Quantum Group Schr\"{o}dinger Field Theory

TL;DR

This paper develops a quantum group Schrödinger field theory in one spatial dimension, establishing its formalism and covariance properties, and generalizes the framework to multiple fields, linking quantum deformation to nonlinear lattice structures.

Contribution

It introduces the formalism of quantum group Schrödinger field theory and demonstrates $SU_q(2)$ covariance for $q$-bosonic and $q$-fermionic fields, extending to multiple fields.

Findings

Quantum deformation corresponds to quantum mechanics on a nonlinear lattice.

Explicit $SU_q(2)$ covariant algebra structures for $q$-fields.

Generalization to multiple quantum fields.

Abstract

We show that a quantum deformation of quantum mechanics given in a previous work is equivalent to quantum mechanics on a nonlinear lattice with step size $\Delta x=~(1-q)x$. Then, based on this, we develop the basic formalism of quantum group Schr\"{o}dinger field theory in one spatial quantum dimension, and explicitly exhibit the $SU_{q}(2)$ covariant algebras satisfied by the $q$-bosonic and $q$-fermionic Schr\"{o}dinger fields. We generalize this result to an arbitrary number of fields.