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QUANTUM SYSTEMS BASED UPON GALOIS FIELDS — FROM SUB-QUANTUM TO SUPER-QUANTUM CORRELATIONS

L. N. CHANG

Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA

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Z. LEWIS

Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA

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D. MINIC

Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA

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, and

T. TAKEUCHI

Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA

Corresponding author.

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https://doi.org/10.1142/S0217751X14300063Cited by:4 (Source: Crossref)

Abstract

In this paper, we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of super-quantum correlations we have constructed in this context. We also discuss the larger questions of the origins and foundations of quantum theory, as well as the relevance of super-quantum theory for the quantum theory of gravity.

Dedicated to Freeman Dyson on the occasion of his 90th birthday.

Keywords:

PACS: 03.65.Aa, 03.65.Ta, 03.65.Ud, 04.60.-m

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References

F. J. Dyson , Disturbing the Universe ( Basic Books , 1981 ) . Google Scholar
L. N. Changet al., Mod. Phys. Lett. B 27, 1350064 (2013), arXiv:1205.4800 DOI: 10.1142/S0217984913500644. Link, Web of Science, Google Scholar
L. N. Changet al., J. Phys. A: Math. Theor. 46, 065304 (2013), arXiv:1206.0064 DOI: 10.1088/1751-8113/46/6/065304. Web of Science, Google Scholar
L. N. Changet al., J. Phys. A: Math. Theor. 46, 485306 (2013), arXiv:1208.5189 DOI: 10.1088/1751-8113/46/48/485306. Web of Science, Google Scholar
T. Takeuchiet al., AIP Conf. Proc. 1508 (2012) p. 502, arXiv:1208.5544. Google Scholar
Y. Nambu, Quantum Field Theory and Quantum Statistics 1, eds. I. A. Batalinet al. (IOP Publishing, 1987) pp. 625–636. Google Scholar
D. R. Finkelstein , Quantum Relativity: A Synthesis of the Ideas of Einstein and Heisenberg ( Springer , 1996 ) . Google Scholar
R. Penrose , The Road to Reality ( Vintage , 2004 ) . Google Scholar
B. Schumacher and M. D. Westmoreland, Found. Phys. 42, 918 (2012), arXiv:1010.2929 DOI: 10.1007/s10701-012-9650-z. Web of Science, ADS, Google Scholar
J. S. Bell, Physics 1, 195 (1964). Google Scholar
J. S. Bell , Speakable and Unspeakable in Quantum Mechanics ( Cambridge University Press , 1987 ) . Google Scholar
L. N. Chang, D. Minic and T. Takeuchi, Mod. Phys. Lett. A 25, 2947 (2010), arXiv:1004.4220 DOI: 10.1142/S0217732310034286. Link, Web of Science, Google Scholar
L. N. Chang, Z. Lewis, D. Minic and T. Takeuchi, to appear in Int. J. Mod. Phys. D, [gr-qc] , arXiv:1305.3313 . Google Scholar
L. N. Changet al., Adv. High Energy Phys. 2011, 593423 (2011), arXiv:1104.3359. Google Scholar
L. N. Changet al., Adv. High Energy Phys. 2011, 493514 (2011), arXiv:1106.0068. Google Scholar
L. Freidel, R. G. Leigh and D. Minic, [hep-th] , arXiv:1307.7080 . Google Scholar
J. F. Clauseret al., Phys. Rev. Lett. 23, 880 (1969), DOI: 10.1103/PhysRevLett.23.880. Web of Science, ADS, Google Scholar
B. S. Cirel'son, Lett. Math. Phys. 4, 93 (1980), DOI: 10.1007/BF00417500. Web of Science, ADS, Google Scholar
L. J. Landau, Phys. Lett. A 120, 52 (1987), DOI: 10.1016/0375-9601(87)90075-2. Google Scholar
S. Popescu and D. Rohrlich, Found. Phys. 24, 379 (1994), DOI: 10.1007/BF02058098. Web of Science, ADS, Google Scholar
E. C. G. Stueckelberg, Helv. Phys. Acta 33, 727 (1960). Google Scholar
M. Gunaydin and D. Minic, [hep-th] , arXiv:1304.0410 . Google Scholar
S. L. Adler , Quaternionic Quantum Mechanics and Quantum Fields ( Oxford University Press , 1995 ) . Google Scholar
J. W. P. Hirschfeld , Projective Geometries over Finite Fields , 2nd edn. ( Oxford University Press , 1998 ) . Crossref, Google Scholar
J. W. P. Hirschfeld , G. Korchmáros and F. Torres , Algebraic Curves over a Finite Field ( Princeton University Press , 2008 ) . Google Scholar
V. I. Arnold , Dynamics, Statistics and Projective Geometry of Galois Fields ( Cambridge University Press , 2011 ) . Google Scholar
D. M. Greenberger, M. A. Horne and A. Zeilinger, [quant-ph] , arXiv:0712.0921v1 . Google Scholar
D. M. Greenbergeret al., Am. J. Phys. 58, 1131 (1990), DOI: 10.1119/1.16243. Web of Science, ADS, Google Scholar
L. Hardy, Phys. Rev. Lett. 71, 1665 (1993), DOI: 10.1103/PhysRevLett.71.1665. Web of Science, ADS, Google Scholar
J. Tits, Sur les analogues algebriques des groupes semi-simple complexes, Colloque d'Algebre Superieure (Centre Belge de Recherches Math., Gauthier-Villar, Paris, 1957) pp. 261–289. Google Scholar
N. Kurokawa and S. Koyama , Absolute Mathematics ( Nippon Hyōron Sha , 2010 ) . Google Scholar
L. N. Chang, Z. Lewis, D. Minic and T. Takeuchi, [quant-ph] , arXiv:1312.4191 . Google Scholar
T. Curtright and L. Mezincescu, J. Math. Phys. 48, 092106 (2007), arXiv:quant-ph/0507015 DOI: 10.1063/1.2196243. Web of Science, Google Scholar
T. Curtright, L. Mezincescu and D. Schuster, J. Math. Phys. 48, 092108 (2007), arXiv:quant-ph/0603170 DOI: 10.1063/1.2191367. Web of Science, Google Scholar
C. M. Bender, S. Boettcher and P. Meisinger, J. Math. Phys. 40, 2201 (1999), arXiv:quant-ph/9809072 DOI: 10.1063/1.532860. Web of Science, ADS, Google Scholar
C. M. Bender, Contemp. Phys. 46, 277 (2005), arXiv:quant-ph/0501052 DOI: 10.1080/00107500072632. Web of Science, ADS, Google Scholar
W. K. Wootters, Phys. Rev. D 23, 357 (1981), DOI: 10.1103/PhysRevD.23.357. Web of Science, ADS, Google Scholar
S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994), DOI: 10.1103/PhysRevLett.72.3439. Web of Science, ADS, Google Scholar
R. Sorkin, Mod. Phys. Lett. A 9, 3119 (1994), DOI: 10.1142/S021773239400294X. Link, Web of Science, ADS, Google Scholar
U. Sinhaet al., AIP Conf. Proc. 1101 (2009) p. 200, arXiv:0811.2068. Google Scholar
U. Sinhaet al., Science 329, 418 (2010), arXiv:1007.4193v1 DOI: 10.1126/science.1190545. Web of Science, ADS, Google Scholar
W. van Dam, Nonlocality and communication complexity, Chap. 9, D. Phil. thesis, Department of Physics, University of Oxford (2000) . Google Scholar
W. van Dam, Nat. Comput. 12, 9 (2013), arXiv:quant-ph/0501159 DOI: 10.1007/s11047-012-9353-6. Web of Science, Google Scholar
G. Brassardet al., Phys. Rev. Lett. 96, 250401 (2006), arXiv:quant-ph/0508042 DOI: 10.1103/PhysRevLett.96.250401. Web of Science, Google Scholar
G. Brassard, Nature Phys. 1, 2 (2005), DOI: 10.1038/nphys134. Web of Science, ADS, Google Scholar
S. Popescu, Nature Phys. 2, 507 (2006), DOI: 10.1038/nphys374. Web of Science, ADS, Google Scholar
J. Barrett, Phys. Rev. A 75, 032304 (2007), arXiv:quant-ph/0508211 DOI: 10.1103/PhysRevA.75.032304. Web of Science, Google Scholar
N. Brunner and P. Skrzypczyk, Phys. Rev. Lett. 102, 160403 (2009), DOI: 10.1103/PhysRevLett.102.160403. Web of Science, Google Scholar
C. A. Fuchs, [quant-ph], and references therein , arXiv:1003.5182v1 . Google Scholar
S. Banach , Theory of Linear Operations ( Dover , 2009 ) . Google Scholar
N. L. Carothers , A Short Course on Banach Space Theory ( Cambridge University Press , 2004 ) . Google Scholar
A. Pietsch , History of Banach Spaces and Linear Operators ( Birkhäuser , 2007 ) . Google Scholar
P. Hajek et al. , Biorthogonal Systems in Banach Spaces ( Springer , 2007 ) . Google Scholar
O. Aharonyet al., Phys. Rep. 323, 183 (2000), arXiv:hep-th/9905111 DOI: 10.1016/S0370-1573(99)00083-6. Web of Science, ADS, Google Scholar
C. M. Hull, J. High Energy Phys. 9807, 021 (1998), arXiv:hep-th/9806146. ADS, Google Scholar
E. Witten , arXiv:hep-th/0106109 . Google Scholar
V. Balasubramanian, J. de Boer and D. Minic, Class. Quantum Grav. 19, 5655 (2002), arXiv:hep-th/0207245 DOI: 10.1088/0264-9381/19/22/302. Web of Science, Google Scholar
V. Balasubramanian, J. de Boer and D. Minic, Phys. Rev. D 65, 123508 (2002), arXiv:hep-th/0110108 DOI: 10.1103/PhysRevD.65.123508. Web of Science, ADS, Google Scholar
J. M. Maldacena, J. High Energy Phys. 0305, 013 (2003), arXiv:astro-ph/0210603. ADS, Google Scholar
M. Gell-Mann and J. B. Hartle, Phys. Rev. D 47, 3345 (1993), arXiv:gr-qc/9210010 DOI: 10.1103/PhysRevD.47.3345. Web of Science, ADS, Google Scholar
J. B. Hartle, Int. J. Theor. Phys. 45, 1390 (2006), arXiv:gr-qc/0510126 DOI: 10.1007/s10773-006-9134-z. Web of Science, Google Scholar
J. B. Hartle , arXiv:gr-qc/0602013 . Google Scholar
L. Hardy , arXiv:gr-qc/0509120 . Google Scholar
L. Hardy, J. Phys. A 40, 3081 (2007), arXiv:gr-qc/0608043 DOI: 10.1088/1751-8113/40/12/S12. ADS, Google Scholar
V. Balasubramanian, J. de Boer and D. Minic , arXiv:gr-qc/0211003 . Google Scholar
D. Minic and C. H. Tze, Phys. Rev. D 68, 061501 (2003), arXiv:hep-th/0305193 DOI: 10.1103/PhysRevD.68.061501. Web of Science, Google Scholar
D. Minic and C. H. Tze, Phys. Lett. B 581, 111 (2004), arXiv:hep-th/0309239 DOI: 10.1016/j.physletb.2003.11.054. Web of Science, ADS, Google Scholar
D. Minic and C. H. Tze , arXiv:hep-th/0401028 . Google Scholar
V. Jejjala, D. Minic and C. H. Tze, Int. J. Mod. Phys. D 13, 2307 (2004), arXiv:gr-qc/0406037 DOI: 10.1142/S0218271804006371. Link, Web of Science, ADS, Google Scholar
V. Jejjala, M. Kavic and D. Minic, Int. J. Mod. Phys. A 22, 3317 (2007), arXiv:0706.2252 DOI: 10.1142/S0217751X07036981. Link, Web of Science, ADS, Google Scholar
V. Jejjalaet al., Int. J. Mod. Phys. A 25, 2515 (2010), arXiv:0804.3598 DOI: 10.1142/S0217751X10049013. Link, Web of Science, ADS, Google Scholar
L. Hardy , arXiv:quant-ph/0101012 . Google Scholar
L. Hardy, [quant-ph] , arXiv:1104.2066 . Google Scholar
E. Witten, Nucl. Phys. B 268, 253 (1986), DOI: 10.1016/0550-3213(86)90155-0. Web of Science, ADS, Google Scholar
S. J. Freedman and J. F. Clauser, Phys. Rev. Lett. 28, 938 (1972), DOI: 10.1103/PhysRevLett.28.938. Web of Science, ADS, Google Scholar
J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526 (1974), DOI: 10.1103/PhysRevD.10.526. Web of Science, ADS, Google Scholar
A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 47, 460 (1981), DOI: 10.1103/PhysRevLett.47.460. Web of Science, ADS, Google Scholar
A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 49, 91 (1982), DOI: 10.1103/PhysRevLett.49.91. Web of Science, ADS, Google Scholar
A. Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett. 49, 1804 (1982), DOI: 10.1103/PhysRevLett.49.1804. Web of Science, Google Scholar
J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D 28, 679 (1983), DOI: 10.1103/PhysRevD.28.679. Web of Science, ADS, Google Scholar
Planck Collab., Planck 2013 Results. XXIV. Constraints on primordial non-Gaussianity, [astro-ph.CO], and references therein , arXiv:1303.5084 . Google Scholar
D. Edmonds, D. Farrah, C. M. Ho, D. Minic, Y. J. Ng and T. Takeuchi, [astro-ph.CO] , arXiv:1308.3252 . Google Scholar
C. M. Ho, D. Minic and Y. J. Ng, Phys. Lett. B 693, 567 (2010), arXiv:1005.3537 DOI: 10.1016/j.physletb.2010.09.008. Web of Science, ADS, Google Scholar
C. M. Ho, D. Minic and Y. J. Ng, Gen. Relativ. Gravit. 43, 2567 (2011), arXiv:1105.2916 DOI: 10.1007/s10714-011-1200-z. Web of Science, ADS, Google Scholar
C. M. Ho, D. Minic and Y. J. Ng, Phys. Rev. D 85, 104033 (2012), arXiv:1201.2365 DOI: 10.1103/PhysRevD.85.104033. Web of Science, Google Scholar
U. Aydemir, D. Minic and T. Takeuchi, Phys. Lett. B 724, 301 (2013), arXiv:1304.6092 DOI: 10.1016/j.physletb.2013.06.044. Web of Science, ADS, Google Scholar