Algebraic varieties in second quantization

We develop algebraic geometry for coupled cluster theory in second quantization. In quantum chemistry, electronic systems are represented by binary tensors. The creation and annihilation operators of particles generate a Clifford algebra known as the Fermi-Dirac algebra. We present a non-commutative Gröbner basis giving an alternative proof of Wick's theorem, a foundational result in quantum chemistry. In coupled cluster theory, the Schrödinger equation is approximated through a hierarchy of polynomial equations at various levels of truncation. The exponential parameterization gives rise to 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.