# The Geometrical Basis of 𝒫𝒯 Symmetry

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## Abstract

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## 1. Introduction

## 2. Basic Concepts on the Transfer Matrix

## 3. Geometry of Transfer Matrices

## 4. Geometry of $\mathcal{PT}$-Invariant Transfer Matrices

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**) The isometric method for a Möbius transformation of the hyperbolic type. We have plotted the isometric circles for the direct (${C}_{d}$) and the inverse (${C}_{i}$) transformations, with centers in ${O}_{d}$ and ${O}_{i}$, respectively. The fixed points are marked in green and L is the reflection line. z and ${z}^{\prime}$ are the points related by $\mathsf{M}$, whereas ${z}_{d}$ is the point obtained by an inversion of z respect to ${C}_{d}$ and ${z}_{d}^{\prime}$ the reflected of z by L. The inversion of ${z}_{d}^{\prime}$ respect to ${C}_{i}$ gives ${z}^{\prime}$. (

**Right**) Two typical families of fixed lines for the same transformation.

**Figure 2.**Families of fixed lines for elliptic (

**left**) and parabolic (

**right**) Möbius transformations. We show the isometric circles ${C}_{d}$ and ${C}_{i}$ and their centers ${O}_{d}$ and ${O}_{i}$. The fixed points are marked in green.

**Figure 3.**The isometric method as in Figure 2, but now applied to an elliptic $\mathcal{PT}$-invariant transfer matrix $\mathcal{M}$ given in (33). We show again the isometric circle for the direct (${C}_{d}$) and the inverse (${C}_{i}$) transformations, with centers in ${O}_{d}$ and ${O}_{i}$, respectively. The symmetry line L coincides with the real axis, so the corresponding reflection reduces to a complex conjugation.

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**MDPI and ACS Style**

Sánchez-Soto, L.L.; Monzón, J.J. The Geometrical Basis of 𝒫𝒯 Symmetry. *Symmetry* **2018**, *10*, 494.
https://doi.org/10.3390/sym10100494

**AMA Style**

Sánchez-Soto LL, Monzón JJ. The Geometrical Basis of 𝒫𝒯 Symmetry. *Symmetry*. 2018; 10(10):494.
https://doi.org/10.3390/sym10100494

**Chicago/Turabian Style**

Sánchez-Soto, Luis L., and Juan J. Monzón. 2018. "The Geometrical Basis of 𝒫𝒯 Symmetry" *Symmetry* 10, no. 10: 494.
https://doi.org/10.3390/sym10100494