dorsal/arxiv
View SchemaQuantum Time and Spatial Localization: An Analysis of the Hegerfeldt Paradox
| Authors | Francis S. G. Von Zuben |
|---|---|
| Categories | |
| ArXiv ID | physics/9708030 |
| URL | https://arxiv.org/abs/physics/9708030 |
| DOI | 10.1063/1.1286877 |
| Journal | Journal of Mathematical Physics, Vol. 41, pp. 6093-6115 (September 2000) |
Abstract
Two related problems in relativistic quantum mechanics, the apparent superluminal propagation of initially localized particles and dependence of spatial localization on the motion of the observer, are analyzed in the context of Dirac's theory of constraints. A parametrization invariant formulation is obtained by introducing time and energy operators for the relativistic particle and then treating the Klein-Gordon equation as a constraint. The standard, physical Hilbert space is recovered, via integration over proper time, from an augmented Hilbert space wherein time and energy are dynamical variables. It is shown that the Newton-Wigner position operator, being in this description a constant of motion, acts on states in the augmented space. States with strictly positive energy are non-local in time; consequently, position measurements receive contributions from states representing the particle's position at many times. Apparent superluminal propagation is explained by noting that, as the particle is potentially in the past (or future) of the assumed initial place and time of localization, it has time to propagate to distant regions without exceeding the speed of light. An inequality is proven showing the Hegerfeldt paradox to be completely accounted for by the hypotheses of subluminal propagation from a set of initial space-time points determined by the quantum time distribution arising from the positivity of the system's energy. Spatial localization can nevertheless occur through quantum interference between states representing the particle at different times. The non-locality of the same system to a moving observer is due to Lorentz rotation of spatial axes out of the interference minimum.
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"abstract": "Two related problems in relativistic quantum mechanics, the apparent\nsuperluminal propagation of initially localized particles and dependence of\nspatial localization on the motion of the observer, are analyzed in the context\nof Dirac\u0027s theory of constraints. A parametrization invariant formulation is\nobtained by introducing time and energy operators for the relativistic particle\nand then treating the Klein-Gordon equation as a constraint. The standard,\nphysical Hilbert space is recovered, via integration over proper time, from an\naugmented Hilbert space wherein time and energy are dynamical variables. It is\nshown that the Newton-Wigner position operator, being in this description a\nconstant of motion, acts on states in the augmented space. States with strictly\npositive energy are non-local in time; consequently, position measurements\nreceive contributions from states representing the particle\u0027s position at many\ntimes. Apparent superluminal propagation is explained by noting that, as the\nparticle is potentially in the past (or future) of the assumed initial place\nand time of localization, it has time to propagate to distant regions without\nexceeding the speed of light. An inequality is proven showing the Hegerfeldt\nparadox to be completely accounted for by the hypotheses of subluminal\npropagation from a set of initial space-time points determined by the quantum\ntime distribution arising from the positivity of the system\u0027s energy. Spatial\nlocalization can nevertheless occur through quantum interference between states\nrepresenting the particle at different times. The non-locality of the same\nsystem to a moving observer is due to Lorentz rotation of spatial axes out of\nthe interference minimum.",
"arxiv_id": "physics/9708030",
"authors": [
"Francis S. G. Von Zuben"
],
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"doi": "10.1063/1.1286877",
"journal_ref": "Journal of Mathematical Physics, Vol. 41, pp. 6093-6115 (September\n 2000)",
"title": "Quantum Time and Spatial Localization: An Analysis of the Hegerfeldt Paradox",
"url": "https://arxiv.org/abs/physics/9708030"
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