Algebraic Varieties in Second Quantization

We develop algebraic geometry for coupled cluster theory in second quantization. In quantum chemistry, electronic systems are represented by elements in the exterior algebra. The creation and annihilation operators of particles generate a Clifford algebra known as the Fermi-Dirac algebra. We present a non-commutative Gr\"obner basis giving an alternative proof of Wick's theorem, a foundational result in quantum chemistry. In coupled cluster theory, the Schr\"odinger equation is approximated through a hierarchy of polynomial equations at various levels of truncation. The exponential parameterization gives rise to the Fock space truncation varieties. This reveals well-known varieties, such as the Grassmannian, flag varieties and spinor varieties. We offer a detailed study of the truncation varieties and their CC degrees. We classify all cases for when the CC degree is equal to the degree of a graph of the exponential parametrization.