EXACT CANONICALLY CONJUGATE MOMENTA APPROACH TO AN SU(N) QUANTUM SYSTEM

Abstract

The collective field formalism by Jevicki and Sakita is a useful approach to the problem of treating general planar diagrams involved in an SU(N ) symmetric quantum system. To approach this problem, standing on the Tomonaga spirit we also previously developed a collective description of an SU(N ) symmetric Hamiltonian. However, this description has the following difficulties: (i) Collective momenta associated with the time derivatives of collective variables are not exact canonically conjugate to the collective variables; (ii) The collective momenta are not independent of each other. We propose exact canonically conjugate momenta to the collective variables with the aid of the integral equation method developed by Sunakawa et al. A set of exact canonical variables which are derived by the first quantized language is regarded as a natural extension of the Sunakawa et al.'s to the case for the SU(N ) symmetric quantum system. A collective Hamiltonian is represented in terms of the exact canonical variables up to the order of 1 N . * This work extends the approximation given by the previous talk which has been presented in 5535 Int. J. Mod. Phys. A 1998.13:5535-5556. Downloaded from www.worldscientific.com by UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN on 03/11/15. For personal use only.

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

“…The collective field Hamiltonian (6.8) also was given by one of the present author's (S.N.) in his exact canonically conjugate momenta approach to an SU(N) quantum system [19]. Contrary to such collective descriptions, in the beginning we already referred to another way to study of collective motions: Tomonaga first developed a quite different approach to an elementary excitation in a Fermi system [20].…”

“…As shown before, the structures of the commutators among ρ 0,0 k , ρ 1,0 k , π 0,0 k and ρ 1,0 k in (2. 19) have the twisted property in the isospin space (T, T z ). Due to this twisted property, unfortunately, the commutators [Π 1,0 k , Π 1,0 k ′ ] do not vanish.…”

“…One of the present authors (S.N.) gave the exact canonical variables using the discrete integral equation method [19], which is regarded as a natural extension of the Sunakawa's variables to the variables in the SU(N) system. But they are derived in the first quantized language.…”

Exact canonically conjugate momenta to quadrupole-type collective coordinates and derivation of nuclear quadrupole-type collective Hamiltonian

“…A proper treatment of collective variables in nuclear physics [1,2,3,4,5] and of collective field formalisms in QCD [6] and [7] have been attempted in different two ways. In Refs.…”

“…In Ref. [7] one of the present author (S.N.) proposed an exact canonically conjugate momenta approach to a SU(N) quantum system and derived a collective Hamiltonian in terms of the exact canonical variables up to the order of 1 N .…”