Whether the CPT Symmetry Can Be Almighty Even in a Photon

Koji Nagata¹, Tadao Nakamura²

¹Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Korea.
²Department of Information and Computer Science, Keio University, Yokohama, Japan.

DOI: 10.4236/oalib.1101806                

Abstract

As a trial, though thinking of general concepts, of our scientific challenge, we consider whether the Charge-Parity-Time (CPT) symmetry can be almighty even in a photon. This is the main aim of this paper. In what follows, we discuss our argumentations dividing the conjecture into two parts. Rotational invariance of physical laws is an accepted principle in Newton’s theory. We show that it leads to an additional constraint on local realistic theories with mixture of ten-particle Greenberger-Horne-Zeilinger state. This new constraint rules out such theories even in some situations in which standard Bell inequalities allow for explicit construction of such theories. This says new hypothesis to the number of ten. Next, it turns out Zermelo-Fraenkel set theory has contradictions. Further, the von Neumann’s theory has a contradiction by using ±1/. We solve the problem of von Neumann’s theory while escaping from all contradictions made by Zermelo-Fraenkel set theory, simultaneously. We assume that the results of measurements are . We assume that only and are possible. This situation meets a structure made by Zermelo-Fraenkel set theory with the axiom of choice. We result in the fact that it may be kept to perform the Deutsch-Jozsa algorithm even in the macroscopic scale because zero does not exist in this case. Our analysis agrees with recent experimental report.

Keywords

Quantum Measurement Theory, Formalism, Quantum Nonlocality, Quantum Computer

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Yokota, K., Yamamoto, T., Koashi, M. and Imoto, N. (2009) Direct Observation of Hardy’s Paradox by Joint Weak Measurement with an Entangled Photon Pair. New Journal of Physics, 11, Article ID: 033011. http://dx.doi.org/10.1088/1367-2630/11/3/033011
[2]	Kostelecky, V.A. and Mewes, M. (2013) Constraints on Relativity Violations from Gamma-Ray Bursts. Physical Review Letters, 110, Article ID: 201601. http://dx.doi.org/10.1103/PhysRevLett.110.201601
[3]	Nagata, K., Laskowski, W., Wieśniak, M. and Żukowski, M. (2004) RotationalInvariance as an Additional Constraint on Local Realism. Physical Review Letters, 93, Article ID: 230403. http://dx.doi.org/10.1103/PhysRevLett.93.230403
[4]	Nagata, K. and Ahn, J. (2008) Violation of Rotational Invariance of Local Realistic Models with Two Settings. Journal of the Korean Physical Society, 53, 2216.
[5]	Nagata, K. and Ahn, J. (2008) The Conflict between Bell-Żukowski Inequality and Bell-Mermin Inequality. Modern Physics Letters A, 23, 2967. http://dx.doi.org/10.1142/S0217732308028727
[6]	Greenberger, D.M., Horne, M.A. and Zeilinger, A. (1989) Going Beyond Bell’s Theorem. In: Kafatos, M., Ed., Bell’s Theorem, Quantum Theory and Conceptions of the Universe, Kluwer Academic, Dordrecht, 69-72. http://dx.doi.org/10.1007/978-94-017-0849-4_10
[7]	Nagata, K. (2007) Multipartite Omnidirectional Generalized Bell Inequality. Journal of Physics A: Mathematical and Theoretical, 40, 13101. http://dx.doi.org/10.1088/1751-8113/40/43/017
[8]	Redhead, M. (1989) Incompleteness, Nonlocality, and Realism. 2nd Impression, Clarendon Press, Oxford.
[9]	Peres, A. (1993) Quantum Theory: Concepts and Methods. Kluwer Academic, Dordrecht.
[10]	Einstein, A., Podolsky, B. and Rosen, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47, 777. http://dx.doi.org/10.1103/PhysRev.47.777
[11]	Bell, J.S. (1964) On the Einstein Podolsky Rosen Paradox. Physics, 1, 195-200.
[12]	Leggett, A.J. (2003) Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem. Foundations of Physics, 33, 1469-1493. http://dx.doi.org/10.1023/A:1026096313729
[13]	Gröblacher, S., Paterek, T., Kaltenbaek, R., Brukner, Č., Żukowski, M., Aspelmeyer, M. and Zeilinger, A. (2007) An Experimental Test of Non-Local Realism. Nature, 446, 871-875. http://dx.doi.org/10.1038/nature05677
[14]	Paterek, T., Fedrizzi, A., Gröblacher, S., Jennewein, T., Żukowski, M., Aspelmeyer, M. and Zeilinger, A. (2007) Experimental Test of Nonlocal Realistic Theories without the Rotational Symmetry Assumption. Physical Review Letters, 99, Article ID: 210406. http://dx.doi.org/10.1103/PhysRevLett.99.210406
[15]	Branciard, C., Ling, A., Gisin, N., Kurtsiefer, C., Lamas-Linares, A. and Scarani, V. (2007) Experimental Falsification of Leggett’s Nonlocal Variable Model. Physical Review Letters, 99, Article ID: 210407. http://dx.doi.org/10.1103/PhysRevLett.99.210407
[16]	Nagata, K. (2008) Classification of Local Realistic Theories. Journal of Physics A: Mathematical and Theoretical, 41, Article ID: 155308. http://dx.doi.org/10.1088/1751-8113/41/15/155308
[17]	Ekert, A.K. (1991) Quantum Cryptography Based on Bell’s Theorem. Physical Review Letters, 67, 661-663. http://dx.doi.org/10.1103/PhysRevLett.67.661
[18]	Scarani, V. and Gisin, N. (2001) Quantum Communication between N Partners and Bell’s Inequalities. Physical Review Letters, 87, Article ID: 117901. http://dx.doi.org/10.1103/PhysRevLett.87.117901
[19]	Brukner, Č., Żukowski, M., Pan, J.-W. and Zeilinger, A. (2004) Bell’s Inequalities and Quantum Communication Complexity. Physical Review Letters, 92, Article ID: 127901. http://dx.doi.org/10.1103/PhysRevLett.92.127901
[20]	Mermin, N.D. (1990) Extreme Quantum Entanglement in a Superposition of Macroscopically Distinct States. Physical Review Letters, 65, 1838-1840. http://dx.doi.org/10.1103/PhysRevLett.65.1838
[21]	Roy, S.M. and Singh, V. (1991) Tests of Signal Locality and Einstein-Bell Locality for Multiparticle Systems. Physical Review Letters, 67, 2761-2764. http://dx.doi.org/10.1103/PhysRevLett.67.2761
[22]	Ardehali, M. (1992) Bell Inequalities with a Magnitude of Violation That Grows Exponentially with the Number of Particles. Physical Review A, 46, 5375-5378. http://dx.doi.org/10.1103/PhysRevA.46.5375
[23]	Belinskii, A.V. and Klyshko, D.N. (1993) Interference of Light and Bell’s Theorem. Physics-Uspekhi, 36, 653-693. http://dx.doi.org/10.1070/PU1993v036n08ABEH002299
[24]	Werner, R.F. and Wolf, M.M. (2001) All-Multipartite Bell-Correlation Inequalities for Two Dichotomic Observables per Site. Physical Review A, 64, Article ID: 032112. http://dx.doi.org/10.1103/PhysRevA.64.032112
[25]	Werner, R.F. and Wolf, M.M. (2001) Bell Inequalities and Entanglement. Quantum Information & Computation, 1, 1-25.
[26]	Żukowski, M. and Brukner, Ű. (2002) Bell’s Theorem for General N-Qubit States. Physical Review Letters, 88, Article ID: 210401.
[27]	Zermelo-Fraenkel Set Theory—Wikipedia, the Free Encyclopedia.
[28]	Abian, A. (1965) The Theory of Sets and Transfinite Arithmetic. W. B. Saunders, Philadelphia.
[29]	Abian, A. and LaMacchia, S. (1978) On the Consistency and Independence of Some Set-Theoretical Axioms. Notre Dame Journal of Formal Logic, 19, 155-158. http://dx.doi.org/10.1305/ndjfl/1093888220
[30]	Devlin, K. (1996) The Joy of Sets. Springer, New York.
[31]	Fraenkel, A., Bar-Hille, Y. and Levy, A. (1973) Foundations of Set Theory. Fraenkel’s Final Word on ZF and ZFC, North Holland.
[32]	Hatcher, W. (1982) The Logical Foundations of Mathematics. Pergamon, London.
[33]	Jech, T. (2003) Set Theory. The Third Millennium Edition, Revised and Expanded, Springer, Berlin.
[34]	Kunen, K. (1980) Set Theory: An Introduction to Independence Proofs. Elsevier, Amsterdam.
[35]	Montague, R. (1961) “Semantic Closure and Non-Finite Axiomatizability” in Infinistic Methods. Pergamon, London, 45-69.
[36]	Suppes, P. (1972) Axiomatic Set Theory. Dover Reprint. Perhaps the Best Exposition of ZFC before the Independence of AC and the Continuum Hypothesis, and the Emergence of Large Cardinals. Includes Many Theorems.
[37]	Takeuti, G. and Zaring, W.M. (1971) Introduction to Axiomatic Set Theory. Springer Verlag, New York.
[38]	Tarski, A. (1939) On Well-Ordered Subsets of Any Set. Fundamenta Mathematicae, 32, 176-183.
[39]	Tiles, M. (2004) The Philosophy of Set Theory. Dover Reprint. Weak on Metatheory; the Author Is Not a Mathematician.
[40]	Tourlakis, G. (2003) Lectures in Logic and Set Theory. Volume 2, Cambridge University Press, Cambridge.
[41]	van Heijenoort, J. (1967) From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, Cambridge.
[42]	Zermelo, E. (1908) Untersuchungen uber die Grundlagen der Mengenlehre I. Mathematische Annalen, 65, 261-281. http://dx.doi.org/10.1007/BF01449999
[43]	van Heijenoort, J. (1967) Investigations in the Foundations of Set Theory. In: van Heijenoort, J., Ed., From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, 199-215.
[44]	Zermelo, E. (1930) Uber Grenzzablen und Mengenbereiche. Fundamenta Mathematicae, 16, 29-47.
[45]	Nagata, K. and Nakamura, T. (2011) Does Singleton Set Meet Zermelo-Fraenkel Set Theory with the Axiom of Choice? Advanced Studies in Theoretical Physics, 5, 57.
[46]	Nagata, K. and Nakamura, T. (2010) Can von Neumann’s Theory Meet the Deutsch-Jozsa Algorithm? International Journal of Theoretical Physics, 49, 162-170. http://dx.doi.org/10.1007/s10773-009-0189-5
[47]	Nagata, K., Ren, C.-L. and Nakamura, T. (2011) Whether Quantum Computation Can Be Almighty. Advanced Studies in Theoretical Physics, 5, 1-14.
[48]	Hess, K. and Philipp, W. (2002) Exclusion of Time in the Theorem of Bell. Europhysics Letters (EPL), 57, 775-781. http://dx.doi.org/10.1209/epl/i2002-00578-y
[49]	Hess, K. and Philipp, W. (2005) The Bell Theorem as a Special Case of a Theorem of Bass. Foundations of Physics, 35, 1749-1767. http://dx.doi.org/10.1007/s10701-005-6520-y
[50]	Nielsen, M.A. and Chuang, I.L. (2000) Quantum Computation and Quantum Information. Cambridge University Press, Cambridge.
[51]	Gudder, S.P. (1980) Proposed Test for a Hidden Variables Theory. International Journal of Theoretical Physics, 19, 163-168. http://dx.doi.org/10.1007/BF00669767
[52]	Ballentine, L.E. (1990) Quantum Mechanics. Prentice-Hall, Englewood Cliffs.
[53]	Zimba, J.R. and Clifton, R.K. (1998) Valuations on Functionally Closed Sets of Quantum Mechanical Observables and von Neumann’s “No-Hidden-Variables” Theorem. In: Dieks, D. and Vermaas, P., Eds., The Modal Interpretation of Quantum Mechanics, Kluwer Academic Publishers, Dordrecht, 69-101.
[54]	Genovese, M. (2007) Research on Hidden Variable Theories: A Review of Recent Progresses. http://arxiv.org/abs/quant-ph/0701071
[55]	Aspect, A., Dalibard, J. and Roger, G. (1982) Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers. Physical Review Letters, 49, 1804-1807. http://dx.doi.org/10.1103/PhysRevLett.49.1804
[56]	Hellmuth, T., Walther, H., Zajonc, A. and Schleich, W. (1987) Delayed-Choice Experiments in Quantum Interference. Physical Review A, 35, 2532-2541. http://dx.doi.org/10.1103/PhysRevA.35.2532
[57]	Kwiat, P.G. (1998) Experimental and Theoretical Progress in Interaction-Free Measurements. Physica Scripta, T76, 115. http://dx.doi.org/10.1238/Physica.Topical.076a00115
[58]	Cristian, J. (2007) Disproof of Bell’s Theorem by Clifford Algebra Valued Local Variables. http://arxiv.org/abs/quant-ph/0703179