In this paper, we develop a formalism describing in a relativistic way a system which consists of a classical and a quantum part being coupled. The formalism models one particle with spin 1/2 and it is a possible relativistic extension of the Event-Enhanced Quantum Theory. We postulate a covariant algorithm which plays the role of the standard reduction postulate in non-relativistic quantum mechanics. Furthermore, we present an algorithm to simulate detections of the particle.
In this paper, we construct isospectral Hamiltonians without shape invariant potentials for the relativistic quantum mechanical potentials such as the Dirac Oscillator and Hydrogen-like atom.
We show that quantum mechanics can be given a Lorentz-invariant realistic interpretation by applying our recently proposed relativistic extension of the de Broglie-Bohm theory to deduce non-locally correlated, Lorentz-invariant individual particle motions for the Einstein-Podolsky-Rosen experiment and the double-interferometer experiment proposed by Horne, Shimony and Zeilinger.
We discuss an extension of the theory of {\em spin-orbit pendulum} phenomenon given in [1] to relativistic approach. It is done within the so called Dirac Oscillator. Our first results, focusing on circular wave packet motion have been published recently [2]. The scope of this paper is motion of a linear wave packet. In relativistic approach we found {\em Zitterbewegung} in spin-orbit motion (in Dirac representation) due to coupling to negative energy states. This effect is washed out in the Foldy-Wouthuysen representation. Another important change with respect to non-relativistic case is the …
We discuss the appearance of relativistic geometric quantum phases for a Dirac neutral particle based on possible scenarios of the Lorentz symmetry violation background in the CPT-even gauge sector of Standard Model Extension. Relativistic analogues of the Anandan quantum phase [J. Anandan, Phys. Lett. A 138, 347 (1989)] are obtained based on the the parity-even and parity-odd sectors of the tensor $\left(K_{F}\right)_{μναβ}$.
Modal interpretations of quantum mechanics assign definite properties to physical systems and specify single-time joint probabilities of these properties. We show that a natural extension, applying to properties at several times, can be given if a decoherence condition is satisfied. This extension defines "consistent histories" of modal properties. We suggest a new form of the modal scheme, that offers prospects of a more general applicability of the histories concept. Finally, we discuss a possible way of applying these ideas to relativistic quantum field theory.
In quantum theory we refer to the probability of finding a particle between positions $x$ and $x+dx$ at the instant $t$, although we have no capacity of predicting exactly when the detection occurs. In this work, first we present an extended non-relativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between $x$ and $x+dx$ during [$t$,$t+dt$]. Then, we find a Schrödinger-like equation for a "mirror" wave function $ϕ(t,x)$ associated with the probability of measuring the system between $t$ and $t+dt$, given that detection occu…
As an extension of our recent work (Results in Physics 12, 147-152 (2019)), we show, both theoretically and numerically, that the breakdown of agreement between the non-relativistic and relativistic quantum dynamical predictions in the non-relativistic regime also occurs for an electron subjected to a time-dependent force.
We extend the ideas of L.P. Horwitz and C. Piron and we propose a relativistic version of Event Enhanced Quantum Theory, with an event generating algorithm for spin one-half particle detectors. The algorithm is based on proper time formulation of the relativistic quantum theory. Although we use indefinite metric, all the probabilities controlling the random process of the detector clicks are non--negative.
The relativistic angular momentum is introduced as an extension of the non-relativistic analysis of allowed states in the phase space for a quantum particle. The paper shows the conceptual basis of the approach. An interesting feature of the present point of view is that the indistinguishability of identical particles and the Pauli principle are found as corollaries.
We construct a tridiagonal matrix representation for the three dimensions Dirac-Coulomb Hamiltonian that provides for a simple and straightforward relativistic extension of the complex scaling method. Besides the Coulomb interaction, additional vector, scalar, and pseudo-scalar coupling to short-range potentials could also be included in the same representation. Using that, we are able to obtain highly accurate values for the relativistic bound states and resonance energies. A simple program code is developed to perform the calculation for a given charge, angular momentum and potential configu…
We derive approximate analytical solutions for a particle in a homogenous gravitational field and confined between two independently vibrating mirrors. This constitutes an extension of the qBOUNCE experiment in which ultra-cold neutrons are employed for Gravity Resonance Spectroscopy.
A relativistic version of the Aharonov-Bohm time of arrival operator for spin-0 particles was constructed by Razavi in [Il Nuovo Cimento B \textbf{63}, 271 (1969)]. We study the operator in detail by taking its rigged Hilbert space extension. It is shown that the rigged Hilbert space extension of the operator provides more insights into the time of arrival problem that goes beyond Razavi's original results. This allows us to use time of arrival eigenfunctions that exhibit unitary arrival to construct time of arrival distributions. The expectation value is also calculated and shown that particl…
We propose a covariant algorithm for relativistic ideal measurements and for relativistic continuous measurements, its non-relativistic limit results the algorithm of the Event-Enhanced Quantum Theory. Therefore an additional intrinsic parameter, the proper time, is used. As an application we compute the time of arrival of a particle at a detector and find good agreement between the expected values of the time of arrival for weak detectors and the results of the relativistic point-mechanic over a wide range. For very high momentums there is a small probability for a negative time of arrival, s…
Yes, there is. - A new kind of gauge theory is introduced, where the minimal coupling and corresponding covariant derivatives are defined in the space of functions pertaining to the functional Schroedinger picture of a given field theory. While, for simplicity, we study the example of an U(1) symmetry, this kind of gauge theory can accommodate other symmetries as well. We consider the resulting relativistic nonlinear extension of quantum mechanics and show that it incorporates gravity in the (0+1)-dimensional limit, where it leads to the Schroedinger-Newton equations. Gravity is encoded here i…
Relativistic free-motion time-of-arrival theory for massive spin-1/2 particles is systematically developed. Contrary to the nonrelativistic time-of-arrival operator studied thoroughly in previous literatures, the relativistic time-of-arrival operator possesses self-adjoint extensions because of the particle-antiparticle symmetry. The nonrelativistic limit of our theory is in agreement with the nonrelativistic time-of-arrival theory.
The Suarez-Scarani model is a non-covariant (frame-dependent) relativistic (time-ordered causal) nonlocal extension of quantum theory, which is not refuted by the experiments proposed in Reference [arXiv:1005.5173v1]. The covariant extensions considered in this Reference are actually self-contradictory and do not require experimental falsification.
We consider the Dirac equation in 1+1 space-time dimension with vector, scalar and pseudo-scalar coupling. In the traditional spin (or pseudo-spin) symmetry, the difference between (or sum of) the scalar and vector potentials is a constant. Here, however, we introduce an alternative symmetry where the scalar or pseudo-scalar potential is proportional to the vector potential. This leads to a model with significant extensions to supersymmetric quantum mechanics. We present a formal solution of the problem but give explicit analytic results for specific examples.
Recurrence formulae for arbitrary hydrogenic radial matrix elements are obtained in the Dirac form of relativistic quantum mechanics. Our approach is inspired on the relativistic extension of the second hypervirial method that has been succesfully employed to deduce an analogous relationship in non relativistic quantum mechanics. We obtain first the relativistic extension of the second hypervirial and then the relativistic recurrence relation. Furthermore, we use such relation to deduce relativistic versions of the Pasternack-Sternheimer rule and of the virial theorem.
Relativistic free-motion time-of-arrival theory for massive spin-1/2 particles is systematically developed. Contrary to the nonrelativistic time-of-arrival operator studied thoroughly in previous literatures, the relativistic time-of-arrival operator possesses self-adjoint extensions because of the particle-antiparticle symmetry. The nonrelativistic limit of our theory is in agreement with the nonrelativistic time-of-arrival theory. By comparing the time-of-arrival operator with the Hamiltonian operator of a free Dirac particle, one can show a duality between position space and momentum space.
The extension of Standard Model of the Fundamental Particles which consider the Lorentz Symmetry Violation governed by a background tensor field presented by Colladay and Kostelecky consider the hypothesis of having a privileged direction in the space-time. Here we investigated a extension of the Schrödinger Equation inspired on that model taking into account a background scalar field. We analyze the behavior of this modified Schrödinger equation on a particle trapped in a well and quantum harmonic oscillator.
In order to solve the time-independent three-dimensional Schrödinger equation, one can transform the time-dependent Schrödinger equation to imaginary time and use a parallelized iterative method to obtain the full three-dimensional eigenstates and eigenvalues on very large lattices. In the case of the non-relativistic Schrödinger equation, there exists a publicly available code called quantumfdtd which implements this algorithm. In this paper, we (a) extend the quantumfdtd code to include the case of the relativistic Schrödinger equation and (b) add two optimized FFT-based kinetic energy terms…
It is shown that quantum entanglement is the only force able to maintain the fourth state of matter, possessing fixed shape at an arbitrary volume. Accordingly, a new relativistic Schrodinger equation is derived and transformed further to the relativistic Bohmian mechanics via the Madelung transformation. Three dissipative models are proposed as extensions of the quantum relativistic Hamilton-Jacobi equation. The corresponding dispersion relations are obtained.
A relativistic topological insulator model in three spatial dimensions which is a non trivial extension of the non-abelian Landau problem is proposed. The model is exactly soluble and energy levels have both a discrete and a continuous degeneracy. The chromomagnetic field is strong and the fermions are confined in a plane and the physical effects that appear reflects the ${\bm Z}_2$ symmetry.
A new kind of gauge theory is introduced, where the minimal coupling and corresponding covariant derivatives are defined in the space of functions pertaining to the functional Schroedinger picture of a given field theory. While, for simplicity, we study the example of a U(1) symmetry, this kind of gauge theory can accommodate other symmetries as well. We consider the resulting relativistic nonlinear extension of quantum mechanics and show that it incorporates gravity in the (0+1)-dimensional limit, similar to recently studied Schroedinger-Newton equations. Gravity is encoded here into a univer…
We consider a 1-parameter family of self-adjoint extensions of the Hamiltonian for a particle confined to a finite interval with perfectly reflecting boundary conditions. In some cases, one obtains negative energy states which seems to violate the Heisenberg uncertainty relation. We use this as a motivation to derive a generalized uncertainty relation valid for an arbitrarily shaped quantum dot with general perfectly reflecting walls in $d$ dimensions. In addition, a general uncertainty relation for non-Hermitean operators is derived and applied to the non-Hermitean momentum operator in a quan…
We handle all the self-adjoint extensions of the minimal Schroedinger operator for the non-relativistic electron living in the one-dimensional configuration space with a junction. We are interested in every boundary condition corresponding to the individual self-adjoint extension. Thus, we clarify all the types of those boundary conditions of the wave functions of the non-relativistic electron. We find a tunnel-junction formula for the non-relativistic electron passing through the junction. Using this tunnel-junction formula, we propose a mathematical possibility of a tunnel-junction device fo…
The non-relativistic hydrogen atom enjoys an accidental $SO(4)$ symmetry, that enlarges the rotational $SO(3)$ symmetry, by extending the angular momentum algebra with the Runge-Lenz vector. In the relativistic hydrogen atom the accidental symmetry is partially lifted. Due to the Johnson-Lippmann operator, which commutes with the Dirac Hamiltonian, some degeneracy remains. When the non-relativistic hydrogen atom is put in a spherical cavity of radius $R$ with perfectly reflecting Robin boundary conditions, characterized by a self-adjoint extension parameter $γ$, in general the accidental $SO(4…
In this paper, the relativistic quantum dynamics of a scalar particle under the effect of Lorentz symmetry violation determined by a tensor $(K_{F})_{μ\,ν\,α\,β}$ out of the Standard Model Extension is investigated. We see that the bound-state solution of the modified Klein-Gordon equation can be obtained, and the spectrum of energy and the wave function depends on the Lorentz symmetry breaking parameters
The planar quantum dynamics of spin-1/2 neutral particle interacting with electrical fields is considered. A set of first order differential equations are obtained directly from the planar Dirac equation with nonminimum coupling. New solutions of this system, in particular, for the Aharonov-Casher effect, are found and discussed in detail. Pauli equation is also obtained by studying the motion of the particle when it describes a circular path of constant radius. We also analyze the planar dynamics in the full space, including the $r=0$ region. The self-adjoint extension method is used to obtai…
