p-Adic Analysis in Mathematical Physics

p-Adic analysis introduces non-Archimedean number fields into the toolkit of theoretical physics, offering a framework in which hierarchical and ultrametric structures are treated on the same footing as conventional real or complex geometries. In this approach, the field of p-adic numbers replaces or augments real coordinates, leading to novel formulations of quantum field theories, string amplitudes and statistical models. The essential feature of p-adic metrics—where distances satisfy a strong triangle inequality—naturally encodes tree-like or multiscale interactions. This has yielded fresh insights into renormalisation group flows, critical phenomena and the regularisation of divergences in amplitudes. Beyond high-energy contexts, p-adic methods have been applied to fluid transport in porous media, dynamics on Cayley trees and the modelling of complex networks. Recent advances exploit p-adic pseudo-differential operators and wavelet bases to solve analogues of the Navier–Stokes equation, while multivariate local zeta functions provide rigorous control over non-local interactions. Together, these developments have established p-adic analysis as a versatile bridge between pure number theory and a wide array of physical systems, from open string theory to condensed-matter hierarchies.

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Research from all publishers

Work on p-adic cellular neural networks has extended classical neural architectures to infinitely many hierarchical layers. These networks are modelled by integro-differential equations over p-adic spatial variables, and their solvability and numerical schemes demonstrate how hierarchical signal processing can be naturally captured in an ultrametric setting. In parallel, investigations into the p-adic Potts–Bethe mapping have revealed rich dynamical behaviour on Cayley trees, including the emergence of Julia sets and chaotic regimes. This underscores the deep links between non-Archimedean statistical models and complex dynamics, with implications for phase transitions in hierarchical lattices. Foundational studies of mixed real–p-adic field theories have constructed scalar and gauge models defined on products of Archimedean and non-Archimedean spaces. These frameworks exhibit lines of Gaussian fixed points that join onto Wilson–Fisher branches under tuning of dynamical exponents, and uncover novel oscillatory propagators whose amplitudes can be controlled via p-adic scales. Collectively, these contributions demonstrate the breadth of p-adic analysis in probing critical phenomena, non-locality and hierarchical interactions across physics.

p-Adic Analysis in Mathematical Physics publication trend

The graph below shows the total number of articles in p-adic analysis in mathematical physics across all publications each year (not limited to Nature Index journals).

Technical terms

p-Adic number: A number expressed in a base-p expansion allowing infinite digits to the left; distances are measured by divisibility by the prime p, yielding a non-Archimedean norm.

Ultrametric space: A metric space in which the triangle inequality strengthens to d(x,z) ≤ max{d(x,y), d(y,z)}, encoding hierarchical clustering.

Pseudo-differential operator: An operator generalising differentiation via symbols in momentum space, here defined over p-adic fields to capture non-local interactions.

Renormalisation group: A mathematical framework describing how physical systems change under scale transformations, here used to analyse fixed points in p-adic field theories.

References

1. Mixed field theory. Journal of High Energy Physics (2019).
2. Modeling Fluid’s Dynamics with Master Equations in Ultrametric Spaces Representing the Treelike Structure of Capillary Networks. Entropy (2016).
3. Chaotic behavior of the P-adic Potts-Bethe mapping. Discrete and Continuous Dynamical Systems (2018).
4. Solvability of the p-Adic Analogue of Navier–Stokes Equation via the Wavelet Theory. Entropy (2019).
5. p-Adic open string amplitudes with Chan-Paton factors coupled to a constant B-field. Nuclear Physics B (2020).
6. p-adic Cellular Neural Networks. Journal of Nonlinear Mathematical Physics (2022).
7. Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra. Transactions of the American Mathematical Society (2007).