Generalization of the Lie-Trotter Product Formula for q-Exponential Operators

TL;DR

This paper extends the Lie-Trotter product formula to q-exponential operators, providing a new mathematical tool for nonextensive quantum systems and related fields.

Contribution

It generalizes the Lie-Trotter formula for q-exponentials and proves its validity, enabling applications in nonextensive quantum mechanics.

Findings

Derived the generalized product formula for q-exponentials.

Proved the limit relation for the q-exponential operators.

Potential applications in nonextensive quantum systems.

Abstract

The Lie-Trotter formula $e^{\hat{A}+\hat{B}} = \lim_{N\to \infty} (e^{\hat{A}/N} e^{\hat{B}/N})^N$ is of great utility in a variety of quantum problems ranging from the theory of path integrals and Monte Carlo methods in theoretical chemistry, to many-body and thermostatistical calculations. We generalize it for the q-exponential function $e_q (x) = [1+ (1-q) x]^{(1/(1-q))}$ (with $e_1(x)=e^x$), and prove $e_q(\hat{A}+\hat{B}+(1-q) [\hat{A}\hat{B}+\hat{B}\hat{A}] /2) = \lim_{N\to \infty} {[e_{1-(1-q)N}(\hat{A}/N)] [e_{1-(1-q)N}(\hat{B}/N)]}^N$. This extended formula is expected to be similarly useful in the nonextensive situations