Bundle Structure of Massless Unitary Representations of the Poincaré Group

authored by: Norbert Dragon
Abstract: Reviewing the construction of induced representations of the Poincaré group of four-dimensional spacetime we find all massive representations, including the ones on interacting many-particle states. Massless momentum wavefunctions of non-vanishing helicity turn out to be more precisely sections of a U(1)-bundle over the massless shell, a property which to date was overlooked in quantum field theory and in bracket notation. Our traditional notation of states in Hilbert space enables questions about square integrability and smoothness. Their answers complete the picture of relativistic quantum physics. Frobenius reciprocity prohibits massless one-particle states with total angular momentum less than the modulus of the helicity. There is no two-photon state with J=1, explaining the longevity of orthopositronium. Partial derivatives of the momentum wave functions are no operators in the space of massless states with nonvanishing helicity. They allow only for covariant, noncommuting derivatives. The massless shell has a noncommutative geometry with helicity being its topological charge.
Organisation(s): Institute of Theoretical Physics
Type: Article
Journal: International Journal of Theoretical Physics
Volume: 63
No. of pages: 23
ISSN: 0020-7748
Publication date: 14.06.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: General Mathematics, Physics and Astronomy (miscellaneous)
Electronic version(s): https://doi.org/10.1007/s10773-024-05612-z (Access: Open)

BibTeX

@article{064e102011c048f4b506fe6b46b3778c,
title = "Bundle Structure of Massless Unitary Representations of the Poincar{\'e} Group",
abstract = "Reviewing the construction of induced representations of the Poincar{\'e} group of four-dimensional spacetime we find all massive representations, including the ones on interacting many-particle states. Massless momentum wavefunctions of non-vanishing helicity turn out to be more precisely sections of a U(1)-bundle over the massless shell, a property which to date was overlooked in quantum field theory and in bracket notation. Our traditional notation of states in Hilbert space enables questions about square integrability and smoothness. Their answers complete the picture of relativistic quantum physics. Frobenius reciprocity prohibits massless one-particle states with total angular momentum less than the modulus of the helicity. There is no two-photon state with J=1, explaining the longevity of orthopositronium. Partial derivatives of the momentum wave functions are no operators in the space of massless states with nonvanishing helicity. They allow only for covariant, noncommuting derivatives. The massless shell has a noncommutative geometry with helicity being its topological charge.",
keywords = "Bundle, Frobenius, Helicity, Interacting representation, Massless, Smoothness of generators",
author = "Norbert Dragon",
note = "Publisher Copyright: {\textcopyright} The Author(s) 2024.",
year = "2024",
month = jun,
day = "14",
doi = "10.1007/s10773-024-05612-z",
language = "English",
volume = "63",
journal = "International Journal of Theoretical Physics",
issn = "0020-7748",
publisher = "Springer New York",
number = "6",
}