Quantum Cellular Automaton Theory of Light

Foundations of Physics, 2015

Recent advances on quantum foundations achieved the derivation of free quantum field theory from general principles, without referring to mechanical notions and relativistic invariance. From the aforementioned principles a quantum cellular automata (QCA) theory follows, whose relativistic limit of small wave-vector provides the free dynamics of quantum field theory. The QCA theory can be regarded as an extended quantum field theory that describes in a unified way all scales ranging from an hypothetical discrete Planck scale up to the usual Fermi scale. The present paper reviews the elementary automaton theory for the Weyl field, and the composite automata for Dirac and Maxwell fields. We then give a simple analysis of the dynamics in the momentum space in terms of a dispersive differential equation for narrowband wave-packets, and some account on the position space description in terms of a discrete path-integral approach. We then review the phenomenology of the free-field automaton and consider possible visible effects arising from the discreteness of the framework. We conclude introducing the consequences of the automaton distorted dispersion relation, leading to a deformed Lorentz covariance and to possible effects on the thermodynamics of ideal gases.

Annals of Physics, 2015

We present a quantum cellular automaton model in one space-dimension which has the Dirac equation as emergent. This model, a discrete-time and causal unitary evolution of a lattice of quantum systems, is derived from the assumptions of homogeneity, parity and time-reversal invariance. The comparison between the automaton and the Dirac evolutions is rigorously set as a discrimination problem between unitary channels. We derive an exact lower bound for the probability of error in the discrimination as an explicit function of the mass, the number and the momentum of the particles, and the duration of the evolution. Computing this bound with experimentally achievable values, we see that in that regime the QCA model cannot be discriminated from the usual Dirac evolution. Finally, we show that the evolution of one-particle states with narrow-band in momentum can be efficiently simulated by a dispersive differential equation for any regime. This analysis allows for a comparison with the dynamics of wave-packets as it is described by the usual Dirac equation. This paper is a first step in exploring the idea that quantum field theory could be grounded on a more fundamental quantum cellular automaton model and that physical dynamics could emerge from quantum information processing. In this framework, the discretization is a central ingredient and not only a tool for performing non-perturbative calculation as in lattice gauge theory. The automaton model, endowed with a precise notion of local observables and a full probabilistic interpretation, could lead to a coherent unification of an hypothetical discrete Planck scale with the usual Fermi scale of high-energy physics.

Foundations of Physics, 2015

After leading to a new axiomatic derivation of quantum theory, the new informational paradigm is entering the domain of quantum field theory, suggesting a quantum automata framework that can be regarded as an extension of quantum field theory to including an hypothetical Planck scale, and with the usual quantum field theory recovered in the relativistic limit of small wave-vectors. Being derived from simple principles (linearity, unitarity, locality, homogeneity, isotropy, and minimality of dimension), the automata theory is quantum ab-initio, and does not assume Lorentz covariance and mechanical notions. Being discrete it can describe localized states and measurements (unmanageable by quantum field theory), solving all the issues plaguing field theory originated from the continuum. These features make the theory an ideal framework for quantum gravity, with relativistic covariance and space-time emergent solely from the interactions, and not assumed a priori. The paper presents a synthetic derivation of the automata theory, showing how from the principles lead to a description in terms of a quantum automaton over a Cayley graph of a group. Restricting to Abelian groups we show how the automata recover the Weyl, Dirac and Maxwell dynamics in the relativistic limit. We conclude with some new routes about the more general scenario of non-Abelian Cayley graphs.

Physical Review A, 2020

It has been shown that certain quantum walks give rise to relativistic wave equations, such as the Dirac and Weyl equations, in their long-wavelength limits. This intriguing result raises the question of whether something similar can happen in the multi-particle case. We construct a one-dimensional quantum cellular automaton (QCA) model which matches the quantum walk in the single particle case, and which approaches the quantum field theory of free fermions in the long-wavelength limit. However, we show that this class of constructions does not generalize to higher spatial dimensions in any straightforward way, and that no construction with similar properties is possible in two or more spatial dimensions. This rules out the most common approaches based on QCAs. We suggest possible methods to overcome this barrier while retaining locality.

Physical Review A

Quantum walks on lattices can give rise to relativistic wave equations in the long-wavelength limit, but going beyond the single-particle case has proven challenging, especially in more than one spatial dimension. We construct quantum cellular automata for distinguishable particles based on two different quantum walks, and show that by restricting to the antisymmetric and symmetric subspaces, respectively, a multiparticle theory for free fermions and bosons in three spatial dimensions can be produced. This construction evades a no-go theorem that prohibits the usual fermionization constructions in more than one spatial dimension. In the long-wavelength limit, these recover Dirac field theory and Maxwell field theory, i.e., free QED.

Physical Review A

Quantum walks on lattices can give rise to one-particle relativistic wave equations in the longwavelength limit. In going to multiple particles, quantum cellular automata (QCA) are natural generalizations of quantum walks. In one spatial dimension, the quantum walk can be "promoted" to a QCA that, in the long-wavelength limit, gives rise to the Dirac quantum field theory (QFT) for noninteracting fermions. This QCA/QFT correspondence has both theoretical and practical applications, but there are obstacles to similar constructions in two or more spatial dimensions. Here we show that a method of construction employing distinguishable particles confined to the completely antisymmetric subspace yields a QCA in two spatial dimensions that gives rise to the 2D Dirac QFT. Generalizing to 3D will entail some additional complications, but no conceptual barriers. We examine how this construction evades the "no go" results in earlier work.

Nuclear Physics B, 1992

A certain class of cellular automata in 1 space + 1 time dimension is shown to be closely related to quantum field theories containing Dirac fermions. In the massless case this relation can be studied analytically, while the introduction of Dirac mass requires numerical simulations. We show that in the last case the cellular automaton describes the corresponding field theory only approximately.

The Zitterbewegung model of an electron offers a classical interpretation for interference and diffraction of electrons. The idea is very intuitive because it incorporates John Wheeler's idea of mass without mass: we have an indivisible naked charge that has no properties but its charge and its size (the classical electron radius) and it is easy to understand that the electromagnetic oscillation that keeps this tiny circular current going -like a perpetual current ring in some superconducting material -cannot be separated from it. In contrast, we keep wondering: what keeps a photon together? Hence, the real challenge for any realist interpretation of quantum mechanics is to explain the quantization of light: what are these photons?

Annales de la Fondation Louis de Broglie, 1997

A unique cellular automaton, the quantum cellular automaton (QCA), is advanced as a candidate process for describing basic quantum mechanics in real space and time. The QCA mimics a zitterbewegung motion arising from the Dirac free particle equation for fermions in a confined lattice sapce-time. It emerges from employing simple QCA calculational rules that a series of scaled autopoietic (self-forming) processes can be used to describe diverse states such as atoms, nuclei and elementary particles when scaled in the 3+1D state. fractal features associated with the QCA hint at an intimate link between chaos/fractal properties and the fundamental efforts to understand the roots of quantum physics in real space and real time. The QCA describes a quantum process world striving to survive in space and time and this picture is distinct from the particulate and wave views endemic in elementary quantum explanations at present.

This paper replaces the hypothetical 'object' called 'Light' (wave/photon). This sixth report on a new research programme that is investigating the electromagnetic (EM) interaction. This paper analyzes the effects of interactions arising from multiple, remote electrons on one or several, local 'target' electrons. These interactions are the result of the new quantized form of the EM impulse introduced in the previous paper. This model is used to re-interpret various optical effects that have previously required the existence of a fundamental object known as 'LIGHT': a basic entity, considered to be either a particle or a wave (or even both?-the 'photon') that travels across space. In contrast, this new EM model is constructed upon the key role of the 'light' emission processes, categorized as either oscillatory (as in antenna) or transitory (as within atoms). These real emission processes are now integrated into the asynchronous action-at-a-distance model of the EM interaction that is the basis of this new theory. Mathematically, this new model describes algebraically how variable or periodic phenomena (that have been assumed require the use of waves) can be explained by periodic, asynchronous, remote interactions between point particles without any use of differential equations (including the wave equation). This paper now extends the earlier pair-wise interaction between two electrons into the many-body world of macroscopic reality. The two key ideas of interaction saturation and selection are now introduced, which totally differentiate this theory from all other theories constructed around universal, continuous interaction (or 'force') models. By eliminating all the ray, wave and photon models of 'light' this paper now extends the original Newtonian mechanical philosophy of nature to the major domain of optics: both classical and quantum. The emphasis is on the electrons and on the relationship between electrons and not on some hypothetical 'carrier' that travels between them – this is the Newtonian action-at-a-distance particulate model extended to multiple times. The idea of selection leads to the introduction of information waves that identify the location and velocities of all other electrons that might participate in a ray-like exchange of momentum between pairs of electrons (saturation) that always act like particles (real trajectories across space). These supra-luminal waves do not carry momentum but ensure that the interaction minimizes the exchange of action across a non-local region of space. This new model resolves the long-time paradox of electrons as waves or particles: electrons are seen here as real point particles that interact periodically (rather than continuously) together; the focus is on the relationship between them that can be described by the discrete mathematics of particles or the periodic mathematics usually associated with waves. This paper includes the first analytical solution to the 3D scattering of two electrons – in the center-of-mass frame of reference both electrons are shown to go in quantized spiraling, conical motions: towards each other and then away from each other. The present theory provides an alternative to Feynman's mathematical approach to " the mysterious properties of light " while providing a physical explanation for some of the calculational diagrams introduced by Feynman in his approach to quantum electrodynamics (QED). This now replaces all field theories of 'light' without introducing the concept of the photon or virtual particles and so eliminates all QED infinities in the physical properties associated with the interactions of electrons arising from the false idea of vacuum polarization, returning the vacuum to its Newtonian role as the passive, empty space between real particles. This new EM theory establishes a firm foundation for a new quantum theory that covers all scales of nature from the macroscopic to the heart of the atomic nucleus, while covering the complete range of interaction sets from a pair of electrons to the myriads of electrons existing in macroscopic objects. The next (companion) paper will explain the wave-like properties of electrons while providing a new, comprehensive theory of quantum measurement. This next paper will finally establish the critical link between the realistic model of the micro-world introduced so far and the macroscopic world of scientific measurements.