Generalization of the Lie–Trotter product formula for q-exponential operators

Abstract

The Lie–Trotter formula e A ̂ + B ̂ = lim N→∞ e A ̂ /N e 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( A ̂ + B ̂ +(1−q)[ A ̂ B ̂ + B ̂ A ̂ ]/2)= lim N→∞ e 1−(1−q)N A ̂ /N e 1−(1−q)N B ̂ /N N . This extended formula is expected to be similarly useful in the nonextensive situations.