariant regularizations of cosmological
vacuum energy. The problem disappears, however, if the cutoff is a spatial
point separation in a "neutral" direction parallel to the boundary. Here we
demonstrate these claims in detail, first for a single flat reflecting wall
intersected by a test boundary, then more rigorously for a region of finite
cross section surrounded by four reflecting walls. We also show how the
moment-expansion theorem can be applied to the distributional limits of the
source and the solution of the Einstein equation, resulting in a mathematically
consistent differential equation where cutoff-dependent coefficients have been
identified as renormalizations of properties of the boundary. A number of
issues surrounding the interpretation of these results are aired.Comment: 22 pages, 2 figures, 1 table; PACS 03.70.+k, 04.20.Cv, 11.10.G
WKB Approximation to the Power Wall
We present a semiclassical analysis of the quantum propagator of a particle
confined on one side by a steeply, monotonically rising potential. The models
studied in detail have potentials proportional to for ; the
limit would reproduce a perfectly reflecting boundary, but at
present we concentrate on the cases and 2, for which exact
solutions in terms of well known functions are available for comparison. We
classify the classical paths in this system by their qualitative nature and
calculate the contributions of the various classes to the leading-order
semiclassical approximation: For each classical path we find the action ,
the amplitude function and the Laplacian of . (The Laplacian is of
interest because it gives an estimate of the error in the approximation and is
needed for computing higher-order approximations.) The resulting semiclassical
propagator can be used to rewrite the exact problem as a Volterra integral
equation, whose formal solution by iteration (Neumann series) is a
semiclassical, not perturbative, expansion. We thereby test, in the context of
a concrete problem, the validity of the two technical hypotheses in a previous
proof of the convergence of such a Neumann series in the more abstract setting
of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses
are violated when caustics develop in the classical dynamics; this opens up the
interesting future project of extending the methods to momentum space.Comment: 30 pages, 8 figures. Minor corrections in v.
Mathematical Aspects of Vacuum Energy on Quantum Graphs
We use quantum graphs as a model to study various mathematical aspects of the
vacuum energy, such as convergence of periodic path expansions, consistency
among different methods (trace formulae versus method of images) and the
possible connection with the underlying classical dynamics.
We derive an expansion for the vacuum energy in terms of periodic paths on
the graph and prove its convergence and smooth dependence on the bond lengths
of the graph. For an important special case of graphs with equal bond lengths,
we derive a simpler explicit formula.
The main results are derived using the trace formula. We also discuss an
alternative approach using the method of images and prove that the results are
consistent. This may have important consequences for other systems, since the
method of images, unlike the trace formula, includes a sum over special
``bounce paths''. We succeed in showing that in our model bounce paths do not
contribute to the vacuum energy. Finally, we discuss the proposed possible link
between the magnitude of the vacuum energy and the type (chaotic vs.
integrable) of the underlying classical dynamics. Within a random matrix model
we calculate the variance of the vacuum energy over several ensembles and find
evidence that the level repulsion leads to suppression of the vacuum energy.Comment: Fixed several typos, explain the use of random matrices in Section
Quantum Stability of Accelerated Black Holes
We study quantum aspects of the accelerated black holes in some detail.
Explicitly shown is the fact that a uniform acceleration stabilizes certain
charged black holes against the well-known thermal evaporation. Furthermore, a
close inspection of the geometry reveals that this is possible only for
near-extremal black holes and that most nonextremal varieties continue to
evaporate with a modified spectrum under the acceleration. We also introduce a
two-dimensional toy model where the energy-momentum flow is easily obtained for
general accelerations, and find the behavior to be in accordance with the
four-dimensional results. After a brief comparison to the classical system of a
uniformly accelerated charge, we close by pointing out the importance of this
result in the WKB expansion of the black hole pair-creation rate.Comment: LaTeX, 22 pages, 5 uuencoded figures (minor errors corrected, more
discussions on the case with black holes formed by gravitational collapse.
Four dimensional R^4 superinvariants through gauge completion
We fully compute the N=1 supersymmetrization of the fourth power of the Weyl
tensor in d=4 x-space with the auxiliary fields. In a previous paper, we showed
that their elimination requires an infinite number of terms; we explicitely
compute those terms to order \kappa^4 (three loop). We also write, in
superspace notation, all the possible N=1 actions, in four dimensions, that
contain pure R^4 terms (with coupling constants). We explicitely write these
actions in terms of the \theta components of the chiral density \epsilon and
the supergravity superfields R, G_m, W_{ABC}. Using the method of gauge
completion, we compute the necessary \theta components which allow us to write
these actions in x-space. We discuss under which circumstances can these extra
R^4 correction terms be reabsorbed in the pure supergravity action, and their
relevance to the quantum supergravity/string theory effective actions.Comment: 20 pages, no figures. Sec. 3 clarified; typos correcte
Systematics of the Relationship between Vacuum Energy Calculations and Heat Kernel Coefficients
Casimir energy is a nonlocal effect; its magnitude cannot be deduced from
heat kernel expansions, even those including the integrated boundary terms. On
the other hand, it is known that the divergent terms in the regularized (but
not yet renormalized) total vacuum energy are associated with the heat kernel
coefficients. Here a recent study of the relations among the eigenvalue
density, the heat kernel, and the integral kernel of the operator
is exploited to characterize this association completely.
Various previously isolated observations about the structure of the regularized
energy emerge naturally. For over 20 years controversies have persisted
stemming from the fact that certain (presumably physically meaningful) terms in
the renormalized vacuum energy density in the interior of a cavity become
singular at the boundary and correlate to certain divergent terms in the
regularized total energy. The point of view of the present paper promises to
help resolve these issues.Comment: 19 pages, RevTeX; Discussion section rewritten in response to
referees' comments, references added, minor typos correcte
Type II and heterotic one loop string effective actions in four dimensions
We analyze the reduction to four dimensions of the R^4 terms which are part
of the ten-dimensional string effective actions, both at tree level and one
loop. We show that there are two independent combinations of R^4 present, at
one loop, in the type IIA four dimensional effective action, which means they
both have their origin in M-theory. The d=4 heterotic effective action also has
such terms. This contradicts the common belief thathere is only one R^4 term in
four-dimensional supergravity theories, given by the square of the Bel-Robinson
tensor. In pure N=1 supergravity this new R^4 combination cannot be directly
supersymmetrized, but we show that, when coupled to a scalar chiral multiplet
(violating the U(1) -symmetry), it emerges in the action after elimination
of the auxiliary fields.Comment: v2: 22 pages. Discussion on the new R^4 term and extended
supergravity has been abridged and improved. Published versio
Finite Size Effects in Thermal Field Theory
We consider a neutral self-interacting massive scalar field defined in a
d-dimensional Euclidean space. Assuming thermal equilibrium, we discuss the
one-loop perturbative renormalization of this theory in the presence of rigid
boundary surfaces (two parallel hyperplanes), which break translational
symmetry. In order to identify the singular parts of the one-loop two-point and
four-point Schwinger functions, we use a combination of dimensional and
zeta-function analytic regularization procedures. The infinities which occur in
both the regularized one-loop two-point and four-point Schwinger functions fall
into two distinct classes: local divergences that could be renormalized with
the introduction of the usual bulk counterterms, and surface divergences that
demand countertems concentrated on the boundaries. We present the detailed form
of the surface divergences and discuss different strategies that one can assume
to solve the problem of the surface divergences. We also briefly mention how to
overcome the difficulties generated by infrared divergences in the case of
Neumann-Neumann boundary conditions.Comment: 31 pages, latex, to appear in J. Math. Phy
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