On Mathematical Structure of Effective Observables

TL;DR

This paper introduces a mathematical framework for decomposing Hilbert spaces and constructing effective observables that preserve symmetries, enabling maximum information extraction about quantum systems.

Contribution

It presents a novel method for decomposing Hilbert spaces and defining effective observables that retain symmetry properties and facilitate system analysis.

Findings

Effective observables can determine eigenvalues and matrix elements within a model space.

Symmetries of the Hamiltonian are preserved in effective representatives.

Complete sets of effective representatives enable maximum information retrieval.

Abstract

We decompose the Hilbert space of wave functions into two subspaces, and assign to a given observable two effective representatives that act in the model space. The first serves to determine some of the eigenvalues of the full observable, while the second serves to determine its matrix elements, in any basis in one of the subspaces, in terms of quantities pertaining to the model space. We also show that if the Hamiltonian of a physical system possesses symmetries then these symmetries continue to hold for its effective representatives of the first type. Maximum information about the system can be obtained in terms of two sets of effective representatives. The first set of representatives is complete. Other observables that do not commute with all members of the complete set have only one type of representative.