Vol 10, No 3 (2018)

This paper, which is a summary (in which considerable creative license has been taken) of the author’s talk at the sixth international conference on p-adic mathematical physics and its applications (CINVESTAV, Mexico City, October 2017), reviews some recent work connecting field theories defined on the p-adic numbers and ideas from the AdS/CFT correspondence. Some results are included, along with general discussion of the utility and interest of p-adic analogues of Lagrangian field theories, at least from the author’s perspective. A few challenges, shortcomings, and ideas for future work are also discussed.

For any prime number p let Ω_p be the p-adic counterpart of the complex numbers C. In this paper we investigate the class of p-adic UHF Banach algebras. A p-adic UHF Banach algebra is any unital p-adic Banach algebra A of the form \(A = \overline {U{M_n}} \), where (M_n) is an increasing sequence of p-adic Banach subalgebras of M such that each M_n is generated over Ω_p by an algebraic system of matrix units {e_ij^{( n)} | 1 ≤ i, j ≤ p_n }. The main result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra.

In this paper, we provide necessary and sufficient conditions for 1-Lipschitz functions that are uniformly differentiable mod p on ℤ_p to be measure-preserving, in Mahler’s expansion, and show that these conditions can be modified to guarantee the existence of a root of p-adic Lipschitz functions.

The differential phase shift quantum key distribution protocol is of high interest due to its relatively simple practical implementation. This protocol uses trains of coherent pulses and allows the legitimate users to resist individual attacks. In this paper, a new attack on this protocol is proposed which is based on the idea of information extraction from the part of each coherent state and then making decision about blocking the rest part depending on the amount of extracted information.