Quantum ${{\mathbb{F}}_{{\rm un}}}$: theq= 1 limit of Galois field quantum mechanics, projective geometry and the field with one element

Introduction. In a paper called “A Theorem in Finite Projective Geometry and some Applications to Number Theory” [Trans. Amer. Math. Soc, vol. 43 (1938), 377-385], J. Singer proved that the finite projective geometry PG(s — 1,pn), that is the projective geometry of dimension s — 1 whose coordinate field is the Galois field GF(pn), admits a collineation L of period q = (psn — 1)/ (pn — 1). Since this q is the number of points of PG(s — 1, pn), Singer's result states that the points of PG(s — 1, pn) are cyclically arranged. Singer's construction of L uses the notion of a “primitive irreducible polynomial of degree 5 belonging to a field GF(pn) which defines a PG(s — 1, pn).”

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GALOIS FIELD QUANTUM MECHANICS

Modern Physics Letters B ◽

10.1142/s0217984913500644 ◽

2013 ◽

Vol 27 (10) ◽

pp. 1350064 ◽

Cited By ~ 8

Author(s):

LAY NAM CHANG ◽

ZACHARY LEWIS ◽

DJORDJE MINIC ◽

TATSU TAKEUCHI

Keyword(s):

Quantum Mechanics ◽

Vector Space ◽

Galois Field ◽

Bell’S Inequality ◽

Variable Theory ◽

Bell's Inequality ◽

Hidden Variable

We construct a discrete quantum mechanics (QM) using a vector space over the Galois field GF(q). We find that the correlations in our model do not violate the Clauser–Horne–Shimony–Holt (CHSH) version of Bell's inequality, despite the fact that the predictions of this discrete QM cannot be reproduced with any hidden variable theory.

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Some Results on Quadrics in Finite Projective Geometry Based on Galois Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-1962-010-2 ◽

1962 ◽

Vol 14 ◽

pp. 129-138 ◽

Cited By ~ 26

Author(s):

D. K. Ray-Chaudhuri

Keyword(s):

Projective Geometry ◽

Galois Field ◽

Interesting Property ◽

Combinatorial Problems ◽

Galois Fields ◽

Cambridge Philosophical Society ◽

Finite Projective Geometry

In a paper (5) published in the Proceedings of the Cambridge Philosophical Society, Primrose obtained the formulae for the number of points contained in a non-degenerate quadric in PG(n, s), the finite projective geometry of n dimensions based on a Galois field GF(s). In § 3 of the present paper the formulae for the number of p-flats contained in a non-degenerate quadric in PG(n, s) are obtained. In § 4 an interesting property of a non-degenerate quadric in PG(2k, 2m) is proved. These properties of a quadric will be used in solving some combinatorial problems of statistical interest in a later paper.

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Quadrics in finite geometries

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100026645 ◽

1951 ◽

Vol 47 (2) ◽

pp. 299-304 ◽

Cited By ~ 27

Author(s):

E. J. F. Primrose

Keyword(s):

Projective Geometry ◽

Galois Field ◽

Finite Geometries

The object of this paper is to obtain some properties of non-degenerate quadric primals in the projective geometry in [s] over a Galois field of order n. It will be shown that, if s is even, there is only one type of quadric, but that, if s is odd, there are two types of quadric. The number of points on a quadric of each type, and the number of quadrics of each type, will be found. Finally, a possible application to statistics will be indicated.

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