# Spontaneous Emergence of a Causal Time Axis in Euclidean Space from a Gauged Rotational Symmetry Theory

## Abstract

**:**

## 1. Introduction

## 2. The CDG Decomposition—A Review

#### The Cho Connection

## 3. Rotational Gauge Symmetry in Euclidean Space

#### 3.1. The CDG Decomposition of $SO\left(4\right)$ Gauge Theory

#### 3.2. Monopole Field Strength

#### 3.3. Invariant Transformations of CDG Decomposition of Gauged Rotational Theory

## 4. Vacuum State at One Loop

#### 4.1. Magnitude of Monopole Condensate

#### 4.2. Stability of the Minkowski Condensate

## 5. The Emergence of Effective Minkowski Space

#### 5.1. Identifying the Local Time Axis

#### 5.2. The Constant Speed of Light

#### 5.3. Emergence of Lorentz Transformations

- 1.
- Linearity;
- 2.
- An internal composition law;
- 3.
- Reflection invariance.

## 6. Discussion

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

QCD | Quantum ChromoDynamics |

MAG | Maximal Abelian Gauge |

DOFs | Degrees of Freedom |

CDG | Cho-Duan-Ge |

## Appendix A. Gauge Degrees of Freedom in SU(2)

## Appendix B. General Derivation of Lorentz Transformation

- 1.
- Linearity;
- 2.
- An internal composition law;
- 3.
- Reflection invariance.

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**Figure 1.**Feynman diagrams of one-loop contributions to the effective action of $SU\left(2\right)$ QCD. The loops on the left of the figure are due to gluons, while those on right are due to ghosts. Note that the contributions of the tadpole diagrams in the bottom row are removed by dimensional renormalisation.

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**MDPI and ACS Style**

Walker, M.L.
Spontaneous Emergence of a Causal Time Axis in Euclidean Space from a Gauged Rotational Symmetry Theory. *Symmetry* **2024**, *16*, 4.
https://doi.org/10.3390/sym16010004

**AMA Style**

Walker ML.
Spontaneous Emergence of a Causal Time Axis in Euclidean Space from a Gauged Rotational Symmetry Theory. *Symmetry*. 2024; 16(1):4.
https://doi.org/10.3390/sym16010004

**Chicago/Turabian Style**

Walker, Michael Luke.
2024. "Spontaneous Emergence of a Causal Time Axis in Euclidean Space from a Gauged Rotational Symmetry Theory" *Symmetry* 16, no. 1: 4.
https://doi.org/10.3390/sym16010004