Heisenberg versus the Covariant String

authored by: Norbert Dragon, Florian Oppermann
Abstract: A Poincaré multiplet of mass eigenstates (P²- m²) Ψ = 0 cannot be a subspace of a space with a D-vector position operator X= (X, ⋯ X_D_-₁) : the Heisenberg algebra [P^m, X_n] = i δ^m_n implies by a simple argument that each Poincaré multiplet of definite mass vanishes. The same conclusion follows from the Stone-von Neumann theorem. In a quantum theory the constraint of an absolutely continuous spectrum to a lower dimensional submanifold yields zero even if Dirac’s treatment of the corresponding classical constraint defines a symplectic submanifold with a consistent corresponding quantum model. Its Hilbert space is not a subspace of the unconstrained theory. Hence the operator relations of the unconstrained model need not carry over to the constrained model. Our argument excludes quantized worldline models of relativistic particles and the physical states of the covariant quantum string. We correct misconceptions about the generators of Lorentz transformations acting on particles.
Organisation(s): Institute of Theoretical Physics
Type: Article
Journal: International Journal of Theoretical Physics
Volume: 63
No. of pages: 9
ISSN: 0020-7748
Publication date: 04.01.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: General Mathematics, Physics and Astronomy (miscellaneous)
Electronic version(s): https://doi.org/10.48550/arXiv.2212.07256 (Access: Open)
https://doi.org/10.1007/s10773-023-05529-z (Access: Open)

BibTeX

@article{4fa7f968b2c44a0597d7a62933eb625f,
title = "Heisenberg versus the Covariant String",
abstract = "A Poincar{\'e} multiplet of mass eigenstates (P2- m2) Ψ = 0 cannot be a subspace of a space with a D-vector position operator X= (X, ⋯ XD-1) : the Heisenberg algebra [Pm, Xn] = i δmn implies by a simple argument that each Poincar{\'e} multiplet of definite mass vanishes. The same conclusion follows from the Stone-von Neumann theorem. In a quantum theory the constraint of an absolutely continuous spectrum to a lower dimensional submanifold yields zero even if Dirac{\textquoteright}s treatment of the corresponding classical constraint defines a symplectic submanifold with a consistent corresponding quantum model. Its Hilbert space is not a subspace of the unconstrained theory. Hence the operator relations of the unconstrained model need not carry over to the constrained model. Our argument excludes quantized worldline models of relativistic particles and the physical states of the covariant quantum string. We correct misconceptions about the generators of Lorentz transformations acting on particles.",
keywords = "Constrained system, Continuous spectrum, Covariant string, Heisenberg algebra, Mass shell condition, Stone-von Neumann theorem",
author = "Norbert Dragon and Florian Oppermann",
note = "Funding Information: Open Access funding enabled and organized by Projekt DEAL. ",
year = "2024",
month = jan,
day = "4",
doi = "10.48550/arXiv.2212.07256",
language = "English",
volume = "63",
journal = "International Journal of Theoretical Physics",
issn = "0020-7748",
publisher = "Springer New York",
}