Exponential Formulas, Normal Ordering and the Weyl-Heisenberg Algebra

Abstract

We consider a class of exponentials in the Weyl-Heisenberg algebra with exponents of type at most linear in coordinates and arbitrary functions of momenta. They are expressed in terms of normal ordering where coordinates stand to the left from momenta. Exponents appearing in normal ordered form satisfy differential equations with boundary conditions that could be solved perturbatively order by order. Two propositions are presented for the Weyl-Heisenberg algebra in 2 dimensions and their generalizations in higher dimensions. These results can be applied to arbitrary noncommutative spaces for construction of star products, coproducts of momenta and twist operators. They can also be related to the BCH formula.

References in corpus (18)

• The quantum structure of spacetime at the Planck scale and quantum fields
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Twisted Statistics in kappa-Minkowski Spacetime
• Towards Quantum Noncommutative $\kappa$-deformed Field Theory
• Scalar Field Theory on Non-commutative Snyder Space-Time
• Modification of Heisenberg uncertainty relations in non-commutative Snyder space-time geometry
• Kappa-deformed Snyder spacetime
• Kappa Snyder deformations of Minkowski spacetime, realizations and Hopf algebra
• K-Poincare-Hopf algebra and Hopf algebroid structure of phase space from twist
• $\kappa$-Deformed Phase Space, Hopf Algebroid and Twisting
• Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach
• Remarks on simple interpolation between Jordanian twists
• $\kappa$-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations
• Twisted statistics and the structure of Lie-deformed Minkowski spaces
• Exponential Formulas and Lie Algebra Type Star Products
• Quantum Twist-Deformed D=4 Phase Spaces with Spin Sector and Hopf Algebroid Structures
• Interpolations between Jordanian Twists Induced by Coboundary Twists
• Lie algebraic deformations of Minkowski space with Poincare algebra

Cited by in corpus (3)

• Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications