A Spin-Statistics Theorem for Certain Topological Geons

We review the mechanism in quantum gravity whereby topological geons, particles made from non-trivial spatial topology, are endowed with nontrivial spin and statistics. In a theory without topology change there is no obstruction to ``anomalous'' spin-statistics pairings for geons. However, in a sum-over-histories formulation including topology change, we show that non-chiral abelian geons do satisfy a spin-statistics correlation if they are described by a wave function which is given by a functional integral over metrics on a particular four-manifold. This manifold describes a topology changing process which creates a pair of geons from $R^3$.Comment: 21 pages, Plain TeX with harvmac, 3 figures included via eps

The Random Walk in Generalized Quantum Theory

One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we ``quantize'' the classical random walk by finding, subject to a certain condition of ``strong positivity'', the most general Markovian, translationally invariant ``decoherence functional'' with nearest neighbor transitions.Comment: 25 pages, no figure

Large Fluctuations in the Horizon Area and what they can tell us about Entropy and Quantum Gravity

We evoke situations where large fluctuations in the entropy are induced, our main example being a spacetime containing a potential black hole whose formation depends on the outcome of a quantum mechanical event. We argue that the teleological character of the event horizon implies that the consequent entropy fluctuations must be taken seriously in any interpretation of the quantal formalism. We then indicate how the entropy can be well defined despite the teleological character of the horizon, and we argue that this is possible only in the context of a spacetime or ``histories'' formulation of quantum gravity, as opposed to a canonical one, concluding that only a spacetime formulation has the potential to compute --- from first principles and in the general case --- the entropy of a black hole. From the entropy fluctuations in a related example, we also derive a condition governing the form taken by the entropy, when it is expressed as a function of the quantal density-operator.Comment: 35 pages, plain Tex, needs mathmacros.tex and msmacros.te

Quantum measures and integrals

We show that quantum measures and integrals appear naturally in any $L_2$-Hilbert space $H$. We begin by defining a decoherence operator $D(A,B)$ and it's associated $q$-measure operator $\mu (A)=D(A,A)$ on $H$. We show that these operators have certain positivity, additivity and continuity properties. If $\rho$ is a state on $H$, then D_\rho (A,B)=\rmtr\sqbrac{\rho D(A,B)} and $\mu_\rho (A)=D_\rho (A,A)$ have the usual properties of a decoherence functional and $q$-measure, respectively. The quantization of a random variable $f$ is defined to be a certain self-adjoint operator \fhat on $H$. Continuity and additivity properties of the map f\mapsto\fhat are discussed. It is shown that if $f$ is nonnegative, then \fhat is a positive operator. A quantum integral is defined by \int fd\mu_\rho =\rmtr (\rho\fhat\,). A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.Comment: 16 page

Energy extremality in the presence of a black hole

We derive the so-called first law of black hole mechanics for variations about stationary black hole solutions to the Einstein--Maxwell equations in the absence of sources. That is, we prove that $\delta M=\kappa\delta A+\omega\delta J+VdQ$ where the black hole parameters $M, \kappa, A, \omega, J, V$ and $Q$ denote mass, surface gravity, horizon area, angular velocity of the horizon, angular momentum, electric potential of the horizon and charge respectively. The unvaried fields are those of a stationary, charged, rotating black hole and the variation is to an arbitrary `nearby' black hole which is not necessarily stationary. Our approach is 4-dimensional in spirit and uses techniques involving Action variations and Noether operators. We show that the above formula holds on any asymptotically flat spatial 3-slice which extends from an arbitrary cross-section of the (future) horizon to spatial infinity.(Thus, the existence of a bifurcation surface is irrelevant to our demonstration. On the other hand, the derivation assumes without proof that the horizon possesses at least one of the following two (related)properties: ($i$) it cannot be destroyed by arbitrarily small perturbations of the metric and other fields which may be present, ($ii$) the expansion of the null geodesic generators of the perturbed horizon goes to zero in the distant future.)Comment: 30 pages, latex fil

Two-Site Quantum Random Walk

We study the measure theory of a two-site quantum random walk. The truncated decoherence functional defines a quantum measure $\mu_n$ on the space of $n$-paths, and the $\mu_n$ in turn induce a quantum measure $\mu$ on the cylinder sets within the space $\Omega$ of untruncated paths. Although $\mu$ cannot be extended to a continuous quantum measure on the full $\sigma$-algebra generated by the cylinder sets, an important question is whether it can be extended to sufficiently many physically relevant subsets of $\Omega$ in a systematic way. We begin an investigation of this problem by showing that $\mu$ can be extended to a quantum measure on a "quadratic algebra" of subsets of $\Omega$ that properly contains the cylinder sets. We also present a new characterization of the quantum integral on the $n$-path space.Comment: 28 page