Gauss Optics and Gauss Sum on an Optical Phenomena

Abstract

In the previous article (Found Phys. Lett. 16 325-341), we showed that a reciprocity of the Gauss sums is connected with the wave and particle complementary. In this article, we revise the previous investigation by considering a relation between the Gauss optics and the Gauss sum based upon the recent studies of the Weil representation for a finite group.

On a commutative ring structure in quantum mechanics

“…The multi-photon absorption in quantum optics is a sure sign of these excitations. It implies one of answers why the quantum mechanics provides a connection with number theory and p-adic structure [M,M1,M2,MO,V1,V2,VVZ].…”

“…There prime numbers play the crucial roles whereas they basically have no meaning in an additive group. However number theory and quantum mechanics sometimes are connected [M,M1,M2,MO,V1,V2,VVZ]. I attempted to answer a question why quantum mechanics is connected with integer theory in [M2,MO].…”

“…However number theory and quantum mechanics sometimes are connected [M,M1,M2,MO,V1,V2,VVZ]. I attempted to answer a question why quantum mechanics is connected with integer theory in [M2,MO]. This article is one of the attempts.…”

“…In [BK,M2,MO], it was showed that the Gauss sum which is a number theoretic object plays the central roles in an interference phenomenon, the fractional Talbot phenomenon [WW]. As I investigated the algebraic structures behind the connection between wave physics and number theory, there are SL(2, R) and SL(2, Z) ⊂ SL(2, R) [M2].…”

“…This commutative ring structure is the key structure in the fast Fourier transformation. I consider that it also plays the crucial roles in the connection in the interference phenomenon [BK,MO,M2], though the property (1.1) comes from the primitive fact that the set of the integers naturally has a commutative ring structure. In other words, the Fourier series as the representation space of the additive group brings the commutative ring structure to the interference phenomenon and contributes to the connection between number theory and quantum mechanics.…”