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Journal of Mathematical Physics

Group content of the Foldy–Wouthuysen transformation and the nonrelativistic limit for arbitrary spin

Juan León, Mariano Quirós C, J. Ramírez Mittelbrunn

Published Jun 1, 1979• 9 citations

Abstract

The relationship between Foldy–Wouthuysen and Lorentz transformations has been clarified throughout this paper. We propose a generalized FW transformation connecting two particular realizations of the (m,j) representation of the Poincaré group: the covariant realization and a canonical realization acting on relativistic probability amplitudes. Fermions and bosons must be considered separately because the intrinsic parity of the particle–antiparticle systems is (−1)2j. Thus for fermions we can directly take the 2(2j+1) - dimensional Joos–Weinberg covariant realization, while for bosons we must double it to get a reducible 4(2j+1) - dimensional realization where particles and antiparticles lie in orthogonal subspaces. In short, in momentum space the FW transformation is the matrix representing a Lorentz boost times the factor (m/p0)1/2, while in configuration space the FW transformation does not belong to the Poincaré group. The last part of the paper is devoted to getting quantum-mechanical representations of the Galileo group as a contraction of Poincaré group representations by using mathematical methods earlier developed by Mickelsson and Niederle. The relevance of our generalized FW transformation for getting a smooth, well defined, nonrelativistic limit is a remarkable result.

DOI: 10.1063/1.524183

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The relationship between Foldy–Wouthuysen and Lorentz transformations has been clarified throughout this paper. We propose a generalized FW transformation connecting two particular realizations of the (m,j) representation of the Poincaré group: the covariant realization and a canonical realization acting on relativistic probability amplitudes. Fermions and bosons must be considered separately because the intrinsic parity of the particle–antiparticle systems is (−1)2j. Thus for fermions we can directly take the 2(2j+1) - dimensional Joos–Weinberg covariant realization, while for bosons we must double it to get a reducible 4(2j+1) - dimensional realization where particles and antiparticles lie in orthogonal subspaces. In short, in momentum space the FW transformation is the matrix representing a Lorentz boost times the factor (m/p0)1/2, while in configuration space the FW transformation does not belong to the Poincaré group. The last part of the paper is devoted to getting quantum-mechanical representations of the Galileo group as a contraction of Poincaré group representations by using mathematical methods earlier developed by Mickelsson and Niederle. The relevance of our generalized FW transformation for getting a smooth, well defined, nonrelativistic limit is a remarkable result.

We construct the Foldy-Wouthuysen transformation operator for wave equations describing two interacting particle systems, and possessing Poincare invariance and manifest covariance. We consider systems composed of one spin-1/2 fermion and one spin-0 boson, and also of one spin-1/2 fermion and one spin-1/2 antifermion. We first construct the above operator in the free case. Then, we show that for several classes of interaction potential it can also be constructed in a compact form. As compared to the one-particle case, manifest covariance facilitates here the construction of the Foldy-Wouthuysen transformation operator, although the number of particles is larger On construit l'operateur de la transformation de Foldy-Wouthuysen pour des equations d'onde decrivant des systemes de deux particules en interaction, et possedant l'invariance de Poincare et la covariance manifeste. On considere des systemes composes d'un fermion de spin 1/2 et d'un boson de spin O, et aussi d'un fermion de spin 1/2 et d'un antifermion de spin 1/2. On construit d'abord cet operateur dans le cas libre. Puis on montre que plusieurs classes de potentiels d'interaction, il peut aussi etre construit sous une forme compacte. En comparaison avec le cas a une particule, la covariance manifeste facilite ici la construction de l'operateur de la transformation de Foldy-Wouthuysen, bien que le nombre de particules soit plus grand

An examination is made of the consequences for the quantum mechanics of spinning particles of equations characteristic of Lorentz-covariant position variables. These equations are commutator analogs of the Poisson bracket equations that express the familiar transformation properties of space-time events in classical mechanics. For a particle of zero spin it is found that the usual canonical coordinate is the unique solution of these equations. For a particle with positive spin there is no position operator which satisfies these equations and has commuting components. For a particle and antiparticle there is a unique solution with commuting components which is valid for all values of the spin and reduces for zero spin to the canonical coordinate. For spin \textonehalf{} this is the Foldy-Wouthuysen transform of the position operator of the Dirac equation. A generalization of the inverse Foldy-Wouthuysen transformation, valid for any value of the spin, appears as a unique unitary transformation which takes this generalized Dirac position to the canonical coordinate. The application of this transformation to the canonical form of the Hamiltonian gives a generalization of the Dirac equation Hamiltonian. This is developed and compared with the literature for spin 1. It gives a nonlocal equation as the spin 1 analog of the Dirac equation.

The relativistic invariant Lagrange approach for the spinor field in Foldy – Wouthuysen representation is constructed. The analogy with classical mechanics for the system with arbitrary number degrees of freedom is used for this construction. The dynamical variables for the spinor field in this representation are found. The conservation laws in terms of quantum mechanical particle-antiparticle momentum-spin amplitudes are presented. In this way the direct physical sense of the conserved quantities is demonstrated. 12 important additional conserved quantities are obtained together with 10 well-known conservation laws for the spinor field, which are the consequences of the Poincare symmetry of the theory. The additional dynamical variables are the consequences of the fact that both orbital and spin parts of the total 4-angular momentum of the spinor field are conserved independently.

It has long been speculated that the Dirac or, more generally, the Dirac-Pauli spinor in the Foldy-Wouthuysen (FW) representation should behave like a classical relativistic spinor in the low-energy limit when the probability of particle-antiparticle pair creation and annihilation is negligible. In the weak-field limit of static and homogeneous electromagnetic fields, by applying the method of direct perturbation theory inductively on the orders of

1/c

in the power series, we rigorously prove that it is indeed the case: the FW transformation of the Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.

The spin reduced matrix defined by taking partial trace with the momentum variable has been shown not to transform covariantly under Lorentz transformations. In this paper, I suggest another representation space of the Poincare group, the elements of which are in a sense apparent wave functions. In this new Hilbert space, the naive spin observable

\frac{1}{2}\boldsymbol{\tau}

from nonrelativistic quantum mechanics on the old Hilbert space is represented as the Newton-Wigner spin operator. Also on this space, one can naturally associate to each state a Lorentz covariant spin reduced density matrix, which is then given a new interpretation.