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Vladimir S. Anashin

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In the paper, we show that matter waves can be derived from discreteness and causality. Namely we show that matter waves can naturally be ascribed to finite discrete causal systems, the Mealy automata having binary input/output which are bit sequences. If assign real numerical values (‘measured quantities’) to bit sequences, the waves arise as a correspondence between the numerical values of input sequences (‘impacts’) and output sequences (‘system-evoked responses’). We show that among all discrete causal systems with arbitrary (not necessarily binary) inputs/outputs, only the ones with binary input/output can be ascribed to matter waves ψ(x, t) = ei(kx−ωt).


matter waves, finite discrete causal systems, binary bit sequences

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C. D. Aliprantis and O. Burkinshaw: Principles of real analysis. Academic Press, Inc., third ed., 1998.

J.-P. Allouche and J. Shallit: Automatic Sequences. Theory, Applications, Generalizations. Cambridge Univ. Press, 2003.

V. Anashin: The non-Archimedean theory of discrete systems. Math. Comp. Sci., 6 (2012), pp. 375–393.

: Quantization causes waves: Smooth finitely computable functions are affine. p-Adic Numbers, Ultrametric Analysis Appl., 7 (2015), pp. 169–227.

V. Anashin and A. Khrennikov: Applied Algebraic Dynamics, vol. 49 of de Gruyter Expositions in Mathematics. Walter de Gruyter GmbH & Co., Berlin—N.Y., 2009.

T. Bastin and C. W. Kilmister: Combinatorial Physics, vol. 9 of Series on Knots and Everything. World Scientific, Singapore, 1995.

A. Bokulich and G. Jaeger, eds.: Philosophy of Quantum Information and Entaglement, Cambridge Univ. Press, 2010.

W. Brauer: Automatentheorie. B. G. Teubner, Stuttgart, 1984. 9. M. Burgin: Theory of Information, vol. 1 of World Scientific Series in Information Studies. World Scientific, Singapore, 2010.

J. Carroll and D. Long: Theory of Finite Automata. Prentice-Hall Inc., 1989.

A. N. Cherepov: On approximation of continuous functions by determinate functions with delay. Discrete Math. Appl., 22 (2010), pp. 1–24.

: Approximation of continuous functions by finite automata. Discrete Math. Appl., 22 (2012), pp. 445–453.

R. Crowell and R. Fox: Introduction to the Knot Theory. Ginu and Co., Boston, 1963.

B. G. Dragovic, P. H. Frampton, and B. V. Urošević: Classical p-adic spacetime. Mod. Phys. Lett., A5 (1990), pp. 1521–1528.

B. Dragovich, A. Y. Khrennikov, S. V. Kozyrev, and I. V. Volovich: On p-adic mathematical physics. p-adic Numbers, Ultrametric Anal. Appl., 1 (2009), pp. 1–17.

D. Dubischar, V. M. Gundlach, O. Steinkamp, and A. Khrennikov: The interference phenomenon, memory effects in the equipment and random dynamical systems over the fields of p-adic numbers. Nuovo Cimento B, 114 (1999), pp. 373–382.

B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov: Modern Geometry - Methods and Applications, vol. II. Springer-Verlag, NY–Berlin-Heidelberg-Tokyo, 1985.

S. Eilenberg: Automata, Languages, and Machines, vol. A. Academic Press, 1974.

D. R. Finkelstein: Quantum Relativity. Texts and Monographs in Physics, Springer, 1996.

B. R. Frieden: Physics from Fisher Information. Cambridge Univ. Press, 1998.

C. Frougny and K. Klouda: Rational base number systems for p-adic numbers. RAIPO Theor. Inform. Appl., 46 (2012), pp. 87–106.

F. Q. Gouvea: p-adic Numbers, An Introduction. Springer-Verlag, Berlin– Heidelberg–New York, second ed., 1997.

R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskii: Automata, dynamical systems, and groups. Proc. Steklov Institute Math., 231 (2000), pp. 128– 203.

B. Hasselblatt and A. Katok: A First Course in Dynamics. Cambridge Univ. Press, Cambridge, etc., 2003. 25.

G. ’t Hooft: Determinism Beneath Quantum Mechanics. In: Quo Vadis Quantum Mechanics? (A. Elitzur, S. Dolev, and N. Kolenda, eds.), Springer, 2005, pp. 99–111.

: Classical cellular automata and quantum field theory, in Proceedings of the Conference in Honor of Murray Gell-Mann’s 80th Birthday “Quantum Mechanics, Elementary Particles, Quantum Cosmology and Complexity”, H. Fritzsch and K. K. Phua, eds., Singapore, 2010, World Scientific, pp. 397–408.

: Relating the quantum mechanics of discrete systems to standard canonical quantum mechanics, Found. Phys., 44 (2014), pp. 406–425.

: The Cellular Automaton Interpretation of Quantum Mechanics, arXiv:1405.1548v3, December 2015.

S. Katok: p-adic analysis in comparison with real. Mass. Selecta, American Mathematical Society, 2003.

J. G. Kemeny and J. L. Snell: Finite Markov Chains. Springer-Verlag, 1976.

A. Khrennikov: Quantum mechanics from time scaling and random fluctuations at the “quick time scale”. Nuovo Cimento B, 121 (2006), pp. 1005–1021.

: To quantum averages through asymptotic expansion of classical averages on infinite-dimensional space. Math. Phys., 48 (2007). Art. No. 013512.

: Ultrametric Hilbert space representation of quantum mechanics with a finite exactness. Found. Physics, 26 (1996), pp. 1033–1054.

: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers, Dordrecht, 1997.

N. Koblitz: p-adic numbers, p-adic analysis, and zeta-functions, vol. 58 of Graduate texts in math. Springer-Verlag, second ed., 1984.

S. Koke, C. Grebing, H. Frei, A. Anderson, A. Assion, and G. Steinmeyer: Direct frequency comb synthesis with arbitrary offset and shot-noise-limited phase noise. Nature Photonic, 4 (2010), pp. 462–465.

M. Konecny: Real functions computable by finite automata using affine representations. Theor. Comput. Sci., 284 (2002), pp. 373–396.

R. Landauer: Information is a physical entity. Physica A, 263 (1999), pp. 63–67.

E. Lerner: Private communication. (Paper in preparation).

L. P. Lisovik and O. Y. Shkaravskaya: Real functions defined by transducers. Cybernetics and System Analysis, 34 (1998), pp. 69–76.

M. Lothaire: Algebraic Combinatorics on Words. Cambridge Univ. Press, 2002.

K. Mahler: p-adic numbers and their functions. Cambridge Univ. Press, 1981. (2nd edition).

V. Mansurov: Knot theory. Chapman & Hall/CRC, Boca Raton - London - NY - Washington, 2004.

N. Margolus: Looking at nature as a computer. Int. J. Theor. Phys., 42 (2003), pp. 309–327.

A. Mishchenko and A. Fomenko: A course of differential geometry and topology. Mir, Moscow, 1988.

A. F. Monna: Sur une transformation simple des nombres p-adiques en nombres r´eels. Indag. Math., 14 (1952), pp. 1–9.

H. P. Noyes: Bit-String Physics, vol. 27 of Series on Knots and Everything. World Scientific, Singapore, 2001. 48. A. F. Parker-Rhodes: The Theory of Inditinguishables. D. Reidel Publ. Co., Dordrecht, 1981.

W. Parry: On the β-expansions of real numbers. Acta Math. Acad. Sci. Hung., 11 (1960), pp. 401–416.

A. Renyi: Representation for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung., 8 (1957), pp. 477–493.

W. H. Schikhof: Ultrametric calculus. Cambridge University Press, 1984.

O. Y. Shkaravskaya: Affine mappings defined by finite transducers. Cybernetics and System Analysis, 34 (1998), pp. 781–783.

N. Sidorov: Arithmetic dynamics. In: Topics in dynamics and ergodic theory (S. Bezuglyi and S. Kolyada, eds.), vol. 310 of London Math. Soc. Lecture Note Series. Cambridge University Press, Cambridge, 2003, pp. 145–189.

V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov: p-adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994.

C. F. von Weizsaker: The Structure of Physics, vol. 154 of Fundamental Theories of Physics. Springer, 2006.

J. Vuillemin: On circuits and numbers. IEEE Trans. on Computers, 43 (1994), pp. 868–879.

: Finite digital synchronous circuits are characterized by 2-algebraic truth tables. In: Advances in computing science - ASIAN 2000, vol. 1961 of Lecture Notes in Computer Science, 2000, pp. 1–7.

: Digital algebra and circuits. In: Verification:Theory and Practice, vol. 2772 of Lecture Notes in Computer Science, 2003, pp. 733–746.

J. A. Wheeler: Information, physics, quantum: The search for links. In: Complexity, Entropy, and the Physics of Information (W. H. Zurek, ed.) Redwood City, Calif., 1990, Addison-Wesley Pub. Co., pp. 309–336.

S. V. Yablonsky: Introduction to discrete mathematics. Mir, Moscow, 1989.

A. Zeilinger: A foundational principle of quantum mechanics. Found. Phys., 21 (1999), pp. 631–643.

K. Zuce: Calculating Space. MIT, Cambridge, Mass., 1970. Translation from German of “Rechnender Raum”. Schr. Daten., Vol.1, Veiweg & Son, Braunschweig, 1969.



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