In this paper the relativistic quantum mechanics is considered in the framework of the nonstandard synchronization scheme for clocks. Such a synchronization preserves Poincar{é} covariance but (at least formally) distinguishes an inertial frame. This enables to avoid the problem of a noncausal transmision of information related to breaking of the Bell's inequalities in QM. Our analysis has been focused mainly on the problem of existence of a proper position operator for massive particles. We have proved that in our framework such an operator exists for particles with arbitrary spin. It fulfills all the requirements: it is Hermitean and covariant, it has commuting components and moreover its eigenvectors (localised states) are also covariant. We have found the explicit form of the position operator and have demonstrated that in the preferred frame our operator coincides with the Newton--Wigner one. We have also defined a covariant spin operator and have constructed an invariant spin square operator. Moreover, full algebra of observables consisting of position operators, fourmomentum operators and spin operators is manifestly Poincaré covariant in this framework. Our results support expectations of other authors (Bell, Eberhard) that a consistent formulation of quantum mechanics demands existence of a preferred frame.
The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz gro…
We present a consistent, generally covariant quantization of light for non-vacuum birefringent, Lorentz-symmetry breaking electrodynamics in the context of the Standard Model Extension. We find that the number of light quanta in the field is not frame independent, and that the interaction of the quantized field with matter is necessarily birefringent. We also show that the conventional Lorenz gauge condition used to restrict the photon-mode basis to solutions of the Maxwell equations must be weakened to consistently describe Lorentz symmetry violation.
In two recent papers exact Hermite-Gaussian solutions to relativistic wave equations have been obtained for both electromagnetic and particle beams that include Gouy phase. The solutions for particle beams correspond to those of the Schrödinger equation in the non-relativistic limit. Here, distinct canonical and kinetic 4-momentum operators will be defined for quantum particles in matter wave beams. The kinetic momentum is equal to the canonical momentum minus the fluctuating terms resulting from the transverse localization of the beam. Three results are obtained. First, the total energy of a …
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…
Two alternative interpretations of the quantum collapse are proposed: a time-ordered and a timeless one. The time-ordered interpretation implies that the speed of light can be defined in an absolute way, while the timeless quantum collapse implies relativity and can be reckoned as covariant too. An experiment is proposed to decide between these two interpretations, which may also be considered a test of Bohm's "preferred frame" assumption.
We have shown the covariant relativistic spin operator is equivalent to the spin operator commuting with the free Dirac Hamiltonian. This implies that the covariant relativistic spin operator is a good quantum observable. The covariant relativistic spin operator has the pure quantum contribution which does not exist in the classical covariant spin operator. Based on this equivalence reduced spin states can be claerly defined. We have shown the change in the entropy of a reduced spin density matrix sweeps through the whole range according to the relative motion of an observer.
The Galilean symmetry and the Poincare symmetry are usually taken as the fundamental (relativity) symmetries for `nonrelativistic' and `relativistic' physics, respectively, quantum or classical. Our fully group theoretical formulation approach to the theories, together with its natural companion of mechanics from symplectic geometry, ask for different perspectives. We present a sketch of the full picture here, emphasizing aspects which are different from the more familiar picture. The letter summarizes our earlier presented formulation while focusing on the part beyond, with an adjusted, or co…
The earlier treatments of Lorentz covariant harmonic oscillator have brought to light various difficulties, such as reconciling Lorentz symmetry with the full Fock space, and divergence issues with their functional representations. We present here a full solution avoiding those problems. The complete set of Fock states is obtained, together with the corresponding explicit wavefunction and their inner product integrals free from any divergence problem and the Lorentz symmetry fully maintained without additional constraints imposed. By a simple choice of the pseudo-unitary representation of the …
A Lorenz-covariant system of wave equations is formulated for a quantum-mechanical two-body system in one space dimension, comprised of one electron and one photon. Manifest Lorentz covariance is achieved using Dirac's formalism of multi-time wave functions, i.e., wave functions $Ψ(\mathbf{x}_{ph},\mathbf{x}_{el})$ where $\mathbf{x}_{el},\mathbf{x}_{ph}$ are the generic spacetime events of the electron and photon, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifold $\{\mathbf{x}_{el}=\mathbf{x}_{ph}\}$, c…
We present in the article the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg-Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors under the Lorentz symmetry. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group $C^*$-algebra giving the algebra of observables. The key feature of the formulation is that it is not unitary but pseudo-un…
As in the case of the hydrogen atom, bound-state wave functions are needed to generate hadronic spectra. For this purpose, in 1971, Feynman and his students wrote down a Lorentz-invariant harmonic oscillator equation. This differential equation has one set of solutions satisfying the Lorentz-covariant boundary condition. This covariant set generates Lorentz-invariant mass spectra with their degeneracies. Furthermore, the Lorentz-covariant wave functions allow us to calculate the valence parton distribution by Lorentz-boosting the quark-model wave function from the hadronic rest frame. However,…
In classical relativistic mechanics, a "preferred" proper direction in spacetime for each particle is determined by the direction of its 4-momentum. Analogously, for each quantum particle we find a local direction uniquely determined by the many-particle wave function, which for each particle defines the proper foliation of spacetime. This can be used to formulate a relativistic-covariant version of Bohmian mechanics, with equivariant probability density on proper hypersurfaces.
