602 research outputs found
Repulsive Casimir Pistons
Casimir pistons are models in which finite Casimir forces can be calculated
without any suspect renormalizations. It has been suggested that such forces
are always attractive. We present three scenarios in which that is not true.
Two of these depend on mixing two types of boundary conditions. The other,
however, is a simple type of quantum graph in which the sign of the force
depends upon the number of edges.Comment: 4 pages, 2 figures; RevTeX. Minor additions and correction
Distributional Asymptotic Expansions of Spectral Functions and of the Associated Green Kernels
Asymptotic expansions of Green functions and spectral densities associated
with partial differential operators are widely applied in quantum field theory
and elsewhere. The mathematical properties of these expansions can be clarified
and more precisely determined by means of tools from distribution theory and
summability theory. (These are the same, insofar as recently the classic
Cesaro-Riesz theory of summability of series and integrals has been given a
distributional interpretation.) When applied to the spectral analysis of Green
functions (which are then to be expanded as series in a parameter, usually the
time), these methods show: (1) The "local" or "global" dependence of the
expansion coefficients on the background geometry, etc., is determined by the
regularity of the asymptotic expansion of the integrand at the origin (in
"frequency space"); this marks the difference between a heat kernel and a
Wightman two-point function, for instance. (2) The behavior of the integrand at
infinity determines whether the expansion of the Green function is genuinely
asymptotic in the literal, pointwise sense, or is merely valid in a
distributional (cesaro-averaged) sense; this is the difference between the heat
kernel and the Schrodinger kernel. (3) The high-frequency expansion of the
spectral density itself is local in a distributional sense (but not pointwise).
These observations make rigorous sense out of calculations in the physics
literature that are sometimes dismissed as merely formal.Comment: 34 pages, REVTeX; very minor correction
Wedges, Cones, Cosmic Strings, and the Reality of Vacuum Energy
One of J. Stuart Dowker's most significant achievements has been to observe
that the theory of diffraction by wedges developed a century ago by Sommerfeld
and others provided the key to solving two problems of great interest in
general-relativistic quantum field theory during the last quarter of the
twentieth century: the vacuum energy associated with an infinitely thin,
straight cosmic string, and (after an interchange of time with a space
coordinate) the apparent vacuum energy of empty space as viewed by an
accelerating observer. In a sense the string problem is more elementary than
the wedge, since Sommerfeld's technique was to relate the wedge problem to that
of a conical manifold by the method of images. Indeed, Minkowski space, as well
as all cone and wedge problems, are related by images to an infinitely sheeted
master manifold, which we call Dowker space. We review the research in this
area and exhibit in detail the vacuum expectation values of the energy density
and pressure of a scalar field in Dowker space and the cone and wedge spaces
that result from it. We point out that the (vanishing) vacuum energy of
Minkowski space results, from the point of view of Dowker space, from the
quantization of angular modes, in precisely the way that the Casimir energy of
a toroidal closed universe results from the quantization of Fourier modes; we
hope that this understanding dispels any lingering doubts about the reality of
cosmological vacuum energy.Comment: 28 pages, 16 figures. Special volume in honor of J. S. Dowke
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.
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