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\begin{abstract}
    The finite-volume QCD spectrum in the $I=\frac{1}{2}$, $S=-1$, parity-even, zero-momentum sector containing the $\kappa$ meson and the $I=1$, $S=0$, parity-even, $G$-parity-odd, zero-momentum sector containing the $a_0(980)$ meson is studied with the inclusion of tetraquark operators using lattice QCD. Several hundred tetraquark operators of different flavor, color, and spatial structures are included in this study. We find that the inclusion of tetraquark operators is crucial for determining the spectrum in each sector, and that there is a state in each sector with sizeable tetraquark component that is missed without the use of tetraquark operators. The spectrum of excited $\Sigma$ baryons in the $I=1$, $S=-1$, parity-even and parity-odd sectors is also studied using large bases of single- and two-hadron operators, for the first time in lattice QCD. We find qualitative agreement with experiment, and near-agreement with a previous study that neglects two-hadron operators. All calculations are performed using 412 gauge field configurations with clover-improved Wilson fermions on a $32^3\times 256$ anisotropic lattice with $m_\pi \approx 230$ MeV, and quark propagation is treated using the Stochastic LapH method. 
\end{abstract}

\chapter*{Acknowledgements}
There is no self-made man, and I am no exception. This work would not have been possible without the love and support of my dear wife, Dana Martin. To all of my family and friends who have supported me, especially my parents Janet and Nasser Darvish, I owe everything to you and you have my deepest gratitude. To my academic mentors and colleagues, especially Ruair\'i Brett, Andrew Hanlon, Jacob Fallica, and Colin Morningstar, you have been invaluable resources and have greatly contributed to my success in research and my understanding of physics. It is difficult to imagine finishing this degree without all of the enlightening conversation, insightful perspective, and education you have provided me.
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    \textit{``There are decades where nothing happens, and\\
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    -- Vladamir Lenin (apocryphal)
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\chapter{Introduction}\label{ch:introduction}
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\input{chapters/introduction}

\chapter{Lattice QCD}\label{ch:latticeqcd}
\input{chapters/latticeQCD}

\chapter{Building Operators for Finite-Volume Spectroscopy}\label{ch:operators}
\input{chapters/operators}

\chapter{Calculation of Correlators using the Stochastic LapH Method}\label{ch:montecarlo}
\input{chapters/slaph}

\chapter{Correlator Analysis: Determining the Finite-Volume Spectrum}\label{ch:analysis}
\input{chapters/analysis}

\chapter{Investigating the Tetraquark Content of the Light Scalar Mesons $\kappa$ and $a_0(980)$}\label{ch:tetraquarks}
\input{chapters/tetraquarks}

\chapter{$\Sigma$ Baryon Spectroscopy}\label{ch:sigmas}
\input{chapters/sigmas}

\chapter{Conclusions}\label{ch:conclusions}
\input{chapters/conclusions}

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