It is known that Lorentz covariance fixes uniquely the current and the associated guidance law in the trajectory interpretation of quantum mechanics for spin particles. In the non-relativistic domain this implies a guidance law for the electron which differs by an additional spin-dependent term from that originally proposed by de Broglie and Bohm. In this paper we explore some of the implications of the modified guidance law. We bring out a property of mutual dependence in the particle coordinates that arises in product states, and show that the quantum potential has scalar and vector components which implies the particle is subject to a Lorentz-like force. The conditions for the classical limit and the limit of negligible spin are given, and the empirical sufficiency of the model is demonstrated. We then present a series of calculations of the trajectories based on two-dimensional Gaussian wave packets which illustrate how the additional spin-dependent term plays a significant role in structuring both the individual trajectories and the ensemble. The single packet corresponds to quantum inertial motion. The distinct features encountered when the wavefunction is a product or a superposition are explored, and the trajectories that model the two-slit experiment are given. The latter paths exhibit several new characteristics compared with the original de Broglie-Bohm ones, such as crossing of the axis of symmetry.
We formalize a notion of discrete Lorentz transforms for Quantum Walks (QW) and Quantum Cellular Automata (QCA), in (1 + 1)-dimensional discrete spacetime. The theory admits a diagrammatic representation in terms of a few local, circuit equivalence rules. Within this framework, we show the first-order-only covariance of the Dirac QW. We then introduce the Clock QW and the Clock QCA, and prove that they are exactly discrete Lorentz covariant. The theory also allows for non-homogeneous Lorentz transforms, between non-inertial frames.
Quantum correlations of observables for two particle states have demonstrated the nonlocal character of the quantum mechanics. However nonlocality can be exhibited even for noncommuting observables of a single particle system. In this paper we show nonlocality of position-momentum correlations of a single particle in the double-slit experiment modeled by an initially correlated Gaussian wavepacket. The positivity or negativity of the Wigner function for the state of the particle at the detection screen is related with the $σ_{xp}$ covariances. A Bell's inequality is constructed from the Wigner…
In the textbook proofs of Lorentz covariance of the Dirac equation, one treats the wave function as a spinor and gamma matrices as scalars, leading to a quite complicated formalism with several pedagogic drawbacks. As an alternative, I propose to teach Dirac equation and its Lorentz covariance by using a much simpler, but physically equivalent formalism, in which these drawbacks do not appear. In this alternative formalism, the wave function transforms as a scalar and gamma matrices as components of a vector, such that the standard physically relevant bilinear combinations do not change their …
The last two decades have witnessed an increasing effort of the scientific community toward pursuing a better framework for quantum resource covariance, with the focus predominantly posed on quantum entanglement. In this work, we move the discussion one step further by analyzing the behavior of both genuine multipartite entanglement and quantum coherence under Lorentz boosts. Specifically, we conduct a case study for the problem of an electron-positron pair created in a superposed multipartite pure state. Our approach is different from the standard treatments also in that we consider all the c…
The difference between Lorentz invariance and Lorentz covariance is discussed in detail. A covariant formalism is developed for the internal space-time symmetry of extended particles, especially in connection with the insightful observations Feynman made during the period 1969-72. A Lorentz-group formalism is presented for the harmonic oscillator model of Feynman, Kislinger and Ravndal, which was originally based on hadronic mass spectra. This covariant version allows us to construct a parton distribution function by Lorentz-boosting the oscillator wave function of a hadron at rest. The role o…
The Lorentz covariance of a non-linear, time-dependent relativistic wave equation is demonstrated; the equation has recently been shown to have highly interesting and significant empirical consequences. It is established here that an operator already exists which ensures the relativistic properties of the equation. Furthermore, we show that the time-dependent equation is gauge invariant. The equation however, breaks Poincare symmetry via time translation in a way consistent with its physical interpretation. It is also shown herein that the Casimir invariant PmuPmu of the Poincare group, which …
Lorentz-covariant harmonic oscillator wave functions are constructed from the Lorentz-invariant oscillator differential equation of Feynman, Kislinger, and Ravndal for a two-body bound state. The wave functions are not invariant but covariant. As the differential equation contains the time-separation variable, the wave functions contain the same time-separation variable which does not exist in Schrödinger wave functions. This time-separation variable can be shown to belong to Feynman's rest of the universe, and can thus be eliminated from the density matrix. The covariant probability interpret…
Quantum correlations encoded in photonic Laguerre-Gaussian modes were shown to be related to the Gouy phase shifts (D. Kawase et al., Phys. Rev. Lett. 101, 050501 (2008)) allowing for a non-destructive manipulation of photonic quantum states. In this work we exploit the relation between phase space correlations of biphotons produced by spontaneously parametric down conversion (SPDC) as encoded in the logarithmic negativity (LN) and the Gouy phase as they are diffracted through an asymmetrical double slit setup. Using an analytical approach based on a double-gaussian approximation for type-I SP…
Lattice gauge theory's discretization of spacetime suffers from a drawback in that Lorentz covariance is lost because the axes of the lattice create preferred directions in spacetime. Smaller and smaller lattice spacings decrease the effect but fail to eliminate it completely. It has been argued recently that detecting such a set of preferred directions or similar constraints would indicate whether the universe itself has an underlying lattice, i.e. the digital universe hypothesis. In this paper, I demonstrate a technique for accomplishing lattice gauge theory simulations while maintaining exa…
A great number of problems of relativistic position in quantum mechanics are due to the use of coordinates which are not inherent objects of spacetime, cause unnecessary complications and can lead to misconceptions. We apply a coordinate-free approach to rule out such problems. Thus it will be clear, for example, that the Lorentz covariance of position, required usually on the analogy of Lorentz covariance of spacetime coordinates, is not well posed and we show that in a right setting the Newton--Wigner position is Poincaré covariant, in contradiction with the usual assertions.
The Young double-slit interference pattern produced by quantum objects, like photons, that move through a double-slit is regarded, by the conventional Copenhagen interpretation of Quantum Mechanics, as the evidence of the wave-like behaviour potentially contained in the wave function. On the contrary, a more realistic view of this phenomenon considers the quantum object a particle accompanied by a pilot wave which would be the cause of the interference fringes. This paper proposes a feasible experiment, based on an easy variation of the nowadays common double-slit experimental set-ups, aimed a…
We show that local Lorentz covariance arises canonically as the group of transformations between local thermal states in the framework of Local Quantum Physics, given the following three postulates: (i) Local observable algebras are finite-dimensional. (ii) Minimal local observable algebras are isomorphic to $\mathbb{M}_2(\mathbb{C})$, the observable algebra of a single qubit. (iii) The vacuum restricted to any minimal local observable algebra is a non-maximally mixed thermal state. The derivation reveals a new and surprising relation between spacetime structure and local quantum states. In pa…
It is known that the Schroedinger equation is not covariant under Galilei boosts, unless the phase of its solutions are shifted simultaneously. It is argued that the phase shift is not a coordinate transformation, because it depends on the mass of the Schroedinger particle. The phase shift also cannot be derived from low speed Lorentz boost. It is proposed to extend the Galilei boost with two terms of order v^2/c^2 to avoid these issues and to guarantee covariance of the Schroedinger kinetic energy and momentum. The extensions imply that proper time and relativity of simultaneity are essential…
The covariance of the Schrödinger equation under Galilei boosts and the compatibility of nonrelativistic quantum mechanics with Einstein's equivalence principle have been constrained for so long to the existence of a superselection rule which would prevent a quantum particle from being found in superposition states of different masses. In an effort to avoid this expedient, and thus allow nonrelativistic quantum mechanics to account for unstable particles, recent works have suggested that the usual Galilean transformations are inconsistent with the nonrelativistic limit implied by the Lorentz t…
In 1905, Einstein formulated his special relativity for point particles. For those particles, his Lorentz covariance and energy-momentum relation are by now firmly established. How about the hydrogen atom? It is possible to perform Lorentz boosts on the proton assuming that it is a point particle. Then what happens to the electron orbit? The orbit could go through an elliptic deformation, but it is not possible to understand this problem without quantum mechanics, where the orbit is a standing wave leading to a localized probability distribution. Is this concept consistent with Einstein's Lore…
A discrete action principle for general quantum automata is proposed. This action principle is particularized to Discrete Time Quantum Walks (DTQWs) and then extended into an energy and momentum preserving, manifestly covariant formulation. Space-time coordinates are introduced as new variables of the action and their equations of motion enforce energy and momentum conservation. This guarantees that the proposed action can be used to build future, DTQW-based self-consistent models of spinors interacting with gauge fields. A discrete stress-energy tensor for the DTQW is also obtained by functio…
The influence of the relativistic covariance requirement on the optimality of the symmetric state-dependent 1 -> 2 cloning machine is studied. Namely, given a photonic qubit whose basis is formed from the momentum-helicity eigenstates, the change to the optimal cloning fidelity is calculated taking into account the Lorentz covariance unitarily represented by Wigner's little group. To pinpoint some of the interesting results, we found states for which the optimal fidelity of the cloning process drops to 2/3 which corresponds to the fidelity of the optimal classical cloner. Also, an implication …
There are several possible applications of quantum electrodynamics in dielectric media which require a quantum description for the electromagnetic field interacting with matter fields. The associated quantum models can refer to macroscopic electromagnetic fields or, in alternative, to mesoscopic fields (polarization fields) describing an effective interaction between electromagnetic field and matter fields. We adopt the latter approach, and focus on the Hopfield model for the electromagnetic field in a dielectric dispersive medium in a framework in which space-time dependent mesoscopic paramet…
The Casimir force between two conducting planes is considered in both the electromagnetic and scalar field cases. This is done by the usual summation over energy eigenmodes of the system as well as by a calculation of the stress tensor in the region between the planes. The latter case requires that careful attention be given to singular operator products, an issue which is accommodated here by invoking the point separation method in conjunction with a scalar cutoff. This is shown to yield cutoff dependent and divergent contributions to the Casimir pressure which are dependent on the separation…
We construct new relativistic linear differential equation in $d$ dimensions generalizing Dirac equation by employing the Clifford algebra of the cubic polynomial associated to Klein-Gordon operator multiplied by the mass parameter. Unlike the Dirac case where the spin content is unique and Lorentz covariance is manifest, here the spin as well as Lorentz covariance of the theory are related to the choice of representation of the Clifford algebra. One of the considered explicit matrix representations gives rise to anyon-like fields in $d=1+1$. Coupling to a U(1) gauge field is discussed and com…
The descripition of in a Hermitian setting seemingly nonlocal and nonperturbative phenomena like confinement or superconductivity is most conveniently performed by generalizing quantum theory to a non-Hermitian regime where these phenomena appear perturbative and local. The short presentation provides a clue how this can be done on the basis of Lorentz covariance while preserving the analyticity of the theory. After deriving with the help of Lorentz covariance a quantum scalar product without making any use of metric or complex conjugation we sketch how the formalism of scattering theory can b…
According to some generalized correspondence principle the classical limit of a non-Hermitian Quantum theory describing quantum degrees of freedom is expected to be well known classical mechanics of classical degrees of freedom in the complex phase space, i.e., some phase space spanned by complex-valued space and momentum coordinates. As special relativity has been developed by Einstein merely for real-valued space-time and four-momentum we will try to understand how special relativity and covariance can be extended to complex-valued space-time and four-momentum. Our considerations will lead u…
In most approaches to fundamental physics, spacetime symmetries are postulated a priori and then explicitly implemented in the theory. This includes Lorentz covariance in quantum field theory and diffeomorphism invariance in quantum gravity, which are seen as fundamental principles to which the final theory has to be adjusted. In this paper, we suggest, within a much simpler setting, that this kind of reasoning can actually be reversed, by taking an operational approach inspired by quantum information theory. We consider observers in distinct laboratories, with local physics described by the l…
When Einstein formulated his special relativity in 1905, he established the law of Lorentz transformations for point particles. It is now known that particles have internal space-time structures. Particles, such as photons and electrons, have spin variables. Protons and other hadrons are regarded as bound states of more fundamental particles called quarks which have their internal variables. It is still one of the most outstanding problems whether these internal space-time variables are transformed according to Einstein's law of Lorentz transformations. It is noted that Wigner, Dirac, and Feyn…
We consider the consistent deformation of the relativistic quantum mechanics introducing the noncommutativity of the space-time and preserving the Lorentz symmetry. The relativistic wave equation describing the spinning particle on coordinate dependent noncommutative space-time (noncommutative Dirac equation) is proposed. The fundamental properties of this equation, like the Lorentz covariance and the continuity equation for the probability density are verified. To this end using the properties of the star product we derive the corresponding probability current density and prove its conservati…
