Geometry Over a Finite Field

Abstract

^> The development of certain aspects of a physically interpretable geometry defined over a finite field is presented. The concepts of order, norm, metric, inner product, etc. are developed over a subset of the total field. It is found that the finite discrete space behaves ..locally, not globally, like the conventional "continuous" spaces. The implications of this behavior for mathematical induction and the limit procedure are discussed, and certain radical conclusions are reached. Among these are: (a) mathematical induction ultimately fails for. a finite system and further extension leads to the introduction of formal indeterminancy; (b) finite space-time operations have inherent formal properties like those heretofore attributed to the substantive physical universe, and (c) certain formal properties attributed to continuous spaces cannot be developed from successive embedding in finite space of finer resolution-but must be based on indc7c: lent axiomatic (nontestable) assumptions. It is suggested that a finite field representation should be used as the fundamental basis of a physical representation. Jj ■ ; i ; ' , i Rapioduod by NATIONAL TECHNICAL .NFORMATION SERVICE Sprin»fl.W. V.. MtSI DISTRIBUTION STATEMENT Thi* document ha« been approved (or public reloam« and sole; its distribution is unlimited. Dm-2 ~i to 200 x 10 and declare the answer to be Ik x 10. In this way, replacement permits our mathematical operations to continue and avoids cessation due to uncomputability or lack of performable instructions. One couid also construct his system to reset itself to some arbitrary point-say zero-whenever an impasse is reached. However, the rationale for replacement is clearly preferable because it seeks a "nearby" problem and we shall adopt it.

Discrete quantum theories

“…We next discuss an approach using finite complexifiable fields that conditionally resolves the inner product condition (C), which is violated by the theory just presented. A possible path is suggested by the work of Reisler and Smith [20]. The general idea is that while the cyclic properties of arithmetic in finite fields make it impossible to globally obtain the desired properties of the conventional Hilbert space inner product, it is possible to recover them locally, thereby restoring, with some restrictions, all the usual properties of the inner product needed for conventional quantum mechanics and conventional quantum computing.…”

“…As the size of the discrete field becomes large, the size of the locally valid computational framework grows as well, leading to the effective emergence of conventional quantum theory. We next briefly outline such a context for local orderable subspaces of a finite field, and introduce an improvement on the original method [20] suggested by recent number theory resources [21].…”

“…However, it is impossible to achieve this re-weighting precisely in a discrete theory because the square roots cannot be calculated exactly in finite fields. We can, however, produce successively more accurate approximations of square roots with bigger and bigger fields using a prescription suggested by Reisler and Smith [20]. We denote the approximate square root of m > 0 in a finite field F p by √ m. This approximate square root is calculated by taking the usual square root of the smallest element in the ordered range S 0 (k) that is greater than m and that is a quadratic residue.…”

“…For computational purposes, this sequence is preferable to the one proposed by Reisler and Smith[20] because it produces smaller primes. Their work showed that a sufficient condition on finite fields to produce sequences of quadratic residues is to further constrain the underlying prime numbers to be of the form 8 m i=1 q i − 1, where q i is the ith odd prime.…”