Hegerfeldt's theorem

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 probabilty 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 1998.[6][7]

See also

References
  1. 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.
  2. 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. doi:10.1007/s00023-022-01215-8. PMID 36817968.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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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