We present a new p-adic version of the Jackiw-Rebbi model. In the new model, the real numeric line is replaced by a p-adic line (the field of p-adic numbers Q_{p}), and the Dirac Hamiltonian is replaced by a non-local operator acting on complex-valued functions defined on Q_{p}. These Hamiltonians admit localized wavefunctions and allow long-range interactions, so spooky action at a distance is allowed. These features are not present in the original model. The new model gives the same predictions as the standard one. The p-adic line serves as a discrete model for the physical space; in this type of space, non-locality emerges naturally.
We present a new p-adic version of the Jackiw-Rebbi model. In the new model, the real numeric line is replaced by a p-adic line (the field of p-adic numbers Q_{p}), and the Dirac Hamiltonian is replaced by a non-local operator acting on complex-valued functions defined on Q_{p}. These Hamiltonians admit localized wavefunctions and allow long-range interactions, so spooky action at a distance is allowed. These features are not present in the original model. The new model gives the same predictions as the standard one. The p-adic line serves as a discrete model for the physical space; in this type of space, non-locality emerges naturally.
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.
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 propose a model of a quantum N-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of p-adic numbers. As in the standard complex setting, states and observables of a p-adic quantum system are implemented by suitable linear operators in a p-adic Hilbert space. In particular, owing to the distinguishing features of p-adic probability theory, the states of an N-dimensional p-adic quantum system are implemented by p-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)—a suitable p-adic counterpart of a POVM in a complex Hilbert space—as a convenient mathematical tool describing the physical observables of a p-adic quantum system. Eventually, we focus on the special case where N=2, thus providing a description of p-adic qubit states and 2-dimensional SOVMs. The analogies—but also the non-trivial differences—with respect to the qubit states of standard quantum mechanics are then analyzed.
We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit i is systematically replaced by the temporal Hilbert transform operator Ht, which satisfies the anti-involution property Ht2=−I and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field ψ and its Hilbert-transformed partner ϕ=Ht[ψ], whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation ψa=ψ−iϕ, under which the standard U(1) gauge symmetry emerges naturally as a global SO(2) rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density ρ=ψ†ψ+ϕ†ϕ=∣ψa∣2 is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.
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.
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.