Conformal bridge transformation, 𝒫𝒯- and super- symmetry (original) (raw)

Conformal bridge transformation, mathcalPT\mathcal{PT}mathcalPT- and supersymmetry

Journal of High Energy Physics

Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by i and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a mathcalPT\mathcal{PT}mathcalPT PT -invariant supersymmetric model with N subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic N = 2 supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the mathcalPT\mathcal{PT}mathcalPT PT -invariant supersymmetric systems that transform into the...

Supersymmetries of the spin-1/2

2010

The quantum nonrelativistic spin-1/2 planar systems in the presence of a perpendicular magnetic field are known to possess the N = 2 supersymmetry. We consider such a system in the field of a magnetic vortex, and find that there are just two self-adjoint extensions of the Hamiltonian that are compatible with the standard N = 2 supersymmetry. We show that only in these two cases one of the subsystems coincides with the original spinless Aharonov-Bohm model and comes accompanied by the super-partner Hamiltonian which allows a singular behavior of the wave functions. We find a family of additional, nonlocal integrals of motion and treat them together with local supercharges in the unifying framework of the trisupersymmetry. The inclusion of the dynamical conformal symmetries leads to an infinitely generated superalgebra, that contains several representations of the superconformal osp(2|2) symmetry. We present the application of the results in the framework of the two-body model of identical anyons. The nontrivial contact interaction and the emerging N = 2 linear and nonlinear supersymmetries of the anyons are discussed.

Extended supersymmetric quantum mechanics

Physics Letters B, 1999

A parametrization of the Hamiltonian of the generalized Witten model of the SUSY QM by a single arbitrary function in d = 1 has been obtained for an arbitrary number of the supersymmetries N = 2N. Possible applications of this formalism have been discussed. It has been shown that the N = 1 and 2 conformal SUSY QM is generalized for any N .

Novel symmetries in supersymmetric quantum mechanical models

Annals of Physics, 2013

We demonstrate the existence of a novel set of discrete symmetries in the context of N = 2 supersymmetric (SUSY) quantum mechanical model with a potential function f (x) that is a generalization of the potential of the 1D SUSY harmonic oscillator. We perform the same exercise for the motion of a charged particle in the X − Y plane under the influence of a magnetic field in the Z-direction. We derive the underlying algebra of the existing continuous symmetry transformations (and corresponding conserved charges) and establish its relevance to the algebraic structures of the de Rham cohomological operators of differential geometry. We show that the discrete symmetry transformations of our present general theories correspond to the Hodge duality operation. Ultimately, we conjecture that any arbitrary N = 2 SUSY quantum mechanical system can be shown to be a tractable model for the Hodge theory.

Superconformal mechanics and nonlinear supersymmetry

Journal of High Energy Physics, 2003

We show that a simple change of the classical boson-fermion coupling constant, 2alphato2alphan2\alpha \to 2\alpha n 2alphato2alphan, ninNn\in \NninN, in the superconformal mechanics model gives rise to a radical change of a symmetry: the modified classical and quantum systems are characterized by the nonlinear superconformal symmetry. It is generated by the four bosonic integrals which form the so(1,2) x u(1) subalgebra, and by the 2(n+1) fermionic integrals constituting the two spin-n/2 so(1,2)-representations and anticommuting for the order n polynomials of the even generators. We find that the modified quantum system with an integer value of the parameter alpha\alphaalpha is described simultaneously by the two nonlinear superconformal symmetries of the orders relatively shifted in odd number. For the original quantum model with ∣alpha∣=p|\alpha|=p∣alpha∣=p, pinNp\in \NpinN, this means the presence of the order 2p nonlinear superconformal symmetry in addition to the osp(2|2) supersymmetry.

Extension of Conformal (Super)Symmetry using Heisenberg and Parabose operators

2007

In this paper we investigate a particular possibility to extend C(1,3) conformal symmetry using Heisenberg operators, and a related possibility to extend conformal supersymmetry using parabose operators. The symmetry proposed is of a simple mathematical form, as is the form of necessary symmetry breaking that reduces it to the conformal (super)symmetry. It turns out that this extension of conformal superalgebra can be obtained from standard non-extended conformal superalgebra by allowing anticommutators Qeta,Qxi\{Q_\eta, Q_\xi\}Qeta,Qxi and barQdoteta,barQdotxi\{\bar Q_{\dot \eta}, \bar Q_{\dot \xi}\}barQdoteta,barQdotxi to be nonzero operators and then by closing the algebra. In regard of the famous Coleman and Mandula theorem (and related Haag-Lopuszanski-Sohnius theorem), the higher symmetries that we consider do not satisfy the requirement for finite number of particles with masses below any given constant. However, we argue that in the context of theories with broken symmetries, this constraint may be unnecessarily strong.

Pseudo supersymmetric partners for the generalized Swanson model

Journal of Physics A: Mathematical and Theoretical, 2008

New non Hermitian Hamiltonians are generated, as isospectral partners of the generalized Swanson model, viz., H − = A † A + αA 2 + βA † 2 , where α , β are real constants, with α = β, and A † and A are generalized creation and annihilation operators. It is shown that the initial Hamiltonian H − , and its partner H + , are related by pseudo supersymmetry, and they share all the eigen energies except for the ground state. This pseudo supersymmetric extension enlarges the class of non Hermitian Hamiltonians H ± , related to their respective Hermitian counterparts h ± , through the same similarity transformation operator ρ : H ± = ρ −1 h ± ρ. The formalism is applied to the entire class of shape-invariant models.

Bosons, fermions and anyons in the plane, and supersymmetry

Annals of Physics, 2010

Universal vector wave equations allowing for a unified description of anyons, and also of usual bosons and fermions in the plane are proposed. The existence of two essentially different types of anyons, based on unitary and also on non-unitary infinite-dimensional half-bounded representations of the (2+1)D Lorentz algebra is revealed. Those associated with non-unitary representations interpolate between bosons and fermions. The extended formulation of the theory includes the previously known Jackiw-Nair (JN) and Majorana-Dirac (MD) descriptions of anyons as particular cases, and allows us to compose bosons and fermions from entangled anyons. The theory admits a simple supersymmetric generalization, in which the JN and MD systems are unified in N = 1 and N = 2 supermultiplets. Two different non-relativistic limits of the theory are investigated. The usual one generalizes Lévy-Leblond's spin 1/2 theory to arbitrary spin, as well as to anyons. The second, "Jackiw-Nair" limit (that corresponds to Inönü-Wigner contraction with both anyon spin and light velocity going to infinity), is generalized to boson/fermion fields and interpolating anyons. The resulting exotic Galilei symmetry is studied in both the non-supersymmetric and supersymmetric cases.