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arXiv:2202.10393 · Reports on Mathematical Physics

Time of arrival operator in the momentum space

Agnieszka Matylda Schlichtinger, Arkadiusz Jadczyk

Published Jun 1, 2023• 3 citations
arXivHTMLPDF
DOI
10.1016/s0034-4877(23)00037-x

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Relativistic free-motion time-of-arrival theory for massive spin-1/2 particles is systematically developed. Contrary to the nonrelativistic time-of-arrival operator studied thoroughly in previous literatures, the relativistic time-of-arrival operator possesses self-adjoint extensions because of the particle-antiparticle symmetry. The nonrelativistic limit of our theory is in agreement with the nonrelativistic time-of-arrival theory. By comparing the time-of-arrival operator with the Hamiltonian operator of a free Dirac particle, one can show a duality between position space and momentum space.
The position-momentum quasidistribution obtained from an Arthurs and Kelly joint measurement model is used to obtain indirectly an ``operational'' time-of-arrival (TOA) distribution following a quantization procedure proposed by Kocha\ifmmode \acute{n}\else \'{n}\fi{}ski and W\'odkiewicz [Phys. Rev. A 60, 2689 (1999)]. This TOA distribution is not time covariant. The procedure is generalized by using other phase-space quasidistributions, and sufficient conditions are provided for time covariance that limit the possible phase-space quasidistributions essentially to the Wigner function, which, however, provides a nonpositive TOA quasidistribution. These problems are remedied with a different quantization approach which, on the other hand, does not guarantee normalization. Finally, an Arthurs and Kelly measurement model for TOA and energy is worked out, valid also for arbitrary conjugate variables when one of the variables is bounded from below. The marginal TOA distribution so obtained, a distorted version of Kijowski's distribution, is time covariant, positive, and normalized.