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.

我们回顾了关于标量场量子理论数学基础的一些最新结果，标量场是非阿基米德场的二次扩展$p$-进数。在我们的方法中，我们受到这个想法的启发——首先在[IV Volovich，$p$-adic 字符串，类。量子重力 4 (1987) L83–L87]——在适当的小尺度以下，该空间不会表现为连续体，因此，应该建模为完全断开的可度量拓扑空间，由满足强三角不等式的度量来统治。我们构建的第一步是对 a 进行适当的定义$p$-adic 希尔伯特空间。接下来，在介绍了所有必要的数学工具之后——特别是线性算子中的各种类别$p$-adic Hilbert 空间——我们考虑物理状态的代数定义$p$- 超量子力学。相应的可观察量，其定义完成了理论的统计解释，被引入为 SOVM，$p$-与复数上的标准量子系统相关联的 POVM 的 adic 对应项。有趣的是，事实证明，标准量子系统状态空间的典型凸几何被替换了，$p$-adic 设置，具有仿射几何形状；因此，a 的对称变换$p$-adic量子系统可以被定义为保留这种仿射几何的映射。因此，我们认为，a 的所有对称变换组成的群$p$-相对于复数标准量子力学的情况，adic 量子系统具有更丰富的结构。