# Integration of Dirac’s Efforts to Construct a Quantum Mechanics Which is Lorentz-Covariant

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## Abstract

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## 1. Introduction

- In 1927, Dirac pointed out that the time-energy uncertainty should be considered if the system is to be Lorentz-covariant [19].
- In 1945, Dirac said the Gaussian form could serve as a representation of the Lorentz group [20].
- In 1949, when Dirac introduced both his instant form of quantum mechanics and his light-cone coordinate system [21], he clearly stated that finding a representation of the inhomogeneous Lorentz group was the task of Lorentz-covariant quantum mechanics.
- In 1963, Dirac used the symmetry of two coupled oscillators to construct the $O(3,\phantom{\rule{0.166667em}{0ex}}2)$ group [22].

## 2. Dirac’s Efforts to Make Quantum Mechanics Lorentz-Covariant

#### 2.1. Dirac’s C-Number Time-Energy Uncertainty Relation

#### 2.2. Dirac’s Four-Dimensional Oscillators

#### 2.3. Dirac’s Light-Cone Coordinate System

## 3. Scattering and Bound States

## 4. Lorentz-Covariant Picture of Quantum Bound States

## 5. Lorentz-Covariant Quark Model

#### 5.1. Proton Form Factor

#### 5.2. Feynman’S Parton Picture

- When protons move with velocity close to that of light, the parton picture is valid.
- Partons behave as free independent particles when the interaction time between the quarks becomes dilated.
- Partons have a widespread distribution of momentum as the proton moves quickly.
- There seems to be an infinite number of partons or a number much larger than that of quarks.

#### 5.3. Historical Note

#### 5.4. Lorentz-Invariant Uncertainty Products

## 6. O(3,2) Symmetry Derivable from Two-Photon States

## 7. Contraction of O(3, 2) to the Inhomogeneous Lorentz Group

## 8. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Quantum mechanics represented in terms of space-time. As can be seen, there are no excitations along the time-like direction, but quantum excitations along the space-like longitudinal direction are allowed.

**Figure 2.**Dirac’s four-dimensional oscillators localized in a closed space-time region. This is not a Lorentz-invariant concept. How about Lorentz covariance?

**Figure 3.**The light-cone coordinate system pictured with a Lorentz boost. Not only does the boost squeeze the square into a rectangle, but it traces a point along the hyperbola.

**Figure 4.**Feynman, in an effort to combine quantum mechanics with special relativity, gave us this roadmap. Feynman’s diagrams provide, in Einstein’s world, a satisfactory resolution for scattering states. Thus they work for running waves. Feynman suggested that harmonic oscillators should be used as a first step for representing standing waves trapped inside an extended hadron.

**Figure 6.**Electron-proton scattering in the Breit frame. The outgoing momentum of the proton is opposite in sign but equal in magnitude to that of the incoming proton.

**Figure 7.**The momentum-energy wave functions are Lorentz squeezed in the form factor calculation. The two wave functions become separated as the momentum transfer increases. However, in the relativistic case, the wave functions maintain an overlapping region. In the non-relativistic calculation, the wave functions become completely separated. The unacceptable behavior of the form factor is caused by this lack of overlapping region.

**Figure 8.**Lorentz-squeezed wave functions in space-time and in momentum-energy variables. Both wave functions become concentrated along their respective positive light-cone axes as the speed of the proton approaches that of light. All the peculiarities of Feynman’s parton picture are presented in these light-cone concentrations.

**Figure 9.**Bohr and Einstein, and then Gell-Mann and Feynman. There are no records indicating that Bohr and Einstein discussed how the hydrogen looks to moving observers. After 1950, with particle accelerators, the physics world started producing protons with relativistic speeds. Furthermore, the proton became a bound state sharing the same quantum mechanics with the hydrogen atom. The problem of fast-moving hydrogen became that of the proton. How would the proton appear when it moves with a speed close to that of light? This is the quark-parton puzzle.

**Figure 10.**Scattering and bound states. These days, Feynman diagrams are used for scattering problems. For bound-state problems, it is possible to construct Lorentz-covariant harmonic oscillators by integrating the papers written by Dirac. Feynman diagrams and the covariant oscillators are both two different representations of the inhomogeneous Lorentz group. Then is it possible to derive Einstein’s special relativity from the Heisenberg brackets? This problem is addressed in Section 6 and Section 7.

**Figure 11.**Contraction of $O(3,\phantom{\rule{0.166667em}{0ex}}2)$ to the inhomogeneous Lorentz group. The extra time variable s becomes a constant, as shown by a flat line in this figure.

**Table 1.**Lorentz covariance of particles both massive and massless. The little group of Wigner unifies, for massive and massless particles, the internal space-time symmetries. The challenge for us is to find another unification: that which unifies, in the physics of the high-energy realm, both the quark and parton pictures. In this paper, we achieve this purpose by integrating Dirac’s three papers. A similar table was published in Ref. [11].

Massive, Slow | COVARIANCE | Massless, Fast | |
---|---|---|---|

Energy-Momentum | Einstein’s | ||

$E={p}^{2}/2m$ | $E=\sqrt{{\left(cp\right)}^{2}+{\left(\right)}^{m}2}$ | $E=cp$ | |

Internal | ${S}_{3}$ | ${S}_{3}$ | |

Space-time | Wigner’s | Gauge | |

Symmetry | ${S}_{1},\phantom{\rule{0.166667em}{0ex}}{S}_{2}$ | Little Groups | Transformation |

Relativistic | Integration | ||

Extended | Quark Model | of Dirac’s papers | Parton Model |

Particles | 1927, 1945, 1949 |

**Table 2.**Three generators of the rotations in the five-dimensional space of $(x,\phantom{\rule{0.166667em}{0ex}}y,\phantom{\rule{0.166667em}{0ex}}z,\phantom{\rule{0.166667em}{0ex}}t,\phantom{\rule{0.166667em}{0ex}}s)$. The time-like s and t coordinates are not affected by the rotations in the three-dimensional space of $(x,\phantom{\rule{0.166667em}{0ex}}y,\phantom{\rule{0.166667em}{0ex}}z)$.

Generators | Differential | Matrix | |||
---|---|---|---|---|---|

${J}_{1}$ | $-i\left(\right)open="("\; close=")">y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& -i& 0& 0\\ 0& i& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$ | |||

${J}_{2}$ | $-i\left(\right)open="("\; close=")">z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}$ | $\left(\begin{array}{ccccc}0& 0& i& 0& 0\\ 0& 0& 0& 0& 0\\ -i& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$ | |||

${J}_{3}$ | $-i\left(\right)open="("\; close=")">x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}$ | $\left(\begin{array}{ccccc}0& -i& 0& 0& 0\\ i& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ \phantom{\rule{4pt}{0ex}}0& 0& 0& 0& 0\end{array}\right)$ |

**Table 3.**Three generators of Lorentz boosts with respect the time variable t. The s coordinate is not affected by these boosts.

Generators | Differential | Matrix | |||
---|---|---|---|---|---|

${K}_{1}$ | $-i\left(\right)open="("\; close=")">x\frac{\partial}{\partial t}+t\frac{\partial}{\partial x}$ | $\left(\begin{array}{ccccc}0& 0& 0& i& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ i& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$ | |||

${K}_{2}$ | $-i\left(\right)open="("\; close=")">y\frac{\partial}{\partial t}+t\frac{\partial}{\partial y}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& i& 0\\ 0& 0& 0& 0& 0\\ 0& i& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$ | |||

${K}_{3}$ | $-i\left(\right)open="("\; close=")">z\frac{\partial}{\partial t}+t\frac{\partial}{\partial z}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& i& 0\\ 0& 0& i& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$ |

**Table 4.**The $O(3,\phantom{\rule{0.166667em}{0ex}}2)$ group has four additional generators. Note that the generators in this table have non-zero elements only in the fifth row and the fifth column. This is unlike those given in Table 2 and Table 3. Here the s variable is contained in every differential operator.

Generators | Differential | Matrix | |||
---|---|---|---|---|---|

${Q}_{1}$ | $-i\left(\right)open="("\; close=")">x\frac{\partial}{\partial s}+s\frac{\partial}{\partial x}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& i\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ i& 0& 0& 0& 0\end{array}\right)$ | |||

${Q}_{2}$ | $-i\left(\right)open="("\; close=")">y\frac{\partial}{\partial s}+s\frac{\partial}{\partial y}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& i\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& i& 0& 0& 0\end{array}\right)$ | |||

${Q}_{3}$ | $-i\left(\right)open="("\; close=")">z\frac{\partial}{\partial s}+s\frac{\partial}{\partial z}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& i\\ 0& 0& 0& 0& 0\\ 0& 0& i& 0& 0\end{array}\right)$ | |||

${S}_{0}$ | $-i\left(\right)open="("\; close=")">t\frac{\partial}{\partial s}-s\frac{\partial}{\partial t}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -i\\ 0& 0& 0& i& 0\end{array}\right)$ |

**Table 5.**Here the generators of translations are given in the four-dimensional Minkowski space. It is of interest to convert the four generators in the $O(3,\phantom{\rule{0.166667em}{0ex}}2)$ group in Table 4 into the four translation generators.

Generators | Differential | Matrix | |||
---|---|---|---|---|---|

${Q}_{1}\to {P}_{1}$ | $-i\frac{\partial}{\partial x}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& i\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$ | |||

${Q}_{2}\to {P}_{2}$ | $-i\frac{\partial}{\partial y}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& i\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$ | |||

${Q}_{3}\to {P}_{3}$ | $-i\frac{\partial}{\partial z}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& i\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$ | |||

${S}_{0}\to {P}_{0}$ | $i\frac{\partial}{\partial t}$ | $\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -i\\ 0& 0& 0& 0& 0\end{array}\right)$ |

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Kim, Y.S.; Noz, M.E.
Integration of Dirac’s Efforts to Construct a Quantum Mechanics Which is Lorentz-Covariant. *Symmetry* **2020**, *12*, 1270.
https://doi.org/10.3390/sym12081270

**AMA Style**

Kim YS, Noz ME.
Integration of Dirac’s Efforts to Construct a Quantum Mechanics Which is Lorentz-Covariant. *Symmetry*. 2020; 12(8):1270.
https://doi.org/10.3390/sym12081270

**Chicago/Turabian Style**

Kim, Young S., and Marilyn E. Noz.
2020. "Integration of Dirac’s Efforts to Construct a Quantum Mechanics Which is Lorentz-Covariant" *Symmetry* 12, no. 8: 1270.
https://doi.org/10.3390/sym12081270