Algebraic Varieties in Second Quantization

TL;DR

This paper introduces an algebraic geometric framework for Fock space coupled cluster theory, connecting quantum chemistry, algebraic varieties, and Clifford algebra to analyze electron states and their polynomial approximations.

Contribution

It develops a novel algebraic geometric approach to coupled cluster theory, describing Fock space truncation varieties and classifying cases where their degrees match exponential parameterizations.

Findings

Explicit descriptions of Fock space truncation varieties.

Classification of cases where coupled cluster degree equals exponential map degree.

Connection between algebraic varieties and quantum state approximations.

Abstract

We develop an algebraic geometric framework for Fock space coupled cluster theory in second quantization. In quantum chemistry, many-electron states are represented as elements of the exterior algebra. The fermionic creation and annihilation operators generate the Fermi-Dirac algebra, which can be realized as a Clifford algebra acting on the exterior algebra. We present a non-commutative Gr\"obner basis for the Fermi-Dirac algebra; offering an alternative proof of Wick's theorem, a fundamental result in quantum field theory. In coupled cluster theory, eigenpairs of the Schr\"odinger equation are approximated by a hierarchy of polynomial equations corresponding to different levels of truncation. The coupled cluster exponential parameterization of quantum states gives rise to 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, providing an explicit description of their defining equations and dimension. Furthermore, we classify all cases in which their coupled cluster degree coincides with the degree of the graph of the exponential parameterization - most notably for singleton truncations such as CCD and for the Schubert like truncation varieties such as the Grassmannian, flag variety and the spinor variety.