Exchange operator formalism for integrable systems of particles.

We formulate one dimensional many body integrable systems in terms of a new set of phase space variables involving exchange operators. The hamiltonian in these variables assumes a decoupled form. This greatly simplifies the derivation of the conserved charges and the proof of their commutativity at the quantum level. In one spatial dimension a class of integrable many-body systems is known, referred to as the Calogero-Sutherland-Moser systems1−3. They constitute of many identical nonrelativistic particles interacting through two-body potentials of the inverse square type and its generalizations, namely the inverse sine square and the Weierstrass two-body potentials. These models are related to root systems of An algebras 4. Corresponding systems related to root systems of other algebras exist, but their two-body potentials are not translationally and/or permutation invariant5. We will restrict ourselves to the An systems. For a comprehensive review of these systems see ref. 5. Many of the above systems admit a matrix formulation5,6. Using this formulation, a generalization of these systems was found recently where the particles also feel external potentials of particular types7. These systems, apart from their purely mathematical interest, are also of significant physical interest, since they are relevant to fractional statistics and anyons8, spin chain models9, soliton wave propagation10 and, indirectly, to nonperturbative two-dimensional quantum gravity11. The purpose of this paper is to present an “exchange operator” formalism for these systems which renders their integrable structure explicit. Specifically, we will write generalized momentum operators in terms of which the integrals of motion assume a “decoupled” form. This will allow for an easy proof of commutativity at the quantum level. Let {xi, pi}, i = 1, . . .N be the coordinates and momenta of N one-dimensional quantum mechanical particles, obeying canonical commutation relations, and let Mij be the particle permutation operators, obeying Mij =Mji =M † ij , M 2 ij = 1 (1) MijAj = AiMij , MijAk = AkMij , for k 6= i, j (2) where Ai is any operator (including Mij themselves) carrying one or more particle indices. Then define the “coupled” momentum operators πi = pi + i ∑ j 6=i VijMij , Vij ≡ V (xi − xj) (3) with V (x) an as yet undetermined function. Note that the πi are “good” one-particle operators, that is they satisfy (2), since the remaining particle indices in (3) appear in a permutation symmetric way. If we impose the hermiticity condition on πi πi = π † i (4) 2 then V (x) must obey V (x)† = −V (−x) (5) Consider now a hamiltonian for the system which takes a free form in terms of πi’s, that is, H = 1 2 ∑ i π2 i (6) In terms of the original phase space variables, H takes the form H = 1 2 ∑