Fock space dualities

TL;DR

This paper reformulates Howe's duality theorem within second quantization, focusing on orthogonal-orthogonal duality, and introduces Young diagrams to analyze irreducible representations, revealing near-perfect symmetry between dual partners.

Contribution

It extends Howe's duality theorem to a second quantization framework and provides a detailed analysis of orthogonal-orthogonal duality using character calculations and Young diagrams.

Findings

Orthogonal-orthogonal duality is characterized in detail.

Young diagrams effectively describe irreducible representations.

Reflection properties support symmetry between dual partners.

Abstract

A general theorem due to Howe of dual action of a classical group and a certain non-associative algebra on a space of symmetric or alternating tensors is reformulated in a setting of second quantization, and familiar examples in atomic and nuclear physics are discussed. The special case of orthogonal-orthogonal duality is treated in detail. It is shown that, like it was done by Helmers more than half a century ago in the analogous case of symplectic-symplectic duality, one can base a proof of the orthogonal-orthogonal duality theorem and a precise characterization of the relation between the equivalence classes of the dually related irreducible representations on a calculation of characters by combining it with an analysis of the representation of a reflection. Young diagrams for the description of equivalence classes of irreducible representations of orthogonal Lie algebras are introduced. The properties of a reflection of the number non-conserving part in the dual relationship between orthogonal Lie algebras corroborate a picture of an almost perfect symmetry between the partners.