Quantum Mechanics for Non-Archimedean Wave Functions

Abstract

The quantum mechanics in which wave functions assume values in quadratic extensions of non-Archimedean fields was suggested in [72], [64], [58]. Representations of Bargmann—Fock and Schrödinger were constructed in these papers in the spaces of functions square summable with respect to the non-Archimedean Gauss and Lebesgue distributions. For quantization use was made of the calculus of pseudodifferential operators acting in the spaces of functions of a non-Archimedean argument with non-Archimedean values (this calculus is a natural generalization of the calculuses of pseudo-differential operators in an infinite-dimensional case [43], [45], [47], [48], [53], [54] and on a superspace [49], [50], [52], [55], [56]). What is common for all these calculuses is the absence of the Lebesgue measure. Here everything is realized in the sense of the theory of distributions. The principle of correspondence of the quantum mechanics and the classical non-Archimedean-valued mechanics was proved in the framework of calculus of pseudo-differential operators (the non-Archimedean deformation parameter was considered).