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Journal of Physics A Mathematical and General

Position operators for the relativistic non-interacting n-particle system

Ulrich Mutze

Published Apr 1, 1978• 3 citations
Publisher

Abstract

The subject under consideration is a system of n non-interacting particles with spin, being described by a tensor product of n unitary irreducible massive representations of the Poincare group. A unitary transformation that effects a separation of centre-of-mass variables and internal variables is constructed. By means of this transformation, for any of the n particles there are constructed two position operators both acting on the internal variables only. The first of these operators (called the internal position) commutes with all spatial translations and with the centre-of-mass position, the second one (called the impact position) commutes moreover with all time translations. It is argued that the impact position may be interpreted as the internal position taken at the instant when the n particles are most closely together. By considering spatial separation properties, it is shown that both the internal position and the impact position of any particle depend essentially on the location of all particles of the system.

DOI
10.1088/0305-4470/11/4/007

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The subject under consideration is a system of n non-interacting particles with spin, being described by a tensor product of n unitary irreducible massive representations of the Poincare group. A unitary transformation that effects a separation of centre-of-mass variables and internal variables is constructed. By means of this transformation, for any of the n particles there are constructed two position operators both acting on the internal variables only. The first of these operators (called the internal position) commutes with all spatial translations and with the centre-of-mass position, the second one (called the impact position) commutes moreover with all time translations. It is argued that the impact position may be interpreted as the internal position taken at the instant when the n particles are most closely together. By considering spatial separation properties, it is shown that both the internal position and the impact position of any particle depend essentially on the location of all particles of the system.
The problem of describing a quantum mechanical system is considered. For a system which is invariant under the Poincaré (inhomogeneous Lorentz) transformations this description is provided by the generators of those transformations, which satisfy the usual commutation relations. General expressions for the centre of mass position and internal angular momentum operators, in terms of these generators, are obtained. The generators of the Poincaré transformations are written in terms of the fundamental dynamical variables for several systems. The systems considered are those consisting of a single free spinless particle, a single free particle with spin, a single free Dirac particle, n noninteracting particles, and ft interacting particles. In each case, the centre of mass position and internal angular momentum are given in terms of the fundamental dynamical variables of the system. For the first two systems listed above, these two operators are found to be equal to the Cartesian coordinates and spin of the particle, respectively. In the case of the Dirac particle, these operators are seen to be related to the Cartesian coordinates and spin of the particle via the Pryce-Foldy-Wouthuysen transformation. Following Bakamjian and Thomas, interaction is introduced to the n particle system via a single operator which depends only on internal variables. The condition of "asymptotic covariance" of the scattering operator is discussed for two particle scattering. The scattering operator for a two particle system with no bound states and with Poincaré generators of the Bakamjian-Thomas form is seen to be asymptotically covariant.
We consider a massive particle of arbitrary spin and the basis vectors that carry the unitary, irreducible representations of the Poincaré group. From the complex coefficients in normalizable superpositions of these basis vectors, we identify momentum/spin-component probability amplitudes with the same interpretation as in the nonrelativistic theory. We find the relativistic transformations of these amplitudes, which are unitary in that they preserve the modulus-squared of scalar products from frame to frame. Space inversion and time reversal are also treated. We reconsider the Newton- Wigner construction of eigenvectors of position and the position operator. Position/spin-component probability amplitudes are also identified and their relativistic, unitary, transformations derived. Again, space inversion and time reversal are considered. For reference, we show how to construct positive energy solutions of the Klein-Gordon and Dirac equations in terms of probability amplitudes. We find the boost transformation of the position operator in the spinless case and present some results on the relativity of position measurements. We consider issues surrounding the classical concept of causality as it applies in quantum mechanics. We briefly examine the relevance of the results presented here for theories of interaction.
The definition of the total momentum, position, and spin for Galilean- or Lorentz-invariant two-free-particle systems is discussed by using the requirement that the generators of the respective invariance groups should have the same form expressed in terms of them as for a single particle. Internal c.m. dynamical variables are introduced by applying the singular transformation due to Gartenhaus and Schwartz on the basic single-particle dynamical variables, the transformation mapping the whole Hilbert space onto the c.m. subspace spanned by states of zero total momentum. The form of the internal c.m. position operator is given for particles with spin, for what appears to be the first time. Using these dynamical variables, it is shown how an interaction can be introduced while maintaining Galilean or Lorentz invariance and satisfying the asymptotic condition of freely propagating particles for large separations.
A review is given of a new relativistic classical mechanics for N-particle systems compatible with relativistic bound states. It is formulated in global non-inertial frames in Minkowski space-time by means of parametrized Minkowski theories and then restricted to the intrinsic inertial rest frame of the isolated system. There is a complete control on the relativistic collective variables (Newton-Wigner center of mass, Fokker-Pryce center of inertia, M/oller center of energy) and on the realization of the Poincare' algebra (with the explicit form of the interaction-dependent Lorentz boosts). The particle world-lines are found to correspond to the ones of predictive mechanics and localization problems are clarified. The model can be consistently quantized avoiding the instantaneous spreading of the center-of-mass wave packets (Hegerfeldt theorem), because the non-local non-covariant center of mass is a non-measurable quantity. This implies that the Hilbert space must be the tensor product of the center-of-mass Hilbert space with the one of relative motions. This spatial non-separability (due to the Lorentz signature of space-time) makes relativistic entanglement much more involved than the non-relativistic one.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.