# Summing over Spacetime Dimensions in Quantum Gravity

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Quantum-Gravitational Properties of Inertial Propagators: Euclidean Space

#### 2.1. An Ultraviolet Completion

#### 2.2. An Identity Satisfied by the ${K}_{\nu}\left(z\right)$

#### 2.3. Summing over Dimensions

## 3. Quantum-Gravitational Properties of Inertial Propagators: Minkowski Space

## 4. An Alternative Derivation due to Padmanabhan

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Padmanabhan, T. Thermality of the Rindler Horizon: A Simple Derivation from the Structure of the Inertial Propagator. arXiv
**2019**, arXiv:1905.08263. [Google Scholar] [CrossRef] [Green Version] - Rajeev, K.; Padmanabhan, T. Exploring the Rindler Vacuum and the Euclidean Plane. arXiv
**2019**, arXiv:1906.09278. [Google Scholar] - Padmanabhan, T. A Measure for Quantum Paths, Gravity and Spacetime Microstructure. arXiv
**2019**, arXiv:1908.10872. [Google Scholar] [CrossRef] [Green Version] - Padmanabhan, T. Duality and Zero–Point Length of Spacetime. Phys. Rev. Lett.
**1997**, 78, 1854. [Google Scholar] [CrossRef] [Green Version] - Padmanabhan, T. Hypothesis of Path Integral Duality. I. Quantum Gravitational Corrections to the Propagator. Phys. Rev.
**1998**, 57, 6206. [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.
**2009**, 80, 044005. [Google Scholar] [CrossRef] [Green Version] - Gradshteyn, I.; Ryzhik, I. Table of Integrals, Series and Products; Academic Press: New York, NY, USA, 1965. [Google Scholar]
- Watson, G. A Treatise on the Theory of Bessel Functions; Cambridge University Press: Cambridge, UK, 1952. [Google Scholar]
- Crowther, K.; Linnemann, N. Renormalizability, Fundamentality and a Final theory: The Role of UV-completion in the Search for Quantum Gravity. Br. J. Philos. Sci.
**2019**, 70, 377. [Google Scholar] [CrossRef] [Green Version] - Padmanabhan, T. 1997; unpublished.
- Blau, M.; Theisen, S. String Theory as a Theory of Quantum Gravity: A Status Report. Gen. Rel. Grav.
**2009**, 41, 743. [Google Scholar] [CrossRef] [Green Version]

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Curiel, E.; Finster, F.; Isidro, J.M.
Summing over Spacetime Dimensions in Quantum Gravity. *Symmetry* **2020**, *12*, 138.
https://doi.org/10.3390/sym12010138

**AMA Style**

Curiel E, Finster F, Isidro JM.
Summing over Spacetime Dimensions in Quantum Gravity. *Symmetry*. 2020; 12(1):138.
https://doi.org/10.3390/sym12010138

**Chicago/Turabian Style**

Curiel, Erik, Felix Finster, and Jose Maria Isidro.
2020. "Summing over Spacetime Dimensions in Quantum Gravity" *Symmetry* 12, no. 1: 138.
https://doi.org/10.3390/sym12010138