# Symmetries Shared by the Poincaré Group and the Poincaré Sphere

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## Abstract

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**PACS**03.65.Fd; 03.67.-; 05.30.-d

## 1. Introduction

## 2. Poincaré Group and Wigner’s Little Groups

#### 2.1. Two-by-Two Representation of the Lorentz Groups

#### 2.2. Wigner’s Little Groups

**Table 1.**Wigner’s Little Groups. The little groups are the subgroups of the Lorentz group, whose transformations leave the four-momentum of a given particle invariant. Thus, the little groups define the internal space-time symmetries of particles. The four-momentum remains invariant under the rotation around it. In addition, the four-momentum remains invariant under the following transformations. These transformations are different for massive, massless and imaginary-mass particles.

Particle mass | Four-momentum | Transform matrices | ||
---|---|---|---|---|

Massive | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}cos(\theta /2)& -sin(\theta /2)\\ sin(\theta /2)& cos(\theta /2)\end{array}\right)$ | ||

Massless | $\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$ | $\left(\begin{array}{cc}1& \gamma \\ 0& 1\end{array}\right)$ | ||

Imaginary mass | $\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ | $\left(\begin{array}{cc}cosh(\lambda /2)& sinh(\lambda /2)\\ sinh(\lambda /2)& cosh(\lambda /2)\end{array}\right)$ |

## 3. Lorentz Completion of Wigner’s Little Groups

#### 3.1. Large-Momentum Limit

#### 3.2. Small-Mass Limit

**Table 2.**Covariance of the energy-momentum relation and covariance of the internal space-time symmetry groups. The γ parameter for the massless case has been studied in earlier papers in the four-by-four matrix formulation [6]. It corresponds to a gauge transformation. Among the three spin components, ${S}_{3}$ is along the direction of the momentum and remains invariant. It is called the “helicity".

Massive, Slow | Covariance | Massless, Fast | ||
---|---|---|---|---|

$E={p}^{2}/2m$ | Einstein’s $E=m{c}^{2}$ | $E=cp$ | ||

${S}_{3}$ | Helicity | |||

Wigner’s Little Group | ||||

${S}_{1},{S}_{2}$ | Gauge Transformation |

## 4. Jones Vectors and Stokes Parameters

**Table 3.**Polarization optics and special relativity sharing the same mathematics. Each matrix has its clear role in both optics and relativity. The determinant of the Stokes or the four-momentum matrix remains invariant under Lorentz transformations. It is interesting to note that the decoherence parameter (least fundamental) in optics corresponds to the mass (most fundamental) in particle physics.

Polarization Optics | Transformation Matrix | Particle Symmetry | ||
---|---|---|---|---|

Phase shift δ | $\left(\begin{array}{cc}{e}^{\delta /2}& 0\\ 0& {e}^{-i\delta /2}\end{array}\right)$ | Rotation around z | ||

Rotation around z | $\left(\begin{array}{cc}cos(\theta /2)& -sin(\theta /2)\\ sin(\theta /2)& cos(\theta /2)\end{array}\right)$ | Rotation around y | ||

Squeeze along x and y | $\left(\begin{array}{cc}{e}^{\eta /2}& 0\\ 0& {e}^{-\eta /2}\end{array}\right)$ | Boost along z | ||

Squeeze along 45° | $\left(\begin{array}{cc}cosh(\lambda /2)& sinh(\lambda /2)\\ sinh(\lambda /2)& cosh(\lambda /2)\end{array}\right)$ | Boost along x | ||

${\left(ab\right)}^{2}{sin}^{2}\chi $ | Determinant | (mass)^{2} |

## 5. Geometry of the Poincaré Sphere

**Figure 1.**Radius of the Poincaré sphere. The radius, R, takes its maximum value, ${S}_{0}$, when the decoherence angle, χ, is zero. It becomes smaller as χ increases. It becomes minimum when the angle reaches 90°. Its minimum value is ${S}_{3}$, as is illustrated in Figure a. The degree of polarization is maximum when $R={S}_{0}$ and is minimum when $R={S}_{3}$. According to Equation (65), ${S}_{3}$ becomes zero when $a=b$, and the minimum value of R becomes zero, as is indicated in Figure 1b. Its maximum value is still ${S}_{0}$. This maximum radius can become larger because b becomes larger to make $a=b$.

## 6. Concluding Remarks

## Acknowledgments

## Conflict of Interest

## Appendix

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Kim, Y.S.; Noz, M.E.
Symmetries Shared by the Poincaré Group and the Poincaré Sphere. *Symmetry* **2013**, *5*, 233-252.
https://doi.org/10.3390/sym5030233

**AMA Style**

Kim YS, Noz ME.
Symmetries Shared by the Poincaré Group and the Poincaré Sphere. *Symmetry*. 2013; 5(3):233-252.
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**Chicago/Turabian Style**

Kim, Young S., and Marilyn E. Noz.
2013. "Symmetries Shared by the Poincaré Group and the Poincaré Sphere" *Symmetry* 5, no. 3: 233-252.
https://doi.org/10.3390/sym5030233