Commutation properties of anticommuting self-adjoint operators,spin representation and Dirac operators

LetA andB be anticommuting self-adjoint operators in a Hilbert space . It is proven thatiAB is essentially self-adjoint on a suitable domain and its closureC(A, B) anticommutes withA andB. LetU_s be the partial isometry associated with the self-adjoint operatorsS, i.e., the partial isometry defined by the polar decompositionS=U_S|S|. LetP_S be the orthogonal projection onto (KerS). Then the following are proven: (i) The operatorsU_A,U_B,U_C(A,B),P_A,P_B, andP_AP_B multiplied by some constants satisfy a set of commutation relations, which may be regarded as an extension of that satisfied by the standard basis of the Lie algebra of the special unitary groupSU(2); (ii) There exists a Lie algebra associated with those operators; (iii) If is separable andA andB are injective, then gives a completely reducible representation of with each irreducible component being the spin representation of the Clifford algebra associated with ³; This result can be extended to the case whereA andB are not necessarily injective. Moreover, some properties ofA+B are discussed. The abstract results are applied to Dirac operators.