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Journal of Mathematical Physics

A Schwinger disentangling theorem

Daniel J. Cross, Robert Gilmore

Published Oct 1, 2010• 4 citations
Publisher

Abstract

Baker–Campbell–Hausdorff formulas are exceedingly useful for disentangling operators so that they may be more easily evaluated on particular states. We present such a disentangling theorem for general bilinear and linear combinations of multiple boson creation and annihilation operators. This work generalizes a classical result of Schwinger.

DOI
10.1063/1.3501027

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Baker–Campbell–Hausdorff formulas are exceedingly useful for disentangling operators so that they may be more easily evaluated on particular states. We present such a disentangling theorem for general bilinear and linear combinations of multiple boson creation and annihilation operators. This work generalizes a classical result of Schwinger.
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Eigenstates of the linear combinations a2+betaadagger2a^2+\\beta a^{\\dagger2}a2+betaadagger2 and $ab+\\beta a^\\dagger b^\\dagger$ of two boson creation and annihilation operators are presented. The algebraic procedure given here is based on the work of Shanta et al. [Phys. Rev. Lett. {\\bf 72}, 1447, 1994] for constructing eigenstates of generalized annihilation operators.
The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.