# Exotic Particle Dynamics Using Novel Hermitian Spin Matrices

**—**Theory and Applications)

## Abstract

**:**

## 1. Introduction

## 2. Analogue Spin Matrices

## 3. Spin Projection Operators

_{1}, is expressed as follows:

_{1}are ${\lambda}_{1}=\frac{1}{2}\hslash $ and ${\lambda}_{2}\approx \hslash \left(-\frac{1}{2}+3.92523\times {10}^{-17}i\right)$. The eigenvectors for S

_{1}are ${v}_{1}=[\left(\sqrt{2}+\sqrt{2}i\right),1]$ and ${v}_{2}=[\left(-\sqrt{2}-\sqrt{2}i\right),1]$. The eigenspinor representations are then as follows:

_{2}, is expressed as follows:

_{2}are ${v}_{1}=[0,1]$ and ${v}_{2}=[1,0]$, and the eigenspinor representations are then as follows:

_{3}, is expressed as follows:

_{3}are ${v}_{1}=[1,0]$ and ${v}_{2}=[0,1]$, and the eigenspinor representations are then as follows:

_{1}, the $\left|\downarrow \right.\u27e9$ spin state contains a small imaginary contribution of 3.92523 × 10

^{−17}i.

## 4. Electrodynamics

## 5. Simulation with Spin Chains

^{N}. Considering the coupling constant, $J=({J}_{x},{J}_{y},{J}_{z})$, to be real-valued, the Hamiltonian operator is represented as follows:

^{th}Pauli matrix on the j

^{th}lattice point with periodic boundary conditions. The Pauli matrices, ${\sigma}_{j}^{k}$, is defined as ${\sigma}_{j}^{k}={I}_{2}^{\otimes j-1}\otimes {\sigma}_{j}^{k}\otimes {I}_{2}^{\otimes N-j}$. In this work, the Heisenberg XXX model with $J>0$ is employed, where the coupling constants conform to the following equivalence: ${J=J}_{x}={J}_{y}={J}_{z}$. The simulation is carried out using the Python programming language. To simplify the simulation, a qubit system is considered where the number of spins in the chain is limited to two. The following parameters are fixed in the simulation: coupling constant, J = 1 × 10

^{−19}and B = 1 × 10

^{−19}T. The simulation is performed at three temperature values: T

_{1}= 10 K (low), T

_{2}= 1000 K (medium), T

_{4}= 10,000 K (high) and T

_{4}= 100,000 K (very high). The spectrum of the Hamiltonian (i.e., energy states, $\lambda \left(\mathit{\sigma}\right)=H\left(\mathit{\sigma}\right))$, is then obtained for each spin state: $\mathit{\sigma}=\left\{\left(1,1\right),\left(1,-1\right),\left(-1,1\right),\left(1,1\right)\right\}$. The configuration probability for each spin state, $P\left(\mathit{\sigma}\right)$, is determined as follows:

_{B}is the Boltzmann constant:

Algorithm 1: One-dimensional half-spin Heisenberg model |

START- Initialize model parameters:
- spin-1/n: set n = 2
- number of particles: 2
- exchange coupling constant: J = 1 × 10
^{−19} - external magnetic field strength: B = 1 × 10
^{−19}T - Boltzmann constant and simulation temperature
- Randomly generate initial spin states.
- Initialize and construct the Hamiltonian matrix.
- Define Zeeman term to account for an external magnetic field.
- Compute the energy of the system and determine quantum states (eigenvalues).
- Compute partition function and determine density matrix (state probabilities).
- Apply Bures metric to determine the difference in the generated density states.
- Output: Print and plot state probabilities and energy levels.
END |

## 6. Analysis

^{−17}i. The electrodynamics exploration conducted in Section 4 using the Schrödinger–Pauli equation and the proposed spin matrices yielded different Hamiltonian expressions compared to that using Pauli matrices. The computation of the rest energy of a theoretical fermion using the proposed gamma matrices is consistent with the analysis of the Dirac equation performed using conventional gamma matrices. As per this line of reasoning, it is also possible to construct higher spin systems (e.g., for bosons) using the proposed spin matrices. For instance, a set of Hermitian spin-1 matrices for triplet states using the proposed spin matrices at n = 1 is constructed. The spin projection operator, S

_{1}, is expressed as follows:

_{1}are ${\lambda}_{1}=-\hslash \sqrt{2}$, ${\lambda}_{2}=\hslash \sqrt{2}$ and ${\lambda}_{3}=0$. The eigenvectors for S

_{1}are ${v}_{1}=[i,-i-1,1]$, ${v}_{2}=[i,i+1,1]$ and ${v}_{3}=[-i,0,1]$. The eigenspinor representations are then as follows:

_{2}, is expressed as follows:

_{2}are ${\lambda}_{1}=-\hslash $, ${\lambda}_{2}=\hslash $ and ${\lambda}_{3}=0$. The eigenvectors for S

_{1}are ${v}_{1}=[0,0,1]$, ${v}_{2}=[1,0,0]$ and ${v}_{3}=[0,1,0]$. The eigenspinor representations are then as follows:

_{3}, is expressed as follows:

_{3}are ${\lambda}_{1}=-\hslash $, ${\lambda}_{2}=\hslash $ and ${\lambda}_{3}=0$. The eigenvectors for S

_{1}are ${v}_{1}=[1,0,0]$, ${v}_{2}=[0,0,1]$ and ${v}_{3}=[0,1,0]$. The eigenspinor representations are then as follows:

## 7. Conclusions & Future Work

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Vector Properties

**Theorem A1.**

**Proof.**

**Theorem A2.**

**Proof.**

**Theorem A3.**

**a**be an arbitrary vector, $\mathit{a}=\u2329{a}_{1},{a}_{2},{a}_{3}\u232a$. Then, for the 3-vector, ${\mathit{\sigma}}^{\prime}=\u2329{{\sigma}^{\prime}}_{0},{{\sigma}^{\prime}}_{1},{{\sigma}^{\prime}}_{2}\u232a$, defined using Relations III and IV, the dot product $\sqrt{2}\mathit{a}\xb7{\mathit{\sigma}}^{\prime}=$ $\sqrt{2}$ $({a}_{1}{{\sigma}^{\prime}}_{0}+{a}_{2}\overline{{{\sigma}^{\prime}}_{0}}+{a}_{3}{{\sigma}^{\prime}}_{2})$exists.

**Proof.**

**Theorem A4.**

**a**be an arbitrary vector, $\mathit{a}=\u2329{a}_{1},{a}_{2},{a}_{3}\u232a$. Then, for the 3-vector, ${\mathit{\sigma}}^{\prime}=\u2329{{\sigma}^{\prime}}_{0},{{\sigma}^{\prime}}_{1},{{\sigma}^{\prime}}_{2}\u232a$, defined using Relations III and IV, the dot product ${\left(\sqrt{2}\mathit{a}\xb7{\mathit{\sigma}}^{\prime}\right)}^{2}=$ (${a}_{1}^{2}+{a}_{2}^{2}+{a}_{3}^{2}){I}_{2}$ exists.

**Proof.**

**Theorem A5.**

**a**be an arbitrary vector,

**a**, such that $\left|\mathit{a}\right|=1$ and the 3-vector ${\mathit{\sigma}}^{\prime}=\u2329{{\sigma}^{\prime}}_{0},{{\sigma}^{\prime}}_{1},{{\sigma}^{\prime}}_{2}\u232a$. For Relations I-IV, the exponential vector (Euler’s formula analogue) with an arbitrary angle, $\theta $, follows the relation exp$\left(i\theta \sqrt{2}\mathit{a}\xb7{\mathit{\sigma}}^{\prime}\right)={I}_{2}\mathit{cos}\theta +i\left(\sqrt{2}\mathit{a}\xb7{\mathit{\sigma}}^{\prime}\right)\mathit{sin}\theta $.

**Proof.**

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**Figure 1.**State probabilities from simulations using the proposed spin matrices and Pauli matrices at different temperatures.

**Figure 2.**Bures distance for the density states generated using the proposed matrices versus the Pauli matrices.

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**MDPI and ACS Style**

Ganesan, T.
Exotic Particle Dynamics Using Novel Hermitian Spin Matrices. *Axioms* **2023**, *12*, 1052.
https://doi.org/10.3390/axioms12111052

**AMA Style**

Ganesan T.
Exotic Particle Dynamics Using Novel Hermitian Spin Matrices. *Axioms*. 2023; 12(11):1052.
https://doi.org/10.3390/axioms12111052

**Chicago/Turabian Style**

Ganesan, Timothy.
2023. "Exotic Particle Dynamics Using Novel Hermitian Spin Matrices" *Axioms* 12, no. 11: 1052.
https://doi.org/10.3390/axioms12111052