Disentangling q-Exponentials: A General Approach
نویسنده
چکیده
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, Eq(A + B) = Eqα0 (A)Eqα1 (B) ∏ ∞ i=2 Eqαi (Ci). 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 qαi , i = 0, 1, 2, . . . , of the q-exponentials in the infinite product. An explicit calculation of the operators Ci in the successive factors, carried out up to sixth order, also shows that the simplest q-Zassenhaus formula is obtained for α0 = α1 = 1, α2 = 2, and α3 = 3. This confirms and reinforces a result of Sridhar and Jagannathan, based on fourth-order calculations. Running head: Disentangling q-Exponentials Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium; e-mail: [email protected] 1
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