Relativistic Position Operators Research Articles

Ehrenfest theorem in relativistic quantum theory* *The B W Shore Memorial Issue of the Journal of Physics B.

Ehrenfest theorem is extended to the relativistic quantum theory of charged particles, moving under the influence of a classical electromagnetic field. In order to extend the original Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton’s second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws for the energy and momentum.

Proper relativistic position operators in 1+1 and 2+1 dimensions

We have revisited the Dirac theory in [Formula: see text] and [Formula: see text] dimensions by using the covariant representation of the parity-extended Poincaré group in their native dimensions. The parity operator plays a crucial role in deriving wave equations in both theories. We studied two position operators, a canonical one and a covariant one that becomes the particle position operator projected onto the particle subspace. In [Formula: see text] dimensions the particle position operator, not the canonical position operator, provides the conserved Lorentz generator. The mass moment defined by the canonical position operator needs an additional unphysical spin-like operator to become the conserved Lorentz generator in [Formula: see text] dimensions. In [Formula: see text] dimensions, the sum of the orbital angular momentum given by the canonical position operator and the spin angular momentum becomes a constant of motion. However, orbital and spin angular momentum do not conserve separately. On the other hand the orbital angular momentum given by the particle position operator and its corresponding spin angular momentum become a constant of motion separately.

Spin and localization of relativistic fermions and uncertainty relations

We discuss relations between several relativistic spin observables and derive a Lorentz-invariant characteristic of a reduced spin density matrix.A relativistic position operator that satisfies all the properties of its nonrelativistic analog does not exist. Instead we propose two causality-preserving positive operator-valued measures (POVMs) that are based on projections onto one-particle and antiparticle spaces, and on the normalized energy density. They predict identical expectation values for position. The variances differ by less than a quarter of the squared de Broglie wavelength and coincide in the nonrelativistic limit. Since the resulting statistical moment operators are not canonical conjugates of momentum, the Heisenberg uncertainty relations need not hold. Indeed, the energy density POVM leads to a lower uncertainty. We reformulate the standard equations of the spin dynamics by explicitly considering the charge-independent acceleration, allowing a consistent treatment of backreaction and inclusion of a weak gravitational field.

Relativistic dynamics, Green function and pseudodifferential operators

The central role played by pseudodifferential operators in relativistic dynamics is known very well. In this work, operators like the Schrodinger one (e.g., square root) are treated from the point of view of the non-local pseudodifferential Green functions. Starting from the explicit construction of the Green (semigroup) theoretical kernel, a theorem linking the integrability conditions and their dependence on the spacetime dimensions is given. Relativistic wave equations with arbitrary spin and the causality problem are discussed with the algebraic interpretation of the radical operator and their relation with coherent and squeezed states. Also we perform by means of pure theoretical procedures (based in physical concepts and symmetry) the relativistic position operator which satisfies the conditions of integrability: it is a non-local, Lorentz invariant and does not have the same problems as the “local”position operator proposed by Newton and Wigner. Physical examples, as zitterbewegung and rogue waves, are presented and deeply analyzed in this theoretical framework.

Factorization method for Schrödinger equation in relativistic configuration space and q-deformations

Review paper is devoted to the relativistic configuration space (RCS) concept, a version of the relativistic Quantum Mechanics in RCS, the generalization of the Dirac-Infeld-Hall factorization method in the framework of the noncommutative differential calculus natural for RCS, different versions of the deformed oscillators, emerging as the generalization of the harmonic oscillator for RCS. In the formulation of the Newton-Wigner postulates for the relativistic localized states the hypothesis of commutativity of the position operator components is silently accepted as an evident fact. In the present work it is shown that commutativity is not necessary condition and the alternative (noncommutative) approach to the relativistic position operator and localization concept can be realized in a framework of the physically as well as mathematically comprehensive scheme. The different generalizations of the Dirac-Infeld-Hall factorization method for this case are constructed. This method enables us to find out all possible generalizations of the most important nonrelativistic integrable case—the harmonic oscillator. It is shown also that the relativistic oscillator = q—oscillator.

The Newton-Wigner localization concept and noncommutative space

In the formulation of the Newton-Wigner postulates for the relativistic localized states the hypothesis of commutativity of the position operator components is silently accepted as an evident fact. In the present work it is shown that commutativity is not necessary condition and the alternative (noncommutative) approach to the relativistic position operator and localization concept can be realized in a framework of the physically as well as mathematically comprehensive scheme.

Q-oscillators and relativistic position operators

We investigate the multi-dimensional q-oscillator whose commutation relations are invariant under the quantum group. The no-interaction limit corresponds to a contraction of the q-oscillator algebra and yields relativistic position operators which can be expressed in terms of the generators of the Poincaré group. This leads to the interpretation of the interacting q-oscillator as an relativistic quantum system and results in a hamiltonian whose spectrum is exactly exponential.

Group manifold analysis of the structure of relativistic quantum dynamics

We present a group law, derived as a contraction of the conformal group, from which we obtain by using a canonical procedure a relativistic quantum system with an invariant evolution parameter (the proper time) and where the position operator belongs to the Lie algebra of the group. The restriction of the theory to the mass shell breaks part of the symmetry; of the previous 15 generators, only 10 remain which generate an action of the Poincaré group defining an orbit in the former group manifold. Some comments on the relativistic position operator are also made.

Multiparticle eigenvectors for relativistic position operators

In the first part of the paper, the so-called generalized position operators proposed by Fleming are investigated. Compact expressions for these operators are given (a so-called transversal angular momentum operator is introduced for this purpose), and the properties of their algebraic structure are discussed. The second part is devoted to multiparticle eigenvectors for two of the three kinds of the position operators. The obtained solutions permit one to separate property the global motion of a physical system from its internal motions. The obtained internal variables are relativistic counterparts of relative velocities. The considerations are carried out in the general framework of quantum field theory and are based on the reconstruction theorem of the axiomatic quantum field theory. This also allows us to use the results for interacting systems. The discussion of Lorentz covariance of the localized states is carried out.

Position Operators and Proper Time in Relativistic Quantum Mechanics

Covariant four-vector position operators ${X}^{\ensuremath{\mu}}$ are proposed, which form a natural operator generalization of the four-position in relativistic classical mechanics. These ${X}^{\ensuremath{\mu}}$ are defined by specifying commutation relations of the ${X}^{\ensuremath{\mu}}$ with the Poincar\'e generators ${P}^{\ensuremath{\mu}}$ and ${M}^{\ensuremath{\mu}\ensuremath{\nu}}$, and thereby extending the Poincar\'e algebra to a larger algebra whose representations are subsequently found. The ${X}^{\ensuremath{\mu}}$ are shown to be acceptable relativistic position operators within a proper-time dynamical framework. A single Hamiltonian is used for all spins, with a covariant proper-time description. The dynamics is capable of describing the time evolution of states which are not mass eigenstates. An automatic Foldy-Wouthuysen-type diagonalization is achieved for all spin representations, with spin and orbital angular momentum being separately conserved. The connection with the standard theory is made via the specific field equations. In making this connection to the standard theory of half-integral spins, the origin of Zitterbewegung and the nonseparate conservation of spin and angular momentum are clarified. The connection to Maxwell's equations provides an interesting statement of those equations and of gauge invariance. The unphysical representations of negative and imaginary mass and continuous spin are not present in this formalism. Other features are discussed.

The «anomalous zitterbewegung» of composite particles

It is shown that the range of the zitterbewegung for generalized infinite-component theories describing realistic composite particles is not the Compton wavelength, but the analog of the Bohr radius. This range is quite different from that in the case of simple Majorana-type equations where it is equal to the Compton wavelength for the ground state. The problems associated with the consistent definition of relativistic position operators for composite particles are discussed.

Localized states and group contraction in 4+2 de Sitter space

A detailed study is made of the problem of imbedding spacetime in 4+2 de Sitter space. It is shown that such an imbedding invariably leads to a contraction of the groupO 4,2 and to a restriction of the Poincare group operations to a spacelike surface in space-time. Localized particle states are constructed as normalizable eigenfunctions of the mass-squared operator. Explicit expressions for relativistic position operators are derived from the contraction of the algebra of the groupO 4,2 introduced by a certain limit-taking involved in the space-time imbedding process.

The relativistic position operator at subatomic level

A certain class of Lie algebras which contain the Lie algebra of the inhomogeneous Lorentz group and the relativistic position operators is derived, and it is shown how the Lie algebra of quantum mechanics is connected with the Lie algebra of the groupSO 4,2.