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arXiv:1804.00826

Spreading of relativistic probability densities and Lorentz contraction

Scott E. Hoffmann

Published Apr 3, 2018• 0 citations
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Abstract

We find the laws for the spreading of the spatial widths (parallel and transverse to the direction of average motion) of the relativistic position probability density for a massive, spinless particle. We find that when the momentum width of the wavepacket is small compared to the average momentum, there is a long time over which spreading is minimal. This result may be useful in particle accelerator design. We also demonstrate the Lorentz contraction of a wavepacket using relativistic probability amplitudes.

DOI
10.48550/arxiv.1804.00826

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We find the laws for the spreading of the spatial widths (parallel and transverse to the direction of average motion) of the relativistic position probability density for a massive, spinless particle. We find that when the momentum width of the wavepacket is small compared to the average momentum, there is a long time over which spreading is minimal. This result may be useful in particle accelerator design. We also demonstrate the Lorentz contraction of a wavepacket using relativistic probability amplitudes.
Abstract We challenge the widespread belief, originated by Newton and Wigner (1949 Rev. Mod. Phys. 21 400) that the incorporation of special relativity into quantum mechanics implies that a massive particle cannot be localized within an arbitrarily small spatial extent, that there is a minimum width approximately equal to the Compton wavelength. Our argument is in four parts. First, the scalar function used by Newton and Wigner as a measure of localization is not a position probability amplitude. The correct relativistic position probability amplitude becomes a delta function for a state vector localized according to the criteria of Newton and Wigner. Second, the possibility of Lorentz contraction as observed from a boosted frame means that the wavepacket width in the boost direction can take arbitrarily small values. Third, we refer to the work of Almeida and Jabs (1984 Am. J. Phys. 52 921) who show that the long time wavepacket spreading rate for relativistic position probability amplitudes is always less than the speed of light no matter how small the initial width of the wavepacket. Lastly, we show that it is a simple matter to construct scalar amplitudes with spatial widths smaller than the supposed minimum.
We study the evolutions of selected quasi-(1+1) dimensional wavepacket solutions to the Klein–Gordon equation for a relativistic charged particle in uniform motion or accelerated by a uniform electric field in Minkowski space . We explore how good the charge density of a Klein–Gordon wavepacket can be approximately described by a Gaussian state with the single-particle interpretation. We find that the minimal initial width of a wavepacket for a good Gaussian approximation in position space is about the Compton wavelength of the particle divided by its Lorentz factor at the initial moment. This can be considered as a manifestation of relativistic length contraction, which is also observed numerically in the spreading of the wavepacket’s charge density.
The author has earlier defined the quantum two-time localization problem as the minimizing of a particle's position spread about specified points at two distinct times. In the present article optimum localization is found for relativistic massive free particles. For short time intervals, spreading necessarily occurs at the speed of light while for long times the previously found diffusion-like behaviour is recovered. In defining relativistic localization, use is made of the work of Newton and Wigner (1949); in particular, their restriction to the positive energy hyperboloid is found to be necessary to recover the nonrelativistic limit of wavepacket spreading.
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.
For a wavepacket representing a particle subject to a conservative force consideration is given to the evolution in time of the mean position and momentum and to the evolution of the spread of these quantities as measured by the mean square deviation from the mean. A closed system of equations involving these spreads is obtained by expanding the potential in powers about the mean position and by neglecting terms of third order or higher in the deviations from the mean, an approximation appropriate when the force does not vary too much over the width of the packet. These equations are solved in terms of the trajectories of a classical time-dependent oscillator. These trajectories can be found by differentiation of the trajectories for the force under consideration. In more than one dimension, or for more than one particle, the appropriate generalisation of the spread is the set of second-order correlations.