Quantum measures and the coevent interpretation

This paper first reviews quantum measure and integration theory. A new representation of the quantum integral is presented. This representation is illustrated by computing some quantum (Lebesgue)${}^2$ integrals. The rest of the paper only considers finite spaces. Anhomomorphic logics are discussed and the classical domain of a coevent is studied. Pure quantum measures and coevents are considered and it is shown that pure quantum measures are strictly contained in the extremal elements for the set of quantum measures bounded above by one. Moreover, we prove that any quantum measure on a finite event space \ascript can be transferred to an ordinary measure on an anhomomorphic logic \ascript ^*. In this way, the quantum dynamics on \ascript can be described by a classical dynamics on the larger space \ascript ^*.Comment: one file submitte

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

Quantum measure and integration theory

This article begins with a review of quantum measure spaces. Quantum forms and indefinite inner-product spaces are then discussed. The main part of the paper introduces a quantum integral and derives some of its properties. The quantum integral's form for simple functions is characterized and it is shown that the quantum integral generalizes the Lebesgue integral. A bounded, monotone convergence theorem for quantum integrals is obtained and it is shown that a Radon-Nikodym type theorem does not hold for quantum measures. As an example, a quantum-Lebesgue integral on the real line is considered.Comment: 28 page

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

Curvature and Quantum Mechanics on Covariant Causal Sets

This article begins by reviewing the causal set approach in discrete quantum gravity. In our version of this approach a special role is played by covariant causal sets which we call $c$-causets. The importance of $c$-causets is that they support the concepts of a natural distance function, geodesics and curvature in a discrete setting. We then discuss curvature in more detail. By considering $c$-causets with a maximum and minimum number of paths, we are able to find $c$-causets with large and small average curvature. We then briefly discuss our previous work on the inflationary period when the curvature was essentially zero. Quantum mechanics on $c$-causets is considered next. We first introduce a free wave equation for $c$-causets. We then show how the state of a particle with a specified mass (or energy) can be derived from the wave equation. It is demonstrated for small examples that quantum mechanics predicts that particles tend to move toward vertices with larger curvature.Comment: 19 page

An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity

We consider a covariant causal set approach to discrete quantum gravity. We first review the microscopic picture of this approach. In this picture a universe grows one element at a time and its geometry is determined by a sequence of integers called the shell sequence. We next present the macroscopic picture which is described by a sequential growth process. We introduce a model in which the dynamics is governed by a quantum transition amplitude. The amplitude satisfies a stochastic and unitary condition and the resulting dynamics becomes isometric. We show that the dynamics preserves stochastic states. By "doubling down" on the dynamics we obtain a unitary group representation and a natural energy operator. These unitary operators are employed to define canonical position and momentum operators.Comment: 18 pages, 1 figur

A Theory of Entanglement

This article presents the basis of a theory of entanglement. We begin with a classical theory of entangled discrete measures in Section~1. Section~2 treats quantum mechanics and discusses the statistics of bounded operators on a Hilbert space in terms of context coefficients. In Section~3 we combine the work of the first two sections to develop a general theory of entanglement for quantum states. A measure of entanglement called the entanglement number is introduced. Although this number is related to entanglement robustness, its motivation is not the same and there are some differences. The present article only involves bipartite systems and we leave the study of multipartite systems for later work.Comment: 20 page

Spooky Action at a Distance

This article studies quantum mechanical entanglement. We begin by illustrating why entanglement implies action at a distance. We then introduce a simple criterion for determining when a pure quantum state is entangled. Finally, we present a measure for the amount of entanglement for a pure state.Comment: A survey of entanglement for students and general reader. 13 page