Quantum mechanics on a p-adic Hilbert space: Foundations and prospects

Abstract

We review some recent results on the mathematical foundations of a quantum theory over a scalar field that is a quadratic extension of the non-Archimedean field of $p$-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, $p$-adic string, Class. Quantum Grav.4 (1987) L83–L87] — that space, below a suitably small scale, does not behave as a continuum and, accordingly, should be modeled as a totally disconnected metrizable topological space, ruled by a metric satisfying the strong triangle inequality. The first step of our construction is a suitable definition of a $p$-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a $p$-adic Hilbert space — we consider an algebraic definition of physical states in $p$-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a $p$-adic counterpart of the POVMs associated with a standard quantum system over the complex numbers. Interestingly, it turns out that the typical convex geometry of the space of states of a standard quantum system is replaced, in the $p$-adic setting, with an affine geometry; therefore, a symmetry transformation of a $p$-adic quantum system may be defined as a map preserving this affine geometry. We argue that, as a consequence, the group of all symmetry transformations of a $p$-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.