Duality relations in the auxiliary field method

TL;DR

This paper explores duality relations in the auxiliary field method, showing how approximate eigenenergies of N-particle systems relate to those of p-particle systems across various potentials and kinematics, with some relations holding approximately for exact energies.

Contribution

It establishes numerous exact duality relations linking eigenenergy approximations for different particle numbers and potentials within the auxiliary field framework.

Findings

Duality relations connect eigenenergies across different particle numbers and potentials.

Approximate duality relations often hold approximately for exact eigenenergies.

The auxiliary field method provides accurate eigenenergy approximations that respect these dualities.

Abstract

The eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$ of a system of $N$ identical particles with a mass $m$ are functions of the various radial quantum numbers $n_i$ and orbital quantum numbers $l_i$. Approximations $E^{(N)}(m;Q)$ of these eigenenergies, depending on a principal quantum number $Q(\{n_i,l_i\})$, can be obtained in the framework of the auxiliary field method. We demonstrate the existence of numerous exact duality relations linking quantities $E^{(N)}(m;Q)$ and $E^{(p)}(m';Q')$ for various forms of the potentials (independent of $m$ and $N$) and for both nonrelativistic and semirelativistic kinematics. As the approximations computed with the auxiliary field method can be very close to the exact results, we show with several examples that these duality relations still hold, with sometimes a good accuracy, for the exact eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$.