Disentangling q-exponentials: A general approach

Abstract

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson $q$-exponential of the sum of two non-$q$-commuting operators as an (in general) infinite product of $q$-exponential operators involving repeated $q$-commutators of increasing order, $E_q(A+B)=E_{q^{{\alpha }_0}}(A)E_{q^{{\alpha }_1}}(B){\prod }_{i=2}^{\infty }{E}_{q^{{\alpha }_i}}(C_i)$. By systematically transforming the $q$-exponentials into exponentials of series and using the conventional Baker-Campbell-Hausdorff formula, we prove that one can make any choice for the bases $q^{{\alpha }_i}$, $i=0$, 1, 2, ..., of the $q$-exponentials in the infinite product. An explicit calculation of the operators $C_i$ in the successive factors, carried out up to sixth order, also shows that the simplest $q$-Zassenhaus formula is obtained for ${\alpha }_0={\alpha }_1=1$, ${\alpha }_2=2$, and ${\alpha }_3=3$. This confirms and reinforces a result of Sridhar and Jagannathan, based on fourth-order calculations.

LaTeX 2e, 19 pages, no figure

References in corpus (1)

• On the q-analogues of the Zassenhaus formula for dientangling exponential operators

Cited by in corpus (18)

• Efficient computation of the Zassenhaus formula
• More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum
• A note on the Zassenhaus product formula
• Twisted dendriform algebras and the pre-Lie Magnus expansion
• New connection formulae for the q-orthogonal polynomials via a series expansion of the q-exponential
• Rigid rotators and diatomic molecules via Tsallis statistics
• On the deformed oscillator and the deformed derivative associated with the Tsallis q-exponential
• Closed forms of the Zassenhaus formula
• Nonextensive statistics of relativistic ideal gas
• Quantum mechanical evolution operator in the presence of a scalar linear potential: discussion on the evolved state
• Quantum field theory of a hyper-complex scalar field on a commutative ring
• On the structure and convergence of the symmetric Zassenhaus formula
• On multi-variable Zassenhaus formula
• A class of energy based ensembles in Tsallis statistics
• Generalized boson algebra and its entangled bipartite coherent states
• Statistics on Graphs, Exponential Formula and Combinatorial Physics
• New connection formulae for some q-orthogonal polynomials in q-Askey scheme
• Non-local imprints of gravity on quantum theory