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Journal of Physics A Mathematical and General

Anticommutator analogues of certain identities involving repeated commutators

Istok P. Mendaš, P Milutinovic

Published Feb 21, 1990• 5 citations
Publisher

Abstract

The generalization of the Baker-Hausdorff lemma and its anticommutator analogue is formulated. Additionally, the anticommutator analogues of several well known operator identities involving repeated commutators are derived. It is pointed out that these are more convenient for application whenever the operators in question satisfy simpler repeated anticommutator relations (and, in particular, when they anticommute) than the repeated commutator relations. The diagonalization of two spin-1 Hamiltonians, in which the anticommutator analogue of the Baker-Hausdorff lemma is used to good advantage, is presented.

DOI
10.1088/0305-4470/23/4/021

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The generalization of the Baker-Hausdorff lemma and its anticommutator analogue is formulated. Additionally, the anticommutator analogues of several well known operator identities involving repeated commutators are derived. It is pointed out that these are more convenient for application whenever the operators in question satisfy simpler repeated anticommutator relations (and, in particular, when they anticommute) than the repeated commutator relations. The diagonalization of two spin-1 Hamiltonians, in which the anticommutator analogue of the Baker-Hausdorff lemma is used to good advantage, is presented.
Two operator identities involving a q-commutator, [A,B]?AB+qBA, where A and B are two arbitrary (generally noncommuting) linear operators acting on the same linear space and q is a variable that commutes with these two operators, are formulated, proved and discussed. The first identity is a direct generalization of the Baker?Hausdorff lemma, whereas the second involves the time derivative of the exponential function of a time-dependant operator. It is indicated how these two identities can be used to good advantage to treat the Foldy?Wouthuysen transformation of the Dirac Hamiltonian for a particle in an external electromagnetic field.
A nonrelativistic proof of the spin-statistics theorem is given in terms of the field operators satisfying commutation and anticommutation relations, which are introduced here in the coordinate space as a means to build the permutation symmetry into the brackets of identical particles. An eigenvalue problem of a πππ-rotation for a product of two annihilation operators is combined with an analysis on its rotational property to prove the connection that the field operators for integral-spin and half-integral-spin particles obey the commutation and anticommutation relations, respectively.
A formulation of supersymmetric quantum mechanics is given and superunitary and generalized canonical transformations are defined acting in a module. Next it is assumed that there are operators that give an irreducible representation of the canonical commutation and anticommutation relations, respectively, and it is proved that two such representations are connected by a uniquely determined superunitary transformation, under suitable domain assumptions. This extends the well-known uniqueness theorem of von Neumann to canonical (anti−) commutation relations using anticommuting parameters.