Details
| Original language | English |
|---|---|
| Article number | 271 |
| Number of pages | 13 |
| Journal | International Journal of Theoretical Physics |
| Volume | 62 |
| Publication status | Published - 26 Dec 2023 |
Abstract
The polynomials in the generators of a unitary representation of the Poincaré group constitute an algebra which maps the dense subspace D of smooth, rapidly decreasing wavefunctions to itself. This mathematical result is highly welcome to physicists, who previously just assumed their algebraic treatment of unbounded operators be justified. The smoothness, however, has the side effect that a rough operator R, which does not map a dense subspace of D to itself, has to be shown to allow for some other dense domain which is mapped to itself both by R and all generators. Otherwise their algebraic product, their concatenation, is not defined. Canonical quantization of the light cone string postulates operators - i X1 and P-= (P- Pz) / 2 and as their commutator the multiplicative operator R= P1/ (P+ Pz) . This is not smooth but rough on the negative z- axis of massless momentum. Using only the commutation relations of Pm with the generators - i Miz of rotations in the Pi - Pz -plane we show that on massless states the operator R is inconsistent with a unitary representation of SO (D- 1) . This makes the algebraic determination of the critical dimension, D= 26 , of the bosonic string meaningless: if the massless states of the light cone string admit R then they do not admit a unitary representation of the subgroup SO (D- 1) of the Poincaré group. With analogous arguments we show: Massless multiplets are inconsistent with a translation group of the spatial momentum which is generated by a self-adjoint spatial position operator X .
Keywords
- Domain of generators, Gårding space, Light cone string, Massless states, Rough operators, Schwartz space, Spatial position operator
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. 62, 271, 26.12.2023.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The Rough with the Smooth of the Light Cone String
AU - Dragon, Norbert
AU - Oppermann, Florian
N1 - Funding Information: Open Access funding enabled and organized by Projekt DEAL.
PY - 2023/12/26
Y1 - 2023/12/26
N2 - The polynomials in the generators of a unitary representation of the Poincaré group constitute an algebra which maps the dense subspace D of smooth, rapidly decreasing wavefunctions to itself. This mathematical result is highly welcome to physicists, who previously just assumed their algebraic treatment of unbounded operators be justified. The smoothness, however, has the side effect that a rough operator R, which does not map a dense subspace of D to itself, has to be shown to allow for some other dense domain which is mapped to itself both by R and all generators. Otherwise their algebraic product, their concatenation, is not defined. Canonical quantization of the light cone string postulates operators - i X1 and P-= (P- Pz) / 2 and as their commutator the multiplicative operator R= P1/ (P+ Pz) . This is not smooth but rough on the negative z- axis of massless momentum. Using only the commutation relations of Pm with the generators - i Miz of rotations in the Pi - Pz -plane we show that on massless states the operator R is inconsistent with a unitary representation of SO (D- 1) . This makes the algebraic determination of the critical dimension, D= 26 , of the bosonic string meaningless: if the massless states of the light cone string admit R then they do not admit a unitary representation of the subgroup SO (D- 1) of the Poincaré group. With analogous arguments we show: Massless multiplets are inconsistent with a translation group of the spatial momentum which is generated by a self-adjoint spatial position operator X .
AB - The polynomials in the generators of a unitary representation of the Poincaré group constitute an algebra which maps the dense subspace D of smooth, rapidly decreasing wavefunctions to itself. This mathematical result is highly welcome to physicists, who previously just assumed their algebraic treatment of unbounded operators be justified. The smoothness, however, has the side effect that a rough operator R, which does not map a dense subspace of D to itself, has to be shown to allow for some other dense domain which is mapped to itself both by R and all generators. Otherwise their algebraic product, their concatenation, is not defined. Canonical quantization of the light cone string postulates operators - i X1 and P-= (P- Pz) / 2 and as their commutator the multiplicative operator R= P1/ (P+ Pz) . This is not smooth but rough on the negative z- axis of massless momentum. Using only the commutation relations of Pm with the generators - i Miz of rotations in the Pi - Pz -plane we show that on massless states the operator R is inconsistent with a unitary representation of SO (D- 1) . This makes the algebraic determination of the critical dimension, D= 26 , of the bosonic string meaningless: if the massless states of the light cone string admit R then they do not admit a unitary representation of the subgroup SO (D- 1) of the Poincaré group. With analogous arguments we show: Massless multiplets are inconsistent with a translation group of the spatial momentum which is generated by a self-adjoint spatial position operator X .
KW - Domain of generators
KW - Gårding space
KW - Light cone string
KW - Massless states
KW - Rough operators
KW - Schwartz space
KW - Spatial position operator
UR - http://www.scopus.com/inward/record.url?scp=85180678022&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2212.14822
DO - 10.48550/arXiv.2212.14822
M3 - Article
AN - SCOPUS:85180678022
VL - 62
JO - International Journal of Theoretical Physics
JF - International Journal of Theoretical Physics
SN - 0020-7748
M1 - 271
ER -