# On Born’s Reciprocal Relativity, Algebraic Extensions of the Yang and Quaplectic Algebra, and Noncommutative Curved Phase Spaces

## Abstract

**:**

## 1. Introduction: Born’s Reciprocal Relativity Theory

**c**-valued (classical) variables which are all boosted (rotated) into each other, were given by [2,3,4] based on the group $U(1,3)$, which is the Born version of the Lorentz group $SO(1,3)$. The $U(1,3)=SU(1,3)\times U\left(1\right)$ group transformations leave invariant the symplectic two-form $\mathsf{\Omega}=-\phantom{\rule{3.33333pt}{0ex}}dT\wedge dE+{\delta}_{ij}d{X}^{i}\wedge d{P}^{j};i,j=1,2,3$ and also the following Born–Green line interval in the $flat$ 8D phase space

## 2. The Deformed Quaplectic Group and Complex Gravity

## 3. The Yang Algebra versus the Deformed Quaplectic Algebra

#### 3.1. The Yang Algebra and Its Extension via Generalized Angular Momentum Operators in Higher Dimensions

#### 3.2. Realization of the Deformed Quaplectic Algebra and its Extensions

## 4. Curved Phase Space Due to Noncommutative Coordinates and Momenta

#### 4.1. Mapping of ${x}^{\mu},{p}^{\mu}$ to the ${Y}^{A},{\mathsf{\Pi}}^{A}$ Variables in Flat Phase Space

#### 4.2. Embedding an 8D Curved Phase Space into a 12D Flat Phase Space

**c**-numbers) $x,p$. A more rigorous notation in the previous section would have been to assign “hats” to operators ${\widehat{x}}^{\mu},{\widehat{p}}^{\mu};{\widehat{Y}}^{A},{\widehat{\mathsf{\Pi}}}^{A}$. For the sake of simplicity, we avoided it. The geometry of the cotangent bundle of spacetime (phase space) can be best-explored within the context of Lagrange–Finsler, Hamilton–Cartan geometry [15,16,17,18]. The line element in the 8D curved phase space is

## 5. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## Notes

1 | Strictly speaking, $U(1,4)$ is a pseudo-unitary group. After performing the Weyl unitary “trick” via an analytical continuation $U(1,4)\to U\left(5\right)$, one obtains the unitary group $U\left(5\right)$ comprised of $5\times 5$ unitary matrices obeying ${U}^{\u2020}={U}^{-1}$. A unitary matrix can be written as $U={e}^{A}$, where A is an anti-Hermitian matrix ${A}^{\u2020}=-A$, and any anti-Hermitian matrix A can be written as $A=\pm iH$, where H is Hermitian; therefore, all group elements can be written in the form $U={e}^{\pm i{\theta}^{AB}{Z}_{AB}}$, where ${\theta}^{AB}$ are the corresponding parameters associated to every generator. |

2 | We choose a different signature than the one in the Introduction. |

3 | Our choice differs by a minus sign from the conventional definition. |

4 | A simple inspection reveals that a correspondence of the form $\frac{{x}^{\mu}}{{L}_{P}}={a}_{1}{J}^{\mu 5}+{b}_{1}{J}^{\mu 6};\mathcal{L}{p}^{\mu}={a}_{2}{J}^{\mu 5}+{b}_{2}{J}^{\mu 6}$ will automatically lead to ${b}_{1}=0,{a}_{2}=0$; or ${b}_{2}=0,{a}_{1}=0$ resulting from the antisymmetry of the commutators $[{x}^{\mu},{x}^{\nu}],[{p}^{\mu},{p}^{\nu}]$ |

5 | Since ${\lambda}_{l}{\lambda}_{p}=1$, in units of $\hslash =1$, the powers of ${\lambda}_{l},{\lambda}_{p}$ decouple explicitly from Equation (44a,b) |

6 | Note that one must not confuse ${Z}_{ab}\equiv \frac{1}{2}({M}_{ab}+{L}_{ab})$ with ${Z}_{\left[{a}_{1}{a}_{2}\right]}$ defined by Equation (55a) |

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Perelman, C.C.
On Born’s Reciprocal Relativity, Algebraic Extensions of the Yang and Quaplectic Algebra, and Noncommutative Curved Phase Spaces. *Universe* **2023**, *9*, 144.
https://doi.org/10.3390/universe9030144

**AMA Style**

Perelman CC.
On Born’s Reciprocal Relativity, Algebraic Extensions of the Yang and Quaplectic Algebra, and Noncommutative Curved Phase Spaces. *Universe*. 2023; 9(3):144.
https://doi.org/10.3390/universe9030144

**Chicago/Turabian Style**

Perelman, Carlos Castro.
2023. "On Born’s Reciprocal Relativity, Algebraic Extensions of the Yang and Quaplectic Algebra, and Noncommutative Curved Phase Spaces" *Universe* 9, no. 3: 144.
https://doi.org/10.3390/universe9030144