Symmetry Algebra of Dynamical and Discrete Calogero Models
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HYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
Symmetry Algebra of Dynamical and Discrete Calogero Models Tigran Hakobyana, *, ** and Seda Vardanyana, *** a
Yerevan State University, Yerevan, 0025 Armenia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected]
Received November 15, 2019; revised January 15, 2020; accepted February 28, 2020
Abstract—The symmetries of the generalized Calogero model and its discrete version are related to the wellknown unitary symmetries of the isotropic oscillator. The correspondence is provided by the Dunkl-operator deformation of the standard quantum momentum. DOI: 10.1134/S1547477120050179
1. CALOGERO MODEL WITH PARTICLE EXCHANGES
The advantage from the included exchanges is the oscillator-type description of the system,
In this letter we briefly describe the symmetry of N -particle quantum Calogero model with the particle exchange operators (sometimes called a generalized Calogero model) and the related discrete system, which may be reduced to the Polychronakos–Frahm spin chain. Using the Dunkl-operator approach, both systems may be considered as certain deformations of the N -dimensional isotropic oscillator. Following this analogy, we present their symmetries, respectively, as deformations of the usual U (N ) generators. The latter are responsible for the hidden symmetries and maximal superintegrability of the isotropic quantum oscillator.
2 (2) Hˆ = 1 π ˆ i2 + xi , 2 i =1 with the momentum operator deformed by means of the exchange operator [2, 3], g (3) M ij . ˆ i = pˆi + π j ≠ i xi − x j
The article is based mainly on our recent work [1], to which we refer for more details and references. We have supplemented it with the description of the symmetry of the two-dimensional Calogero model based on the dihedral group. Consider a quantum mechanical system of bound one-dimensional particles interacting by an inversesquare potential [2, 3],
Hˆ = 1 2
N
i =1
2 ˆ p i
+
xi2
+
i< j
g( g − M ij ) . ( xi − x j )2
N
It is based on the Dunkl operator, ∇i = πi [6], and results in a deformed Heisenberg algebra, (4) [π ˆ i, π ˆ j] = 0, [ xi , π ˆ j] = Sˆij, (5) Sˆij = (δij − 1)gM ij + δij ( + g M ik ).
k ≠i
A Dunkl-operator analog of lowering-rising operators is defined in the standard way [2, 3, 7]. They commute mutually as in the usual oscillator case while the mixed commutator is deformed as
ˆi xi π (6) , [aˆi , aˆ+j ] = Sˆij . 2 The generalized Hamiltonian can be expressed in terms of them, aˆi± =
Hˆ = (1)
The potential is not local but contains the nonlocal operator M ij , which exchanges the coordinates of ith and jth particles. Clearly, for identical particles, M ij = ±1, depending on whether are bosons or fermions, and the above Hamiltonian is reduced to the well known Calogero model [4] (for the reviews, see [5]).
aˆ aˆ + 2N − S, + i i
(7)
i
where we have used the permutation-group invariant element
S = −g
M . i< j
ij
(8)
As a result, such operators obey
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