Hegerfeldt's theorem is a no-go theorem that demonstrates the incompatibility of the existence of spatially localized discrete particles with the combination of the principles of quantum mechanics and special relativity. A crucial requirement is that the states of single particle have positive energy. It has been used to support the conclusion that reality must be described solely in terms of field-based formulations.[1][2] However, it is possible to construct localization observables in terms of positive-operator valued measures that are compatible with the restrictions imposed by the Hegerfeldt theorem.[3]
Specifically, Hegerfeldt's theorem refers to a free particle whose time evolution is determined by a positive Hamiltonian. If the particle is initially confined in a bounded spatial region, then the spatial region where the probability to find the particle does not vanish, expands superluminarly, thus violating Einstein causality by exceeding the speed of light.[4][5] Boundedness of the initial localization region can be weakened to a suitably exponential decay of the localization probability at the initial time. The localization threshold is provided by twice the Compton length of the particle. As a matter of fact, the theorem rules out the Newton-Wigner localization.
The theorem was developed by Gerhard C. Hegerfeldt and first published in 1974.[6][7][8]
See also
editReferences
edit- ^ Halvorson, Hans; Clifton, Rob (November 2002). "No place for particles in relativistic quantum theories?". Ontological Aspects of Quantum Field Theory. pp. 181–213. arXiv:quant-ph/0103041. doi:10.1142/9789812776440_0010. ISBN 978-981-238-182-8. S2CID 8845639.
- ^ Finster, Felix; Paganini, Claudio F. (2022-09-16). "Incompatibility of Frequency Splitting and Spatial Localization: A Quantitative Analysis of Hegerfeldt's Theorem". Annales Henri Poincaré. 24 (2): 413–467. arXiv:2005.10120. doi:10.1007/s00023-022-01215-8. PMID 36817968.
- ^ Moretti, Valter (2023-06-07). "On the relativistic spatial localization for massive real scalar Klein–Gordon quantum particles". Letters in Mathematical Physics. 66. doi:10.1007/s11005-023-01689-5. hdl:11572/379089.
- ^ Barat, N.; Kimball, J. C. (February 2003). "Localization and Causality for a free particle". Physics Letters A. 308 (2–3): 110–115. arXiv:quant-ph/0111060. Bibcode:2003PhLA..308..110B. doi:10.1016/S0375-9601(02)01806-6. S2CID 119332240.
- ^ Hobson, Art (2013-03-01). "There are no particles, there are only fields". American Journal of Physics. 81 (3): 211–223. arXiv:1204.4616. Bibcode:2013AmJPh..81..211H. doi:10.1119/1.4789885. S2CID 18254182.
- ^ Hegerfeldt, Gerhard C. (1974-11-15). "Remark on causality and particle localization". Physical Review D. 10 (10): 3320–3321. doi:10.1103/PhysRevD.10.3320. ISSN 0556-2821.
- ^ Hegerfeldt, Gerhard C. (1998). "Causality, particle localization and positivity of the energy". Irreversibility and Causality Semigroups and Rigged Hilbert Spaces. Lecture Notes in Physics. Vol. 504–504. pp. 238–245. arXiv:quant-ph/9806036. doi:10.1007/BFb0106784. ISBN 978-3-540-64305-0. S2CID 119463020.
- ^ Hegerfeldt, G.C. (December 1998). "Instantaneous spreading and Einstein causality in quantum theory". Annalen der Physik. 510 (7–8): 716–725. arXiv:quant-ph/9809030. doi:10.1002/andp.199851007-817. S2CID 248267636.
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