AUTHOR: STAN GUDDER (John Evans Professor of Mathematical Physics, University of Denver, USA) -- -/- We consider a discrete scalar, quantum field theory based on a cubic 4-dimensional lattice. We mainly investigate a discrete scattering operator S(x0,r) where x0 and r are positive integers representing time and maximal total energy, respectively. The operator S(x0,r) is used to define transition amplitudes which are then employed to compute transition probabilities. These probabilities are conditioned on the time-energy (x0,r). In order to maintain total (...) unit probability, the transition probabilities need to be reconditioned at each (x0,r). This is roughly analogous to renormalization in standard quantum field theory, except no infinities or singularities are involved. We illustrate this theory with a simple scattering experiment involving a common interaction Hamiltonian. We briefly mention how discreteness of space-time might be tested astronomically. Moreover, these tests may explain the existence of dark energy and dark matter. (shrink)

The formalism of general probabilistic theories provides a universal paradigm that is suitable for describing various physical systems including classical and quantum ones as particular cases. Contrary to the usual no-restriction hypothesis, the set of accessible meters within a given theory can be limited for different reasons, and this raises a question of what restrictions on meters are operationally relevant. We argue that all operational restrictions must be closed under simulation, where the simulation scheme involves mixing and classical post-processing of (...) meters. We distinguish three classes of such operational restrictions: restrictions on meters originating from restrictions on effects; restrictions on meters that do not restrict the set of effects in any way; and all other restrictions. We fully characterize the first class of restrictions and discuss its connection to convex effect subalgebras. We show that the restrictions belonging to the second class can impose severe physical limitations despite the fact that all effects are accessible, which takes place, e.g., in the unambiguous discrimination of pure quantum states via effectively dichotomic meters. We further demonstrate that there are physically meaningful restrictions that fall into the third class. The presented study of operational restrictions provides a better understanding on how accessible measurements modify general probabilistic theories and quantum theory in particular. (shrink)

We show that an effect algebra E possess an order-determining set of states if and only if E is semiclassical; that is, E is essentially a classical effect algebra. We also show that if E possesses at least one state, then E admits hidden variables in the sense that E is homomorphic to an MV-algebra that reproduces the states of E. Both of these results indicate that we cannot distinguish between a quantum mechanical effect algebra and a classical one. Hereditary (...) properties of sharpness and coexistence are discussed and the existence of {0,1} and dispersion-free states are considered. We then discuss a stronger structure called a sequential effect algebra (SEA) that we believe overcomes some of the inadequacies of an effect algebra. We show that a SEA is semiclassical if and only if it possesses an order-determining set of dispersion-free states. (shrink)

This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the σ-effect algebra of effects (fuzzy events) $\mathcal{E}{\text{ }}\left( {\Omega ,\mathcal{A}} \right)$ and the set of probability measures $M_1^ + {\text{ }}\left( {\Omega ,\mathcal{A}} \right)$ on a measurable space $\left( {\Omega ,\mathcal{A}} \right)$ . An observable $X:\mathcal{B} \to {\text{ }}\mathcal{E}{\text{ }}\left( {\Omega ,\mathcal{A}} \right)$ is defined, where $\begin{gathered} X:\mathcal{B} \to {\text{ }}\mathcal{E}{\text{ }}\left( {\Omega ,\mathcal{A}} \right) \\ \left( {\Lambda ,{\text{ }}\mathcal{B}} \right) (...) \\ \end{gathered} $ is the value space of X. It is noted that there exists a one-to-one correspondence between states on $\mathcal{E} \left( {\Omega ,\mathcal{A} } \right)$ and elements of $M_1^ + {\text{ }}\left( {\Omega ,\mathcal{A} } \right)$ and between observables $X:\mathcal{B} \to \mathcal{E} \left( {\Omega ,\mathcal{A} } \right)$ and σ-morphisms from $\mathcal{E} \left( {\Lambda , \mathcal{B}} \right)$ to $\mathcal{E} \left( {\Omega , \mathcal{A}} \right)$ . Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map $M_1^ + {\text{ }}\left( {\Lambda ,\mathcal{B} } \right)$ is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived. (shrink)

Although a few new results are presented, this is mainly a review article on the relationship between finite-dimensional quantum mechanics and finite groups. The main motivation for this discussion is the hidden subgroup problem of quantum computation theory. A unifying role is played by a mathematical structure that we call a Hilbert *-algebra. After reviewing material on unitary representations of finite groups we discuss a generalized quantum Fourier transform. We close with a presentation concerning position-momentum measurements in this framework.

We summarize a recent search for quantum reality. The full anhomomorphic logic of coevents for an event set is introduced. The quantum integral over an event with respect to a coevent is defined. Reality filters such as preclusivity and regularity of coevents are considered. A quantum measure that can be represented as a quantum integral with respect to a coevent is said to 1-generate that coevent. This gives a stronger filter that may produce a unique coevent called the “actual reality” (...) for a physical system. What we believe to be a more general filter is defined in terms of a double quantum integral and is called 2-generation. It is noted that ordinary measures do not 1 or 2-generate coevents except in a few simple cases. Various general results are stated. (shrink)

A transition effect matrix (TEM) is a quantum generalization of a classical stochastic matrix. By employing a TEM we obtain a quantum generalization of a classical Markov chain. We first discuss state and operator dynamics for a quantum Markov chain. We then consider various types of TEMs and vector states. In particular, we study invariant, equilibrium and singular vector states and investigate projective, bistochastic, invertible and unitary TEMs.