Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences· 2026Q1
Dirac wave functions of positive energy with arbitrarily small position uncertainty
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- Q1SCImago
- 2026year
Short summary
Wave functions from the positive-energy subspace of the free Dirac Hamiltonian can have arbitrarily small position uncertainty, disproving a long-standing hypothesis.
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Key points
- Wave functions from the positive-energy subspace (H+) of the free Dirac Hamiltonian are analyzed.
- The hypothesis that these wave functions have a positive lower bound to position uncertainty (σx) is investigated.
- A sequence of wave functions, previously introduced by Bracken and Melloy, is used to show this hypothesis is false.
- The paper provides a corrected proof that such states can be arbitrarily narrow in position.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract We consider wave functions in the Hilbert space H=L2(R3,C4) of a single Dirac particle, specifically from the positive-energy subspace H+ of the free Dirac Hamiltonian. Over the decades, various authors hypothesized that for wave functions from H+, there is a positive lower bound to the position uncertainty σx; in other words, that such states cannot be arbitrarily narrow in x. Using a sequence of wave functions introduced by Bracken and Melloy, we show that this hypothesis is false. (In fact, they already stated that it is false, but their proof that their sequence is a counter-example had a gap.)
The authors' abstract, as published at the source. Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics