## Details

Original language | English |
---|---|

Article number | 5 |

Number of pages | 9 |

Journal | International Journal of Theoretical Physics |

Volume | 63 |

Publication status | Published - 4 Jan 2024 |

## Abstract

A Poincaré multiplet of mass eigenstates (P^{2}- m^{2}) 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.

## Keywords

- Constrained system, Continuous spectrum, Covariant string, Heisenberg algebra, Mass shell condition, Stone-von Neumann theorem

## ASJC Scopus subject areas

**Mathematics(all)**- Physics and Astronomy(all)
**Physics and Astronomy (miscellaneous)**

## Cite this

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**Heisenberg versus the Covariant String.**/ Dragon, Norbert; Oppermann, Florian.

In: International Journal of Theoretical Physics, Vol. 63, 5, 04.01.2024.

Research output: Contribution to journal › Article › Research › peer review

*International Journal of Theoretical Physics*, vol. 63, 5. https://doi.org/10.48550/arXiv.2212.07256, https://doi.org/10.1007/s10773-023-05529-z

*International Journal of Theoretical Physics*,

*63*, Article 5. https://doi.org/10.48550/arXiv.2212.07256, https://doi.org/10.1007/s10773-023-05529-z

}

TY - JOUR

T1 - Heisenberg versus the Covariant String

AU - Dragon, Norbert

AU - Oppermann, Florian

N1 - Funding Information: Open Access funding enabled and organized by Projekt DEAL.

PY - 2024/1/4

Y1 - 2024/1/4

N2 - 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, ⋯ XD-1) : the Heisenberg algebra [Pm, Xn] = i δmn 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.

AB - 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, ⋯ XD-1) : the Heisenberg algebra [Pm, Xn] = i δmn 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.

KW - Constrained system

KW - Continuous spectrum

KW - Covariant string

KW - Heisenberg algebra

KW - Mass shell condition

KW - Stone-von Neumann theorem

UR - http://www.scopus.com/inward/record.url?scp=85181229594&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2212.07256

DO - 10.48550/arXiv.2212.07256

M3 - Article

AN - SCOPUS:85181229594

VL - 63

JO - International Journal of Theoretical Physics

JF - International Journal of Theoretical Physics

SN - 0020-7748

M1 - 5

ER -