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Einstein’s E = mc^{2} Derivable from Heisenberg’s Uncertainty Relations

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## Abstract

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## 1. Introduction

## 2. Symmetries of the Single-Mode States

## 3. Symmetries from Two Oscillators

## 4. Contraction of $\mathit{O}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{2}\mathbf{)}$ to the Inhomogeneous Lorentz Group

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Dirac’s three papers. His 1927 and 1945 papers can be described by a circle in the longitudinal space-like and time-like coordinate. Dirac introduced the light-cone coordinate system in 1949. In this system, the Lorentz boost is a squeeze transformation. It is then natural to synthesize these two figures to a squeezed circle or an ellipse. Figure A2 will illustrate how this elliptic squeeze manifests itself in the real world.

**Figure A2.**In the harmonic-oscillator regime, the momentum–energy wave function takes the same mathematical form as that of the space-time wave functions. This figure shows that the quark model and the parton model are two different aspects of one Lorentz-covariant entity. In 1969 [17], Feynman observed that the fast-moving proton appears as a collection of a large number of light-like partons with a wide-spread momentum distribution, and short interaction time with the external signal. This figure is a graphical illustration of the 1977 paper by Kim and Noz [19]. This figure is from a recent book by the present authors [20].

**Figure A3.**The Bohr–Einstein issue is 100 years old. Fifty years later, it became the quark–parton puzzle, based on observations made in high-energy laboratories.

## References

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**Figure 1.**The Inönü–Wigner contraction procedure interpreted as squeeze transformations. In (

**a**), the square becomes a narrow rectangle during the squeeze process. When the rectangle becomes narrow enough, the point A can be moved to the horizontal axis. Then, the inverse squeeze brings back the rectangle to the original shape. The point A remains on the horizontal axis. In (

**b**), both the hyperbola and the circle become flattened to the horizontal axis, during the initial squeeze. The point on the curve moves to the horizontal axis. This point moves back to its finite position during the inverse squeeze.

**Figure 2.**According to Dirac’s 1949 paper, the task of constructing quantum mechanics is essentially constructing a representation of the inhomogeneous Lorentz group. In his 1963 paper, Dirac constructed the Lie algebra of the $O(3,2)$ de Sitter group from the algebra of two harmonic oscillators, which is a direct consequence of Heisenberg’s uncertainty commutation relations. It is possible to derive the Lie algebra of the inhomogeneous Lorentz group from that of $O(3,2)$ using the group-contraction procedure of Inönü and Wigner [3].

**Figure 3.**Rotations and squeezes in the phase space produced by the $Sp\left(2\right)$ transformations. The squeeze along the x direction corresponds to the Lorentz boost along the z direction, while the squeeze along the 45${}^{\circ}$ angle corresponds to the boost along the x direction. The rotation by 45${}^{\circ}$ corresponds to the rotation by 90${}^{\circ}$ around the y axis.

**Table 1.**Transformation for the Gaussian function, in terms of harmonic oscillators, two-dimensional phase space, and the four-dimensional Minkowski space.

Generators | Oscillator | Phase Space | Lorentz |
---|---|---|---|

${J}_{2}$ | $\frac{1}{2}\left(a{a}^{\u2020}+{a}^{\u2020}a\right)$ | $\frac{1}{2}{\sigma}_{2}$ | $\left(\begin{array}{cccc}0& 0& i& 0\\ 0& 0& 0& 0\\ i& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$ |

${K}_{1}$ | $\frac{1}{2i}\left({a}^{\u2020}{a}^{\u2020}+aa\right)$ | $\frac{i}{2}{\sigma}_{1}$ | $\left(\begin{array}{cccc}0& 0& 0& i\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ i& 0& 0& 0\end{array}\right)$ |

${K}_{3}$ | $\frac{1}{2}\left({a}^{\u2020}{a}^{\u2020}-aa\right),$ | $\frac{i}{2}{\sigma}_{3}$ | $\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& i\\ 0& 0& i& 0\end{array}\right)$ |

Generators | Two Oscillators | Phase Space |
---|---|---|

${J}_{1}$ | $\frac{1}{2}\left({a}_{1}^{\u2020}{a}_{2}+{a}_{2}^{\u2020}{a}_{1}\right)$ | $-\frac{1}{2}\left(\begin{array}{cc}0& I\\ I& 0\end{array}\right){\sigma}_{2}$ |

${J}_{2}$ | $\frac{1}{2i}\left({a}_{1}^{\u2020}{a}_{2}-{a}_{2}^{\u2020}{a}_{1}\right)$ | $\frac{i}{2}\left(\begin{array}{cc}0& -I\\ I& 0\end{array}\right)I$ |

${J}_{3}$ | $\frac{1}{2}\left({a}_{1}^{\u2020}{a}_{1}-{a}_{2}^{\u2020}{a}_{2}\right),$ | $\frac{1}{2}\left(\begin{array}{cc}-I& 0\\ 0& I\end{array}\right){\sigma}_{2}$ |

${S}_{0}$ | $\frac{1}{2}\left({a}_{1}^{\u2020}{a}_{1}+{a}_{2}{a}_{2}^{\u2020}\right),$ | $\frac{1}{2}\left(\begin{array}{cc}I& 0\\ 0& I\end{array}\right){\sigma}_{2}$ |

${K}_{1}$ | $-\frac{1}{4}\left({a}_{1}^{\u2020}{a}_{1}^{\u2020}+{a}_{1}{a}_{1}-{a}_{2}^{\u2020}{a}_{2}^{\u2020}-{a}_{2}{a}_{2}\right)$ | $\frac{i}{2}\left(\begin{array}{cc}I& 0\\ 0& -I\end{array}\right){\sigma}_{1}$ |

${K}_{2}$ | $+\frac{i}{4}\left({a}_{1}^{\u2020}{a}_{1}^{\u2020}-{a}_{1}{a}_{1}+{a}_{2}^{\u2020}{a}_{2}^{\u2020}-{a}_{2}{a}_{2}\right)$ | $\frac{i}{2}\left(\begin{array}{cc}I& 0\\ 0& I\end{array}\right){\sigma}_{3}$ |

${K}_{3}$ | $\frac{1}{2}\left({a}_{1}^{\u2020}{a}_{2}^{\u2020}+{a}_{1}{a}_{2}\right)$ | $-\frac{i}{2}\left(\begin{array}{cc}0& I\\ I& 0\end{array}\right){\sigma}_{1}$ |

${Q}_{1}$ | $-\frac{i}{4}\left({a}_{1}^{\u2020}{a}_{1}^{\u2020}-{a}_{1}{a}_{1}-{a}_{2}^{\u2020}{a}_{2}^{\u2020}+{a}_{2}{a}_{2}\right)$ | $-\frac{i}{2}\left(\begin{array}{cc}I& 0\\ 0& -I\end{array}\right){\sigma}_{3}$ |

${Q}_{2}$ | $-\frac{1}{4}\left({a}_{1}^{\u2020}{a}_{1}^{\u2020}+{a}_{1}{a}_{1}+{a}_{2}^{\u2020}{a}_{2}^{\u2020}+{a}_{2}{a}_{2}\right)$ | $\frac{i}{2}\left(\begin{array}{cc}I& 0\\ 0& I\end{array}\right){\sigma}_{1}$ |

${Q}_{3}$ | $\frac{i}{2}\left({a}_{1}^{\u2020}{a}_{2}^{\u2020}-{a}_{1}{a}_{2}\right)$ | $\frac{1}{2}\left(\begin{array}{cc}I& 0\\ 0& I\end{array}\right){\sigma}_{2}$ |

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**MDPI and ACS Style**

Başkal, S.; Kim, Y.S.; Noz, M.E. Einstein’s *E* = *mc*^{2} Derivable from Heisenberg’s Uncertainty Relations. *Quantum Rep.* **2019**, *1*, 236-251.
https://doi.org/10.3390/quantum1020021

**AMA Style**

Başkal S, Kim YS, Noz ME. Einstein’s *E* = *mc*^{2} Derivable from Heisenberg’s Uncertainty Relations. *Quantum Reports*. 2019; 1(2):236-251.
https://doi.org/10.3390/quantum1020021

**Chicago/Turabian Style**

Başkal, Sibel, Young S. Kim, and Marilyn E. Noz. 2019. "Einstein’s *E* = *mc*^{2} Derivable from Heisenberg’s Uncertainty Relations" *Quantum Reports* 1, no. 2: 236-251.
https://doi.org/10.3390/quantum1020021