Abstract
In a series of recent papers [1–4] it has been shown how free quantum field theory can be derived without using mechanical primitives (including space-time, special relativity, quantization rules, etc.), but only considering the easiest quantum algorithm encompassing a countable set of quantum systems whose network of interactions satisfies the simple principles of unitarity, homogeneity, locality, and isotropy. This has opened the route to extending the axiomatic information-theoretic derivation of the quantum theory of abstract systems [5, 6] to include quantum field theory. The inherent discrete nature of the informational axiomatization leads to an extension of quantum field theory to a quantum cellular automata theory, where the usual field theory is recovered in a regime where the discrete structure of the automata cannot be probed. A simple heuristic argument sets the scale of discreteness to the Planck scale, and the customary physical regime where discreteness is not visible is the relativistic one of small wavevectors.
In this paper we provide a thorough derivation from principles that in the most general case the graph of the quantum cellular automaton is the Cayley graph of a finitely presented group, and showing how for the case corresponding to Euclidean emergent space (where the group resorts to an Abelian one) the automata leads to Weyl, Dirac and Maxwell field dynamics in the relativistic limit. We conclude with some perspectives towards the more general scenario of non-linear automata for interacting quantum field theory.
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G. M. D’Ariano and P. Perinotti, Derivation of the Dirac equation from principles of information processing, Phys. Rev. A 90(6), 062106 (2014)
A. Bisio, G. M. D’Ariano, and A. Tosini, Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension, Ann. Phys. 354, 244 (2015)
A. Bisio, G. M. D’Ariano, and A. Tosini, Dirac quantum cellular automaton in one dimension: Zitterbewegung and scattering from potential, Phys. Rev. A 88(3), 032301 (2013)
A. Bisio, G. M. D’Ariano, and P. Perinotti, Quantum cellular automaton theory of light, Ann. Phys. 368, 177 (2016)
G. Chiribella, G. D’Ariano, and P. Perinotti, Informational derivation of quantum theory, Phys. Rev. A 84(012311), 012311 (2011)
G. M. D’Ariano and P. Perinotti, Quantum theory is an Information Theory, Found. Phys. 46, 269 (2016)
L. Hardy, Quantum theory from five reasonable axioms, arXiv: quant-ph/0101012 (2001)
C. A. Fuchs, Quantum Mechanics as Quantum Information (and only a little more), arXiv: quant-ph/0205039 (2002)
G. M. D. Ariano, On the missing axiom of quantum mechanicss, AIP Conf. Proc. 810(1), 114 (2006)
G. M. D’Ariano, Probabilistic Theories: What is Special about Quantum Mechanics? in: Philosophy of Quantum Information and Entanglement, Eds. A. Bokulich and G. Jaeger, Cambridge UK: Cambridge University Press, 2010, p. 85
G. Chiribella, G. M. D’Ariano, and P. Perinotti, Probabilistic theories with purification, Phys. Rev. A 81(6), 062348 (2010)
J. D. Bekenstein, Black Holes and Entropy, Phys. Rev. D 7(8), 2333 (1973)
S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43(3), 199 (1975)
R. Bousso, Light Sheets and Bekenstein’s Entropy Bound, Phys. Rev. Lett. 90(12), 121302 (2003)
R. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21(6), 467 (1982)
G. Grössing and A. Zeilinger, Quantum cellular automata, Complex Syst. 2(2), 197 (1988)
Y. Aharonov, L. Davidovich, and N. Zagury, Quantum random walks, Phys. Rev. A 48(2), 1687 (1993)
A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Wa-trous, in: Proceedings of the thirty-third annual ACM sym-posium on Theory of computing (ACM, 2001), pp 37–49
I. Bialynicki-Birula, Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata, Phys. Rev. D 49(12), 6920 (1994)
D. Meyer, From quantum cellular automata to quantum lattice gases, J. Stat. Phys. 85(5), 551 (1996)
J. Yepez, Relativistic path integral as a lattice-based quantum algorithm, Quantum Inform. Process. 4(6), 471 (2006)
G. M. D’Ariano, A computational grand-unified theory, 2010
G. D’Ariano, CP1232 quantum theory, Reconsideration of Foundations 5, 3 (2010)
G. D’Ariano, Advances in quantum theory, AIP Conf. Proc. 1327, 7 (2011)
G. D’Ariano, The Dirac quantum automaton: A preview, arXiv: 1211.2479 (2012)
G. M. D’Ariano, The quantum field as a quantum computer, Phys. Lett. A 376(5), 697 (2012)
G. M. D’Ariano, A quantum-digital universe, Adv. Sci. Lett. 17(1), 130 (2012)
G. M. D’Ariano, A quantum digital universe, Il Nuovo Saggiatore 28, 13 (2012)
P. Arrighi, V. Nesme, and M. Forets, The Dirac equation as a quantum walk: higher dimensions, observational convergence, J. Phys. A 47(46), 465302 (2014)
T. C. Farrelly and A. J. Short, Discrete spacetime and relativistic quantum particles, arXiv: 1312.2852 (2013)
G. M. D’Ariano and P. Perinotti, The Dirac quantum automaton: A short review, Phys. Scr. 2014, 014014 (2014)
A. Bibeau-Delisle, A. Bisio, G. M. D’Ariano, P. Perinotti, and A. Tosini, Doubly special relativity from quantum cellular automata, EPL (Europhys. Lett.) 109(5), 50003 (2015)
A. Bisio, G. M. D’Ariano, and P. Perinotti, How important is i" in QFT? arXiv: 1503.0101 (2015)
M. Erba, Non-Abelian quantum walks and renormalization, Master Thesis, 2014
S. B. Bravyi and A. Y. Kitaev, Fermionic quantum computation, Ann. Phys. 298(1), 210 (2002)
G. M. D’Ariano, F. Manessi, P. Perinotti, and A. Tosini, The Feynman problem and fermionic entanglement: Fermionic theory versus qubit theory, Int. J. Mod. Phys. A 29(17), 1430025 (2014)
G. M. D’Ariano, F. Manessi, P. Perinotti, and A. Tosini, Fermionic computation is non-local tomographic and violates monogamy of entanglement, EPL (Europhys. Lett.) 107(2), 20009 (2014)
P. de la Harpe, Topics in Geometric Group Theory, The University of Chicago Press, 2Ed., 2003
M. Gromov, in: Proc. International Congress of Mathematicians, Vol. 1, p. 2, 1984
In the present case local systems are local Fermionic modes. For a detailed description of the Fermionic theory, see Refs. [36, 37].
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D’Ariano, G.M., Perinotti, P. Quantum cellular automata and free quantum field theory. Front. Phys. 12, 120301 (2017). https://doi.org/10.1007/s11467-016-0616-z
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DOI: https://doi.org/10.1007/s11467-016-0616-z