# Introduction to a Quantum Theory over a Galois Field

## Abstract

**:**

**PACS**02.10.De; 03.65.Ta; 11.30.Fs; 11.30.Ly

## 1. Motivation

- Quantum states are represented by elements of a linear projective space over a Galois field and physical quantities are represented by linear operators in that space.

## 2. Modular IRs of the sp(2) Algebra

## 3. Modular IRs of the so(2,3) Algebra

## 4. Massless Particles and Dirac Singletons

## 5. Matrix Elements of Representation Operators

## 6. Quantization and AB Symmetry

## 7. Physical and Nonphysical States

## 8. AdS Symmetry Breaking

## 9. Dirac Vacuum Energy Problem

## 10. Neutral Particles and Spin-Statistics Theorem

- i) it is a particle coinciding with its antiparticle
- ii) it is a particle which does not coincide with its antiparticle but they have the same properties

## 11. Modular IRs of the osp(1,4) Superalgebra

**define**the so(2,3) generators as follows:

- If ${q}_{2}>1$ and $s\ne 0$ (massive IRs), the osp(1,4) supermultiplets contain four IRs of the so(2,3) algebra characterized by the values of the mass and spin $(m,s),(m+1,s+1),(m+1,s-1),(m+2,s).$
- If ${q}_{2}>1$ and $s=0$ (collapsed massive IRs), the osp(1,4) supermultiplets contain three IRs of the so(2,3) algebra characterized by the values of the mass and spin $(m,s),(m+1,s+1),(m+2,s).$
- If ${q}_{2}=1$ (massless IRs) the osp(1,4) supermultiplets contains two IRs of the so(2,3) algebra characterized by the values of the mass and spin $(2+s,s),(3+s,s+1)$
- Dirac supermultiplet containing two Dirac singletons (see Section 4).

## 12. Discussion

## Acknowledgements

## References

- Weinberg, S. The Quantum Theory of Fields; Cambridge University Press: Cambridge, UK, 1999; Volume I, p. 602. [Google Scholar]
- Weinberg, S. Living with infinities. arXiv:0908.1964 (hep-th).2009. [Google Scholar]
- Bogolubov, N.N.; Logunov, A.A.; Oksak, A.I.; Todorov, I.T. General Principles of Quantum Field Theory; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1987; p. 682. [Google Scholar]
- Adler, S.L. Quaternionic Quantum Mechanics and Quantum Fields; Oxford University Press: Oxford, UK, 1995; p. 608. [Google Scholar]
- Dragovich, B.; Khrennikov, A.Y.; Kozyrev, S.V.; Volovich, I.V. On p-adic mathematical physics. P-Adic Num. Ultra. Anal. Appl.
**2009**, 1, 1–17. [Google Scholar] [CrossRef] - Coish, H.R. Elementary particles in a finite world geometry. Phys. Rev.
**1959**, 114, 383–393. [Google Scholar] [CrossRef] - Shapiro, I.S. Weak interactions in the theory of elementary particles with finite space. Nucl. Phys.
**1960**, 21, 474–491. [Google Scholar] [CrossRef] - Nambu, Y. Field Theory Of Galois Fields. In Quantum Field Theory and Quantum Statistics; Batalin, I.A., Isham, C.J., Eds.; Adam Hilger: Bristol, UK, 1987; pp. 625–636. [Google Scholar]
- Vourdas, A. Quantum systems with finite Hilbert space. Rep. Progr. Phys.
**2004**, 67, 267–320. [Google Scholar] [CrossRef] - Doughty, H. Hints of Finiteness. From the instant proceedings of the 89-92 meetings of Alternative Natural Philosophy Association. Private Communication of June 24th, 2004. [Google Scholar]
- Vourdas, A.; Banderier, C. Symplectic transformations and quantum tomography in finite quantum systems. J. Phys.
**2010**, 43, 042001: 1-18. [Google Scholar] [CrossRef] - Volovich, I.V. Number theory as the ultimate physical theory. P-Adic Num. Ultra. Anal. Appl.
**2010**, 2, 77–87. [Google Scholar] [CrossRef] - Planat, M.; Saniga, M. Finite geometries in quantum theory: From Galois (fields) to Hjelmslev (rings). J. Mod. Phys.
**2006**, B20, 1885–1892. [Google Scholar] - Planat, M. Huyghens, Bohr, Riemann and Galois: Phase-locking. Int. J. Mod. Phys.
**2006**, B20, 1833–1850. [Google Scholar] [CrossRef] - Saniga, M.; Havlicek, H.; Planat, M.; Pracna, P. Twin "Fano-Snowflakes" over the smallest ring of ternions. SIGMA
**2008**, 4, 050: 1-7. [Google Scholar] [CrossRef] - Rosen, J. Time, c and nonlocality: A glimpse beneath the surface. Phys. Essays
**1994**, 7, 335–339. [Google Scholar] [CrossRef] - Rosen, J. Symmetry Rules: How Science and Nature Are Founded on Symmetry; Springer: Berlin, Germany, 2008; p. 305. [Google Scholar]
- Rickles, D. Time and structure in canonical gravity. In The Structural Foundations of Quantum Gravity; Rickles, D., French, S., Saatsi, J.T., Eds.; Oxford University Press: Oxford, UK, 2006; pp. 152–195. [Google Scholar]
- Newton, T.D.; Wigner, E.P. Localized states for elementary systems. Rev. Mod. Phys.
**1949**, 21, 400–405. [Google Scholar] [CrossRef] - Berestetsky, V.B.; Lifshits, E.M.; Pitaevsky, L.P. Relativistic Quantum Theory; Nauka: Moscow, Russia, 1968; Volume IV, Part 1, p. 480. [Google Scholar]
- Rickles, D. Who’s afraid of background independence? Philos. Found. Phys.
**2008**, 4, 133–152. [Google Scholar] - Van der Waerden, B.L. Algebra I; Springer-Verlag: New York, NY, USA, 1967; p. 648. [Google Scholar]
- Ireland, K.; Rosen, M. A Classical Introduction to Modern Number Theory; Springer-Verlag: New York, NY, USA, 1987; p. 394. [Google Scholar]
- Davenport, H. The Higher Arithmetic; Cambridge University Press: Cambridge, UK, 1999; p. 241. [Google Scholar]
- Lev, F.M. Modular representations as a possible basis of finite physics. J. Math. Phys.
**1989**, 30, 1985–1998. [Google Scholar] [CrossRef] - Lev, F.M. Finiteness of Physics and Its Possible Consequences. J. Math. Phys.
**1993**, 34, 490–527. [Google Scholar] [CrossRef] - Lev, F.M. Quantum Theory on a Galois Field. hep-th/0403231, 2004.
- Lev, F.M. Why is quantum theory based on complex numbers? Finite Fields Appl.
**2006**, 12, 336–356. [Google Scholar] [CrossRef] - Dirac, P.A.M. Forms of relativistic dynamics. Rev. Mod. Phys.
**1949**, 21, 392–399. [Google Scholar] [CrossRef] - Weinberg, S. What is quantum field theory, and what did we think it is? hep-th/9702027. 1997. [Google Scholar]
- Wigner, E.P. On Unitary Representations of the Inhomogeneous Lorentz Group. Ann. Math.
**1939**, 40, 149–204. [Google Scholar] [CrossRef] - Zassenhaus, H. The Representations of lie algebras of prime characteristic. Proc. Glasgow Math. Assoc.
**1954**, 2, 1–36. [Google Scholar] [CrossRef] - Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurement of omega and lambda from H42 high-redshift supernovae. Astrophys. J.
**1999**, 517, 565–586. [Google Scholar] [CrossRef] - Melchiorri, A.; Ade, P.A.R.; de Bernardis, P.; Bock, J.J.; Borrill, J.; Boscaleri, A.; Crill, B.P.; De Troia, G.; Farese, P.; Ferreira, P. G.; et al. A Measurement of omega from the north American rest flight of boomerang. Astrophys. J.
**2000**, 536, L63–L66. [Google Scholar] [CrossRef] - Witten, E. Quantum gravity in de Sitter space. hep-th/0106109. 2001. [Google Scholar]
- Evans, N.T. Discrete series for the universal covering group of the 3+2 de Sitter group. J. Math. Phys.
**1967**, 8, 170–184. [Google Scholar] [CrossRef] - Braden, B. Restricted representatins of classical lie algebras of types A
_{2}and B_{2}. Bull. Amer. Math. Soc.**1967**, 73, 482–486. [Google Scholar] [CrossRef] - Lev, F.M.; Mirmovich, E.G. Some aspects of de Sitter invariant theory. VINITI
**1984**. No 6099 Dep. [Google Scholar] - Dirac, P.A.M. A remarkable representation of the 3 + 2 de Sitter group. J. Math. Phys.
**1963**, 4, 901–909. [Google Scholar] [CrossRef] - Lev, F. Massless elementary particles in a quantum theory over a Galois field. Theor. Math. Phys.
**2004**, 138, 208–225. [Google Scholar] [CrossRef] - Ikeda, N.; Fukuyuama, T. Fermions in (anti) de Sitter Gravity in four dimensions. Prog. Theor. Phys.
**2009**, 122, 339–353. [Google Scholar] [CrossRef] - Inonu, E.; Wigner, E.P. Representations of the Galilei group. Il Nuovo Cimento
**1952**, 9, 705–718. [Google Scholar] [CrossRef] - Flato, M.; Fronsdal, C. One massles particle equals two Dirac singletons. Lett. Math. Phys.
**1978**, 2, 421–426. [Google Scholar] [CrossRef] - Fetter, A.L.; Walecka, J.D. Quantum Theory of Many-Particle Systems; Dover Publications Inc.: Mineola, NY, USA, 2003; p. 645. [Google Scholar]
- Weinberg, S. The Quantum Theory of Fields, Volume III Supersymmetry; Cambridge University Press: Cambridge, UK, 2000; p. 419. [Google Scholar]
- Heidenreich, W. All linear unitary irreducible representations of de Sitter supersymmetry with positive energy. Phys. Lett.
**1982**, B110, 461–464. [Google Scholar] [CrossRef] - Fronsdal, C. Dirac supermultiplet. Phys. Rev.
**1982**, D26, 1988–1995. [Google Scholar] [CrossRef] - Lev, F.M. Supersymmetry in a quantum theory over a Galois field. hep-th/0209229. 2002. [Google Scholar]
- Giulini, D. Superselection Rules. arXiv:0710.1516v2 (quant-ph).2009. [Google Scholar]
- Lev, F.M. Could only fermions be elementary? J. Phys.
**2004**, A37, 3285–3304. [Google Scholar] [CrossRef]

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Lev, F.M.
Introduction to a Quantum Theory over a Galois Field. *Symmetry* **2010**, *2*, 1810-1845.
https://doi.org/10.3390/sym2041810

**AMA Style**

Lev FM.
Introduction to a Quantum Theory over a Galois Field. *Symmetry*. 2010; 2(4):1810-1845.
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**Chicago/Turabian Style**

Lev, Felix M.
2010. "Introduction to a Quantum Theory over a Galois Field" *Symmetry* 2, no. 4: 1810-1845.
https://doi.org/10.3390/sym2041810