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 (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.
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)
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In: International Journal of Theoretical Physics, Vol. 63, 5, 04.01.2024.
Research output: Contribution to journal › Article › Research › peer review
}
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 -