Preface

Summary

For a few hundred years theoretical physics has been developed on the basis of real and, later, complex numbers. This mathematical model of physical reality survived even in the process of the transition from classical to quantum physics – complex numbers became more important than real, but not essentially more so than in the Fourier analysis which was already being used, e.g., in classical electrodynamics and acoustics. However, in the last 20 years the field of p-adic numbers ℚ_p (as well as its algebraic extensions, including the field of complex p-adic numbers ℂ_p) has been intensively used in theoretical and mathematical physics (see [1]–[3], [10]–[15], [24], [35], [38], [43], [44], [55], [64], [65], [77]–[79], [89], [94], [115]–[122], [123], [124], [133], [157], [158], [174], [198], [241]–[246] and the references therein). Thus, notwithstanding the fact that p-adic numbers were only discovered by K. Hensel around the end of the nineteenth century, the theory of p-adic numbers has already penetrated intensively into several areas of mathematics and its applications.

The starting point of applications of p-adic numbers to theoretical physics was an attempt to solve one of the most exciting problems of modern physics, namely, to combine consistently quantum mechanics and gravity, thus to create a theory of quantum gravity. In spite of the considerable success of somemodels, there is still no satisfactory general theory.