# Dirac Matrices and Feynman’s Rest of the Universe

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## Abstract

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## 1. Introduction

## 2. Dirac Matrices in the Majorana Representation

## 3. Dirac’s Coupled Oscillators

#### 3.1. Wigner Phase-Space Representation

#### 3.2. Translation into Four-by-Four Matrices

## 4. Extension to $O(3,3)$ Symmetry

**Table 1.**$SL(4,r)$ and Dirac matrices. Two sets of rotation generators and three sets of boost generators. There are 15 generators.

First component | Second component | Third component | |

Rotation | ${L}_{1}=\frac{-i}{2}{\gamma}_{0}$ | ${L}_{2}=\frac{-i}{2}{\gamma}_{5}{\gamma}_{0}$ | ${L}_{3}=\frac{-1}{2}{\gamma}_{5}$ |

Rotation | ${S}_{1}=\frac{i}{2}{\gamma}_{2}{\gamma}_{3}$ | ${S}_{2}=\frac{i}{2}{\gamma}_{1}{\gamma}_{2}$ | ${S}_{3}=\frac{i}{2}{\gamma}_{3}{\gamma}_{1}$ |

Boost | ${K}_{1}=\frac{-i}{2}{\gamma}_{5}{\gamma}_{1}$ | ${K}_{2}=\frac{1}{2}{\gamma}_{1}$ | ${K}_{3}=\frac{i}{2}{\gamma}_{0}{\gamma}_{1}$ |

Boost | ${Q}_{1}=\frac{i}{2}{\gamma}_{5}{\gamma}_{3}$ | ${Q}_{2}=\frac{-1}{2}{\gamma}_{3}$ | ${Q}_{3}=-\frac{i}{2}{\gamma}_{0}{\gamma}_{3}$ |

Boost | ${G}_{1}=\frac{-i}{2}{\gamma}_{5}{\gamma}_{2}$ | ${G}_{2}=\frac{1}{2}{\gamma}_{2}$ | ${G}_{3}=\frac{i}{2}{\gamma}_{0}{\gamma}_{2}$ |

#### 4.1. Non-Canonical Transformations in Classical Mechanics

**Figure 1.**Expanding and contracting phase spaces. Canonical transformations leave the area of each phase space invariant. Non-canonical transformations can change them, yet the product of these two areas remains invariant.

#### 4.2. Local Isomorphism between O(3,3) and SL(4,r)

**Table 2.**Three-by-three matrices constituting the two-by-two representation of generators of the $O(3,3)$ group.

i = 1 | i = 2 | i = 3 | |

${A}_{i}$ | $\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -i\\ 0& i& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& i\\ 0& 0& 0\\ -i& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& -i& 0\\ i& 0& 0\\ 0& 0& 0\end{array}\right)$ |

${B}_{i}$ | $\left(\begin{array}{ccc}i& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& 0\\ i& 0& 0\\ 0& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ i& 0& 0\end{array}\right)$ |

${C}_{i}$ | $\left(\begin{array}{ccc}0& i& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& 0\\ 0& i& 0\\ 0& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& i& 0\end{array}\right)$ |

${D}_{i}$ | $\left(\begin{array}{ccc}0& 0& i\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& i\\ 0& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& i\end{array}\right)$ |

## 5. Feynman’s Rest of the Universe

#### 5.1. Canonical Approach

**Figure 2.**Two-dimensional Gaussian form for two-coupled oscillators. One of the variables is observable while the second variable is not observed. It belongs to Feynman’s rest of the universe.

#### 5.2. Non-Canonical Approach

#### 5.3. Entropy and the Expanding Wigner Phase Space

## 6. Concluding Remarks

## Acknowledgments

## References

- Dirac, P.A.M. A remarkable representation of the 3 + 2 de Sitter Group. J. Math. Phys.
**1963**, 4, 901–909. [Google Scholar] [CrossRef] - Yuen, H.P. Two-photon coherent states of the radiation field. Phys. Rev. A
**1976**, 13, 2226–2243. [Google Scholar] [CrossRef] - Yurke, B.S.; McCall, S.L.; Klauder, J.R. SU(2) and SU(1,1) interferometers. Phys. Rev. A
**1986**, 33, 4033–4054. [Google Scholar] [CrossRef] [PubMed] - Kim, Y.S.; Noz, M.E. Phase Space Picture of Quantum Mechanics; World Scientific Publishing Company: Singapore, 1991. [Google Scholar]
- Han, D.; Kim, Y.S.; Noz, M.E.; Yeh, L. Symmetries of two-mode squeezed states. J. Math. Phys.
**1993**, 34, 5493–5508. [Google Scholar] [CrossRef] - Han, D.; Kim, Y.S.; Noz, M.E. O(3,3)-like symmetries of coupled harmonic oscillators. J. Math. Phys.
**1995**, 36, 3940–3954. [Google Scholar] [CrossRef] - Lee, D.-G. The Dirac gamma matrices as “relics” of a hidden symmetry?: As fundamental representation of the algebra Sp(4,r). J. Math. Phys.
**1995**, 36, 524–530. [Google Scholar] [CrossRef] - Han, D.; Kim, Y.S.; Noz, M.E. Illustrative example of Feynman’s rest of the universe. Am. J. Phys.
**1999**, 67, 61–66. [Google Scholar] [CrossRef] - Feynman, R.P. Statistical Mechanics; Benjamin/Cummings: Reading, MA, USA, 1972. [Google Scholar]
- Majorana, E. Relativistic theory of particles with arbitrary intrinsic angular momentum. Nuovo Cimento
**1932**, 9, 335–341. [Google Scholar] [CrossRef] - Itzykson, C.; Zuber, J.B. Quantum Field Theory; MaGraw-Hill: New York, NY, USA, 1980. [Google Scholar]
- Goldstein, H. Classical Mechanics, 2nd ed.; Addison-Wesley: Reading, MA, USA, 1980. [Google Scholar]
- Abraham, R.; Marsden, J.E. Foundations of Mechanics, 2nd ed.; Benjamin/Cummings: Reading, MA, USA, 1978. [Google Scholar]
- Kim, Y.S.; Li, M. Squeezed states and thermally excited states in the Wigner phase-space picture of quantum mechanics. Phys. Lett. A
**1989**, 139, 445–448. [Google Scholar] [CrossRef] - Davies, R.W.; Davies, K.T.R. On the Wigner distribution function for an oscillator. Ann. Phys.
**1975**, 89, 261–273. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Statistical Physics; Pergamon Press: London, UK, 1958. [Google Scholar]
- Yurke, B.; Potasek, M. Obtainment of thermal noise from a pure state. Phys. Rev. A
**1987**, 36, 3464–3466. [Google Scholar] [CrossRef] [PubMed] - Ekert, A.K.; Knight, P.L. Correlations and squeezing of two-mode oscillations. Am. J. Phys.
**1989**, 57, 692–697. [Google Scholar] [CrossRef] - Barnett, S.M.; Phoenix, S.J.D. Information theory, squeezing and quantum correlations. Phys. Rev. A
**1991**, 44, 535–545. [Google Scholar] [CrossRef] [PubMed] - Kim, Y.S.; Noz, M.E.; Oh, S.H. A simple method for illustrating the difference between the homogeneous and inhomogeneous Lorentz Groups. Am. J. Phys.
**1979**, 47, 892–897. [Google Scholar] [CrossRef] - Kim, Y.S.; Noz, M.E. Theory and Applications of the Poincaré Group; Reidel: Dordrecht, the Netherlands, 1986. [Google Scholar]
- Giedke, G.; Wolf, M.M.; Krueger, O.; Werner, R.F.; Cirac, J.J. Entanglement of formation for symmetric Gaussian states. Phys. Rev. Lett.
**2003**, 91, 107901–107904. [Google Scholar] [CrossRef] [PubMed] - Han, D.; Kim, Y.S.; Noz, M.E. Lorentz-squeezed hadrons and hadronic temperature. Phys. Lett. A
**1990**, 144, 111–115. [Google Scholar] [CrossRef] - von Neumann, J. Mathematical Foundation of Quantum Mechanics; Princeton University: Princeton, NJ, USA, 1955. [Google Scholar]
- Fano, U. Description of states in quantum mechanics by density matrix and operator techniques. Rev. Mod. Phys.
**1957**, 29, 74–93. [Google Scholar] [CrossRef] - Blum, K. Density Matrix Theory and Applications; Plenum: New York, NY, USA, 1981. [Google Scholar]
- Kim, Y.S.; Wigner, E.P. Entropy and Lorentz transformations. Phys. Lett. A
**1990**, 147, 343–347. [Google Scholar] [CrossRef] - Feynman, R.P.; Kislinger, M.; Ravndal, F. Current matrix elements from a relativistic Quark Model. Phys. Rev. D
**1971**, 3, 2706–2732. [Google Scholar] [CrossRef] - Dirac, P.A.M. Principles of Quantum Mechanics, 4th ed.; Oxford University: London, UK, 1958. [Google Scholar]
- Hussar, P.E.; Kim, Y.S.; Noz, M.E. Three-particle symmetry classifications according to the method of Dirac. Am. J. Phys.
**1980**, 48, 1038–1042. [Google Scholar] [CrossRef] - Başkal, S.; Kim, Y.S. Lorentz Group in ray and polarization optics. In Mathematical Optics: Classical, Quantum and Imaging Methods; Lakshminarayanan, V., Calvo, M.L., Alieva, T., Eds.; CRC Press: New York, NY, USA, 2012. [Google Scholar]
- Gilmore, R. Lie Groups, Lie Algebras, and Some of Their Applications; Wiley: New York, NY, USA, 1974. [Google Scholar]

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Kim, Y.S.; Noz, M.E.
Dirac Matrices and Feynman’s Rest of the Universe. *Symmetry* **2012**, *4*, 626-643.
https://doi.org/10.3390/sym4040626

**AMA Style**

Kim YS, Noz ME.
Dirac Matrices and Feynman’s Rest of the Universe. *Symmetry*. 2012; 4(4):626-643.
https://doi.org/10.3390/sym4040626

**Chicago/Turabian Style**

Kim, Young S., and Marilyn E. Noz.
2012. "Dirac Matrices and Feynman’s Rest of the Universe" *Symmetry* 4, no. 4: 626-643.
https://doi.org/10.3390/sym4040626