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

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

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

The Universe and The Quantum Computer

It is first pointed out that there is a common mathematical model for the universe and the quantum computer. The former is called the histories approach to quantum mechanics and the latter is called measurement based quantum computation. Although a rigorous concrete model for the universe has not been completed, a quantum measure and integration theory has been developed which may be useful for future progress. In this work we show that the quantum integral is the unique functional satisfying certain basic physical and mathematical principles. Since the set of paths (or trajectories) for a quantum computer is finite, this theory is easier to treat and more developed. We observe that the sum of the quantum measures of the paths is unity and the total interference vanishes. Thus, constructive interference is always balanced by an equal amount of destructive interference. As an example we consider a simplified two-slit experimentComment: 15 pages, IQSA 2010 proceeding

Compactness of the space of causal curves

We prove that the space of causal curves between compact subsets of a separable globally hyperbolic poset is itself compact in the Vietoris topology. Although this result implies the usual result in general relativity, its proof does not require the use of geometry or differentiable structure.Comment: 15 page

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

The Random Walk in Generalised 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

A domain of spacetime intervals in general relativity

Beginning from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic spacetime in a purely order theoretic manner. The ultimate reason for this is that globally hyperbolic spacetimes belong to a category that is equivalent to a special category of domains called interval domains.Comment: 25 page

Multiple mechanisms collectively regulate clathrin-mediated endocytosis of the epidermal growth factor receptor

Four independent mechanisms for uptake of activated EGFR are identified by a combination of receptor mutagenesis and RNA interference approaches

Noncommutative Geometry as a Regulator

We give a perturbative quantization of space-time $R^4$ in the case where the commutators $C^{{\mu}{\nu}}=[X^{\mu},X^{\nu}]$ of the underlying algebra generators are not central . We argue that this kind of quantum space-times can be used as regulators for quantum field theories . In particular we show in the case of the ${\phi}^4$ theory that by choosing appropriately the commutators $C^{{\mu}{\nu}}$ we can remove all the infinities by reproducing all the counter terms . In other words the renormalized action on $R^4$ plus the counter terms can be rewritten as only a renormalized action on the quantum space-time $QR^4$ . We conjecture therefore that renormalization of quantum field theory is equivalent to the quantization of the underlying space-time $R^4$ .Comment: Latex, 30 pages, no figures,typos corrected,references added . Substantial amount of rewriting of the last section . Final interesting remarks added at the end of the pape