# Revisiting the Okubo–Marshak Argument

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction: String-Localized Fields

#### Massless Bosonic SLF: General Theory

## 2. Dealing with String Independence

**e**$=({e}_{1},\dots ,{e}_{m})$ of string coordinates, with m the maximum number of SLFs appearing in a sub-monomial of ${S}_{1}$. For $n\ge 2$, the ${S}_{n}$ are time-ordered products that need to be constructed. Two sets of strings cannot be ordered, after chopping them into segments if necessary [33], if and only if they touch each other (see the Appendix A for details). The resulting exceptional set

**string independence**principle: colloquially, the strings “ought not to be seen”. In this paper, it will replace the “gauge principle” with advantage.

#### 2.1. The Aste–Scharf Argument Recast in SLF Theory

**Proposition**

**1.**

**Proof**

**of**

**Proposition 1.**

#### 2.2. Dealing with String Independence at Second Order: Preliminaries

#### 2.3. The Jacobi Identity Emerges

#### 2.4. The Quartic Term

## 3. Discussion

#### 3.1. Story of Two Principles

#### 3.2. Reassessing the Okubo–Marshak Argument

#### 3.3. Coda

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Time Ordering Outside the String Diagonal

## References and Note

- Okubo, S.; Marshak, R.E. Argument for the non-existence of the “Strong CP problem” in QCD. Prog. Theor. Phys.
**1992**, 87, 1159–1162. [Google Scholar] [CrossRef] - Kugo, T.; Ojima, I. Local covariant operator formalism of non-Abelian gauge theories and quark confinement problem. Prog. Theor. Phys. Suppl.
**1979**, 66, 1–130. [Google Scholar] [CrossRef][Green Version] - Dirac, P.A.M. Gauge-invariant formulation of quantum electrodynamics. Can. J. Phys.
**1955**, 33, 650–660. [Google Scholar] [CrossRef] - Mandelstam, S. Quantum electrodynamics without potentials. Ann. Phys.
**1962**, 19, 1–24. [Google Scholar] [CrossRef] - Buchholz, D.; Fredenhagen, K. Locality and the structure of particle states. Commun. Math. Phys.
**1982**, 84, 1–54. [Google Scholar] [CrossRef] - Steinmann, O. A Jost–Schroer theorem for string fields. Commun. Math. Phys.
**1982**, 87, 259–264. [Google Scholar] [CrossRef] - Steinmann, O. Perturbative QED in terms of gauge invariant fields. Ann. Phys.
**1984**, 157, 232–254. [Google Scholar] [CrossRef] - Steinmann, O. Gauge Invariant Fields in Nonabelian Gauge Theories; Preprint BI-TP-85/4; Bielefeld Universität: Bielefeld, Germany, 1985. [Google Scholar]
- Mund, J.; Schroer, B.; Yngvason, J. String-localized quantum fields from Wigner representations. Phys. Lett. B
**2004**, 596, 156–162. [Google Scholar] [CrossRef][Green Version] - Mund, J.; Schroer, B.; Yngvason, J. String-localized quantum fields and modular localization. Commun. Math. Phys.
**2006**, 268, 621–672. [Google Scholar] [CrossRef][Green Version] - Borchers, H.J. On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys.
**2000**, 41, 3604–3673. [Google Scholar] [CrossRef][Green Version] - Connes, A. Noncommutative Geometry; Academic Press: London, UK; San Diego, CA, USA, 1994. [Google Scholar]
- Bisognano, J.; Wichmann, E. On the duality condition for quantum fields. J. Math. Phys.
**1976**, 17, 303–321. [Google Scholar] [CrossRef][Green Version] - Mund, J. The Bisognano–Wichmann theorem for massive theories. Ann. Henri Poincaré
**2001**, 2, 907–926. [Google Scholar] [CrossRef][Green Version] - Dütsch, M.; Schroer, B. Massive vector bosons and gauge theory. J. Phys. A
**2000**, 33, 4317–4356. [Google Scholar] [CrossRef] - Fassarella, L.; Schroer, B. Wigner particles and local quantum physics. J. Phys. A
**2002**, 35, 9123–9164. [Google Scholar] [CrossRef] - Brunetti, R.; Guido, D.; Longo, R. Modular localization and Wigner particles. Rev. Math. Phys.
**2002**, 14, 759–785. [Google Scholar] [CrossRef][Green Version] - Yngvason, J. Zero-mass infinite spin representations of the Poincaré group and quantum field theory. Commun. Math. Phys.
**1970**, 18, 195–203. [Google Scholar] [CrossRef] - Weinberg, S. The Quantum Theory of Fields I; Cambridge University Press: Cambridge, MA, USA, 1995. [Google Scholar]
- Mund, J.; Rehren, K.-H.; Schroer, B. Helicity decoupling in the massless limit of massive tensor fields. Nucl. Phys. B
**2017**, 924, 699–727. [Google Scholar] [CrossRef] - Van Dam, T.; Veltman, M. Massive and massless Yang–Mills and gravitational fields. Nucl. Phys. B
**1970**, 22, 397–411. [Google Scholar] [CrossRef][Green Version] - Zakharov, V.I. Linearized graviton theory and the graviton mass. JETP Lett.
**1970**, 12, 312–313. [Google Scholar] - Mund, J.; Rehren, K.-H.; Schroer, B. Relations between positivity, localization and degrees of freedom: The Weinberg–Witten theorem and the van Dam–Veltman–Zakharov discontinuity. Phys. Lett. B
**2017**, 773, 625–631. [Google Scholar] [CrossRef] - Rehren, K.-H. Pauli–Lubański limit and stress-energy tensor for infinite-spin fields. J. High Energy Phys.
**2017**, 11, 130. [Google Scholar] [CrossRef][Green Version] - Velo, G.; Zwanziger, D. Noncausality and other defects of interaction Lagrangians for particles with spin one and higher. Phys. Rev.
**1969**, 188, 2218. [Google Scholar] [CrossRef] - Schroer, B. The role of positivity and causality in interactions involving higher spin. Nucl. Phys. B
**2019**, 941, 91–144. [Google Scholar] [CrossRef] - Mund, J.; Rehren, K.-H.; Schroer, B. Gauss’ law and string-localized quantum field theory. J. High Energy Phys.
**2020**, 2020, 1. [Google Scholar] [CrossRef][Green Version] - Epstein, H.; Glaser, V.J. The role of locality in perturbation theory. Ann. Inst. Henri Poincaré A
**1973**, 19, 211–295. [Google Scholar] - Gracia-Bondía, J.M.; Mund, J.; Várilly, J.C. The chirality theorem. Ann. Henri Poincaré
**2018**, 19, 843–874. [Google Scholar] [CrossRef][Green Version] - Aste, A.; Scharf, G. Non-abelian gauge theories as a consequence of perturbative quantum gauge invariance. Int. J. Mod. Phys. A
**1999**, 14, 3421–3434. [Google Scholar] [CrossRef][Green Version] - Dütsch, M.; Scharf, G. Perturbative gauge invariance: The electroweak theory. Ann. Phys.
**1999**, 8, 359–387. [Google Scholar] [CrossRef] - Grigore, D.R. The standard model and its generalizations in the Epstein–Glaser approach to renormalization theory. J. Phys. A Math. Gen.
**2000**, 33, 8443–8476. [Google Scholar] [CrossRef] - Cardoso, L.T.; Mund, J.; Várilly, J.C. String chopping and time-ordered products of linear string-localized quantum fields. Math. Phys. Anal. Geom.
**2018**, 21, 3. [Google Scholar] [CrossRef][Green Version] - Gaß, C. Renormalization in string-localized field theories: A microlocal analysis. arXiv
**2021**, arXiv:2107.12834. [Google Scholar] - Duch, P. Weak adiabatic limit in quantum field theories with massless particles. Ann. Henri Poincaré
**2018**, 19, 875–935. [Google Scholar] [CrossRef][Green Version] - Duistermaat, J.J.; Kolk, J.A.C. Lie Groups; Springer: Berlin, Germany, 1999. [Google Scholar]
- Cornwall, J.M.; Levin, D.N.; Tiktopoulos, G. Derivation of gauge invariance from high-energy unitarity bounds on the S-matrix. Phys. Rev. D
**1974**, 10, 1145–1167. [Google Scholar] [CrossRef] - Schwartz, M.D. Quantum Field Theory and the Standard Model; Cambridge University Press: Cambridge, MA, USA, 2015. [Google Scholar]
- Zwanziger, D. Construction of amplitudes with massless particles and gauge invariance in S-matrix theory. Phys. Rev.
**1964**, B133, 1036–1045. [Google Scholar] [CrossRef] - Marshak, R.E. Conceptual Foundations of Modern Particle Physics; World Scientific: Singapore, 1993. [Google Scholar]
- Gracia-Bondía, J.M. On Marshak’s and Connes’ views of chirality. In A Gift of Prophecy—Essays in Celebration of the Life of Robert Eugene Marshak; Sudarshan, E.C.G., Ed.; World Scientific: Singapore, 1994; pp. 208–217. [Google Scholar]
- Lee, T.D. Foreword to Conceptual Foundations of Modern Particle Physics, op. cit.: See [40].
- Donaldson, S.K. An application of gauge theory to four-dimensional topology. J. Differ. Geom.
**1983**, 18, 279–315. [Google Scholar] [CrossRef] - Alvarez-Gaumé, L.; Witten, E. Gravitational anomalies. Nucl. Phys. B
**1984**, 234, 269–330. [Google Scholar] [CrossRef] - Geng, C.; Marshak, R.E. Reply to “Comment on anomaly cancellation in the Standard Model”. Phys. Rev D
**1990**, 41, 717–718. [Google Scholar] [CrossRef] - Alvarez, E.; Gracia-Bondía, J.M.; Martín, C.P. Anomaly cancellation and gauge group of the Standard Model in NCG. Phys. Lett. B
**1995**, 364, 33–40. [Google Scholar] [CrossRef][Green Version] - Leyland, P.; Roberts, J.; Testard, D. Duality for Quantum Free Fields; Preprint 78/P.1016; Centre de Physique Théorique du CNRS: Marseille, France, 1978. [Google Scholar]
- Schroer, B. Pascual Jordan’s legacy and the ongoing research in quantum field theory. Eur. Phys. J. H
**2010**, 35, 377–434. [Google Scholar] [CrossRef][Green Version] - Raffelt, G.; Seckel, D. Bounds on exotic-particle interactions from SN1987A. Phys. Rev. Lett.
**1988**, 60, 1793–1796. [Google Scholar] [CrossRef] - Witten, E. Notes on some entanglement properties of quantum field theory. Rev. Mod. Phys.
**2018**, 90, 045003. [Google Scholar] [CrossRef][Green Version] - Azcoiti, V. Axial U
_{A}(1) anomaly: A new mechanism to generate massless bosons. Symmetry**2021**, 13, 209. [Google Scholar] [CrossRef] - Cebrián, S. The role of small scale experiments in the direct detection of dark matter. Universe
**2021**, 7, 81. [Google Scholar] [CrossRef] - Nakamura, Y.; Schierholz, G. Does confinement imply CP invariance of the strong interactions? Proc. Sci.
**2020**, 2019, 172. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gaß, C.; Gracia-Bondía, J.M.; Mund, J.
Revisiting the Okubo–Marshak Argument. *Symmetry* **2021**, *13*, 1645.
https://doi.org/10.3390/sym13091645

**AMA Style**

Gaß C, Gracia-Bondía JM, Mund J.
Revisiting the Okubo–Marshak Argument. *Symmetry*. 2021; 13(9):1645.
https://doi.org/10.3390/sym13091645

**Chicago/Turabian Style**

Gaß, Christian, José M. Gracia-Bondía, and Jens Mund.
2021. "Revisiting the Okubo–Marshak Argument" *Symmetry* 13, no. 9: 1645.
https://doi.org/10.3390/sym13091645