Extension of Wick's theorem for many-particle matrix elements

Wick's theorem for the expectation value of an operator between many-partic le states is generalized to the case of a homogeneous linear transformation operator a, which leaves invariant the 2N-dimensional linear space compose d of the annihilation and the creation operators for the N fermion or boson single-particle basis. Such a transformation generally belongs to the grou p SO(2N,C) for the case of fermion, or the group Sp(2N,C) for the case of b oson. This theorem is applied to derive a reduction theorem for the matrix element Of the type [alpha\alpha 1 alpha 2...alpha m (O) over cap beta(m+1) dagger beta(m+2)dagger...beta 2n dagger\beta], whose many-quasiparticle sta tes and vacua appearing in both sides are represented in different quasipar ticle pictures as indicated by alpha and beta. The reduction formula is use ful in calculating any nuclear transition matrix element for the quantum-nu mber projection calculation associated with the constrained Hartree-Fock-Bo goliubov solutions, and also the generator coordinate method with quantum n umber projection. [S0556-2813(99)01405-3].