On the causality in quantum theory with a minimal scale length

In the talk we describe causal functions that are consistent with Volovich postulates which read: (1) only rational numbers can be observed; irrational numbers cannot; (2) distances smaller than Planck length cannot be measured; (3) fundamental physical laws should be invariant with respect to change of number field. Causal functions over discrete time are mappings of one-side infinite sequences over a finite set of `elementary causes' to one-side infinite sequences over a finite set of `elementary effects' such that each i-th term of the image depends only on the terms 0,1,..,i of the pre-image. The causal functions constitute therefore a class of all non-expansive mappings of the non-Archimedean metric spaces; or, which is equivalent, exactly the class of all mappings performed by automata which are sequential machines, namely, the letter-to-letter transducers. The automata can be judged as the least powerful computers in comparison with cellular automata, which are the most powerful ones. Causal functions therefore are also consistent with the `t Hooft's causality postulate which lies in the base of his cellular automaton interpretation of quantum mechanics and reads that 'at its most basic level there is no randomness in Nature ... Every event takes place for a reason, it was caused by the action of physical law, not just by chance'. We describe functions which are causal (thus, continuous) w.r.t. p-adic metrics for all p and which simultaneously are continuous w.r.t. real metric; we argue that namely these functions can be judged as `closed forms' of physical laws which satisfy Volovich postulates (1)—(3). By exploiting these ideas we develop an information model of measurements of open physical systems in a theory with a minimal scale length thus relating the subject of the talk to Wheeler's `it from bit' doctrine. The talk follows the approach outlined in the author's paper «Toward the (non-cellular) automata interpretation of quantum mechanics: Volovich postulates as a roadmap», Int'l J. Modern Phys. A, 2022, DOI 10.1142/S0217751X22430035.