Action principle for cellular automata and the linearity of quantum mechanics

We introduce an action principle for a class of integer valued cellular automata and obtain Hamiltonian equations of motion. Employing sampling theory, these discrete deterministic equations are invertibly mapped on continuum equations for a set of bandwidth limited harmonic oscillators, which encode the Schr\"odinger equation. Thus, the linearity of quantum mechanics is related to the action principle of such cellular automata and its conservation laws to discrete ones.Comment: 6 pages; presented in part in invited talks at "Wigner 111 Scientific Symposium" (11-13 Nov 2013, Budapest) and "EmQM13 - Emergent Quantum Mechanics" (3-6 Oct 2013, Vienna

Open Quantum Systems, Entropy and Chaos

Entropy generation in quantum sytems is tied to the existence of a nonclassical environment (heat bath or other) with which the system interacts. The continuous `measuring' of the open system by its environment induces decoherence of its wave function and entropy increase. Examples of nonrelativistic quantum Brownian motion and of interacting scalar fields illustrate these general concepts. It is shown that the Hartree-Fock approximation around the bare classical limit can lead to spurious semiquantum chaos, which may affect the determination of entropy production and thermalization also in other cases.Comment: 22 pages including 4 figures; LaTex uses epsf and sprocl.sty file. - Invited talk, 5th Rio de Janeiro International Workshop on Relativistic Aspects of Nuclear Physics, August 1997; proceedings to be publ., T. Kodama et al., eds. (World Scientific

On configuration space, Born's rule and ontological states

It is shown how configuration space, possibly encompassing ordinary spatial structures, Born's rule, and ontological states aiming to address an underlying reality beyond Quantum Mechanics relate to each other in models of Hamiltonian cellular automata.Comment: 12 pages; to appear in the FIAS Interdisciplinary Science Series volume dedicated to the memory of Walter Greiner. arXiv admin note: text overlap with arXiv:1711.0032

Are nonlinear discrete cellular automata compatible with quantum mechanics?

We consider discrete and integer-valued cellular automata (CA). A particular class of which comprises "Hamiltonian CA" with equations of motion that bear similarities to Hamilton's equations, while they present discrete updating rules. The dynamics is linear, quite similar to unitary evolution described by the Schroedinger equation. This has been essential in our construction of an invertible map between such CA and continuous quantum mechanical models, which incorporate a fundamental discreteness scale. Based on Shannon's sampling theory, it leads, for example, to a one-to-one relation between quantum mechanical and CA conservation laws. The important issue of linearity of the theory is examined here by incorporating higher-order nonlinearities into the underlying action. These produce inconsistent nonlocal (in time) effects when trying to describe continuously such nonlinear CA. Therefore, in the present framework, only linear CA and local quantum mechanical dynamics are compatible.Comment: 11 pages. Talk presented at the DISCRETE 2014 international conference, King's College, London, 2-6 December 201

Spacetime and Matter - a duality of partial orders

A new kind of duality between the deep structures of spacetime and matter is proposed here, considering two partial orders which incorporate causality, extensity, and discreteness. This may have surprising consequences for the emergence of quantum mechanics, which are discussed.Comment: Fourth prize in the 2009 FQXi essay contest "What is Ultimately Possible in Physics?

Gauge Symmetry of the Third Kind and Quantum Mechanics as an Infrared Phenomenon

We introduce functional degrees of freedom by a new gauge principle related to the phase of the wave functional. Thereby, quantum mechanical systems are seen as dissipatively embedded part of a nonlinear classical structure producing universal correlations. There are a fundamental length and an entropy/area parameter, besides standard couplings. For states that are sufficiently spread over configuration space, quantum field theory is recovered.Comment: 12 pages. To appear in the Festschrift "A Sense of Beauty in Physics" dedicated to Adriano Di Giacomo on occasion of his 70th birthda

Four questions for quantum-classical hybrid theory

Four questions are discussed which may be addressed to any proposal of a quantum-classical hybrid theory which concerns the direct coupling of classical and quantum mechanical degrees of freedom. In particular, we consider the formulation of hybrid dynamics presented recently in Ref.[1]. This linear theory differs from the nonlinear ensemble theory of Hall and Reginatto, but shares with it to fulfil all consistency requirements discussed in the literature, while earlier attempts failed. - Here, we additionally ask: Does the theory allow for superposition, separable, and entangled states originating in the quantum mechanical sector? Does it allow for "Free Will", as introduced, in this context, by Diosi [2]. Is it local? Does it provide hints for the emergence of quantum mechanics from dynamics beneath?Comment: 16 pages - based on talk at the Conference on Emergent Quantum Mechanics / Heinz von Foerster Congress (Vienna University, Nov. 11-13, 2011

Quantum-classical hybrid dynamics - a summary

A summary of a recently proposed description of quantum-classical hybrids is presented, which concerns quantum and classical degrees of freedom of a composite object that interact directly with each other. This is based on notions of classical Hamiltonian mechanics suitably extended to quantum mechanics.Comment: 7 pages; summarizing invited talks at 6th International Workshop DICE2012, Castello Pasquini/Castiglioncello (Tuscany), September 17-21, 2012 and at Marcus Wallenberg Symposium "Quantum Theory: Advances and Problems - QTAP", Linnaeus University, Vaxjoe (Sweden) June 10-13, 201

The Gauge Symmetry of the Third Kind and Quantum Mechanics as an Infrared Limit

We introduce functional degrees of freedom by a new gauge principle related to the phase of the wave functional. Thus, quantum mechanical systems are dissipatively embedded into a nonlinear classical dynamical structure. There is a necessary fundamental length, besides an entropy/area parameter, and standard couplings. For states that are sufficiently spread over configuration space, quantum field theory is recovered.Comment: 7 pages. To appear in International Journal of Quantum Information. Talk at "Advances in Foundations of Quantum Mechanics and Quantum Information with Atoms and Photons", Torino, May 2-5, 200

Multipartite Cellular Automata and the Superposition Principle

Cellular automata can show well known features of quantum mechanics, such as a linear updating rule that resembles a discretized form of the Schr\"odinger equation together with its conservation laws. Surprisingly, a whole class of "natural" Hamiltonian cellular automata, which are based entirely on integer-valued variables and couplings and derived from an Action Principle, can be mapped reversibly to continuum models with the help of Sampling Theory. This results in "deformed" quantum mechanical models with a finite discreteness scale $l$, which for $l\rightarrow 0$ reproduce the familiar continuum limit. Presently, we show, in particular, how such automata can form "multipartite" systems consistently with the tensor product structures of nonrelativistic many-body quantum mechanics, while maintaining the linearity of dynamics. Consequently, the Superposition Principle is fully operative already on the level of these primordial discrete deterministic automata, including the essential quantum effects of interference and entanglement.Comment: 12 pages; accepted for publication - Int. J. Qu. Inf