No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

The full text of this article is not currently available.

Application of abelian holonomy formalism to the elementary theory of numbers

### Abstract

We consider an abelian holonomy operator in two-dimensional conformal field theory with zero-mode contributions. The analysis is made possible by use of a geometric-quantization scheme for abelian Chern-Simons theory on *S* ^{1} × *S* ^{1} × **R**. We find that a purely zero-mode part of the holonomy operator can be expressed in terms of Riemann's zeta function. We also show that a generalization of linking numbers can be obtained in terms of the vacuum expectation values of the zero-mode holonomy operators. Inspired by mathematical analogies between linking numbers and Legendre symbols, we then apply these results to a space of **F** _{ p } = **Z**/*p* **Z**, where *p* is an odd prime number. This enables us to calculate “scattering amplitudes” of identical odd primes in the holonomy formalism. In this framework, the Riemann hypothesis can be interpreted by means of a physically obvious fact, i.e., there is no notion of “scattering” for a single-particle system. Abelian gauge theories described by the zero-mode holonomy operators will be useful for studies on quantum aspects of topology and number theory.

© 2012 American Institute of Physics

Received 24 October 2011
Accepted 26 April 2012
Published online 16 May 2012

Article outline:

I. INTRODUCTION
II. REVIEW OF ABELIAN HOLONOMY OPERATORS
A. Braid groups, KZ equation, and comprehensive gauge fields
B. Abelian holonomy operators
III. GEOMETRIC QUANTIZATION AND LINKING NUMBERS
A. Holonomies of torus
B. Geometric quantization of the toric *U*(1) Chern-Simons theory
C. Realization of linking numbers
IV. ZERO-MODE CONTRIBUTIONS TO THE ABELIAN HOLONOMY OPERATOR
A. Construction of Θ_{ R, γ}(*z*, *a*)
B. Extraction of
C. Integral representation of Riemann's zeta function
D. Generalization of linking numbers
E. Summary
V. APPLICATION TO THE ELEMENTARY THEORY OF NUMBERS
A. Linking numbers, Legendre symbols, and Jacobi symbols
B. Correspondence between knots and primes
C. Quantum realization of linking numbers
D. Analogs of scattering amplitudes for prime numbers
E. The Gauss sum as a prime-creation operator
F. Physical interpretation of the Riemann hypothesis
VI. CONCLUDING REMARKS

/content/aip/journal/jmp/53/5/10.1063/1.4716186

http://aip.metastore.ingenta.com/content/aip/journal/jmp/53/5/10.1063/1.4716186

Article metrics loading...

/content/aip/journal/jmp/53/5/10.1063/1.4716186

2012-05-16

2016-10-26

Full text loading...

###
Most read this month

Article

content/aip/journal/jmp

Journal

5

3

Commenting has been disabled for this content