We present a new non-Archimedean realization of the Fock representation of the q-oscillator algebras where the creation and annihilation operators act on complex-valued functions, which are defined on a non-Archimedean local field of arbitrary characteristic, for instance, the field of p-adic numbers. This new realization implies that many quantum models constructed using q-oscillator algebras are non-Archimedean models, in particular, p-adic quantum models. In this framework, we select a q-deformation of the Heisenberg uncertainty relation and construct the corresponding q-deformed Schrödinger equations. In this way we construct a p-adic quantum mechanics which is a p-deformed quantum mechanics. We also solve the time-independent Schrödinger equations for the free particle, and a particle in a non-Archimedean box. In the last case, we show the existence of a discrete sequence of energy levels. We determine the eigenvalues of Schrödinger operator for a general radial potential. By choosing the potential in a suitable form we recover the energy levels of the q-hydrogen atom.
We present a new non-Archimedean realization of the Fock representation of the q-oscillator algebras where the creation and annihilation operators act on complex-valued functions, which are defined on a non-Archimedean local field of arbitrary characteristic, for instance, the field of p-adic numbers. This new realization implies that many quantum models constructed using q-oscillator algebras are non-Archimedean models, in particular, p-adic quantum models. In this framework, we select a q-deformation of the Heisenberg uncertainty relation and construct the corresponding q-deformed Schrödinger equations. In this way we construct a p-adic quantum mechanics which is a p-deformed quantum mechanics. We also solve the time-independent Schrödinger equations for the free particle, and a particle in a non-Archimedean box. In the last case, we show the existence of a discrete sequence of energy levels. We determine the eigenvalues of Schrödinger operator for a general radial potential. By choosing the potential in a suitable form we recover the energy levels of the q-hydrogen atom.
Within the framework of quantum mechanics over a quadratic extension of the non-Archimedean field of p-adic numbers, we provide a definition of a quantum state relying on a general algebraic approach and on a p-adic model of probability theory. As in the standard complex case, a distinguished set of physical states are related to a notion of trace for a certain class of bounded operators, and in fact, we show that one can define a suitable space of trace class operators in the non-Archimedean setting, as well. The analogies—but also the several (highly non-trivial) differences—with respect to the case of standard quantum mechanics in a complex Hilbert space are analyzed.
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 [Formula: see text]-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, [Formula: see text]-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 [Formula: see text]-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a [Formula: see text]-adic Hilbert space — we consider an algebraic definition of physical states in [Formula: see text]-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a [Formula: see text]-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 [Formula: see text]-adic setting, with an affine geometry; therefore, a symmetry transformation of a [Formula: see text]-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 [Formula: see text]-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.
We present Unified Quantum Mechanics (UQM) by unifying real and standard complex quantum theories via real energy and momentum operators derived from the Hilbert Transform. This paper especially presents a fully real-valued formulation of quantum mechanics, showing that while complex numbers are commonly used in the standard theory, they are not fundamentally necessary for describing quantum phenomena. We develop Real Quantum Mechanics (RQM) by defining the real energy and momentum operators using the temporal Hilbert transform which shifts the phases of wave functions while preserving all essential quantum behavior. These real operators unify the proposed RQM and standard Complex Quantum Mechanics (CQM), as these are valid for both real and complex wavefunctions. We introduce a real-valued algebraic structure constructed from the identity operator and the Hilbert transform operator. This framework is mathematically equivalent to the system of complex numbers, preserving all fundamental algebraic properties while operating entirely within the realm of real operators. The isomorphism provides a novel representation of complex quantum mechanics using only real-valued components, offering potential advantages for both theoretical analysis and physical implementations. Importantly, we establish that the Hilbert transform operator is a generalization of the imaginary unit. The key insight is that the Hilbert transform generalizes the role of imaginary numbers in standard quantum mechanics, allowing us to construct real operators for both real and complex wave functions that produce the same physical predictions. We derive real versions of energy and momentum operators, leading to a modified wave equation that replaces the Schrödinger equation. Despite its non-local time evolution, this framework retains local spatial behavior and reproduces the usual energy-momentum relations. We rigorously test this approach on fundamental quantum systems, including the hydrogen atom and relativistic particles, confirming that it yields identical energy levels and wave functions as conventional quantum mechanics. Notably, we establish a consistent probabilistic framework by introducing real analogues of quantum probabilities via the analytic Born rule. We derive a unified uncertainty principle that coincides with the standard Heisenberg uncertainty bound for complex-valued wave functions, while admitting a vanishing lower bound in the case of real-valued wave functions. The implications of this work are significant: (1) it demonstrates that complex numbers are a mathematical convenience rather than a fundamental requirement in quantum theory, (2) it introduces new tools for analyzing quantum dynamics using signal-processing techniques, (3) it provides insights into non-local quantum evolution, (4) it connects quantum mechanics to real-field theories. Ultimately, RQM reproduces all known quantum predictions while offering a fresh perspective on operator structure, time evolution, and the mathematical foundations of the theory, and (5) it provides a real qubit representation on the modified Bloch sphere for quantum computing.
This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ℚ p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ℚ p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.
We introduce a new class of p-adic Dirac equations, formulated in the standard axiomatic framework of quantum mechanics, in which the ordinary momentum operators are replaced by bounded non-local operators built from arbitrary integrable radial kernels. We diagonalize the free Dirac Hamiltonian in momentum space, construct its plane-wave solutions, and determine its spectrum, which is pure point, with eigenvalues of infinite multiplicity; in the massless case we describe the zero-energy subspace explicitly. We identify the charge-conjugation symmetry of the model: it is given by an antiunitary operator whose square is minus the identity, it exchanges the positive- and negative-energy subspaces, and in an external electromagnetic field it reverses the sign of both the charge and the vector potential. We then discretize the free equation in two ways, both giving genuine continuous-time quantum walks rather than discrete-time, coined walks: one on a countable graph and one on a finite, tree-like graph. In both cases the discretization is exact: the walk is the restriction of the p-adic Dirac evolution to an invariant subspace, and its Hamiltonian is a Dirac-type combination of weighted graph Laplacians with ultrametric weights. We prove that the Born-rule transition probabilities, summed over the internal degrees of freedom and averaged over the initial internal state, form a doubly stochastic matrix at every instant, and that, as for any non-trivial continuous-time quantum walk, this family of matrices is not a Markov semigroup. We discuss how the construction can serve as the foundation of a quantum network with four-component spinor degrees of freedom, complementing earlier, non-relativistic p-adic quantum neural networks, and prove global well-posedness of the associated nonlinear network equation. We close with a discussion of open problems.
The purely algebraic technique associated with the creation and annihilation operators to resolve the radial equation of Hydrogen-like atoms (HLA) for generating the bound energy spectrum and the corresponding wave functions is suitable for many calculations in quantum physics. However, the analytical approach with series is extensively used based on wave mechanics theory in most of quantum textbooks. Indeed, much More complete than the old solution of Schrödinger's time-independent differential equation (TISE), one can simply earn all quantum information of a system by using the operational method. In addition to earlier two models, including the quantum harmonic oscillator and the total angular momentum, it can undoubtedly be a third fundamental model to solve Schrödinger's eigenvalue equation of the HLA systems in Hilbert space similar to the harmonic oscillator. We will illustrate how systematically making an appropriate groundwork to discover the coherent states can lead to providing the energy quantization and normalized radial wave functions attached to the matrix representation without additional assumptions.