$p$-Adic Mathematical Physics: The First 30 Years (1705.04758v1)

Abstract: $p$-Adic mathematical physics is a branch of modern mathematical physics based on the application of $p$-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of $p$-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

Overview of "p-Adic Mathematical Physics: The First 30 Years"

The paper "p-Adic Mathematical Physics: The First 30 Years" by B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, and E. I. Zelenov examines the development and applications of p-adic mathematical physics over the last three decades. This field, which utilizes p-adic numbers and ultrametric spaces, aims to provide insights into complex and hierarchical systems at small spacetime scales.

Mathematical Foundations

The work begins by exploring the mathematical underpinnings that drive p-adic physics, such as the Vladimirov fractional operator and wavelets. The authors methodically detail how p-adic numbers, initially introduced by K. Hensel in 1897, and adeles serve as a non-Archimedean framework for understanding phenomena beyond classical real number analysis. Through tools like p-adic wavelets and pseudodifferential operators, the paper highlights how hierarchies and connectivity in ultrametric spaces can be modeled more efficiently.

Highlighted Applications in Physics

The review comprehensively discusses the utility of p-adic analysis in diverse physical theories:

1. Strings, Fields, and Quantum Theories: p-Adic strings provide an alternative perspective on string theory dynamics, especially useful when exploring aspects such as Veneziano scattering amplitudes. The development of p-adic string theory, effective Lagrangians, and the intriguing connection to the Riemann zeta function are emphasized.
2. p-Adic Quantum Mechanics and Cosmology: p-Adic and adelic formulations offer novel frameworks for quantum mechanics, integrating contributions from all primes p. In cosmology, the potential for p-adic gravity models brings new perspectives on Planck scale physics and suggests fundamental invariance under changes in number fields.
3. Stochastic Processes and Complexity: The paper also explores p-adic stochastic processes and their use in modeling diffusion and Brownian motion. Particularly, the potential of these methods in statistical mechanics of disordered systems and complex systems like spin glasses is noted.

Biological Implications

Significantly, p-adic methods have found applications in modeling complex biological systems. For instance, the dynamics of proteins and genetic code structuring illustrate hierarchical organization, making p-adic spaces effective tools for understanding biological ultrametricity.

Broader Implications and Future Prospects

The paper speculates about future developments in p-adic mathematical physics, especially concerning its interdisciplinary applications. The review suggests promising advancements in data mining, cryptography, and information security, rooted in the continuity offered by p-adic analysis.

In conclusion, the authors present a thorough and dense review that underscores both the theoretical advancements and practical applications of p-adic mathematical physics. They provide a roadmap for future research that bridges mathematical innovation with applied sciences, offering insight into systems that are inherently complex and hierarchical. The paper itself acts as a nexus of past achievements and prospective research trajectories within this niche yet potent area of paper.