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Istituto Lombardo - Accademia di Scienze e Lettere - Rendiconti di Scienze

ESTENSIONE DELLA TEORIA QUANTISTICA DI CAMPO A TEORIA DI AUTOMA CELLULARE QUANTISTICO

Giacomo Mauro D’Ariano

Published Dec 30, 2012• 0 citations
Publisher

Abstract

In the present note I will briefly report recent results on the novel field theory based on the information-theoretical paradigm. This program continues the previous one on the axiomatic derivation of quantum theory from information-theoretical principles, of which I presented a note on January 31st 2008. The passage from the quantum theory of abstract systems to the theory of fields is achieved by adding to the axioms of quantum theory new general principles for the network of interation among a denumerable set of quantum systems. Such principles can be synthesized with the requirement of minimal complexity of the quantum algorithm describing the physical law. From general principles such as homogeneity, isotropy, locality, linearity, and unitarity of the interaction network, along with minimal dimension of the field vector, one derives the free quantum field theory, namely Weyl, Dirac, and Maxwell fields. The principles lead to a description in terms of a quantum cellular automaton on the Cayley graph of a group. The advantages of the new theory are numerous. For example the theory tautologically solves all the problems originating from the continuum, such as the ultraviolet divergencies, the localization problem, and causality violations. The mechanics itself is not assumed, but it is a consequence of the principles, and the theory is quantum ab initio, namely it does not need quantization of a classical field theory. Lorentz covariance is not assumed, but follows itself from the principles, in the relativistic limit of small wavevectors. The notion of reference system is restated in terms of irreducible representation of the quantum automaton. Finally, the new theory, derived from purely mathematical principles–whence with adimensional variables–contains itself the units of measure for distance, time, and mass, which are given in terms of extremal values of the variables, which are in principle experimentally detectable. Also effects with GR flavor emerge from the automaton theory, such as an upper bound for the Dirac particle rest-mass (from the unitariety condition), where the dispersion relation becomes completely flat, in strict analogy with the notion of mini black hole.

DOI
10.4081/scie.2012.179

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In the present note I will briefly report recent results on the novel field theory based on the information-theoretical paradigm. This program continues the previous one on the axiomatic derivation of quantum theory from information-theoretical principles, of which I presented a note on January 31st 2008. The passage from the quantum theory of abstract systems to the theory of fields is achieved by adding to the axioms of quantum theory new general principles for the network of interation among a denumerable set of quantum systems. Such principles can be synthesized with the requirement of minimal complexity of the quantum algorithm describing the physical law. From general principles such as homogeneity, isotropy, locality, linearity, and unitarity of the interaction network, along with minimal dimension of the field vector, one derives the free quantum field theory, namely Weyl, Dirac, and Maxwell fields. The principles lead to a description in terms of a quantum cellular automaton on the Cayley graph of a group. The advantages of the new theory are numerous. For example the theory tautologically solves all the problems originating from the continuum, such as the ultraviolet divergencies, the localization problem, and causality violations. The mechanics itself is not assumed, but it is a consequence of the principles, and the theory is quantum ab initio, namely it does not need quantization of a classical field theory. Lorentz covariance is not assumed, but follows itself from the principles, in the relativistic limit of small wavevectors. The notion of reference system is restated in terms of irreducible representation of the quantum automaton. Finally, the new theory, derived from purely mathematical principles–whence with adimensional variables–contains itself the units of measure for distance, time, and mass, which are given in terms of extremal values of the variables, which are in principle experimentally detectable. Also effects with GR flavor emerge from the automaton theory, such as an upper bound for the Dirac particle rest-mass (from the unitariety condition), where the dispersion relation becomes completely flat, in strict analogy with the notion of mini black hole.
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Quantum Mechanics has taught us a lesson that goes well beyond a set of new physical laws. It has provided us with genuine “theorems of epistemology”. We now know that there are situations when the observation of the world cannot be considered as the mere reading of a reality pre-existing the observation, whereas there are circumstances where the knowledge of the whole do not corresponds to the knowledge of the parts (holism). We know that there are incompatible properties that are complementary. We know that there exist properties of the whole that are incompatible with any property of each part. The notion of “object” defined in terms of its “properties ” is in contrast with its mereologic connotation according to which objects can be composed to form new objects. The particle ontology as localizable unity is in contrast with the Malament theorem in quantum field theory. To the above add the fact that the two more general fundamental theories in physics – the quantum theory of fields and the general relativity of Einstein – are logically incompatible. In order to reconcile the logical coherence of the observations with their theoretical explanation we need a radical change of paradigm. The solution here proposed is to abandon the vision of the world as a “mechanism” and to substitute it with that of “algorithm”. This is the paradigm of the “universe as a huge computer” that have been latent within the community since Richard Feynman, and nowadays is resurrecting in physics, showing its full theoretical power. In the new algorithmic vision the “mechanics” becomes an emergent phenomenology. Without the need of physical primitive, the new “informational” program allows us to found physics on solid axiomatic grounds. The quantum theory of abstract systems along with the free quantum field theory are derived from information-theoretic axioms. The axioms of quantum theory all have an epistemological connotation, and pertain the possibility of falsifying the propositions of the theory. They reconnect holism with reductionism, probabilism and falsifiability, substituting the notion of “object” with that of “system” and of “event”, and logically implying the theorem of quantum theory without the need of using the abstract Hilbert-spaces toolkit. The free quantum field theory (Weyl, Dirac e Maxwell) is obtained by adding axioms of minimization of algorithmic complexity. The informational framework well separates the notions of “experiment”and “theory”: the theory connects input with output, the experiment being identified with the collection of input, output, and objective intermediate events. The objective reality (experiment) is made of “icons” with which we interact: the theory is the underlying algorithm. The paradigm describes the tapestry of reality as pure software, “software”. “Software without hardware”: the dematerialization of physics. “Reality without realism” means pure logical coherentism.
Can we reduce Quantum Field Theory (QFT) to a quantum computation? Can physics be simulated by a quantum computer? Do we believe that a quantum field is ultimately made of a numerable set of quantum systems that are unitarily interacting? A positive answer to these questions corresponds to substituting QFT with a theory of quantum cellular automata (QCA), and the present work is examining this hypothesis. These investigations are part of a large research program on a quantum-digitalization of physics, with Quantum Theory as a special theory of information, and Physics as emergent from the same quantum-information processing. A QCA-based QFT has tremendous potential advantages compared to QFT, being quantum ab-initio and free from the problems plaguing QFT due to the continuum hypothesis. Here I will show how dynamics emerges from the quantum processing, how the QCA can reproduce the Dirac-field phenomenology at large scales, and the kind of departures from QFT that that should be expected at a Planckscale discreteness. I will introduce the notions of linear field quantum automaton and local-matrix quantum automaton, in terms of which I will provide the solution to the Feynman's problem about the possibility of simulating a Fermi field with a quantum computer.
Without using the relativity principle, we show how the Dirac equation in three space dimensions emerges from the large-scale dynamics of the minimal nontrivial quantum cellular automaton satisfying unitarity, locality, homogeneity, and discrete isotropy. The Dirac equation is recovered for small wave vector and inertial mass, whereas Lorentz covariance is distorted in the ultrarelativistic limit. The automaton can thus be regarded as a theory unifying scales from Planck to Fermi. A simple asymptotic approach leads to a dispersive Schr\"odinger equation describing the evolution of narrowband states at all scales.
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Can we reduce Quantum Field Theory (QFT) to a quantum computation? Can physics be simulated by a quantum computer? Do we believe that a quantum field is ultimately made of a numerable set of quantum systems that are unitarily interacting? A positive answer to these questions corresponds to substituting QFT with a theory of quantum cellular automata (QCA), and the present work is examining this hypothesis. These investigations are part of a large research program on a "quantum-digitalization" of physics, with Quantum Theory as a special theory of information, and Physics as emergent from the same quantum-information processing. A QCA-based QFT has tremendous potential advantages compared to QFT, being quantum "ab-initio" and free from the problems plaguing QFT due to the continuum hypothesis. Here I will show how dynamics emerges from the quantum processing, how the QCA can reproduce the Dirac-field phenomenology at large scales, and the kind of departures from QFT that that should be expected at a Planck-scale discreteness. I will introduce the notions of linear field quantum automaton and local-matrix quantum automaton, in terms of which I will provide the solution to the Feynman's problem about the possibility of simulating a Fermi field with a quantum computer.