Quantized representation for Kadomtsev-Petviashvili equation on the soliton sector

TL;DR

This paper develops a quantum operator framework for the KP equation's soliton solutions, linking classical solitons to quantum states in a Fock space, offering a novel perspective on integrable systems.

Contribution

It introduces a quantized operator representation of the KP soliton solutions over a Fock space, bridging classical solutions with quantum operators.

Findings

Classical solitons are expectation values of the quantum operator.

The operator is constructed using particle-number operators.

The approach applies to both bosonic and fermionic particles.

Abstract

Exploiting the known structure of soliton solutions, obtained through the Hirota transformation, a quantized representation of the Kadomtsev-Petviashvili (KP) equation on the soliton sector is constructed over a Fock space of particles, which may be either bosons or fermions. The classical solution is mapped into an operator, which also obeys the KP equation. The operator is constructed in terms of the particle-number operators. Classical soliton solutions are the expectation values of this operator in multi-particle states in the Fock space. The operator is equation-specific and the state in the Fock space is in one-to-one correspondence with the particular soliton solution.