- The paper introduces a novel regularization method for p-adic string amplitudes using multivariate local zeta functions to achieve meromorphic continuation.
- It adapts Hironaka’s resolution of singularities to analyze convergence and compute p-adic N-tachyon integrals effectively.
- By linking p-adic analysis with motivic integration, the work uncovers fresh insights for regularization in quantum field theory and string theory.
Regularization of p-adic String Amplitudes and Multivariate Local Zeta Functions
This paper explores the intricate connections between p-adic string amplitudes and multivariate local zeta functions, often referred to as Igusa's local zeta functions. The primary focus of the work lies in the establishment of meromorphic continuations for integrals associated with p-adic versions of string amplitudes, uncovering formal parallels with their Archimedean counterparts, and regularizing these computations through local zeta functions.
Discussion Overview
At its core, the paper is concerned with extending the framework of p-adic string theory, a domain revitalized by Volovich's conjecture that proposes a non-Archimedean structure of space-time at the Planck scale. By transposing the well-known formalism of Veneziano amplitudes into the p-adic context, the authors provide an approach to regularize integrals of p-adic open string N-tachyon amplitudes. The regularization methodology draws heavily on the analytic properties of multivariate local zeta functions.
Methodology and Results
Key results in the paper revolve around the construction and manipulation of p-adic open string N-point zeta functions, defined for complex parameters associated with vertex insertion points on a Riemann surface. A significant theorem demonstrated in the work is the extension of these zeta functions into entire holomorphic functions in specific domains within the complex plane. This result hinges on the application of Hironaka’s resolution of singularities in algebraic geometry, appropriately adapted for the p-adic setting.
The authors present a detailed analysis of the convergence properties of these integrals and their meromorphic extensions. By unveiling an algorithm for computing such regularizations, the paper enhances the practical toolkit available for p-adic string theory computations. These contributions are made possible by leveraging the intricate properties of zeta functions which, as shown, can be elegantly connected to fundamental questions in theoretical physics regarding particle interactions and quantum field definitions.
Implications and Future Directions
The implications of the paper are multifaceted, affecting both theoretical developments in string theory and practical computations in particle physics. The work underscores the utility of p-adic analysis in the domain of mathematical physics, especially in defining regularized amplitudes where traditional methods may falter. Furthermore, by illustrating a connection to motivic integration, the paper hints at broader theoretical implications for constructing geometrically motivated string amplitudes.
The extension of Igusa's local zeta functions as motivically specialized to account for p-adic conditions suggests enticing avenues for further exploration. These nascent concepts could lead to fresh insights into the link between p-adic and classical models of physical phenomena, especially if the empirical evidence continues to support correspondences as p approaches 1.
Conclusion
This paper rigorously expands the analytical framework applied to p-adic string amplitudes. By establishing regularization techniques and showcasing the amenability of local zeta functions to p-adic approaches, it provides a valuable model for the regularization of divergences encountered in quantum field theories. This intersection between abstract mathematical constructs and real-world physical theories posits exciting research prospects, potentially unraveling new layers of understanding within the structure of the quantum cosmos. As research progresses, further refinement of these theoretical models and potential empirical validation could open substantive insights into the p-adic realms of string and quantum field theories.