Finite geometries in quantum theory

Veeravalli S. Varadarajan

Semiu Oladipupo Oladejo, Adediran Dauda Adeshola, Adedayo David Adeniyi

In this work, join and meet algebraic structure which exists in non-near-linear finite geometry are discussed. Lines in non-near-linear finite geometry were expressed as products of lines in near-linear finite geometry (where p is a prime). An existence of lattice between any pair of near-linear finite geometry of is confirmed. For q|d, a one-to-one correspondence between the set of subgeometry of and finite geometry from the subsets of the set {D(d)} of divisors of d (where each divisor represents a finite geometry) and set of subsystems {∏(q)} (with variables in Zq) of a finite quantum system ∏(d) with variables in Zd and a finite system from the subsets of the set of divisors of d is established.

Yrjö Ahmavaara

The world geometry underlying the theory of elementary particles is investigated. The elementary, non-relativistic geometry of the physical space is given a general formulation, sufficient to yield the quanta of momentum and energy when the group quantization process is applied to the physical space. The elementary geometry is represented by a commutative (finite or continuous) group of translations to which Pontrjagin's theorem concerning the duality between a commutative group and its character group applies. The principle of uncertainty is given a group theoretical formulation in terms of the Pontrjaginian duality. This formulation allows a consequent and meaningful extension of quantum theory to a finite geometry. The role of the relativistic (finite or continuous) geometry of space-time in the theory of elementary particles is investigated. A formulation of the problem of elementary particles is given, and an eventual connection between the observed elementary particles and a geometry of the Lorentz group space is suggested. (auth)

Veeravalli S. Varadarajan

Xun Chun-ling

Using the low density parity check matrices created by finite geometric approach with its dual codes,which are constructed by splitting the rows of the LDPC check matrices,a quantum LDPC codes construction method is proposed based on finite geometries in this paper. Finally,this method is proved by theory deduction adequately.

James Thomas Wheeler