# Loop Representation of Wigner’s Little Groups

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## Abstract

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## 1. Introduction

- As for the massive particle, Wigner worked out his little group in the Lorentz frame where the particle is at rest with zero momentum, resulting in the three-dimensional rotation group. He could have Lorentz-boosted the $O\left(3\right)$-like little group to make the little group for a moving particle.
- While the little group for a massless particle is like $E\left(2\right)$, it is not difficult to associate the rotational degree of freedom to the helicity. However, Wigner did not give physical interpretations to the two translation-like degrees of freedom.
- While the Lorentz group does not allow mass variations, particles with infinite momentum should behave like massless particles. The question is whether the Lorentz-boosted $O\left(3\right)$-like little group becomes the $E\left(2\right)$-like little group for particles with infinite momentum.

- In his original paper [1], Wigner worked out his little group for the massive particle when its momentum is zero. How about moving massive particles? In this paper, we start with a moving particle with non-zero momentum. We then perform rotations and boosts whose net effect does not change the momentum [6,7,8]. This procedure can be applied to the massive, massless, and imaginary-mass cases.
- By now, we have a clear understanding of the group $SL(2,c)$ as the universal covering group of the Lorentz group. The logic with two-by-two matrices is far more transparent than the mathematics based on four-by-four matrices. We shall thus use the two-by-two representation of the Lorentz group throughout the paper [5,9,10,11].

## 2. Lorentz Group and Its Representations

## 3. Two-by-Two Representation of Wigner’s Little Groups

## 4. Loop Representation of Wigner’s Little Groups

#### 4.1. Continuity Problems

#### 4.2. Parity, Time Reversal, and Charge Conjugation

## 5. Dirac Matrices as a Representation of the Little Group

#### 5.1. Polarization of Massless Neutrinos

#### 5.2. Small-Mass Neutrinos

## 6. Scalars, Vectors, and Tensors

- scalar with one state,
- pseudo-scalar with one state,
- four-vector with four states,
- axial vector with four states,
- second-rank tensor with six states.

#### 6.1. Four-Vectors

#### 6.2. Second-Rank Tensor

#### 6.3. Higher Spins

## 7. Concluding Remarks

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**$O\left(3\right)$-like and $E\left(2\right)$-like internal space-time symmetries of massive and massless particles. The sphere corresponds to the $O\left(3\right)$-like little group for the massive particle. There is a plane tangential to the sphere at its north pole, which is $E\left(2\right)$. There is also a cylinder tangent to the sphere at its equatorial belt. This cylinder gives one helicity and one gauge degree of freedom. This figure thus gives a unified picture of the little groups for massive and massless particles [5].

**Figure 2.**Evolution of the Wigner loop. In 1976 [6], Kupersztych considered a rotation followed by a boost whose net result will leave the momentum invariant. In 1981 [7], Han and Kim considered the same problem with simpler forms for boost matrices. In 1988, Han and Kim [8] constructed the Lorentz kinematics corresponding to the Bargmann decomposition [10] consisting of one boost matrix sandwiched by two rotation matrices. In the present case, the two rotation matrices are identical.

**Figure 3.**Non-Lorentzian transformations allowing mass variations. The D matrix of Equation (29) allows us to change the $\chi $ and $\alpha $ analytically within the square region in (

**a**). These variations allow the mass variations illustrated in (

**b**), not allowed in Lorentz transformations. The Lorentz transformations are possible along the hyperbolas given in this figure.

**Figure 4.**Parity, time reversal, and charge conjugation of Wigner’s little groups in the loop representation.

**Figure 5.**Unified picture of massive and massless particles. The gauge transformation is a Lorentz-boosted rotation matrix and is applicable to all massless particles. It is possible to construct higher-spin states starting from the four states of the spin-1/2 particle in the Lorentz-covariant world.

Generators | Two-by-Two | Four-by-Four |
---|---|---|

${J}_{3}=\frac{1}{2}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ | $\left(\begin{array}{cc}exp(i\varphi /2)& 0\\ 0& exp(-i\varphi /2)\end{array}\right)$ | $\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& cos\varphi & -sin\varphi \\ 0& 0& sin\varphi & cos\varphi \end{array}\right)$ |

${K}_{3}=\frac{1}{2}\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)$ | $\left(\begin{array}{cc}exp(\eta /2)& 0\\ 0& exp(-\eta /2)\end{array}\right)$ | $\left(\begin{array}{cccc}cosh\eta & sinh\eta & 0& 0\\ sinh\eta & cosh\eta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$ |

${J}_{1}=\frac{1}{2}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ | $\left(\begin{array}{cc}cos(\theta /2)& isin(\theta /2)\\ isin(\theta /2)& cos(\theta /2)\end{array}\right)$ | $\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& cos\theta & 0& sin\theta \\ 0& 0& 1& 0\\ 0& -sin\theta & 0& cos\theta \end{array}\right)$ |

${K}_{1}=\frac{1}{2}\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)$ | $\left(\begin{array}{cc}cosh(\lambda /2)& sinh(\lambda /2)\\ sinh(\lambda /2)& cosh(\lambda /2)\end{array}\right)$ | $\left(\begin{array}{cccc}cosh\lambda & 0& sinh\lambda & 0\\ 0& 1& 0\\ sinh\lambda & 0& cosh\lambda & 0\\ 0& 0& 0& 1\end{array}\right)$ |

${J}_{2}=\frac{1}{2}\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$ | $\left(\begin{array}{cc}cos(\theta /2)& -sin(\theta /2)\\ sin(\theta /2)& cos(\theta /2)\end{array}\right)$ | $\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& cos\theta & -sin\theta & 0\\ 0& sin\theta & cos\theta & 0\\ 0& 0& 0& 1\end{array}\right)$ |

${K}_{2}=\frac{1}{2}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ | $\left(\begin{array}{cc}cosh(\lambda /2)& -isinh(\lambda /2)\\ isinh(\lambda /2)& cosh(\lambda /2)\end{array}\right)$ | $\left(\begin{array}{cccc}cosh\lambda & 0& 0& sinh\lambda \\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ sinh\lambda & 0& 0& cosh\lambda \end{array}\right)$ |

**Table 2.**The Wigner momentum vectors in the two-by-two matrix representation together with the corresponding transformation matrix. These four-momentum matrices have determinants that are positive, zero, and negative for massive, massless, and imaginary-mass particles, respectively.

Particle Mass | Four-Momentum | Transform Matrix |
---|---|---|

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

Start | Time Reflection | |
---|---|---|

Start | $\begin{array}{c}\mathrm{Start}\mathrm{with}\\ R\left(\alpha \right)S(-2\chi )R\left(\alpha \right)\end{array}$ | $\begin{array}{c}\mathrm{Time}\mathrm{Reversal}\\ R(-\alpha )S\left(2\chi \right)R(-\alpha )\end{array}$ |

$\begin{array}{c}\mathrm{Space}\\ \mathrm{Inversion}\end{array}$ | $\begin{array}{c}\mathrm{Parity}\\ R\left(\alpha \right)S\left(2\chi \right)R\left(\alpha \right)\end{array}$ | $\begin{array}{c}\mathrm{Charge}\mathrm{Conjugation}\\ R(-\alpha )S(-2\chi )R(-\alpha )\end{array}$ |

**Table 4.**Sixteen combinations of the $SL(2,c)$ spinors. In the $SU\left(2\right)$ regime, there are two spinors leading to four bilinear forms. In the $SL(2,c)$ world, there are two undotted and two dotted spinors. These four-spinors lead to sixteen independent bilinear combinations.

Spin 1 | Spin 0 |
---|---|

$uu,\phantom{\rule{1.em}{0ex}}\frac{1}{\sqrt{2}}(uv+vu),\phantom{\rule{1.em}{0ex}}vv,$ | $\frac{1}{\sqrt{2}}(uv-vu)$ |

$\dot{u}\dot{u},\phantom{\rule{1.em}{0ex}}\frac{1}{\sqrt{2}}(\dot{u}\dot{v}+\dot{v}\dot{u}),\phantom{\rule{1.em}{0ex}}\dot{v}\dot{v},$ | $\frac{1}{\sqrt{2}}(\dot{u}\dot{v}-\dot{v}\dot{u})$ |

$u\dot{u},\phantom{\rule{1.em}{0ex}}\frac{1}{\sqrt{2}}(u\dot{v}+v\dot{u}),\phantom{\rule{1.em}{0ex}}v\dot{v},$ | $\frac{1}{\sqrt{2}}(u\dot{v}-v\dot{u})$ |

$\dot{u}u,\phantom{\rule{1.em}{0ex}}\frac{1}{\sqrt{2}}(\dot{u}v+\dot{v}u),\phantom{\rule{1.em}{0ex}}\dot{v}v,$ | $\frac{1}{\sqrt{2}}(\dot{u}v-\dot{v}u)$ |

After the operation of $Q(\eta ,\varphi )$ and $\dot{Q}(\eta ,\varphi )$ | |

${e}^{-i\varphi}{e}^{\eta}uu,\phantom{\rule{1.em}{0ex}}\frac{1}{\sqrt{2}}(uv+vu),\phantom{\rule{1.em}{0ex}}{e}^{i\varphi}{e}^{-\eta}vv,$ | $\frac{1}{\sqrt{2}}(uv-vu)$ |

${e}^{-i\varphi}{e}^{-\eta}\dot{u}\dot{u},\phantom{\rule{1.em}{0ex}}\frac{1}{\sqrt{2}}(\dot{u}\dot{v}+\dot{v}\dot{u}),\phantom{\rule{1.em}{0ex}}{e}^{i\varphi}{e}^{\eta}\dot{v}\dot{v},$ | $\frac{1}{\sqrt{2}}(\dot{u}\dot{v}-\dot{v}\dot{u})$ |

${e}^{-i\varphi}u\dot{u},\phantom{\rule{1.em}{0ex}}\frac{1}{\sqrt{2}}({e}^{\eta}u\dot{v}+{e}^{-\eta}v\dot{u}),\phantom{\rule{1.em}{0ex}}{e}^{i\varphi}v\dot{v},$ | $\frac{1}{\sqrt{2}}({e}^{\eta}u\dot{v}-{e}^{-\eta}v\dot{u})$ |

${e}^{-i\varphi}\dot{u}u,\phantom{\rule{1.em}{0ex}}\frac{1}{\sqrt{2}}(\dot{u}v+\dot{v}u),\phantom{\rule{1.em}{0ex}}{e}^{i\varphi}\dot{v}v,$ | $\frac{1}{\sqrt{2}}({e}^{-\eta}\dot{u}v-{e}^{\eta}\dot{v}u)$ |

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Başkal, S.; Kim, Y.S.; Noz, M.E. Loop Representation of Wigner’s Little Groups. *Symmetry* **2017**, *9*, 97.
https://doi.org/10.3390/sym9070097

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Başkal S, Kim YS, Noz ME. Loop Representation of Wigner’s Little Groups. *Symmetry*. 2017; 9(7):97.
https://doi.org/10.3390/sym9070097

**Chicago/Turabian Style**

Başkal, Sibel, Young S. Kim, and Marilyn E. Noz. 2017. "Loop Representation of Wigner’s Little Groups" *Symmetry* 9, no. 7: 97.
https://doi.org/10.3390/sym9070097