# Constraints on the String T-Duality Propagator from the Hydrogen Atom

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## Abstract

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## 1. Introduction

## 2. Hydrogen Atom Energy Levels

#### 2.1. Conventional Description

#### 2.2. Contribution from T-Duality Propagator

## 3. Constraints on the Zero-Point Length

#### 3.1. Ground State Energy

#### 3.2. Transition Frequency

## 4. Discussion

## 5. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Useful Identities

## References

- Weinberg, S. The Quantum Theory of Fields; Volume II. Modern Applications; Cambridge University Press: Cambridge, UK, 2015; pp. 305–318. [Google Scholar]
- De Rham, C. Massive Gravity. Living Rev. Rel.
**2014**, 17, 7. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Maggiore, M. Gravitational Waves; Volume 1. Theory and Experiments; Oxford University Press: Oxford, UK, 2017; pp. 7–12, 70–74. [Google Scholar]
- Fontanini, M.; Spallucci, E.; Padmanabhan, T. Zero-point length from string fluctuations. Phys. Lett. B
**2006**, 633, 627–630. [Google Scholar] [CrossRef] [Green Version] - Smailagic, A.; Spallucci, E.; Padmanabhan, T. String theory T duality and the zero point length of space-time. arXiv
**2003**, arXiv:hep-th/0308122. [Google Scholar] - Spallucci, E.; Fontanini, M. Zero-point length, extra-dimensions and string T-duality. In New Developments in String Theory Research; Grece, S.A., Ed.; Nova Science Publishers, Inc.: Hauppauge, NY, USA, 2006; pp. 245–270. [Google Scholar]
- Nicolini, P.; Spallucci, E.; Wondrak, M.F. Quantum Corrected Black Holes from String T-Duality. Phys. Lett. B
**2019**, 797, 134888. [Google Scholar] [CrossRef] - Padmanabhan, T. Duality and Zero-Point Length of Spacetime. Phys. Rev. Lett.
**1997**, 78, 1854–1857. [Google Scholar] [CrossRef] [Green Version] - Padmanabhan, T. Hypothesis of path integral duality. I. Quantum gravitational corrections to the propagator. Phys. Rev. D
**1998**, 57, 6206–6215. [Google Scholar] [CrossRef] [Green Version] - Srinivasan, K.; Sriramkumar, L.; Padmanabhan, T. The Hypothesis of path integral duality. II: Corrections to quantum field theoretic results. Phys. Rev. D
**1998**, 58, 044009. [Google Scholar] [CrossRef] [Green Version] - Sriramkumar, L.; Shankaranarayanan, S. Path integral duality and Planck scale corrections to the primordial spectrum in exponential inflation. J. High Energy Phys.
**2006**, 12, 050. [Google Scholar] [CrossRef] [Green Version] - Kothawala, D.; Sriramkumar, L.; Shankaranarayanan, S.; Padmanabhan, T. Path integral duality modified propagators in spacetimes with constant curvature. Phys. Rev. D
**2009**, 80, 044005. [Google Scholar] [CrossRef] [Green Version] - Kothawala, D.; Shankaranarayanan, S.; Sriramkumar, L. Quantum gravitational corrections to the stress-energy tensor around the BTZ black hole. J. High Energy Phys.
**2008**, 09, 095. [Google Scholar] [CrossRef] [Green Version] - Padmanabhan, T. A Measure for Quantum Paths, Gravity and Spacetime Microstructure. arXiv
**2019**, arXiv:gr-qc/1908.10872. [Google Scholar] - Shankaranarayanan, S.; Padmanabhan, T. Hypothesis of path integral duality: Applications to QED. Int. J. Mod. Phys. D
**2001**, 10, 351–366. [Google Scholar] [CrossRef] - Ohanian, H.C. Finite quantum electrodynamics with a gravitationally smeared propagator. Phys. Rev. D
**1997**, 55, 5140–5146. [Google Scholar] [CrossRef] - Ohanian, H.C. Finite quantum electrodynamics and gauge invariance. Nuovo Cim. A
**1997**, 110, 751–756. [Google Scholar] - Ohanian, H.C. Smearing of propagators by gravitational fluctuations on the Planck scale. Phys. Rev. D
**1999**, 60, 104051. [Google Scholar] [CrossRef] - Mohr, P.J.; Newell, D.B.; Taylor, B.N. CODATA Recommended Values of the Fundamental Physical Constants: 2014. Rev. Mod. Phys.
**2016**, 88, 035009. [Google Scholar] [CrossRef] [Green Version] - Brau, F. Minimal length uncertainty relation and hydrogen atom. J. Phys. A
**1999**, 32, 7691–7696. [Google Scholar] [CrossRef] - Akhoury, R.; Yao, Y.P. Minimal length uncertainty relation and the hydrogen spectrum. Phys. Lett. B
**2003**, 572, 37–42. [Google Scholar] [CrossRef] [Green Version] - Hossenfelder, S.; Bleicher, M.; Hofmann, S.; Ruppert, J.; Scherer, S.; Stöcker, H. Signatures in the Planck regime. Phys. Lett. B
**2003**, 575, 85–99. [Google Scholar] [CrossRef] - Antonacci Oakes, T.L.; Francisco, R.O.; Fabris, J.C.; Nogueira, J.A. Ground State of the Hydrogen Atom via Dirac Equation in a Minimal Length Scenario. Eur. Phys. J. C
**2013**, 73, 2495. [Google Scholar] [CrossRef] [Green Version] - Wondrak, M.F.; Nicolini, P.; Bleicher, M. Unparticle contribution to the hydrogen atom ground state energy. Phys. Lett. B
**2016**, 759, 589–592. [Google Scholar] [CrossRef] [Green Version] - Kramida, A.E. A critical compilation of experimental data on spectral lines and energy levels of hydrogen, deuterium, and tritium. Atom. Data Nucl. Data Tables
**2010**, 96, 586–644, Erratum in**2019**, 126, 295. [Google Scholar] [CrossRef] - Greiner, W. Quantenmechanik. Einführung; Wissenschaftlicher Verlag Harri Deutsch: Frankfurt am Main, Germany, 2005; pp. 219–234. [Google Scholar]
- Schwabl, F. Quantenmechanik (QM I); Eine Einführung; Springer: Berlin, Germany, 2007; pp. 121–142, 217–227. [Google Scholar]
- Blumenhagen, R.; Cvetic, M.; Langacker, P.; Shiu, G. Toward realistic intersecting D-brane models. Annu. Rev. Nucl. Part. Sci.
**2005**, 55, 71–139. [Google Scholar] [CrossRef] [Green Version] - Jentschura, U.D.; Kotochigova, S.; LeBigot, E.O.; Mohr, P.J.; Taylor, B.N. The Energy Levels of Hydrogen and Deuterium (Version 2.1); National Institute of Standards and Technology: Gaithersburg, MD, USA, 2005. Available online: http://physics.nist.gov/HDEL (accessed on 30 August 2019).
- Matveev, A.; Parthey, C.G.; Predehl, K.; Alnis, J.; Beyer, A.; Holzwarth, R.; Udem, T.; Wilken, T.; Kolachevsky, N.; Abgrall, M.; et al. Precision Measurement of the Hydrogen 1S-2S Frequency via a 920-km Fiber Link. Phys. Rev. Lett.
**2013**, 110, 230801. [Google Scholar] [CrossRef] - Gradshteyn, I.S.; Ryzhik, I.M. Tables of Integrals, Series, and Products, 7th ed.; Elsevier Academic Press: London, UK, 2007. [Google Scholar]

**Figure 1.**Potential energy. The solid curve (blue) displays the absolute value of the conventional Coulomb energy, $\left|{V}_{0}\right|$, while the dashed line (orange) shows the absolute value of the T-duality corrected energy, $\left|{V}_{\mathrm{Td}}\right|$. The difference of both equals the Hamiltonian contribution ${H}_{\mathrm{Td}}$ (dot-dashed curve, green).

**Figure 2.**Normalized uncertainty in the hydrogen ground state energy. The possible T-duality contribution $\Delta {E}_{1,0}^{\mathrm{Td}}$ (solid, blue) increases with the zero-point length, ${l}_{0}$. The dashed (orange) and dot-dashed (green) lines show the differences between the experimental value on the one hand and the fine-structure corrected or the current theoretical value on the other hand—taking into account the respective standard deviations. The experimental error is presented by the dotted line (red).

**Figure 3.**Normalized uncertainty in the $1{\mathrm{S}}_{1/2}-2{\mathrm{S}}_{1/2}$ hydrogen transition frequency. The possible T-duality contribution $\Delta {\nu}_{1\mathrm{S}-2\mathrm{S}}^{\mathrm{Td}}$ (solid, blue) increases with the zero-point length, ${l}_{0}$. The dashed (orange) line shows the difference between the experimental and the fine-structure corrected value. The dot-dashed (green) line shows the difference between the experimental and the current theoretical value. Both take into account the respective standard deviations. The experimental error is presented by the dotted line (red).

**Table 1.**Theoretical and experimental values of the hydrogen ground state energy. The calculations of ${E}_{\mathrm{th}}^{\mathrm{S}}$ and ${E}_{\mathrm{th}}^{\mathrm{fs}}$ are based on the 2014 CODATA recommended values [19]. When expressed in eV, the actual precision of the current theoretical and experimental value is masked by the less-precisely known Planck constant [19]. For this reason, h is factored out and the values are given also in terms of $\mathrm{MHz}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}h$.

Energy | Description | Value |
---|---|---|

${E}_{\mathrm{th}}^{\mathrm{S}}$ | Schrödinger | $-13.59828715\left(9\right)\mathrm{eV}$ |

${E}_{\mathrm{th}}^{\mathrm{fs}}$ | Schrödinger, incl. fine-structure | $-13.59846818\left(9\right)\mathrm{eV}$ |

${E}_{\mathrm{th}}^{\mathrm{QED}}$ | current theoretical value [29] | $-13.59843449\left(9\right)\mathrm{eV}$ |

$-3288086857.1276\left(31\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{MHz}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$ | ||

${E}_{\mathrm{exp}}$ | current experimental value [25] | $-13.59843448\left(9\right)\mathrm{eV}$ |

$-3288086856.8\left(7\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{MHz}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$ |

**Table 2.**Theoretical and experimental values of the $1{\mathrm{S}}_{1/2}-2{\mathrm{S}}_{1/2}$ hydrogen transition frequency. The calculations of ${\nu}_{\mathrm{th}}^{\mathrm{S}}$ and ${\nu}_{\mathrm{th}}^{\mathrm{fs}}$ are based on the 2014 CODATA recommended values [19].

Frequency | Description | Value |
---|---|---|

${\nu}_{\mathrm{th}}^{\mathrm{S}}$ | Schrödinger | $2466038423\left(32\right)\mathrm{MHz}$ |

${\nu}_{\mathrm{th}}^{\mathrm{fs}}$ | Schrödinger, incl. fine-structure | $2466068517\left(32\right)\mathrm{MHz}$ |

${\nu}_{\mathrm{th}}^{\mathrm{QED}}$ | current theoretical value [29] | $2466061413.187103\left(46\right)\mathrm{MHz}$ |

${\nu}_{\mathrm{exp}}$ | current experimental value [30] | $2466061413.187018\left(11\right)\mathrm{MHz}$ |

Reference Value | Upper Bound on ${\mathit{l}}_{0}$ | Reference Value | Upper Bound on ${\mathit{l}}_{0}$ |
---|---|---|---|

${E}_{\mathrm{exp}}-{E}_{\mathrm{th}}^{\mathrm{fs}}$ | $1.6\times {10}^{-14}$ m | ${\nu}_{\mathrm{exp}}-{\nu}_{\mathrm{th}}^{\mathrm{fs}}$ | $1.6\times {10}^{-14}$ m |

${E}_{\mathrm{exp}}-{E}_{\mathrm{th}}^{\mathrm{QED}}$ | $1.4\times {10}^{-16}$ m | ${\nu}_{\mathrm{exp}}-{\nu}_{\mathrm{th}}^{\mathrm{QED}}$ | $1.5\times {10}^{-18}$ m |

$\Delta {E}_{\mathrm{exp}}$ | $1.1\times {10}^{-16}$ m | $\Delta {\nu}_{\mathrm{exp}}$ | $3.9\times {10}^{-19}$ m |

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Wondrak, M.F.; Bleicher, M.
Constraints on the String T-Duality Propagator from the Hydrogen Atom. *Symmetry* **2019**, *11*, 1478.
https://doi.org/10.3390/sym11121478

**AMA Style**

Wondrak MF, Bleicher M.
Constraints on the String T-Duality Propagator from the Hydrogen Atom. *Symmetry*. 2019; 11(12):1478.
https://doi.org/10.3390/sym11121478

**Chicago/Turabian Style**

Wondrak, Michael F., and Marcus Bleicher.
2019. "Constraints on the String T-Duality Propagator from the Hydrogen Atom" *Symmetry* 11, no. 12: 1478.
https://doi.org/10.3390/sym11121478