#
Entanglement—A Higher Order Symmetry^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

**not mean that signals can be transmitted faster than the speed of light**. There is

**no**violation of the laws of physics. Commuting or partially commuting operators imply shared eigenvectors and shared eigenvectors imply two operators with similar characteristics, both classically and from a relativistic point of view. Singlet state particles conserve Lorentz invariance but as a higher order symmetry, meaning Lorentz invariance requires that a singlet state remains as a singlet state. We now investigate this in more detail and derive Fermi–Dirac statistics as a consequence of this invariance.

## 2. Mathematical Methods: Grasping the Difficulty, the Issue of Hidden Variables

- (1)
- ${\psi}_{1}\otimes {\psi}_{2}\otimes {\psi}_{3}+{\psi}_{2}\otimes {\psi}_{3}\otimes {\psi}_{1}+{\psi}_{3}\otimes {\psi}_{1}\otimes {\psi}_{2}$ $\mathrm{is}\text{}\mathrm{a}\text{}\mathrm{pure}\text{}\mathrm{entangled}\text{}\mathrm{state}$,
- (2)
- ${\psi}_{1}\otimes ({\psi}_{2}\otimes {\psi}_{3}+{\psi}_{3}\otimes {\psi}_{2})$ $\mathrm{is}\text{}\mathrm{a}\text{}\mathrm{mixed}\text{}\mathrm{state}$ with ${\mathsf{\psi}}_{2}$ and ${\mathsf{\psi}}_{3}$ entangled,
- (3)
- ${\psi}_{1}\otimes {\psi}_{2}\otimes {\psi}_{3}$, as a product of factors, is not an entangled state.

**R**$|\psi \rangle $ = $|\psi \rangle $, for

**R**$\u03f5$ SO(2), where SO(2) represents the rotation group. For example, the two states ${\psi}_{1}\otimes {\psi}_{2}-{\psi}_{2}\otimes {\psi}_{1}$ and $\psi \otimes \psi +{\psi}^{\perp}\otimes {\psi}^{\perp}$ defined over a two-dimensional Hilbert space, where $\psi \mathrm{and}{\psi}^{\perp}$ are orthogonal, are rotationally invariant in that

#### 2.1. A Coupling Principle

= P(u,.,d)[P[(.,u,.)|(u,.,d)]+P[(.,d,.)|(u,.,d)]

= P(u,.,d) having summed over the marginal distribution.

#### 2.2. An Intuitive Model

- (1)
- $\psi =\mathrm{cos}\vartheta \text{}{\mathit{e}}_{\mathbf{1}}+\mathrm{sin}\vartheta {\mathit{e}}_{\mathbf{2}}$
- (2)
- $\psi =\mathrm{cos}\vartheta \text{}{\mathit{e}}_{\mathbf{1}}-\mathrm{sin}\vartheta {\mathit{e}}_{\mathbf{2}}$
- (3)
- $\psi =-\mathrm{cos}\vartheta \text{}{\mathit{e}}_{\mathbf{1}}+\mathrm{sin}\vartheta {\mathit{e}}_{\mathbf{2}}$
- (4)
- $\psi =-\mathrm{cos}\vartheta \text{}{\mathit{e}}_{\mathbf{1}}-\mathrm{sin}\vartheta {\mathit{e}}_{\mathbf{2}}$

#### 2.3. Method: What Is Happening Mathematically?

## 3. Results

#### 3.1. Relation between the Pauli Exclusion Principle and the Coupling Principle

**S**(1) and

**S**(2) associated with two different particles commute beyond the light cone when measured at the same angle and anti-commute only when orthogonal. Consequently, Pauli’s derivation is not applicable in this case. Indeed, Pauli notes “that for integral spin the quantization according to the exclusion principle is not possible. For this result it is essential that the use of the ${D}_{1}$function in place of the $D$ function be, for general reasons, discarded” [9]. However, the “general reason” for discarding ${D}_{1}$ is the principle of micro-causality, which is not applicable to entangled operators.

#### 3.2. Incorporating Entanglement into General Relativity

^{2}. In the language of limits as ${\widehat{ds}}_{1}\to {\widehat{ds}}_{2}\to \text{}\widehat{ds}$ then ds₁ds₂ $\to $ ds

^{2}. Moreover, if both bodies have mass and we let m₁$\to 0$ and m₂$\to 0$, we obtain the representation of the singlet state in Minkowski space. A generalized Dirac equation may be obtained by taking the dual of the metric [10].

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

O’Hara, P.
Entanglement—A Higher Order Symmetry. *Phys. Sci. Forum* **2023**, *7*, 4.
https://doi.org/10.3390/ECU2023-14011

**AMA Style**

O’Hara P.
Entanglement—A Higher Order Symmetry. *Physical Sciences Forum*. 2023; 7(1):4.
https://doi.org/10.3390/ECU2023-14011

**Chicago/Turabian Style**

O’Hara, Paul.
2023. "Entanglement—A Higher Order Symmetry" *Physical Sciences Forum* 7, no. 1: 4.
https://doi.org/10.3390/ECU2023-14011