We construct in a rigorous mathematical way interacting quantum field theories on [Formula: see text]-adic spacetime. The main result is the construction of measure function space which allows definition partition function. advantage approach presented here that all perturbation calculations can be carried out standard using functional derivatives, but mathematically way.

We shall find the following necessary and sufficient conditions: I. the space is metrizable (cf. Nagata [l], Smirnof [2]), II. the space is strongly O-dimensional. Property II means that any two closed disjoint sets in the space can be separated (by the empty set). We shall prove furthermore that the conditions I and II are equivalent to the following topological properties: the space is a Haus...

We outline the classification of K-rank 1 groups over non-archimedean local fields K up to strict isogeny, as in [Ti1] and [Ti2]. We outline the classification of absolutely simple algebraic groups over non-archimedean local fields, up to strict isogeny. This is classical, and accounts of it have been written by Tits ([Ti1], [Ti2]) and Satake ([Sa]). Tits compiled tables of ‘admissible indices’...

We explore the distinction between convergence and absolute convergence of series in both Archimedean and non-Archimedean ordered fields and find that the relationship between them is closely connected to sequential (Cauchy) completeness.

Given a reductive group over complete non-Archimedean field, it is well known that techniques from analytic geometry provide an embedding of the corresponding Bruhat–Tits building into space associated with group; by composing maps to suitable proper spaces, this eventually leads various compactifications building. In present paper, we give intrinsic characterization embedding.