With a view toward application of the Pauli-Villars regularization method to the Casimir energy of boundaries, we calculate the expectation values of the components of the stress tensor of a confined massive field in 1+1 space-time dimensions. Previo
us papers by Hays and Fulling are bridged and generalized. The Green function for the time-independent Schrodinger equation is constructed from the Green function for the whole line by the method of images; equivalently, the one-dimensional system is solved exactly in terms of closed classical paths and periodic orbits. Terms in the energy density and in the eigenvalue density attributable to the two boundaries individually and those attributable to the confinement of the field to a finite interval are distinguished so that their physical origins are clear. Then the pressure is found similarly from the cylinder kernel, the Green function associated most directly with an exponential frequency cutoff of the Fourier mode expansion. Finally, we discuss how the theory could be rendered finite by the Pauli-Villars method.
A common tool in Casimir physics (and many other areas) is the asymptotic (high-frequency) expansion of eigenvalue densities, employed as either input or output of calculations of the asymptotic behavior of various Green functions. Here we clarify so
me fine points and potentially confusing aspects of the subject. In particular, we show how recent observations of Kolomeisky et al. [Phys. Rev. A 87 (2013) 042519] fit into the established framework of the distributional asymptotics of spectral functions.
A recent paper (2012 emph{J. Phys. A} textbf{45} 374018) is extended by investigating the behavior of the regularized quantum scalar stress tensor near the axes of cones and their covering manifold, the Dowker space. A cone is parametrized by its ang
le $theta_1$, where $theta_1=2pi$ for flat space. We find that the tensor components have singularities of the type $r^gamma$, but the generic leading $gamma$ equals ${4pi over theta_1} - 2$, which is negative if and only if $theta_1>2pi$, and is a positive integer if $theta_1={2piover N}$. Thus the functions are analytic in those cases that can be solved by the method of images starting from flat space, and they are not divergent in the cases that interpolate between those. As a wedge of angle $alpha$ can be solved by images starting from a cone of angle $2alpha$, a divergent stress can arise in a wedge with $pi <alpha le 2pi$ but not in a smaller one.
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 $x^{alpha}$ for $x>0$; the limit $alphatoinfty
$ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases $alpha =1$ 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 $S$, the amplitude function $A$ and the Laplacian of $A$. (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.
The expectation values of energy density and pressure of a quantum field inside a wedge-shaped region appear to violate the expected relationship between torque and total energy as a function of angle. In particular, this is true of the well-known De
utsch--Candelas stress tensor for the electromagnetic field, whose definition requires no regularization except possibly at the vertex. Unlike a similar anomaly in the pressure exerted by a reflecting boundary against a perpendicular wall, this problem cannot be dismissed as an artifact of an ad hoc regularization.
Rectangular cavities are solvable models that nevertheless touch on many of the controversial or mysterious aspects of the vacuum energy of quantum fields. This paper is a thorough study of the two-dimensional scalar field in a rectangle by the metho
d of images, or closed classical (or optical) paths, which is exact in this case. For each point r and each specularly reflecting path beginning and ending at r, we provide formulas for all components of the stress tensor T_{mu u}(r), for all values of the curvature coupling constant xi and all values of an ultraviolet cutoff parameter. Arbitrary combinations of Dirichlet and Neumann conditions on the four sides can be treated. The total energy is also investigated, path by path. These results are used in an attempt to clarify the physical reality of the repulsive (outward) force on the sides of the box predicted by calculations that neglect both boundary divergences and the exterior of the box. Previous authors have studied piston geometries that avoid these problems and have found the force to be attractive. We consider a pistol geometry that comes closer to the original problem of a box with a movable lid. We find again an attractive force, although its origin and detailed behavior are somewhat different from the piston case. However, the pistol (and the piston) model can be criticized for extending idealized boundary conditions into short distances where they are physically implausible. Therefore, it is of interest to see whether leaving the ultraviolet cutoff finite yields results that are more plausible. We then find that the force depends strongly on a geometrical parameter; it can be made repulsive, but only by forcing that parameter into the regime where the model is least convincing physically.
We offer a clarification of the significance of the indicated paper of H. Cheng. Chengs conclusions about the attractive nature of Casimir forces between parallel plates are valid beyond the particular model in which he derived them; they are likely
to be relevant to other recent literature on the effects of hidden dimensions on Casimir forces.
In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order differenti
al operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a Laplacian) with suitable conditions at vertices. For the case of scale-invariant vertex conditions (i.e., conditions that do not mix the values of functions and of their derivatives), the constant term of the heat-kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the index-theory context by factoring the Laplacian into two first-order operators, H =A*A, and relating the constant term to the index of A. An independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions. It is found also that the algebraic multiplicity of 0 as a root of the secular determinant of H is the sum of the nullities of A and A*.
