[Paper Review] Investigation into Compactified Dimensions: Casimir Energies and Phenomenological Aspects

Richard Obousy|ArXiv.org|Jan 23, 2009

Quantum Electrodynamics and Casimir Effect117 references3 citations

TL;DR

This dissertation investigates the role of compactified extra dimensions in stabilizing higher-dimensional theories and generating small vacuum energy densities via the Casimir effect. It derives a novel Casimir energy for a 5D vector-scalar coupled system, proposes a mechanism for moduli stabilization, and explores speculative propulsion using extra dimensions, while also presenting a new $E_8$ embedding in $SU(9)$ gauge theory with implications for heterotic string phenomenology.

ABSTRACT

The primary focus of this dissertation is the study of the Casimir effect and the possibility that this phenomenon may serve as a mechanism to mediate higher dimensional stability, and also as a possible mechanism for creating a small but non-zero vacuum energy density. In chapter one we review the nature of the quantum vacuum and discuss the different contributions to the vacuum energy density arising from different sectors of the standard model. Next, in chapter two, we discuss cosmology and the introduction of the cosmological constant into Einstein's field equations. In chapter three we explore the Casimir effect and study a number of mathematical techniques used to obtain a finite physical result for the Casimir energy. We also review the experiments that have verified the Casimir force. In chapter four we discuss the introduction of extra dimensions into physics. We begin by reviewing Kaluza Klein theory, and then discuss three popular higher dimensional models: bosonic string theory, large extra dimensions and warped extra dimensions. Chapter five is devoted to an original derivation of the Casimir energy we derived for the scenario of a higher dimensional vector field coupled to a scalar field in the fifth dimension. In chapter six we explore a range of vacuum scenarios and discuss research we have performed regarding moduli stability. Chapter seven explores a novel approach to spacecraft propulsion we have proposed based on the idea of manipulating the extra dimensions of string/M theory. Finally, in chapter 8, we discuss some issues in heterotic string phenomenology derived from the free fermionic approach.

Motivation & Objective

• To investigate whether the Casimir effect can stabilize extra dimensions and generate a small, non-zero vacuum energy density.
• To derive a finite Casimir energy for a 5D vector field coupled to a scalar field in a compactified fifth dimension.
• To explore the implications of extra dimensions for vacuum stability and moduli stabilization in higher-dimensional field theories.
• To propose a novel mechanism for spacecraft propulsion based on manipulating extra dimensions in string/M-theory.
• To present a new $E_8$ embedding in $SU(9)$ gauge group and analyze its implications for heterotic string phenomenology.

Proposed method

• Derives the Casimir energy for a 5D vector-scalar coupled system using quantum field theory techniques in a compactified fifth dimension.
• Applies regularization methods to handle divergences in vacuum energy calculations, ensuring finite physical results.
• Uses Kaluza-Klein theory and higher-dimensional models (bosonic strings, large and warped extra dimensions) as theoretical frameworks.
• Performs a novel $E_8$ embedding in $SU(9)$ by transforming the standard $SO(16)$ root system via a Weyl rotation.
• Analyzes modular invariance and charge lattice structures in heterotic string models using the free fermionic approach.
• Considers the role of $\mathbf{128}$ spinor and $\mathbf{248}$ adjoint representations in the $SU(9)$-based $E_8$ construction.

Experimental results

Research questions

• RQ1 Can the Casimir effect in compactified dimensions provide a mechanism for stabilizing moduli fields in higher-dimensional theories?
• RQ2 What is the finite Casimir energy for a 5D vector field coupled to a scalar field in a compactified fifth dimension?
• RQ3 How does an $E_8$ embedding in $SU(9)$ rather than $SO(16)$ affect the structure of heterotic string models and their phenomenological viability?
• RQ4 Can vacuum energy contributions from extra dimensions be reconciled with the observed small cosmological constant?
• RQ5 Is it possible to engineer propulsion by manipulating the geometry of extra dimensions in M-theory?

Key findings

• A finite Casimir energy is derived for a 5D vector-scalar coupled system in a compactified fifth dimension, providing a quantitative model for vacuum energy in higher dimensions.
• The $E_8$ algebra is successfully embedded in $SU(9)$ via a Weyl rotation, yielding $\mathbf{248} = \mathbf{80} + \mathbf{84} + \overline{\mathbf{84}}$, with implications for gauge sector structure.
• Modular invariance in the $SU(9)$-based model requires the gravitino-producing sector to be linked to the $\mathbf{84} + \overline{\mathbf{84}}$ gauge sector, indicating a non-trivial consistency condition.
• The model starts with $\mathcal{N}=4$ supersymmetry, implying that the $SU(9)$ embedding of $E_8$ is only viable in a supersymmetric context.
• The $SO(16)$ and $SU(9)$ embeddings of $E_8$ are related via a Weyl rotation that connects to theta-function product identities, suggesting deeper mathematical structure.
• The paper identifies a $U(1)$ generator in the $SU(9)$ basis with charge $\frac{1}{3}$ per component, consistent with the $\mathbf{84} + \overline{\mathbf{84}}$ representation.