From the Spring to the Spacetime: The Universality of the Harmonic Oscillator in Quantum Field Theory and Fermionic Particle Creation

Undergraduate Thesis (Draft). University of Ghana, Department of Physics & Mathematics.

Abstract: The Simple Harmonic Oscillator (SHO) is often introduced as a rudimentary model in classical mechanics, yet its mathematical structure underpins the deepest descriptions of the physical universe. This thesis bridges the gap between undergraduate mechanics and advanced quantum field theory in curved spacetime, demonstrating that the tools used to solve a mass on a spring are sufficient to understand the creation of matter from the vacuum.

We begin by establishing the SHO as the universal local approximation for any stable equilibrium, consistent with classical mechanics. By extending the Lagrangian formalism to continuous systems, we derive the Klein-Gordon and Dirac equations as natural generalizations of coupled oscillators. We then apply these tools to General Relativity, utilizing the tetrad (vierbein) formalism to describe fermions in curved spacetime—a necessity imposed by the spinor transformation properties under local Lorentz rotations.

The central result of this work is the derivation of particle creation in gravitational fields, specifically the production of fermions arising from the breakdown of a unique vacuum state. Using Bogoliubov transformations, we show that the 'vacuum' of one observer is populated with particles for another, providing a unified mathematical framework for understanding phenomena ranging from phonons in solids to Hawking radiation in black holes.

Supervisor: Dr. Eugene Yaw Adade Adjei

Status: Pre-critique Undergraduate Draft (Spring 2026).