Field quantization in non-archimedean field theory (Q2640655)

A crucial problem in establishing a quantum field theory admitting prescribed features is always to find - and to properly interpret - a suitable, widely canonical procedure of quantization. This general problem has recently been encountered in the various attempts of creating possible string theories defined over non-archimedean groundfields, in particular over the (ultrametric) p-adic number fields. There are several proposals and approaches to quantization methods in non-archimedean field theories, among them the very first attempts by I. V. Volovich, B. Grossman, P. Freund, M. Olson, E. Witten, and others about five years ago. In these original approaches, which basically deal with the p-adic groundfields \({\mathbb{Q}}_ p\), the formulation of a p-adic bosonic field theory is based on some adelic relation that provides a link between the Veneziano amplitude and the four-point amplitudes \(A_ 4^{(p)}\) for all the p-adic bosonic strings, p being an arbitrary prime number. In the present paper, the author takes up this viewpoint and introduces a canonical quantization procedure for both bosonic and fermionic p-adic string fields. Moreover, he investigates the properties of the operators occuring in the corresponding p-adic (super-) actions and discusses, as for further research, the possible link to other approaches, in particular to a pure operator formulation of p-adic string theory and to the different approach by \textit{L. Chekhov}, \textit{A. Mironov}, \textit{A. Zabrodin}, and \textit{Yu. Zinov'ev} [cf. Commun. Math. Phys. 125, 675-711 (1989; Zbl 0685.22005) and Lett. Math. Phys. 20, No.3, 211-219 (1990; Zbl 0718.22007)], which is based on the theory of Bruhat-Tits trees. Some of the basic ideas and constructions given in the present work have already been discussed in an earlier article of the author's [Phys. Rev., Ser. D 41, 2631-2633 (1990)], e.g., the proposed p-adic superaction in the fermionic case and the nonlocal operators occuring there, which serve as a p-adic substitute for the ordinary derivatives appearing in the classical archimedean case.