New realizations of 𝒩 = 2l-conformal Newton–Hooke superalgebra

Abstract

By applying Niederer-like transformation, we construct a representation of the N = 2 l-conformal Newton-Hooke superalgebra for the case of a negative cosmological constant in terms of linear differential operators as well as its dynamical realization. Another variant of N = 2 supersymmetric Pais-Uhlenbeck oscillator for a particular choice of its frequencies is proposed. The advantages of such realizations as compared to their analogues introduced in [J. Math. Phys. 53 (2012)

The odd-order Pais–Uhlenbeck oscillator

“…where ǫ ij is the Levi-Civitá symbol with ǫ 12 = 1. Symmetry structure intrinsic to the model (1) allows one to construct an N = 2 supersymmetric generalization of this system with the aid of Niederer-like coordinate transformations [15,35]. The action functional associated with this extension reads…”

“…The configuration space of this model involves the real bosonic coordinates x i , the fermionic coordinates ψ i ,ψ i , which are complex conjugates of each otherψ i = ψ * i , and real extra bosonic coordinates z i . The model (2) is invariant under the supersymmetry transformations of the form [15]…”

“…As the next step, we need to generalize the supersymmetry transformations (3) to the case of arbitrary distinct oscillation frequencies. To this end, let us note that the transformations (3) are also available for an N = 2 supersymmetric even-order PU oscillator which exhibits conformal invariance [15]. Therefore, it is natural to expect that both the model (5) and its even-order analogue [13] are invariant with respect to the supersymmetry transformations…”

“…For a particular choice of oscillation frequencies, an N = 2 supersymmetric extension of the odd-order PU oscillator has been derived in Ref. [15]. It has been shown that this extension accommodates conformal symmetry provided frequencies of oscillation form a certain arithmetic sequence.…”

An alternative Hamiltonian formulation for the Pais–Uhlenbeck oscillator

“…Recently, in [20], [21] an N = 2 supersymmetric extension of the PU oscillator has been constructed. It has been shown that the invariance of the model under time translations implies unbounded from below spectrum.…”