Determinants on lens spaces and cyclotomic units

The Laplacian functional determinants for conformal scalars and coexact one-forms are evaluated in closed form on inhomogeneous lens spaces of certain orders, including all odd primes when the essential part of the expression is given, formally as a cyclotomic unitComment: 18 pages, 1 figur

Hyperspherical entanglement entropy

The coefficient of the log term in the entanglement entropy associated with hyperspherical surfaces in flat space-time is shown to equal the conformal anomaly by conformally transforming Euclideanised space--time to a sphere and using already existing formulae for the relevant heat--kernel coefficients after cyclic factoring. The analytical reason for the result is that the conformal anomaly on the lune has an extremum at the ordinary sphere limit. A proof is given. Agreement with a recent evaluation of the coefficient is found.Comment: 7 pages. Final revision. Historical comments amended. Minor remarks adde

The C_2 heat-kernel coefficient in the presence of boundary discontinuities

We consider the heat-kernel on a manifold whose boundary is piecewise smooth. The set of independent geometrical quantities required to construct an expression for the contribution of the boundary discontinuities to the C_{2} heat-kernel coefficient is derived in the case of a scalar field with Dirichlet and Robin boundary conditions. The coefficient is then determined using conformal symmetry and evaluation on some specific manifolds. For the Robin case a perturbation technique is also developed and employed. The contributions to the smeared heat-kernel coefficient and cocycle function are calculated. Some incomplete results for spinor fields with mixed conditions are also presented.Comment: 25 pages, LaTe

Causality in Time-Neutral Cosmologies

Gell-Mann and Hartle (GMH) have recently considered time-neutral cosmological models in which the initial and final conditions are independently specified, and several authors have investigated experimental tests of such models. We point out here that GMH time-neutral models can allow superluminal signalling, in the sense that it can be possible for observers in those cosmologies, by detecting and exploiting regularities in the final state, to construct devices which send and receive signals between space-like separated points. In suitable cosmologies, any single superluminal message can be transmitted with probability arbitrarily close to one by the use of redundant signals. However, the outcome probabilities of quantum measurements generally depend on precisely which past {\it and future} measurements take place. As the transmission of any signal relies on quantum measurements, its transmission probability is similarly context-dependent. As a result, the standard superluminal signalling paradoxes do not apply. Despite their unusual features, the models are internally consistent. These results illustrate an interesting conceptual point. The standard view of Minkowski causality is not an absolutely indispensable part of the mathematical formalism of relativistic quantum theory. It is contingent on the empirical observation that naturally occurring ensembles can be naturally pre-selected but not post-selected.Comment: 5 pages, RevTeX. Published version -- minor typos correcte

Spherical Universe topology and the Casimir effect

The mode problem on the factored 3--sphere is applied to field theory calculations for massless fields of spin 0, 1/2 and 1. The degeneracies on the factors, including lens spaces, are neatly derived in a geometric fashion. Vacuum energies are expressed in terms of the polyhedral degrees and equivalent expressions given using the cyclic decomposition of the covering group. Scalar functional determinants are calculated and the spectral asymmetry function treated by the same approach with explicit forms on one-sided lens spaces.Comment: 33 pages, 1 figure. Typos corrected and one reference adde

Causal continuity in degenerate spacetimes

A change of spatial topology in a causal, compact spacetime cannot occur when the metric is globally Lorentzian. One can however construct a causal metric from a Riemannian metric and a Morse function on the background cobordism manifold, which is Lorentzian almost everywhere except that it is degenerate at each critical point of the function. We investigate causal structure in the neighbourhood of such a degeneracy, when the auxiliary Riemannian metric is taken to be Cartesian flat in appropriate coordinates. For these geometries, we verify Borde and Sorkin's conjecture that causal discontinuity occurs if and only if the Morse index is 1 or n-1.Comment: 34 pages, 11 figures, Latex2e, important references added, introduction and discussions sections reworded slightl

Quantum Dynamics without the Wave Function

When suitably generalized and interpreted, the path-integral offers an alternative to the more familiar quantal formalism based on state-vectors, selfadjoint operators, and external observers. Mathematically one generalizes the path-integral-as-propagator to a {\it quantal measure} $\mu$ on the space $\Omega$ of all ``conceivable worlds'', and this generalized measure expresses the dynamics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such ``histories-based'' schemes new, and more ``realistic'' possibilities open up for resolving the philosophical problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of $\mu$ in its sets of measure zero: such sets are to be ``precluded''. But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of ``quantum logic'', would accommodate the contradictions by dualizing $\Omega$ to a space of ``co-events'' and effectively identifying reality with an element of this dual space.Comment: plainTeX, 24 pages, no figures. To appear in a special volume of {\it Journal of Physics A: Mathematical and General} entitled ``The Quantum Universe'' and dedicated to Giancarlo Ghirardi on the occasion of his 70th birthday. Most current version is available at http://www.physics.syr.edu/~sorkin/some.papers/ (or wherever my home-page may be

The Decay of Magnetic Fields in Kaluza-Klein Theory

Magnetic fields in five-dimensional Kaluza-Klein theory compactified on a circle correspond to ``twisted'' identifications of five dimensional Minkowski space. We show that a five dimensional generalisation of the Kerr solution can be analytically continued to construct an instanton that gives rise to two possible decay modes of a magnetic field. One decay mode is the generalisation of the ``bubble decay" of the Kaluza-Klein vacuum described by Witten. The other decay mode, rarer for weak fields, corresponds in four dimensions to the creation of monopole-anti-monopole pairs. An instanton for the latter process is already known and is given by the analytic continuation of the \KK\ Ernst metric, which we show is identical to the five dimensional Kerr solution. We use this fact to illuminate further properties of the decay process. It appears that fundamental fermions can eliminate the bubble decay of the magnetic field, while allowing the pair production of Kaluza-Klein monopoles.Comment: 25 pages, one figure. The discussion of fermions has been revised: We show how fundamental fermions can eliminate the bubble-type instability but still allow pair creation of monopole

The hybrid spectral problem and Robin boundary conditions

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made. A hemisphere hybrid problem is introduced that involves Robin boundary conditions and leads to logarithmic terms in the heat--kernel expansion which are evaluated explicitly.Comment: 24 pages. Typos and a few factors corrected. Minor comments added. Substantial Robin additions. Substantial revisio