The relativistic semi-classical approximation for a free massive particle is studied using the Wigner-Weyl formalism. A non-covariant Wigner function is proposed using the Newton-Wigner position operator. The perturbative solution for the time evolution is found. Causality is found to be perturbatively respected.
The relativistic phase-space representation by means of the usual position and momentum operators for a class of observables with Weyl symbols independent of charge variable (i.e. with any combination of position and momentum) is proposed. The dynamical equation coincides with its analogue in the non-local theory (generalization of the Newton-Wigner position operator approach) under conditions when particles creation is impossible. Differences reveal themselves in specific constraints on possible initial conditions.
Although the concept of localization in non-relativistic quantum mechanics is mathematically well-defined there is no obvious and unambiguous way to generalize it to relativistic quantum theory. After a brief review of localization in quantum mechanics the Newton-Wigner localization scheme is introduced and it is shown by the example of a massive spinless system how it leads to a position operator with seemingly unphysical properties. This motivated the development of several theorems which claim to rule out the existence of localizable particles. A particularly important one of them is presented and its validity discussed. In the last section however it is shown that if localization is considered with respect to spacelike hyperplanes, the properties of the Newton-Wigner position operators are not necessarily unphysical and localizable particles can exist.
Abstract. We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we re-discuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centered on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves. 1. introduction The Wigner [11] function W is a representation of the quantum density operator ˆρ as a real function on phase space, for a non-relativistic dynamical system with classical-quantum correspondence and n degrees of freedom. Here, all the underlying geometry is euclidean and we can include the Wigner function within the general framework of Weyl representation of operators: Â → A ∈ Ck IR (IR2n), so W(x) = (2π�) −n ρ(x) , where x ≡ (p1,...pn, q1,...qn) denotes a point in phase space. In this context, the quantum evolution of the density operator ˆρt effected by an autonomous external hamiltonian
We study the Classical Particle Limit of Non-Relativistic Quantum Mechanics. We show that the unitary group describing the evolution of the quantum fluctuation around any classical phase orbit has a classical limit as ħ→0 in the strong operator topology for a very large class of time independent scalar and vector potentials, which in practice covers all physically interesting cases. We also show that the mean values of the quantum mechanical position and velocity operators on suitable states, obtained by time evolution of the product of a Weyl operator centred around the large coordinates and momenta ħ −1/2 ξ and ħ −1/2 π and a fixed ħ-independent wave function, converge to the solution of the classical equations with initial data ξ, π as ħ→0 for a broad class of repulsive interactions On demontre que le groupe unitaire qui decrit l'evolution des fluctuations quantiques autour d'une orbite classique quelconque a une limite classique quand ħ→0 dans la topologie forte des operateurs, pour une vaste classe de potentiels qui couvre en pratique tous les cas physiquement interessants. On demontre que les valeurs moyennes des operateurs quantiques position et vitesse, dans des etats convenables obtenus par l'evolution temporelle du produit d'un operateur de Weyl centre autour de grandes coordonnees et de grandes impulsions ħ −1/2 ξ, ħ -1/2 π et d'une fonction d'onde independante de ħ, tendent vers la solution des equations classiques avec donnees initiales ξ, π quand ħ→0 pour une grande classe d'interactions repulsives
Abstract Avoiding the assumption that relativistic scattering be describable by interacting fields we find in the Schrödinger picture relativistic scattering closely analogue to the non-relativistic case. On the space of scattering states the invariant mass operator M′ M ′ of the interacting time evolution has to be unitarily equivalent to the invariant mass M=P2 M = P 2 where P, acting on many-particle states, is the sum of the one-particle four-momenta. For an observer at rest P0 P 0 generates the free time evolution. Poincaré symmetry requires the interacting Hamiltonian H′ H ′ to Lorentz transform as 0-component of a four-vector and to commute with the four-velocity U=P/M U = P / M but not with P, else there is no scattering. Even though H′ H ′ does not commute with P, the scattering matrix does. The four-velocity U generates translations of states as they are seen by shifted observers. Superpositions of nearly mass degenerate particles such as a Klong K long are seen by an inversely shifted observer as a shifted Klong K long with an unchanged relative phase. In contrast, the four-momentum P generates oscillated superpositions e.g. a shifted Kshort K short with a changed relative phase. The probability of scattering of massive particles is shown to be approximately proportional to the spacetime overlap of their position wave functions. This is basic to macroscopic locality and justifies to represent the machinery of actual scattering experiments by the vacuum. In suitable variables the relativistic Hamiltonian of many-particle states is not the sum of a Hamiltonian for the motion of the center and a commuting Hamiltonian for the relative motion but factorizes as their product. They act on different variables of the wave functions.