Exact canonically conjugate momenta approach to a one-dimensional neutron-proton system, I

Introducing collective variables, a collective description of nuclear surface oscillations has been developed with the first quantized language, contrary to the second quantized one in Sunakawa's approach for a Bose system. It overcomes difficulties remaining in the traditional theories of nuclear collective motions: Collective momenta are not exact canonically conjugate to collective coordinates and are not independent. On the contrary to such a description, Tomonaga first gave the basic idea to approach elementary excitations in a one-dimensional Fermi system. The Sunakawa's approach for a Fermi system is also expected to work well for such a problem. In this paper, on the $isospin$ space, we define a density operator and further following Tomonaga, introduce a collective momentum. We propose an $exact$ canonically momenta approach to a one-dimensional neutron-proton (N-P) system under the use of the Grassmann variables.