Dyck language and fermionic second quantization: I. Theory

Abstract

This paper proposes a novel framework connecting fermionic second quantization and Dyck languages. By defining translations of creation and annihilation operators using bracket alphabets, the study establishes nullity criteria for expectation values of chains of second quantization operators. Those translations are designed to reveal sufficient conditions for the nullity of the expectation values relatively to one-determinant states or to the physical vacuum. The nullity criteria are purely syntactic, and simply reduce to the inspection of sequences of opening and closing brackets. Moreover, numbers and transformations in Dyck languages can be imported in the context of fermionic second quantization. One of these numbers, the depth, originally absent from second quantization, can be used to introduce more nullity criteria.