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arXiv:1811.03323 · Journal of Physics A Mathematical and Theoretical

No relativistic Newton–Wigner probability current for any spin

Scott E. Hoffmann

Published Apr 23, 2019• 5 citations
arXivHTMLPDF

Abstract

Abstract We investigate whether the Newton–Wigner position probability density, extended from spinless particles to electrons/positrons and particles of higher spin, can be incorporated as the zero component of a four-component probability current density that transforms locally as a four-vector function of the spacetime coordinates. We find that this is not possible, in all cases.

DOI
10.1088/1751-8121/ab1bc9

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Abstract We investigate whether the Newton–Wigner position probability density, extended from spinless particles to electrons/positrons and particles of higher spin, can be incorporated as the zero component of a four-component probability current density that transforms locally as a four-vector function of the spacetime coordinates. We find that this is not possible, in all cases.
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.
This paper gives formulas for position operators that apply uniformly for all spins of particle. The three-vector X, of the Newton-Wigner and Foldy type, is treated first. Then it is shown that, although X has complicated Lorentz transformation properties, it is linearly related to a certain four-vector ${Y}_{\ensuremath{\mu}}$ which is built up from the Poincar\'e group generators. The four-vector is the generalization of the classical notion of the component of the position four-vector in the direction perpendicular to the world line of the particle.
The connection between the (orthodox) single-particle Newton-Wigner position and the Bohm/de Broglie hidden-particle position in the relativistic regime is investigated. The effect of non-positive excursions of the ensemble density for extreme cases of positive-energy waves is easily computed using an integral of the equations of motion developed here and is interpreted in terms of virtual-like pair creation and annihilation beneath the Compton wavelength. A Bohm-theoretic description of the acausal explosion of Newton-Wigner-localized states is presented in detail for spin-0 particles in attempt to relate Bohm/de Broglie and orthodox positions. Spin-1/2 particles are considered as well with the assumption that the convective current component describes ensemble dynamics as with the spin-0 theory. For spin-1/2 particles, a specific second-rank tensor is defined in terms of Dirac spinors for generalizing from simply a quantum potential to a stress tensor required to account for the force of pilot wave on particle. A simple dependence of the stress tensor on a two-component spin pseudovector field is determined for approaching a non-relativistic limit. Consistency is found with an earlier non-relativistic theory of objects with spin.
A manifestly covariant formalism is developed suitable for a discussion of position operators for relativistic systems. It is shown that the rest-frame center of mass and the frame-dependent center of mass, which were studied previously by Pryce and M\o{}ller, and the local Newton-Wigner operator all transform like the spatial part of a four-vector when appropriately interpreted. The formalism allows an equivalent treatment of the space and time components of a position four-vector operator, and manifestly covariant canonical commutation rules are obtained for the three operators mentioned above. Finally it is suggested that the Newton-Wigner operator be interpreted as a center of spin.
In this paper, a conserved current for Klein--Gordon equation is derived. It is shown, for (1+11+11+1) dimensions, the first component of this current is non-negative and reduces to ${|\ensuremath{\phi}|}^{2}$ in nonrelativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from ${|\ensuremath{\phi}|}^{2}$ only when the uncertainty in momentum is greater than m0c{m}_{0}cm0​c.