Complex $q$-Analysis and Scalar Field Theory on a $q$-Lattice

TL;DR

This paper develops complex q-analysis to solve q-difference equations, connecting to classical Laplace equations, and applies it to quantum mechanics and scalar field theories on q-lattices.

Contribution

It introduces a formalism for complex q-analysis, extending classical analysis to q-deformed settings and applying it to quantum and field theoretical models.

Findings

Solutions to q-difference equations mimic classical Laplace solutions as q→1.

Constructed an inner product space for q-functions.

Applied formalism to Schrödinger and scalar field equations.

Abstract

We develop the basic formalism of complex $q$-analysis to study the solutions of second order $q$-difference equations which reduce, in the $q\rightarrow 1$ limit, to the ordinary Laplace equation in Euclidean and Minkowski space. After defining an inner product on the function space we construct and study the properties of the solutions, and then apply this formalism to the Schr\"{o}dinger equation and two-dimensional scalar field theory.