Hegerfeldt's theorem

{{Short description|Theorem in relativistic quantum mechanics}}

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.{{Cite book |last1=Halvorson |first1=Hans |last2=Clifton |first2=Rob |title=Ontological Aspects of Quantum Field Theory |date=November 2002 |chapter=No place for particles in relativistic quantum theories? |pages=181–213 |doi=10.1142/9789812776440_0010|arxiv=quant-ph/0103041 |isbn=978-981-238-182-8 |s2cid=8845639 }}{{Cite journal |last1=Finster |first1=Felix |last2=Paganini |first2=Claudio F. |date=2022-09-16 |title=Incompatibility of Frequency Splitting and Spatial Localization: A Quantitative Analysis of Hegerfeldt's Theorem |journal=Annales Henri Poincaré |volume=24 |issue=2 |pages=413–467 |doi=10.1007/s00023-022-01215-8 |pmid=36817968 |doi-access=free |pmc=9928833 |arxiv=2005.10120 }} 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.{{Cite journal |last1=Moretti |first1=Valter |date=2023-06-07 |title=On the relativistic spatial localization for massive real scalar Klein–Gordon quantum particles |journal=Letters in Mathematical Physics |volume=66 |issue=3 |doi=10.1007/s11005-023-01689-5 |doi-access=free |arxiv=2304.02133 |bibcode=2023LMaPh.113...66M |hdl=11572/379089 |hdl-access=free }}

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.{{Cite journal |last1=Barat |first1=N. |last2=Kimball |first2=J. C. |date=February 2003 |title=Localization and Causality for a free particle |journal=Physics Letters A |volume=308 |issue=2–3 |pages=110–115 |doi=10.1016/S0375-9601(02)01806-6|arxiv=quant-ph/0111060 |bibcode=2003PhLA..308..110B |s2cid=119332240 }}{{Cite journal |last=Hobson |first=Art |date=2013-03-01 |title=There are no particles, there are only fields |journal=American Journal of Physics |volume=81 |issue=3 |pages=211–223 |doi=10.1119/1.4789885 |arxiv=1204.4616|bibcode=2013AmJPh..81..211H |s2cid=18254182 }} 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.{{Cite journal |last=Hegerfeldt |first=Gerhard C. |date=1974-11-15 |title=Remark on causality and particle localization |url=https://link.aps.org/doi/10.1103/PhysRevD.10.3320 |journal=Physical Review D |language=en |volume=10 |issue=10 |pages=3320–3321 |doi=10.1103/PhysRevD.10.3320 |bibcode=1974PhRvD..10.3320H |issn=0556-2821|url-access=subscription }}{{Cite book |last=Hegerfeldt |first=Gerhard C. |title=Irreversibility and Causality Semigroups and Rigged Hilbert Spaces |date=1998 |chapter=Causality, particle localization and positivity of the energy |series=Lecture Notes in Physics |volume=504-504 |pages=238–245 |doi=10.1007/BFb0106784|arxiv=quant-ph/9806036 |isbn=978-3-540-64305-0 |s2cid=119463020 }}{{Cite journal |last=Hegerfeldt |first=G.C. |date=December 1998 |title=Instantaneous spreading and Einstein causality in quantum theory |journal=Annalen der Physik |language=en |volume=510 |issue=7–8 |pages=716–725 |doi=10.1002/andp.199851007-817 |arxiv=quant-ph/9809030 |s2cid=248267636 }}