Make the dissertation page using this template. The template includes sections for the title page, certificate/declaration, abstract, and sample chapters. You can customize the content and formatting as needed for your dissertation.

Instructions

• Use Times New Roman font for the entire document.
• Use 1.5 spacing.

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Symmetry in Physics

A Dissertation Submitted in Partial Fulfillment
of the Requirements for the Degree of
Master of Science in Physics

By
Rajesh Kumar

Under the Supervision of
Dr. A. Sharma

University Department of Physics
Sido Kanhu Murmu University
Dumka, Jharkhand, India

2026

CERTIFICATE / DECLARATION

Certificate

This is to certify that the dissertation titled "Symmetry in Physics: Mathematical Structures and Quantum Realizations" submitted by Rajesh Kumar is a bona fide record of work carried out under my supervision.

Supervisor Signature: ____________
Name: Dr. A. Sharma
Date: ____________

Declaration

I hereby declare that this dissertation is my original work.

Student Signature: ____________
Name: Rajesh Kumar
Date: ____________

ACKNOWLEDGMENT

I would like to express my sincere gratitude to my supervisor and faculty for their guidance and support.

ABSTRACT

This dissertation explores symmetry in physics, focusing on mathematical structures and quantum applications. It highlights Lie algebra, quantum operators, and supersymmetry.

TABLE OF CONTENTS

1. Introduction
2. Mathematical Foundations
3. Symmetry in Quantum Mechanics
4. Supersymmetry
5. Applications
6. Conclusion
7. References
8. Appendix

Chapter 1: Introduction

1.1 Background

Symmetry governs conservation laws.

1.2 Motivation

Simplifies physical systems.

1.3 Objectives

• Study symmetry
• Analyze QM
• Explore SUSY

Chapter 2: Mathematical Foundations

Translation Generator

$$ \hat{P} = -i\hbar \frac{d}{dx} $$

Rotation Generator

$$ \hat{L} = \mathbf{r} \times \mathbf{p} $$

Chapter 3: Symmetry in Quantum Mechanics

Operators and commutation relations: $$ [\hat{X}, \hat{P}] = i\hbar $$

Chapter 4: Supersymmetry

$$ H = \{Q, Q^\dagger\} $$ $$ \{Q, Q^\dagger\} = H $$

Chapter 5: Applications

Symmetry leads to conservation laws and simplifies physical systems.

Conclusion

Symmetry unifies physical laws.

References

1. Sakurai – Modern Quantum Mechanics
2. Georgi – Lie Algebras in Particle Physics

Appendix

$$ E = mc^2 $$