A correspondence between the free and interacting field theories

TL;DR

This paper establishes a novel correspondence between free and interacting quantum field theories using Hermite function bases, enabling alternative quantization and dimensional reduction, exemplified in Yang-Mills theory leading to the BFSS matrix model.

Contribution

It introduces a new duality framework connecting free and interacting theories via Hermite functions, offering a novel quantization and reduction method.

Findings

Hermite basis transforms free propagator into massive state tower

Duality links plane wave basis to Hermite basis in field theories

Reduction of 3+1D Yang-Mills to BFSS matrix model using Hermite basis

Abstract

We discover a correspondence between the free field and the interacting states. This correspondence is firstly given from the fact that the free propagator can be converted into a tower of propagators for massive states, when expanded with the Hermite function basis. The equivalence of propagators reveals that in this particular case the duality can naturally be regarded as the equivalence of one theory on the plane wave basis to the other on the Hermite function basis. More generally, the Hermite function basis provides an alternative quantization process with the creation/annihilation operators that correspond directly to the interacting fields. Moreover, the Hermite function basis defines an exact way of dimensional reduction. As an illustration, we apply this basis on 3+1 dimensional Yang-Mills theory with three dimensional space being reduced through the Hermite function basis, and if with only the lowest order Hermite function, the equivalent action becomes the Banks-Fischler-Shenker-Susskind (BFSS) matrix model.