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I show that the kernel of the random phase approximation (RPA) matrix basedon a stable Hartree, Hartree-Fock, Hartree-Bogolyubov orHartree-Fock-Bogolyubov mean field solution is decomposed into a subspace witha basis whose vectors are associated, in the equivalent formalism of aclassical Hamiltonian homogeneous of second degree in canonical coordinates,with conjugate momenta of cyclic coordinates (Nambu-Goldstone modes) and asubspace with a basis whose vectors are associated with pairs of a coordinateand its conjugate momentum neither of which enters the Hamiltonian at all. In asubspace complementary to the one spanned by all these coordinates includingthe conjugate coordinates of the Nambu-Goldstone momenta, the RPA matrixbehaves as in the case of a zerodimensional kernel. This result was derivedvery recently by Nakada as a corollary to a general analysis of RPA matricesbased on both stable and unstable mean field solutions. The present proof doesnot rest on Nakada's general results.

Nambu-Goldstone modes in the random phase approximation