Quantum SU (2|1) supersymmetric C N Smorodinsky--Winternitz system
aa r X i v : . [ h e p - t h ] O c t Quantum SU (2 | supersymmetric C N Smorodinsky–Winternitz system
Evgeny Ivanov a ) , Armen Nersessian b ) , a ) , c ) , Stepan Sidorov a ) a ) Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia b ) Yerevan Physics Institute, 2 Alikhanian Brothers St., 0036, Yerevan, Armenia c ) Institute of Radiophysics and Electronics, Alikhanian 1, Ashtarak, 0203, Armenia [email protected], [email protected], [email protected]
Abstract
We study quantum properties of SU(2 |
1) supersymmetric (deformed N = 4, d = 1 supersymmetric) ex-tension of the superintegrable Smorodinsky–Winternitz system on a complex Euclidian space C N . Thefull set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU(2 | |
1) multiplets of the wave functions is given. An equivalent description of the samesystem in terms of superconformal SU(2 | ,
1) quantum mechanics is considered and a new representationof the hidden symmetry generators in terms of the SU(2 | ,
1) ones is found.
PACS: 11.15.-q, 03.50.-z, 03.50.DeKeywords: supersymmetric quantum mechanics, deformation, superconformal symmetry
Introduction
The models of supersymmetric mechanics were initially introduced as toy models for supersym-metric field theories [1]. Respectively, their study was mainly limited to the models exhibitingone-dimensional Poincar´e supersymmetry defined by the relations { Q A , Q B } = 2 δ AB H, [ H, Q A ] = 0 , A = 1 , . . . N , (1.1)where Q A are N real supercharges and H is the Hamiltonian.Some decade ago, there started a wide activity related to the study of field-theoretical modelswith the “rigid supersymmetry on curved superspaces” (see, e.g., [2]). These studies motivatedtwo of us (E. I. and S. S.) to investigate the one-dimensional ( i.e. mechanical) analogs of thesetheories [3], [4]. The main subjects of our interest were systems in which d = 1, N = 4 Poincar´esupersymmetry is deformed by a mass-dimension parameter m into su (2 |
1) supersymmetry, withthe following non-vanishing (anti)commutators : (cid:8) Q i , ¯ Q j (cid:9) = 2 m I ij + 2 δ ij H , (cid:2) I ij , I kl (cid:3) = δ kj I il − δ il I kj , (cid:2) I ij , ¯ Q l (cid:3) = 12 δ ij ¯ Q l − δ il ¯ Q j , (cid:2) I ij , Q k (cid:3) = δ kj Q i − δ ij Q k , (cid:2) H , ¯ Q l (cid:3) = m Q l , (cid:2) H , Q k (cid:3) = − m Q k . (1.2)Here, the generator H is the Hamiltonian (coinciding with the U(1) generator), while the remainingbosonic generators I ij ( i = 1 ,
2) define SU(2) R-symmetry.As one of the results of this study, it was observed that the so-called “weak N = 4 super-symmetric K¨ahler oscillator” models (which are the particle models living on K¨ahler spaces andinteracting with the specific potential field and a constant magnetic field) suggested earlier in [5, 6]supply nice examples of SU(2 |
1) supersymmetric mechanics , and they can be reproduced from theSU(2 |
1) superfield approach worked out in [4]. This observation further entailed the constructionof a few novel SU(2 |
1) supersymmetric superintegrable oscillator-like models specified by the inter-action with a constant magnetic field. They include superextensions of isotropic oscillators on C N and CP N [5], [8], as well as of C N Smorodinsky–Winternitz system in the presence of a constantmagnetic field [9] and CP N Rosochatius system [10].In a recent paper [11] we noticed that switching on an interaction with a constant magneticfield in the ordinary N = 4 supersymmetric mechanics on K¨ahler manifold breaks either N = 4supersymmetry or isometries of the initial bosonic system, including their hidden symmetries (cf.[12]). We demonstrated that this drawback can be overcome by performing, instead of the d = 1Poincar´e supersymmetrization, its deformed variant, i.e. SU(2 |
1) supersymmetrization. Examiningthe SU(2 |
1) supersymmetric models listed above, we observed that the SU(2 |
1) supersymmetrizationpreserves all kinematical symmetries of the initial systems, and, in a number of cases, also the hiddenones (the explicit expressions for the “super-counterparts” of the hidden symmetry generators wereconstructed so far for C N oscillator and C N Smorodinsky–Winternitz system in the presence of aconstant magnetic field). It should be pointed out that the hidden symmetries play an importantrole in the quantum domain: e.g., in the standard supersymmetric quantum mechanics they amountto an additional degeneracy of the (higher) energy levels as compared to N -Poincar´e supersymmetrywhich enhances it by the factor 2 N / .Keeping in mind these reasonings, it is desirable to better understand, on the concrete examples,how SU(2 |
1) supersymmetry affects the energy spectrum of the systems considered. The basicpeculiar feature of the supersymmetric K¨ahler oscillator models is that their Hamiltonian is in factidentified with the U(1)-generator in (1.2). As was shown in [4], an extra U(1) R-charge (“fermionicnumber”) can be introduced in addition to the Hamiltonian in this class of models only in the limit These su (2 |
1) superalgebra relations differ from those in [11] by rescaling Q → √ Q . We mostly follow theconventions of [4]. Another version of this kind of supersymmetric mechanics was studied in [7], with SU(2 |
1) termed as “weaksupersymmetry”. = 0. One can expect that the impossibility to separate the Hamiltonian from the fermionicnumber in such models at m = 0 could have an essential impact on the structure of the relevantenergy spectra.In the present paper we construct SU(2 |
1) supersymmetric extension of the C N Smorodinsky–Winternitz (in what follows, S.-W.) quantum system in the presence of a constant magnetic fieldas an instructive and simple example of the general K¨ahler oscillator models. In particular, wefocus on the interplay of SU(2 |
1) supersymmetry and hidden symmetry in forming the energyspectrum of this system. An analogous analysis of its purely bosonic sector was performed in [9].An interesting unique feature of the SU(2 | C N S.-W. model as compared to other models ofSU(2 |
1) supersymmetric K¨ahler oscillators is its implicit superconformal SU(2 | ,
1) symmetry. Atthe classical level this follows from the results of ref. [13], while here we extend the correspondencewith a complex SU(2 | ,
1) superconformal mechanics to the quantum domain.As a preamble, let us remind that the C N S.-W. system considered in [9] amounts to a sumof N two-dimensional isotropic oscillators with a ring-shaped potential interacting with a constantmagnetic field (orthogonal to the plane). It is defined by the Hamiltonian [9] H = N X a =1 H a , H a = ¯ π a π a + g a z a ¯ z a + ω z a ¯ z a , ω := | ω | , (1.3)with (cid:2) π a , z b (cid:3) = − iδ ba , [ π a , ¯ π b ] = δ a ¯ b B, (1.4)where B is a constant strength of the magnetic field. Besides the standard Liouville integralsof motion, the model possesses the additional ones generated by the so-called Uhlenbeck tensor,and thus provides an example of superintegrable system. The C N S.-W. system interacting with aconstant magnetic field belongs to the class of K¨ahler oscillators with the following K¨ahler potential[11] K = N X a =1 (cid:18) z a ¯ z a + ig a ω log z a − ig a ω log ¯ z a (cid:19) . (1.5)It admits SU(2 |
1) supersymmetric extension given by the superalgebra (1.2), with the deformationparameter m = p ω + B . (1.6)This extension, at the classical level, was described in [11]. It is a particular case of general SU(2 | , , ) [4].It is convenient to summarize here at once the basic results of the present paper. • The energy spectrum of the deformed SU(2 |
1) supersymmetric system constructed revealsan interesting feature: SU(2 |
1) supersymmetry gives rise to the separation of bosonic andfermionic states in the spectrum. Bosonic states are associated with the even levels, while allfermionic states with the odd ones. So the energy spectrum exhibits “even-odd” feature: theadjacent bosonic and fermionic states carry different energies shifted by half-integer numbers.The intrinsic reason for such a splitting is that the Hamiltonian in (1.2) hides in itself thefermionic number operator F and so does not commute with the supercharges. The groundstate belongs to a non-singlet representation of SU(2 | • The system exhibits superconformal SU(2 | ,
1) symmetry with the central charge given bythe sum of the generators of kinematical U(1) symmetries. The SU(2 |
1) supersymmetric This feature is shared by SU(2 | C N isotropic oscillator model, the Hamiltonian of which is a particular case ofSU(2 | C N S.-W. Hamiltonian corresponding to the choice g a = 0 in (1.3) and (1.5) below. Originally, S.-W. system was formulated on the Euclidean R N space [14]. H can be split into a sum of superconformal Hamiltonian and a central chargegenerator accounting for the constant external magnetic field. The whole set of wave functionsis closed under the action of the superconformal algebra su (2 | , • Due to an additional hidden symmetry generated by the proper superanalog of Uhlenbecktensor commuting with the Hamiltonian, the spectrum of quantum SU(2 | C N S.-W. systemreveals an extra degeneracy. The superextended Uhlenbeck tensor is shown to admit a nicerepresentation in terms of bilinears of the SU(2 | ,
1) generators. • Furthermore, a generalization of the super Uhlenbeck tensor was found, such that it commuteswith the full set of the generators of the properly rotated SU(2 |
1) supergroup. It is alsoexpressed through SU(2 | ,
1) generators and gives rise to an additional degeneracy of theeigenvalues of the SU(2 |
1) Casimirs.The paper is organized as follows. In Section 2 we formulate SU(2 |
1) supersymmetric extensionof the superintegrable C S.-W. quantum system and determine its energy spectrum and wavefunctions. We also analyze the space of quantum states of this supersymmetric system from thestandpoint of the SU(2 |
1) representation theory. In Section 3 we show that the system consideredadmits an equivalent description in terms of some quantum superconformal SU(2 | ,
1) mechanics.In Section 4 we define the SU(2 |
1) supersymmetric C N S.-W. quantum system as a sum of N copiesof the C -systems. We reveal its hidden symmetry given by the supersymmetric counterpart of theUhlenbeck tensor and show that it is responsible for an additional degeneracy of the spectrum of theSU(2 |
1) Casimir operators. A new representation for the superextended Uhlenbeck tensor in termsof the superconformal SU(2 | ,
1) generators is found. The summary and outlook are the contentsof Section 5. In Appendices A and B we present some details related to the non-linear algebraof hidden symmetries. In Appendix C, it is briefly discussed how conformal SL(2,R) symmetrymanifests itself in the spectrum of quantum bosonic C N S.-W. system. C Smorodinsky–Winternitz system
We proceed from the SU(2 |
1) supersymmetric S.-W. system on the complex Euclidian space C .The quantum Hamiltonian constructed according to the generic prescription of [4, 11] for the specific N = 1 K¨ahler potential (1.5), is defined by the expression, H = ¯ ππ + g z ¯ z + ω z ¯ z + ig (cid:18) ξ k ξ k z − ¯ ξ k ¯ ξ k z (cid:19) + B ξ k ¯ ξ k , (2.1)with the (anti)commutators [ π, z ] = − i, [¯ π, ¯ z ] = − i, [ π, ¯ π ] = B, (cid:8) ξ i , ¯ ξ j (cid:9) = ξ i ¯ ξ j + ¯ ξ j ξ i = δ ij . (2.2)The operators of bosonic momenta are represented in the standard form π = − i (cid:18) ∂ z + B z (cid:19) , ¯ π = − i (cid:18) ∂ ¯ z − B z (cid:19) , (2.3)while the fermionic operators ξ i , ¯ ξ i can be represented as a 4 × z , ¯ z , ξ i (with ξ i , i = 1 , , being a doublet of Grassmann variables), and represent ¯ ξ j as a differentialoperator, ¯ ξ j = ∂/∂ξ j . (2.4) For reader’s convenience we quote here the dimensions of various quantities (in the length units):[ z ] = cm / , [ ξ k ] = [ g ] = cm , [ B ] = [ m ] = [ ω ] = [ H ] = cm − . ψ ( z, ¯ z ), ξ k ξ k ψ ′ ( z, ¯ z ), and twofermionic ones, Ψ i = ξ i ψ ′′ ( z, ¯ z ).In what follows, it will be convenient to equivalently replace the parameters B and ω by λ and m , where m is a mass-dimension contraction parameter defined in (1.6), and λ is a dimensionlessangle-type parameter defined by the relations B = m cos 2 λ , ω = m sin 2 λ , m = p ω + B > , λ ∈ [0 , π/ . (2.5)In the new notation, the Hamiltonian (2.1) is rewritten as H = − ∂ ¯ z ∂ z + m cos 2 λ L + m z ¯ z g z ¯ z + ig (cid:18) ξ k ξ k z − ¯ ξ k ¯ ξ k z (cid:19) , (2.6)where L is the angular momentum (or U(1)) operator L = z∂ z − ¯ z∂ ¯ z + ξ k ¯ ξ k − . (2.7)The Hamiltonian (2.6) is manifestly invariant under U(1)-transformation z → e iκ z , ξ i → e iκ ξ i generated by this operator. The rest of SU(2 |
1) generators is expressed as Q i = −√ i h cos λ ξ i (cid:16) ∂ z + m z (cid:17) − sin λ ¯ ξ i (cid:16) ∂ ¯ z + m z (cid:17)i + g √ (cid:18) sin λ ξ i z + cos λ ¯ ξ i ¯ z (cid:19) , ¯ Q j = −√ i h cos λ ¯ ξ j (cid:16) ∂ ¯ z − m z (cid:17) + sin λ ξ j (cid:16) ∂ z − m z (cid:17)i + g √ (cid:18) sin λ ¯ ξ j ¯ z − cos λ ξ j z (cid:19) ,I ij = ξ i ¯ ξ j − δ ij ξ k ¯ ξ k . (2.8)It is straightforward to check that they satisfy the su (2 |
1) superalgebra relations (1.2). The operator L commutes with all su (2 |
1) generators, [ L, H ] = (cid:2) L, Q i (cid:3) = (cid:2) L, I ij (cid:3) = 0, i.e. it can be interpretedas a central charge . The SU(2) generators I ij commute with the Hamiltonian (2.6), implying thatthis SU(2) is realized on the quantum states as an exact symmetry. We define super wave functions as a ξ i -expansion of the wave functions depending on ( z, ¯ z, ξ i ), with¯ ξ j being an annihilation operator. This expansion amounts to the fermionic wave function Ψ i ∼ ξ i ψ ′′ ( z, ¯ z ) and two bosonic wave functions ψ ( z, ¯ z ), ξ k ξ k ψ ′ ( z, ¯ z ). As distinct from the fermionicfunction, the bosonic ones are not eigenstates of the Hamiltonian (2.6). The correct eigenstates arerepresented as their proper combinations:Ω ∼ ψ ( z, ¯ z ) + ξ k ξ k ψ ′ ( z, ¯ z ) (2.9)(see eq. (2.24) below).The simplest way to solve the eigenvalue problem is to firstly consider the action of (2.6) on thefermionic wave functions Ψ i ∼ ξ i ψ ′′ ( z, ¯ z ). On the bosonic factor ψ ′′ ( z, ¯ z ) the Hamiltonian (2.6)acts exactly as the bosonic one [cf. (1.3)], and this action reads H = − ∂ ¯ z ∂ z + m cos 2 λ z∂ z − ¯ z∂ ¯ z ) + m z ¯ z g z ¯ z . (2.10)After solving the eigenvalue problem as in the pure bosonic case [9], the fermionic states are writtenas Ψ i ( n,l ) = ξ i ( n − z ¯ z ) ˜ l ( z/ ¯ z ) l e − mz ¯ z L (˜ l ) n − ( mz ¯ z ) , ˜ l = p l + g , (2.11)4here L (˜ l ) n − are generalized Laguerre polynomials, i.e. n = 1 , , . . . is a positive integer . Theinteger number l is an eigenvalue of the angular-momentum operator L , L Ψ i ( n,l ) = l Ψ i ( n,l ) , l = 0 , ± , ± . . . . (2.12)The energy spectrum of the fermionic states is directly calculated to be H Ψ i ( n,l ) = E ( n,l ) Ψ i ( n,l ) , E ( n,l ) = m (cid:20) n + 12 (cid:16) ˜ l + l cos 2 λ − (cid:17)(cid:21) . (2.13)The double degeneracy of the fermionic levels is due to the unbroken su (2) ⊂ su (2 |
1) with respectto which the wave function (2.11) transforms as a doublet.The bosonic wave functions can be now obtained by action of the supercharges on the fermionicwave function Ψ i : Q i Ψ j ( n,l ) = ε ij √ m Ω − ( n,l ) , ¯ Q j Ψ i ( n,l ) = δ ij √ m Ω +( n,l ) . (2.14)They are explicitly expressed asΩ − ( n,l ) = ( n − √ m ( z ¯ z ) ˜ l ( z/ ¯ z ) l e − mz ¯ z (cid:20) z (cid:16) g cos λ − il sin λ + i ˜ l sin λ (cid:17) L (˜ l ) n − ( mz ¯ z )+ 12 z ξ k ξ k (cid:16) g sin λ − il cos λ − i ˜ l cos λ (cid:17) L (˜ l ) n − ( mz ¯ z )+ im (cid:0) z sin λ − ¯ z ξ k ξ k cos λ (cid:1) h L (˜ l ) n − ( mz ¯ z ) − L (˜ l +1) n − ( mz ¯ z ) i (cid:21) , Ω +( n,l ) = ( n − √ m ( z ¯ z ) ˜ l ( z/ ¯ z ) l e − mz ¯ z (cid:20) z (cid:16) g sin λ + il cos λ − i ˜ l cos λ (cid:17) L (˜ l ) n − ( mz ¯ z ) − z ξ k ξ k (cid:16) g cos λ + il sin λ + i ˜ l sin λ (cid:17) L (˜ l ) n − ( mz ¯ z )+ im (cid:0) z cos λ + ¯ z ξ k ξ k sin λ (cid:1) L (˜ l +1) n − ( mz ¯ z ) (cid:21) , L Ω ± ( n,l ) = l Ω ± ( n,l ) . (2.15)On the other hand, the action of supercharges on the bosonic wave functions yields the originalfermionic wave functions Ψ i : Q i Ω +( n,l ) = (cid:20) n + 12 (cid:16) ˜ l + l cos 2 λ (cid:17)(cid:21) √ m Ψ i ( n,l ) , ¯ Q j Ω +( n,l ) = 0 , ¯ Q j Ω − ( n,l ) = − (cid:20) n − (cid:16) ˜ l + l cos 2 λ (cid:17)(cid:21) √ m Ψ j ( n,l ) , Q i Ω − ( n,l ) = 0 . (2.16)Using these expressions, we derive the energy spectrum for the bosonic states Ω ± ( n,l ) and find thatit differs from that for the fermionic states, H Ω ± ( n,l ) = (cid:16) E ( n,l ) ± m (cid:17) Ω ± ( n,l ) . (2.17)Thus SU(2 |
1) supersymmetry creates additional energy levels, with the bosonic and fermionic statesbeing separated.Note that the discrete energy spectrum bounded from below in the model under consideration isensured by the oscillator term ∼ m in (2.10). So the limit m = 0 cannot be taken in the quantumcase (see also Section 3). One could work with the Laguerre polynomials L (˜ l ) n ′ ( mz ¯ z ) where n ′ = 0 , , . . . , but for further convenience wedeal with n = n ′ + 1, such that n = 1 , , . . . (see the next Subsection). .2 SU (2 | representations The SU(2 |
1) representations are specified by the eigenvalues of Casimir operators [16] C = H m − I ij I ji Q k ¯ Q k − ¯ Q k Q k m , (2.18) C = (cid:18) C + 12 (cid:19) H m + 18 m (cid:0) δ ij H − m I ij (cid:1) (cid:0) Q j ¯ Q i − ¯ Q i Q j (cid:1) . (2.19)On the quantum states Ψ j ( n,l ) , Ω ± ( n,l ) these eigenvalues are: C = (cid:20) n + 12 (cid:16) ˜ l + l cos 2 λ (cid:17)(cid:21) (cid:20) n + 12 (cid:16) ˜ l + l cos 2 λ (cid:17) − (cid:21) ,C = (cid:20) n + 12 (cid:16) ˜ l + l cos 2 λ − (cid:17)(cid:21) C . (2.20)The quantum number n uniquely defines SU(2 |
1) representation for a fixed l . Casimir operatorstake non-zero eigenvalues (2.20) on all states. It means that all SU(2 |
1) representations are typicalwith the simplest four-fold degeneracy . Respectively, the SU(2 |
1) multiplets are encompassed bythe sets n Ψ j ( n,l ) , Ω − ( n,l ) , Ω +( n,l ) o and constitute infinite-dimensional “towers” characterized by all values of the angular-momentum L , l = 0 , ± , ± . . . for a fixed n .The minimal energy for a fixed l is always positive and it corresponds to the bosonic state Ω − (1 ,l ) :˜ l = p l + g > l ⇒ E min = m (cid:16) ˜ l + l cos 2 λ (cid:17) > . (2.21)Therefore, the ground states for each l belongs to a non-trivial four-fold SU(2 |
1) multiplet. Thismeans that SU(2 |
1) supersymmetry is spontaneously broken at any l .The quantum states within the given SU(2 |
1) multiplet occupy different energy levels accordingto (2.13) and (2.17). Since the supercharges do not commute with the Hamiltonian (2.6), theyare capable to decrease and increase its eigenvalues. One can designate levels occupied by bosonicstates as even levels (including the zeroth level for the ground state), then the fermionic statesoccupy odd levels. As distinct from the standard d = 1 Poincar´e supersymmetry, there is no anydegeneracy between these two types of the levels. The relevant picture is drawn on Figure 1.One can see that the bosonic states Ω +( n,l ) and Ω − ( n +1 ,l ) have the same energy though they belongto different SU(2 |
1) representations, E ( n,l ) + m E ( n +1 ,l ) − m m (cid:20) n + 12 (cid:16) ˜ l + l cos 2 λ (cid:17)(cid:21) , n = 1 , , . . . . (2.22)For example, the energy level E (1 ,l ) + m/ E (2 ,l ) − m/ ξ i .The wave functions Ω +( n,l ) and Ω − ( n +1 ,l ) for n = 1 , , . . . can be represented asΩ − ( n +1 ,l ) = (cid:16) ˜ l + n (cid:17) g ˜ l h(cid:16) ˜ l + l (cid:17) cos λ + ig sin λ i Ω n,l ) + n g ˜ l h(cid:16) ˜ l − l (cid:17) cos λ − i sin λ i Ω n,l ) , Ω +( n,l ) = (cid:16) ˜ l + n (cid:17) g ˜ l n h(cid:16) ˜ l + l (cid:17) sin λ − ig cos λ i Ω n,l ) + 12 g ˜ l h(cid:16) ˜ l − l (cid:17) sin λ + i cos λ i Ω n,l ) , (2.23) This degeneracy is with respect to the Casimir operators, not to the Hamiltonian. ③③③ × ×× ×× × ③③ E (1 ,l ) − m/ E (1 ,l ) E (1 ,l ) + m/ E (2 ,l ) E (2 ,l ) + m/ E (3 ,l ) H Ψ i ( n,l ) Ω +( n,l ) Ω − ( n,l ) PPPPP✐ Pq ¯ Qj Qi
PPPPP✐ Pq ¯ Qj Qi
PPPPP✐ Pq ¯ Qj Qi
PPPPPPqP✐ Qi ¯ Qj PPPPPPqP✐ Qi ¯ Qj PPPPPPqP✐ Qi ¯ Qj Figure 1: Degeneracy of energy levels with a fixed number l . Bosonic (circles) and fermionic(crosses) states of the 4-fold SU(2 |
1) multiplet, characterized by the number n , fill the energy levels E ( n,l ) ± m/ E ( n,l ) .where Ω n,l ) = n !2 √ m ( z ¯ z ) ˜ l ( z/ ¯ z ) l e − mz ¯ z (cid:20) z (cid:16) ˜ l − l (cid:17) − ig z ξ k ξ k (cid:21) L (˜ l − n ( mz ¯ z ) , Ω n,l ) = ( n − √ m z ¯ z ) ˜ l ( z/ ¯ z ) l e − mz ¯ z h (cid:16) ˜ l + l (cid:17) z + ig ¯ z ξ k ξ k i L (˜ l +1) n − ( mz ¯ z ) . (2.24)These functions have just the structure (2.9) and are eigenfunctions of the Hamiltonian (2.6) withthe eigenvalues (2.22), but they are not eigenfunctions of Casimirs (2.18) and (2.19). The onlyexception is the wave function Ω n,l ) which can be naturally continued to n = 0 asΩ ,l ) = 12 √ m ( z ¯ z ) ˜ l ( z/ ¯ z ) l e − mz ¯ z (cid:20) z (cid:16) ˜ l − l (cid:17) − ig z ξ k ξ k (cid:21) . (2.25)It is directly related to the lowest bosonic state Ω − (1 ,l ) 8 :Ω − (1 ,l ) = (cid:16) ˜ l + l (cid:17) cos λg + i sin λ Ω ,l ) . (2.26)In the next section we will show that the state Ω ,l ) can be interpreted as a singlet ground statewith respect to some new SU(2 |
1) supercharges ˜ Q , ¯˜ Q .Let us summarize the basic peculiarities of the energy spectrum.The system has two fermionic and two bosonic sets of iso-spectral states, with the energy levelsgiven by (2.13) and (2.17), respectively, so that each energy level is doubly degenerated (excepting Note that the apparent singularities in g and l in eqs. (2.23) and (2.26) are fake: no such singularities are presentin the original expressions (2.15) for Ω ± ( n,l ) . B , ω the energy spectrum reads E σ ( n,l ) = p ω + B n − σ + p l + g ! + Bl , (2.27)with n = 1 , , . . . , l = 0 , ± , ± . . . , σ = 0 , / , , where σ = 0 and σ = 1 correspond to thebosonic states Ω +( n,l ) and Ω − ( n,l ) , respectively, and σ = 1 / In this section, following [13], we relate the generic supersymmetric C S.-W. system to the super-conformal mechanics with the Hamiltonian H conf = − ∂ ¯ z ∂ z + m z ¯ z g z ¯ z + ig (cid:18) ξ k ξ k z − ¯ ξ k ¯ ξ k z (cid:19) . (3.1)We will show that the Hamiltonians of these two systems differ by a central charge generator, and,as a consequence, the Hamiltonian of conformal model inherits all the symmetries of the originalHamiltonian (and vice-versa).As was proved in [13], the superconformal Hamiltonian of deformed supersymmetric mechanicsshould be an even function of the deformation mass parameter m : m → − m , H conf → H conf . Inaccord with this proposition we define the superconformal Hamiltonian (3.1) as H conf = H + m Z , with Z = − cos 2 λ L. (3.2)Such a change of the Hamiltonian amounts to the effective elimination of the magnetic field .One can equally choose the basis in which the conformal Hamiltonian is not deformed by theoscillator term, H conf = H conf − m z ¯ z − ∂ ¯ z ∂ z + g z ¯ z + ig (cid:18) ξ k ξ k z − ¯ ξ k ¯ ξ k z (cid:19) . (3.3)Then we introduce the dilatation generator D and the generator of conformal boosts K : D := i z∂ z + ¯ z∂ ¯ z + 1) , K := z ¯ z. (3.4)These generators, together with the Hamiltonian (3.3), close on the conformal algebra so (2 , ∼ sl (2 , R ) [17]: [ H conf , D ] = iH conf , [ K, D ] = − iK, [ H conf , K ] = 2 iD. (3.5)The trigonometric type of (super)conformal mechanics involving the parameter m is defined by thefollowing linear combinations [13]: H conf = H conf + m K, T = H conf − m K − imD, ¯ T = H conf − m K + imD. (3.6)The algebra (3.5) is then rewritten as (cid:2) ¯ T , T (cid:3) = 2 m H conf , (cid:2) H conf , ¯ T (cid:3) = − m ¯ T , [ H conf , T ] = m T. (3.7) With B = 0, i.e. λ = π/
4, the deformation parameter coincides with the frequency, and the central charge Z vanishes: λ = π ⇒ B = 0 , m = 2 ω, Z = 0 . Therefore, at this special choice of parameters the C S.-W. Hamiltonian (2.10) just coincides with the superconformalHamiltonian (3.1), H = H conf . H according to (3.2), but in this case the con-formal algebra will be deformed by a central charge. So, irrespective of whether we deal withsuperconformal symmetry or its bosonic limit, it is appropriate to use the Hamiltonian H conf con-taining no magnetic field. Note that the first relation in (3.6) implies that just H conf with m = 0,as opposed to H conf , is the correct quantum Hamiltonian with the spectrum bounded from below,in accordance with the assertion in the pioneering paper [17].The conformal algebra (3.7) can be extended to the superconformal algebra in the followingway. Applying the discrete transformation m → − m to the supercharges Q i defined by (2.8)we obtain the new fermionic generators which can be identified with the conformal supercharges, S i := Q i ( − m ): S i = −√ i h cos λ ξ i (cid:16) ∂ z − m z (cid:17) − sin λ ¯ ξ i (cid:16) ∂ ¯ z − m z (cid:17)i + g √ (cid:18) sin λ ξ i z + cos λ ¯ ξ i ¯ z (cid:19) , ¯ S j = −√ i h cos λ ¯ ξ j (cid:16) ∂ ¯ z + m z (cid:17) + sin λ ξ j (cid:16) ∂ z + m z (cid:17)i + g √ (cid:18) sin λ ¯ ξ j ¯ z − cos λ ξ j z (cid:19) . (3.8)These new generators extend the superalgebra su (2 |
1) to the centrally extended superalgebra su (2 | ,
1) : (cid:8) Q i , ¯ Q j (cid:9) = 2 m I ij − m δ ij Z + 2 δ ij H conf , (cid:8) S i , ¯ S j (cid:9) = − m I ij + m δ ij Z + 2 δ ij H conf , (cid:8) S i , ¯ Q j (cid:9) = 2 δ ij T, (cid:8) Q i , ¯ S j (cid:9) = 2 δ ij ¯ T , (cid:8) Q i , S j (cid:9) = m ε ij Z +2 , (cid:8) ¯ Q i , ¯ S j (cid:9) = − m ε ij Z − , (cid:2) I ij , I kl (cid:3) = δ kj I il − δ il I kj , (cid:2) ¯ T , T (cid:3) = 2 m H conf , (cid:2) H conf , ¯ T (cid:3) = − m ¯ T , [ H conf , T ] = m T, (cid:2) I ij , ¯ Q l (cid:3) = 12 δ ij ¯ Q l − δ il ¯ Q j , (cid:2) I ij , Q k (cid:3) = δ kj Q i − δ ij Q k , (cid:2) I ij , ¯ S l (cid:3) = 12 δ ij ¯ S l − δ il ¯ S j , (cid:2) I ij , S k (cid:3) = δ kj S i − δ ij S k , (cid:2) H conf , ¯ S l (cid:3) = − m S l , (cid:2) H conf , S k (cid:3) = m S k , (cid:2) H conf , ¯ Q l (cid:3) = m Q l , (cid:2) H conf , Q k (cid:3) = − m Q k , (cid:2) T, Q i (cid:3) = − m S i , (cid:2) T, ¯ S j (cid:3) = − m ¯ Q j , (cid:2) ¯ T , ¯ Q j (cid:3) = m ¯ S j , (cid:2) ¯ T , S i (cid:3) = m Q i . (3.9)The superconformal algebra contains three central charges: Z = − cos 2 λ L, Z ± = sin 2 λ L ± ig. (3.10)Note that the quadratic operator constructed out of the central charges is reduced on the states tothe square of the quantum number ˜ l = p l + g defined in (2.11),( Z ) + Z +2 Z − = L + g := ˜ L . (3.11)When g = 0 , this operator coincides with L . Below we will show that the set of three centralcharges in (3.9) can be reduced to a single central charge ˜ Z = ˜ L . This agrees with the factthat the superalgebra su (2 | ,
1) contains 15 generators: eight supercharges, three su (2) generators,three generators of so (2 ,
1) and one central charge, in accord with the decomposition su (2 | ,
1) = psu (2 | , ⊕ ˜ Z , where psu (2 | ,
1) is a centerless superalgebra.Before proceeding further, we point out that the SU(2 | ,
1) trigonometric superconformal modelof the multiplet ( , , ) with a superpotential term resulting in the Hamiltonian (3.1) was con-structed within a manifestly SU(2 |
1) covariant ( N = 4 deformed) superfield approach in [13]. Inthe N = 2 , d = 1 superfield formalism, the model amounts to a system of the coupled ( , , )and ( , , ) multiplets, with only N = 2 superconformal symmetry SU(1 | , ⊂ SU(2 | ,
1) beingmanifest. In such a formulation, the inverse-square terms with g = 0 in (3.1), (2.1) come outsolely from the coupling of these two multiplets and disappear after decoupling of the fermionic9ultiplet ( , , ). So in this limit (still respecting SU(1 | ,
1) invariance) our model is reduced tothe two-dimensional N = 2 superconformal oscillator model based on the chiral multiplet ( , , ).In ref. [18] (see also [19]) there was considered a different SU(1 | ,
1) superconformal model of themultiplet ( , , ), with the Hamiltonian involving an inverse-square potential induced by some spincoupling. It cannot be obtained as any truncation of our model. Let us consider the quadratic Casimir operator of D (2 , α ) [20]: C ′ ( α )2 = 2 H − T ¯ T − ¯ T T m + αI ij I ji Q k ¯ Q k − ¯ Q k Q k m − S k ¯ S k − ¯ S k S k m − ( α + 1)2 (cid:0) F + C ¯ C (cid:1) . (3.12)The definition of the superalgebra D (2 , α ) in terms of these generators was given in [13], with m = − αµ . The limit α = − D (2 , α = − → psu (2 | , ⊕ su (2) ,where psu (2 | ,
1) is a centerless superalgebra and su (2) is an external automorphism generated bythe generators F , C and ¯ C . Hence, the Casimir operator of psu (2 | ,
1) reads C ′ ( α = − = 2 H − T ¯ T − ¯ T T m − I ij I ji Q k ¯ Q k − ¯ Q i Q i m − S k ¯ S k − ¯ S k S k m . (3.13)The automorphism su (2) generators F , C and ¯ C commute with this operator.In our case we deal with the centrally extended superalgebra (3.9), so we are led to perform analternative way of reaching the limit α = −
1. It implies the following preliminary redefinition ofthe extra su (2) generators: Z = − α + 1) F, Z +2 = − α + 1) C, Z − = − α + 1) ¯ C. (3.14)Then, multiplying (3.12) by the factor ∼ ( α + 1) and taking the limit α = − − α + 1) C ′ ( α )2 α = − = ⇒ ( Z ) + Z +2 Z − . (3.15)The generators (3.14) commute among themselves and with all other generators in the limit con-sidered, so they form the triplet of central charges. Thus we are left with the superalgebra (3.9) forwhich the invariant operator (3.11) is a proper limit of the quadratic Casimir operator (3.12) andit is the genuine Casimir for the centrally extended su (2 | ,
1) superalgebra.Below we will show that SU(2 | ,
1) is a spectrum-generating supersymmetry acting on infinite-dimensional irreducible SU(2 | ,
1) multiplets labeled by the eigenvalues ˜ l of (3.11). However, wemust take into account that the angular momentum operator L can take two eigenvalues l = ±| l | leading to the same (˜ l ) . So, the operator L commuting with all su (2 | ,
1) generators can be treatedas the second Casimir operator of su (2 | ,
1) and the full space of quantum states of the model isthe collection of two copies of SU(2 | ,
1) multiplets, with the same value of ˜ l and two opposite-signvalues of the quantum number l . 10 .2 Passing to the new basis in SU (2 | , It is useful to bring the superconformal algebra (3.9) to the form containing only one central charge.To eliminate two of the original central charges, we perform the rotation˜ Q i = cos ϕ √ (cid:0) Q i − i ¯ S i (cid:1) e i ( λ − π/ − sin ϕ √ (cid:0) ¯ S i − iQ i (cid:1) e − i ( λ − π/ , ¯˜ Q j = cos ϕ √ (cid:0) ¯ Q j − iS j (cid:1) e − i ( λ − π/ + sin ϕ √ (cid:0) S j − i ¯ Q j (cid:1) e i ( λ − π/ , ˜ S i = cos ϕ √ (cid:0) S i − i ¯ Q i (cid:1) e i ( λ − π/ − sin ϕ √ (cid:0) ¯ Q i − iS i (cid:1) e − i ( λ − π/ , ¯˜ S j = cos ϕ √ (cid:0) ¯ S j − iQ j (cid:1) e − i ( λ − π/ + sin ϕ √ (cid:0) Q j − i ¯ S j (cid:1) e i ( λ − π/ , (3.16)where cos 2 ϕ = g p L + g , sin 2 ϕ = L p L + g . (3.17)The only central charge ˜ Z we are left with reads˜ Z = q ( Z ) + Z +2 Z − = p L + g , (3.18)and it takes the value ˜ l = p l + g on the quantum states (see (3.11)). In the new basis, thesuperalgebra is rewritten as n ˜ Q i , ¯˜ Q j o = 2 m I ij − m δ ij ˜ Z + 2 δ ij H conf , n ˜ S i , ¯˜ S j o = − m I ij + m δ ij ˜ Z + 2 δ ij H conf , n ˜ S i , ¯˜ Q j o = 2 δ ij T, n ˜ Q i , ¯˜ S j o = 2 δ ij ¯ T , (cid:2) I ij , I kl (cid:3) = δ kj I il − δ il I kj , (cid:2) ¯ T , T (cid:3) = 2 m H conf , (cid:2) H conf , ¯ T (cid:3) = − m ¯ T , [ H conf , T ] = m T, h I ij , ¯˜ Q l i = 12 δ ij ¯˜ Q l − δ il ¯˜ Q j , h I ij , ˜ Q k i = δ kj ˜ Q i − δ ij ˜ Q k , h I ij , ¯˜ S l i = 12 δ ij ¯˜ S l − δ il ¯˜ S j , h I ij , ˜ S k i = δ kj ˜ S i − δ ij ˜ S k , h H conf , ¯˜ S l i = − m S l , h H conf , ˜ S k i = m S k , h H conf , ¯˜ Q l i = m Q l , h H conf , ˜ Q k i = − m Q k , h T, ˜ Q i i = − m ˜ S i , h T, ¯˜ S j i = − m ¯˜ Q j , h ¯ T , ¯˜ Q j i = m ¯˜ S j , h ¯ T , ˜ S i i = m ˜ Q i . (3.19)The generators ˜ Q i , ¯˜ Q j , I ij and ˜ H = H conf − m Z (3.20)form a different su (2 |
1) subalgebra, with the same (anti)commutators as in (1.2). So one canconstruct the new space of quantum SU(2 |
1) states just with respect to this transformed su (2 | su (2 | ,
1) generators in the original and new bases,taking into account their explicit form, can be realized on the full set of the quantum states definedin Section 2, so that this set is closed under the action of these generators. Thereby, the constructionusing the transformed su (2 |
1) superalgebra will give rise to the same total set of quantum states,although with the energy spectrum calculated with respect to the Hamiltonian ˜ H defined in (3.20). Here, cos 2 ϕ and sin 2 ϕ are non-linear operators expanded as Taylor series over the generator L . Since L is acentral charge, their series expansions take certain constant values. In the case of an arbitrary parameter ϕ suchrotations amount to a particular SO(3) ∼ SU(2) external group rotation preserving the invariant (3.11). i ( nl ) are still defined as in (2.11). Indeed, on the bosonic factor ψ ′′ ( z, ¯ z ) the Hamiltonian ˜ H can be represented as˜ H → H − m (cid:16) ˜ l + l cos 2 λ (cid:17) , (3.21)whence ˜ H Ψ i ( n,l ) = h H − m (cid:16) ˜ l + l cos 2 λ (cid:17)i Ψ i ( nl ) = m (cid:18) n − (cid:19) Ψ i ( n,l ) , (3.22)where we made use of the relation (2.13). Thus Ψ i ( n,l ) are eigenfunctions of both H and ˜ H .The new definition of the bosonic wave functions is as follows˜ Q i Ψ j ( n,l ) = ( n − √ m ε ij ˜Ω − ( n,l ) , ˜ S i Ψ j ( n,l ) = √ m ε ij ˜Ω − ( n +1 ,l ) , ¯˜ Q j Ψ i ( n,l ) = √ m δ ij ˜Ω +( n,l ) , ¯˜ S j Ψ i ( n,l ) = (cid:16) n + ˜ l − (cid:17) √ m δ ij ˜Ω +( n − ,l ) , ˜ Q i ˜Ω +( n,l ) = n √ m Ψ i ( n,l ) , ˜ S i ˜Ω +( n,l ) = √ m Ψ i ( n +1 ,l ) , ¯˜ Q j ˜Ω − ( n,l ) = − ( n − √ m Ψ j ( n,l ) , ¯˜ S j ˜Ω − ( n,l ) = − (cid:16) n + ˜ l − (cid:17) √ m Ψ j ( n − ,l ) , ˜ Q i ˜Ω − ( n,l ) = 0 , ˜ S i ˜Ω − ( n,l ) = 0 , ¯˜ Q j ˜Ω +( n,l ) = 0 , ¯˜ S j ˜Ω +( n,l ) = 0 , (3.23)where n = 0 , , . . . for ˜Ω +( n,l ) , and n = 2 , , . . . for ˜Ω − ( n,l ) . (3.24)They can be expressed through the wave functions (2.24) and (2.25),˜Ω − ( n,l ) = (cos ϕ + sin ϕ ) e − iπ/ √ (cid:16) ˜ l + l (cid:17) Ω n − ,l ) , ˜Ω +( n,l ) = √ ϕ e − iπ/ ˜ l (cid:16) ˜ l + g − l (cid:17) Ω n,l ) , (3.25)and, further, through Ω ± ( n,l ) . The conformal generators T and ¯ T are realized on the new wavefunctions as the creation and annihilation operators: T ˜Ω − ( n,l ) = m ˜Ω − ( n +1 ,l ) , ¯ T ˜Ω − ( n,l ) = ( n − (cid:16) n + ˜ l − (cid:17) m ˜Ω − ( n − ,l ) ,T Ψ i ( n,l ) = m Ψ i ( n +1 ,l ) , ¯ T Ψ i ( n,l ) = ( n − (cid:16) n + ˜ l − (cid:17) m Ψ i ( n − ,l ) ,T ˜Ω +( n,l ) = m ˜Ω +( n +1 ,l ) , ¯ T ˜Ω +( n,l ) = n (cid:16) n + ˜ l − (cid:17) m ˜Ω +( n − ,l ) . (3.26)Now, the full energy spectrum of the Hamiltonian ˜ H = H conf − m ˜ Z / E σ ( n ) = m ( n − σ ) , (3.27)where σ = 0 , n = 0 , , . . . for ˜Ω +( n,l ) ,σ = 1 / , n = 1 , , . . . for Ψ +( n,l ) ,σ = 1 , n = 2 , , . . . for ˜Ω − ( n,l ) (3.28)[cf. (3.22)]. Thus, the bosonic and fermionic states occupy integer and half integer energy values,respectively. In other words, they fill even and odd levels.12he new SU(2 |
1) supersymmetry generated by the supercharges ˜ Q i and ¯˜ Q j is not spontaneouslybroken and the corresponding ground state is given by the SU(2 |
1) singlet ˜Ω +(0 ,l ) . To prove this, letus consider the relevant Casimir operators˜ C = 1 m (cid:16) H conf − m Z (cid:17) − I ij I ji Q i ¯˜ Q i − ¯˜ Q i ˜ Q i m , ˜ C = 1 m (cid:18) ˜ C + 12 (cid:19) (cid:16) H conf − m Z (cid:17) + 18 m (cid:16) δ ij H conf − m δ ij ˜ Z − m I ij (cid:17) (cid:16) ˜ Q j ¯˜ Q i − ¯˜ Q i ˜ Q j (cid:17) . (3.29)On the states ˜Ω ± ( n,l ) and Ψ i ( n,l ) , they take the eigenvalues˜ C = n ( n − , ˜ C = (cid:18) n − (cid:19) ˜ C . (3.30)The state ˜Ω +(0 ,l ) thus corresponds to zero eigenvalues of both Casimir operators:˜ C ˜Ω +(0 ,l ) = 0 , ˜ C ˜Ω +(0 ,l ) = 0 . (3.31)It follows from the relations (3.23) that ˜ Q i ˜Ω +(0 ,l ) = ¯˜ Q j ˜Ω +(0 ,l ) = 0, as it should be for the singletground state.The states with n = 1 correspond to an atypical representation, since Casimir operators are alsozero on these states. This representation is spanned by the three statesΨ i (1 ,l ) , ¯˜ Q j Ψ i (1 ,l ) = √ m δ ij ˜Ω +(1 ,l ) , (3.32)such that ˜ Q j Ψ i (1 ,l ) = 0 . The states n Ψ i (1 ,l ) , ˜Ω +(1 ,l ) o form the fundamental SU(2 |
1) representation. All excited states with n > |
1) representations.Whereas the supersymmetry associated with ˜ Q i and ¯˜ Q j is not broken, the second pair of SU(2 | S i and ¯˜ S j corresponds to the spontaneously broken supersymmetry (see Figure 2), withthe minimal energy ˜ l m as the lowest eigenvalue of the relevant shifted Hamiltonian H conf + m ˜ Z / | ,
1) supercharges mixes all bosonic andfermionic states with a fixed number of the angular momentum l . Thus, all these states at fixed l belong to a single infinite-dimensional SU(2 | ,
1) representation labeled by ˜ l = l + g as the squareof the central charge. The states with − l belong to the other SU(2 | ,
1) representation labeled bythe same ˜ l = l + g . These two representations are distinguished only by the eigenvalue of theoperator L (see discussion in Subsection 3.1) and they exhaust the whole space of the quantumstates of the model. Thus the conformal supergroup su (2 | ,
1) acts as the spectrum-generatingalgebra on this space of quantum states .Taking into account (3.23), we can define the generators B + = 12 m ˜ S k ˜ Q k , B − = 12 m ¯˜ Q k ¯˜ S k , (3.33)which are responsible for the two-fold degeneracy of exited bosonic states: B + ˜Ω +( n,l ) = 2 ˜Ω − ( n +1 ,l ) , n = 1 , , , . . . ,B − ˜Ω − ( n +1 ,l ) = − n (cid:16) ˜ l + n (cid:17) ˜Ω +( n,l ) , n = 1 , , . . . . (3.34) The analogous role of the superconformal algebras D (2 , α ) and su (1 | ,
1) in the quantum Hilbert spaces ofsome superconformal mechanics models was pointed out in [18], [21], [19]. ③③③ × ×× ×× × ③③ m/ m m/ m m/ H conf − m Z Ψ i ( n,l ) ˜Ω +( n,l ) ˜Ω − ( n,l ) PPPPPP✐ PqPPPPPPqP✐PPPPPP✐ Pq P✐PPPPPPqP✐PPPPPq✏✏✏✮ ✏✶ ✏✏✏✮ ✏✶✏✏✏✮ ✏✶✏✏✏✮ ✏✶✏✏✏✮ ✏✶✏✶✏✏✏✮
Figure 2: Degeneracy of energy levels with a fixed number l . The action of the supercharges ˜ Q i and ¯˜ Q j is drawn by solid lines, while dashed lines corresponds to the action of ˜ S i and ¯˜ S j .These generators act on the bosonic wave functions only and form an exotic SU(2) symmetry [3]belonging to the universal enveloping of su (2 | , B + , B − ] = 2 B , [ B , B ± ] = ∓ n (cid:16) ˜ l + n (cid:17) B ± ,B ˜Ω +( n,l ) = 2 n (cid:16) ˜ l + n (cid:17) ˜Ω +( n,l ) , B ˜Ω − ( n +1 ,l ) = − n (cid:16) ˜ l + n (cid:17) ˜Ω − ( n +1 ,l ) , n = 1 , , , . . . . (3.35)The ground state ˜Ω +(0 ,l ) is annihilated by these SU(2) generators. This SU(2) symmetry is respon-sible as well for the double-fold degeneracy of the initial wave functions Ω ± ( n,l ) , since the latter arerelated to ˜Ω ± ( n,l ) via (2.23) and (3.25). The Casimir operator (3.13) allows us to guess the form of some new invariant operators ˜ I , ˜ M ofthe su (2 |
1) superalgebra. They commute with ˜ Q i , ¯˜ Q j , I ij , ˜ Z , H conf and read ˜ I = 2 H − T ¯ T − ¯ T T m − I ij I ji Q k ¯˜ Q k − ¯˜ Q k ˜ Q k m − ˜ S k ¯˜ S k − ¯˜ S k ˜ S k m − H conf ˜ Z m + (cid:16) ˜ Z (cid:17) , ˜ M = T ¯ T m + ˜ S k ¯˜ S k m . (3.36)On the SU(2 |
1) representations generated by ˜ Q i and ¯˜ Q j they take the values˜ I = ˜ l (cid:18) − n (cid:19) , ˜ M = n (cid:16) n + ˜ l − (cid:17) . (3.37)The quadratic SU(2 |
1) Casimir can be written in terms of these operators and the central charge:˜ C = ˜ I + 2 ˜ M − ˜ Z . (3.38) The operator ˜ I becomes the quadratic Casimir (3.13) of psu (2 | ,
1) in the limit ˜ Z = 0, where ˜ Q i ≡ Q i . su (2 |
1) generated by the transformedsupercharges ˜ Q i and ¯˜ Q j . No their analogs commuting with Q i and ¯ Q j exist. The superconformal Hamiltonian H conf , eq. (3.1), differs from the original SU(2 |
1) Hamiltonian H , eq. (2.6), by a central charge generator, eq. (3.2). The whole set of the superconformalSU(2 | ,
1) generators, including H conf , is realized on the original Hilbert space. The latter can beequally restored, starting from a new su (2 | ⊂ su (2 | ,
1) which is related to the original one by thetransformation (3.16), (3.17) and involves the Hamiltonian ˜ H = H conf − m ˜ Z /
2. The new groundstate proves to be SU(2 |
1) singlet and possesses zero energy, so the redefined SU(2 |
1) symmetryis not spontaneously broken, in contrast to the original one. The fermionic wave functions arethe same as in the original SU(2 |
1) model, while the bosonic ones ˜Ω ± , including the new groundstate, are represented by the proper linear combinations of the original wave functions Ω ± . In thenew basis the dependence of the spectrum (3.27) and Casimir’s eigenvalues (3.30) on the originalparameter λ vanishes. This phenomenon is due to superconformal symmetry and generalizes to thequantum case the property observed in [13] at the classical level.To avoid a possible confusion, we point out that the complete quantum consideration of theSU(2 |
1) supersymmetric C N S.-W. system, including energy spectrum, the structure of the Hilbertspace of wave functions and their SU(2 |
1) representation contents, has been already given in Section2. The basic aim of Section 3 was to demonstrate that the same results can be restored, starting froman equivalent description of this model in terms of complex SU(2 | ,
1) superconformal mechanicsassociated with the supermultiplet ( , , ). Many peculiar features of the original formulationbecome simpler in the superconformal formulation, including, e.g., simplifying the formula for theenergy spectrum. The phenomenon of disappearance of the dependence on the parameter λ in thesecond formulation also deserves an attention. C N Smorodinsky–Winternitz system
We define the quantum SU(2 |
1) supersymmetric C N S.-W. system as a sum of N copies of the C system with the Hamiltonians (2.1) involving the same parameters B , ω (equivalently, m , λ ), H = N X a =1 H a , H a = ¯ π a π a + ω z a ¯ z a + ( g a ) z a ¯ z a + ig a " ξ ak ξ ka z a ) − ¯ ξ ka ¯ ξ ak z a ) + B ξ ak ¯ ξ ak . (4.1)The supercharges and R-charges which form, together with the Hamiltonian H , su (2 |
1) superal-gebra, are also defined as sums of the relevant quantities of each particular C -system. Clearly,the generators H a commute with each other, and thus define the constants of motion of the super-symmetric C N S.-W. system. In addition to N commuting integrals H a , this system possesses N manifest U(1) symmetries z a → e iκ z a , ξ ai → e iκ ξ ai , with the generators L a = z a ∂ a − ¯ z a ∂ ¯ a + ξ ak ¯ ξ ak − L a , L b ] = [ L a , H b ] = 0 . (4.2)Hence, these generators provide the system to be integrable.The wave functions of this system are obviously given by the products of those of N one-dimensional copies, and the energy spectrum – by the sum of energies (2.27), E σ ( n,l ) = p ω + B n − σ + ˜ l ! + Bl , (4.3)where n = N X a =1 n a , l = N X a =1 l a , ˜ l = N X a =1 ˜ l a = N X a =1 p l a + g a , σ = N X a =1 σ a . (4.4)15ne observes the same distinction between the spectra of bosonic and fermionic wave functions asin the C case.The SU(2 |
1) supersymmetric C N S.-W. system has an additional degeneracy of the spectrum.It is due to the existence of the additional constants of motion given by the components of thesupersymmetric extension of the Uhlenbeck tensor generating a hidden symmetry in the bosoniccase. The classical version of this supersymmetric Uhlenbeck tensor was constructed in [11], whileits quantum counterpart can be written in the form I ab = 14 ( z a ∂ a + ¯ z a ∂ ¯ a + 1) (cid:0) z b ∂ b + ¯ z b ∂ ¯ b + 1 (cid:1) − (cid:0) z a ¯ z a ∂ b ∂ ¯ b + z b ¯ z b ∂ a ∂ ¯ a (cid:1) + i (cid:20) g b (cid:16) z a ¯ z a z b z b ξ bk ξ kb − z a ¯ z a ¯ z b ¯ z b ¯ ξ kb ¯ ξ kb (cid:17) + g a (cid:16) z b ¯ z b z a z a ξ ak ξ ka − z b ¯ z b ¯ z a ¯ z a ¯ ξ ka ¯ ξ ka (cid:17)(cid:21) + ( g b ) z a ¯ z a z b ¯ z b + ( g a ) z b ¯ z b z a ¯ z a − δ ab z a ∂ a + ¯ z a ∂ ¯ a + 1) , (4.5)where no sum over a and b is assumed. These constants of motion, together with (4.2) and H a ,provide the system with the superintegrability property.It turns out that this tensor admits a convenient representation in terms of the generators ofthe associated superconformal algebra su (2 | , Let us define the superconformal Hamiltonian on C N as a sum of N copies of superconformalHamiltonians on C , H conf = N X a =1 H (conf) a , (4.6)where H (conf) a is given by (3.1), with different parameters g a for each a ( a = 1 . . . N ), but with thecommon parameters λ and m . So we deal with a direct sum of su (2 | ,
1) algebras labeled by theindex a . Then, we take sums of all these generators˜ Q i = N X a =1 (cid:16) ˜ Q a (cid:17) i , ˜ S i = N X a =1 (cid:16) ˜ S a (cid:17) i , ¯˜ Q j = N X a =1 (cid:16) ¯˜ Q a (cid:17) j , ¯˜ S j = N X a =1 (cid:16) ¯˜ S a (cid:17) j ,I ij = N X a =1 ( I a ) ij , T = N X a =1 T a , ¯ T = N X a =1 ¯ T a , ˜ Z = N X a =1 ˜ Z a , (4.7)and obtain, once again, the conformal superalgebra (3.19) with the superconformal Hamiltonian(4.6). Here, ˜ Z a are defined as ˜ Z a = q ( L a ) + g a . (4.8)The wave eigenfunctions of (4.6) are obviously the products of N wave functions correspondingto a = 1 . . . N . Taking into account (4.4), the energy spectrum of the SU(2 |
1) Hamiltonian ˜ H = H conf − m ˜ Z / E σ ( n ) = m ( n − σ ) , (4.9)where σ is a sum of σ a = 0 , / , N .Bosonic and fermionic states still occupy separate levels with integer and half-integer values of theenergy (modulo the overall parameter m ), respectively.The Uhlenbeck tensor (4.5) commutes with the superconformal Hamiltonian (4.6). In terms ofthe generators of the conformal algebra so (2 ,
1) it can be represented in the very simple form I ab = 12 (cid:2) H (conf) a K b + K a H (conf) b (cid:3) − D a D b , (4.10)16r, in the basis (3.6), I ab = 1 m (cid:20) H (conf) a H (conf) b − (cid:0) T a ¯ T b + ¯ T a T b (cid:1)(cid:21) = 1 m (cid:2) H (conf) a H (conf) b − m δ ab H (conf) a (cid:3) − (cid:0) M a ¯ b + M b ¯ a (cid:1) , (4.11)where M a ¯ b := 12 m T a ¯ T b . (4.12)The second form of I ab makes obvious its commutativity with the superconformal Hamiltonian(4.6), as well as with the Hamiltonian of C N S.-W. system (4.1).The non-linear algebra generated by I ab reads[ I ab , I cd ] = δ ac T cbd + δ ad T dbc + δ bc T cad + δ bd T dac , a = b, c = d , [ I aa , I cd ] = 0 (4.13)(no summation over repeated indices), where the function T cbd has a simple representation throughthe generators of conformal algebra: T cbd = 1 m (cid:2) H (conf) c (cid:0) M b ¯ d − M d ¯ b (cid:1) + H (conf) d (cid:0) M c ¯ b − M b ¯ c (cid:1) + H (conf) b (cid:0) M d ¯ c − M c ¯ d (cid:1)(cid:3) . (4.14)Notice that for calculating the commutation relations (4.13) we do not need the explicit expressionsfor I ab in terms of the variables ( z a , ¯ z a , ξ ai , ¯ ξ ai ) as in (4.5), now it suffices to make use of the standardcommutation relations (3.5) or (3.7) of the conformal algebra so (2 ,
1) .Looking at the expressions (4.11) and (4.14), we observe that they involve, apart from N Hamil-tonians H (conf) a , also N bilinear generators M a ¯ b that commute with (4.6) and (4.1). Thus, whatactually matters is the nonlinear closed algebra generated by M a ¯ b and H (conf) b :[ M a ¯ b , M b ¯ c ] = 1 m M a ¯ c H (conf) b , a = b, b = c, c = a, [ M b ¯ b , M b ¯ c ] = 1 m M b ¯ c H (conf) b , b = c, [ M a ¯ b , M b ¯ b ] = 1 m M a ¯ b (cid:16) H (conf) b − m (cid:17) , b = a, (4.15) (cid:2) H (conf) b , M c ¯ d (cid:3) = m (cid:16) δ bc − δ bd (cid:17) M c ¯ d (4.16)(no summation over indices). One can add to this set the U(1) generators L a , which commute witheverything. Note that the symmetric combination M a ¯ b + M b ¯ a can be directly expressed through H (conf) b and I ab from (4.11), but it is not true for the antisymmetric one M a ¯ b − M b ¯ a entering T abc .However, it is possible to express ( M a ¯ b − M b ¯ a ) through the rest of constants of motion:( M a ¯ b − M b ¯ a ) = ( M a ¯ b + M b ¯ a ) − M a ¯ a M b ¯ b , a = b , (4.17) M a ¯ b + M b ¯ a = 1 m H (conf) a H (conf) b − I ab , a = b , (4.18)2 M a ¯ a = 1 m (cid:2) H (conf) a H (conf) a − m H (conf) a (cid:3) − I aa . (4.19)Thus the quantity T cbd defined in (4.14) is a function of the original hidden symmetry generators H (conf) a , I cd and so the relations (4.13), (4.14) constitute a closed non-linear algebra which isequivalent to the algebra (4.15), (4.16). 17ike in the bosonic C model (see (C.10)), the diagonal integrals I aa are yet expressed throughother integrals: I aa = 14 (cid:0) L a + g a (cid:1) − I a A a , (4.20) I a := ( I a ) ij ( I a ) ji , A a := ig a (cid:18) ¯ z a z a ξ ak ξ ak − z a ¯ z a ¯ ξ ak ¯ ξ ak (cid:19) − (cid:0) ξ ak ¯ ξ ak − (cid:1) L a , (4.21)with (cid:2) H (conf) a , I a (cid:3) = (cid:2) H (conf) a , A a (cid:3) = 0 , A a = 14 (cid:0) L a + g a (cid:1) (cid:18) − I a (cid:19) . (4.22)The operators I a are Casimirs for N copies of SU(2) symmetries acting only on fermionic variables.The additional new integrals of motion A a can be written in terms of the superconformal SU(2 | , A a = I a m (cid:20)(cid:16) ˜ S a (cid:17) k (cid:16) ¯˜ S a (cid:17) k − (cid:16) ¯˜ S a (cid:17) k (cid:16) ˜ S a (cid:17) k (cid:21) − m (cid:20)(cid:16) ˜ Q a (cid:17) k (cid:16) ¯˜ Q a (cid:17) k − (cid:16) ¯˜ Q a (cid:17) k (cid:16) ˜ Q a (cid:17) k (cid:21) . (4.23)In Appendix A we adduce some further details on the structure of these extra constants of motion.Degeneracy of the energy spectrum (4.9) can be attributed to any operator commuting withthe Hamiltonian. One can construct many examples of such operators like (4.11), (4.12) or (3.33).Let us illustrate, on the simplest N = 2 example, how an action of the operators (4.12) creates( n + 1)-fold degeneracy of the bosonic wave functions ˜Ω +( n ,l ) ⊗ ˜Ω +( n ,l ) , with n = n + n . Theaction of M on them is simple: M n ˜Ω +( n ,l ) ⊗ ˜Ω +( n ,l ) o = n (cid:16) n + ˜ l − (cid:17) n ˜Ω +( n +1 ,l ) ⊗ ˜Ω +( n − ,l ) o , H conf − m ˜ Z ! n ˜Ω +( n +1 ,l ) ⊗ ˜Ω +( n − ,l ) o = m ( n + n ) n ˜Ω +( n +1 ,l ) ⊗ ˜Ω +( n − ,l ) o . (4.24)It just increases the number n as n → n + 1 and decreases n as n → n −
1, so that the totalnumber n = n + n is not altered. The action of M is opposite: n → n − n → n + 1.A slight modification of the Uhlenbeck tensor (4.11) by other superconformal SU(2 | ,
1) gener-ators yields a generalization of the operator (3.36), such that it commutes also with the SU(2 | Q i , ¯˜ Q j , I ij and ˜ Z defined by (4.7):˜ I ab = 1 m (cid:20) H (conf) a H (conf) b − (cid:0) T a ¯ T b + ¯ T a T b (cid:1)(cid:21) + 14 ˜ Z a ˜ Z b −
12 ( I a ) ij ( I b ) ji + 14 m (cid:20)(cid:16) ˜ Q a (cid:17) k (cid:16) ¯˜ Q b (cid:17) k − (cid:16) ¯˜ Q a (cid:17) k (cid:16) ˜ Q b (cid:17) k (cid:21) − m (cid:20)(cid:16) ˜ S a (cid:17) k (cid:16) ¯˜ S b (cid:17) k − (cid:16) ¯˜ S a (cid:17) k (cid:16) ˜ S b (cid:17) k (cid:21) − m (cid:16) H (conf) a ˜ Z b + H (conf) b ˜ Z a (cid:17) . (4.25)In a similar way, the bilinear operator (4.12) is modified as˜ M a ¯ b = 12 m T a ¯ T b + 14 m (cid:16) ˜ S a (cid:17) i (cid:16) ¯˜ S b (cid:17) i . (4.26)Once again, these invariants can be constructed only for new supercharges ˜ Q i and ¯˜ Q j . No theiranalogs can be defined for su (2 |
1) generated by Q i and ¯ Q j . The algebra of the operators (4.25)and (4.26) is nonlinear and its closure lies in the universal enveloping of the superconformal algebra su (2 | ,
1) (3.19). The non-zero commutators of the generators (4.25) and (4.26) are presented in theAppendix B. It is worth to point out that the crucial property for revealing various degeneracies ofthe su (2 |
1) multiplets of the wave functions is the commutativity of ˜ M a ¯ b and ˜ I ab with the SU(2 | Q i , ¯˜ Q j , I ij and ˜ Z and, hence, with the relevant Casimir operators. The precise structureof the closure of the hidden symmetry generators is not too important from this point of view.18 .2 Products of SU (2 | representations One can consider the degeneracy of eigenvalues of the Casimir operators (3.29) of the N dimensionalsystem, though these eigenvalues cannot be presented by a generic formula and so each particular N ≥ |
1) generators, comes out with respect to the hidden symmetry operators (4.25) and(4.26). Below we present their action as the hidden symmetry operators on the SU(2 |
1) multiplets ofwave functions. We will always deal with the “superconformal” SU(2 |
1) generated by the generators˜ Q i , ˜¯ Q k and the relevant SU(2 |
1) multiplets.The product of N one dimensional SU(2 |
1) representations can be decomposed as a non-trivialsum of irreducible SU(2 |
1) representations [22]. For simplicity we consider here only N = 2 caseand present the decomposition for the levels n = 0 , , (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E , n = n + n , (4.27)where the relevant SU(2 |
1) Casimir operators are given by the expressions (3.29) with the compositegenerators (4.7). The action of the diagonal elements of (4.25) and (4.26) is defined as˜ I (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E = (cid:18) − n (cid:19) ˜ l (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E , ˜ I (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E = (cid:18) − n (cid:19) ˜ l (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E , ˜ M (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E = n (cid:16) n + ˜ l − (cid:17) (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E , ˜ M (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E = n (cid:16) n + ˜ l − (cid:17) (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E . (4.28)We can also define the action of the operator ˜ I + ˜ M + ˜ M (cid:16) ˜ I + ˜ M + ˜ M (cid:17) (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E = n n (cid:12)(cid:12)(cid:12) n, n − n , ˜ C , ˜ C E . (4.29)The remaining operators ˜ I and ˜ M − ˜ M either annihilate a quantum state or change its quantumnumber n − n . The level n = 0 . The lowest state with n = 0 is a product of the single states with n = 0 and n = 0: | i ≡ | , , , i = n ˜Ω +(0 ,l ) ⊗ ˜Ω +(0 ,l ) o . (4.30)This singlet state is just the ground state and the operators ˜ I , ˜ M and ˜ M annihilate it. The level n = 1 . The level n = 1 corresponds to a direct sum of two fundamental representationswhich are spanned by the states | , − , , i = n ˜Ω +(0 ,l ) ⊗ Ψ i (1 ,l ) , ˜Ω +(0 ,l ) ⊗ ˜Ω +(1 ,l ) o , | , , , i = n Ψ i (1 ,l ) ⊗ ˜Ω +(0 ,l ) , ˜Ω +(1 ,l ) ⊗ ˜Ω +(0 ,l ) o . (4.31)The operator ˜ I mixes these representations according to˜ I | , − , , i = − ˜ l | , , , i , ˜ I | , , , i = − ˜ l | , − , , i . (4.32)19he operators ˜ M − ˜ M act as (cid:16) ˜ M − ˜ M (cid:17) | , − , , i = ˜ l | , , , i , (cid:16) ˜ M − ˜ M (cid:17) | , , , i = − ˜ l | , − , , i . (4.33)Both representations | , ± , , i are identical to each other with respect to the action of SU(2 | The level n = 2 . For the excited level n = 2 there exist three options n = 1 , n = 1 ,n = 0 , n = 2 ,n = 2 , n = 0 . (4.34)The product of the fundamental representations with n = 1, n = 1 has the total dimension3 × i (1 ,l ) ⊗ Ψ j (1 ,l ) , Ψ i (1 ,l ) ⊗ ˜Ω +(1 ,l ) , ˜Ω +(1 ,l ) ⊗ Ψ i (1 ,l ) , ˜Ω +(1 ,l ) ⊗ ˜Ω +(1 ,l ) . (4.35)With respect to the generators (4.7) it splits into a sum of 4-dimensional typical representationand 5-dimensional atypical representation. The atypical representation is spanned by the triplet ofbosonic states and the doublet of fermionic states: | , , , i = n Ψ ( i (1 ,l ) ⊗ Ψ j )(1 ,l ) , Ψ i (1 ,l ) ⊗ ˜Ω +(1 ,l ) − ˜Ω +(1 ,l ) ⊗ Ψ i (1 ,l ) o . (4.36)The operator ˜ I annihilates this state. The typical representation encompasses the states | , , , i = n ε ij Ψ i (1 ,l ) ⊗ Ψ j (1 ,l ) , Ψ i (1 ,l ) ⊗ ˜Ω +(1 ,l ) + ˜Ω +(1 ,l ) ⊗ Ψ i (1 ,l ) , ˜Ω +(1 ,l ) ⊗ ˜Ω +(1 ,l ) o . (4.37)The remaining two options associated with (4.34) are given by the states | , − , , i = n ˜Ω +(0 ,l ) ⊗ ˜Ω − (2 ,l ) , ˜Ω +(0 ,l ) ⊗ Ψ i (2 ,l ) , ˜Ω +(0 ,l ) ⊗ ˜Ω +(2 ,l ) o , | , , , i = n ˜Ω − (2 ,l ) ⊗ ˜Ω +(0 ,l ) , Ψ i (2 ,l ) ⊗ ˜Ω +(0 ,l ) , ˜Ω +(2 ,l ) ⊗ ˜Ω +(0 ,l ) o . (4.38)So we face three identical SU(2 |
1) representations | , , , i , | , − , , i and | , , , i on the level n = 2, since they possess the same values of the SU(2 |
1) Casimir operators˜ C = 2 , ˜ C = 3 . (4.39)However, the operators ˜ I bb ˜ Z a − ˜ I aa ˜ Z b = ˜ l a ˜ l b ( n a − n b ) take different values on these SU(2 | I , ˜ M − ˜ M mix them among one another.The further consideration of irreducible SU(2 |
1) representations for higher levels n > I ab and ˜ M a ¯ b . The values of thediagonal operators ˜ I aa uniquely mark each representation in the set of identical ( i.e. , having thesame values of the Casimir operators) SU(2 |
1) irreducible representations, while the off-diagonaloperators mix these representations among themselves as the appropriate packages of bosonic andfermionic states.
In this paper we studied the quantum mechanics of SU(2 |
1) supersymmetric extension of the S.-W.system on the complex Euclidian space C N interacting with an external constant magnetic field209]. This supersymmetric system can be considered as a unification of N non-interacting C S.-W.systems. Accordingly, we first quantized the model on C and then generalized the considerationon the case of C N . We constructed the complete space of the wave functions and found therelevant energy spectrum. We studied how all bosonic and fermionic states are distributed overthe irreducible representations of the supergroup SU(2 | ,
1) (see Appendix C). In the supersymmetric case weredefined the Hamiltonian as (3.2) and showed that it exhibits SU(2 | ,
1) superconformal symmetrywhich serves as the spectrum-generating symmetry on the full set of the quantum states.The wave functions of supersymmetric quantum S.-W. system on C N were constructed as prod-ucts of N wave functions of the C models. Correspondingly, on these products irreducible SU(2 | N = 2 that already amountsto a non-trivial sum of irreducible SU(2 |
1) representations. Also, the generalization to C N revealshidden symmetry generators (4.25) which correspond to a supersymmetrization of Uhlenbeck tensor[9, 11]. It is responsible for the degeneracy of the wave functions belonging to irreducible SU(2 | su (2 | ,
1) were found.It would be interesting to consider, along the same lines, quantum deformed SU(2 |
1) extensionsof other K¨ahler oscillator models, e.g., of the CP N one. These models are not superconformal, sotheir quantum analysis should be similar to what has been performed in Section 2. On the otherhand, a non-trivial multi-particle extension of the SU(2 | C S.-W. model could be a complex N -particle interacting system of the Calogero-Moser type, hopefully preserving the superconformalinvariance of the one-particle C model. Then the whole consideration of Sections 3 and 4 basedon the superconformal group SU(2 | ,
1) could be applicable.Finally, let us notice that the quantum C S.-W. system without magnetic field (also known as a“circular oscillator with ring-shaped potential”) was used in a more phenomenological setting for thestudy of the particle behavior in the two-dimensional quantum ring [23]. Respectively, the C N S.-W. system with coincident parameters g a can be interpreted as an ensemble of N free particles in asingle quantum ring interacting with a constant magnetic field orthogonal to the plane. It would beinteresting to reveal possible physical implications of the quantum SU(2 |
1) supersymmetric versionof this system within such an interpretation.
Acknowledgments
The authors thank Hovhannes Shmavonyan for interest in the work. A.N. thanks Ruben Mkrtchyanfor useful comments. E.I. and S.S. acknowledge support from the RFBR grant No. 18-02-01046and a grant of the Ter-Antonyan–Smorodinsky Program.
A More on the integrals of motion (4.23)
The fermionic integrals of motion A a can further be expressed via other, odd integrals of motion,as follows A a − I a m h (Ξ a ) k (cid:0) ¯ S a (cid:1) k − (cid:0) ¯Ξ a (cid:1) k ( S a ) k − m (cid:0) ¯Ξ a (cid:1) k (Π a ) k i − m h (Π a ) k (cid:0) ¯ Q a (cid:1) k − (cid:0) ¯Π a (cid:1) k ( Q a ) k + m (cid:0) ¯Π a (cid:1) k (Ξ a ) k i , (A.1)21here ( Q a ) i := (cid:16) ˜ Q a (cid:17) i ( T a ) / , (cid:0) ¯ Q a (cid:1) j = (cid:0) ¯ T a (cid:1) / (cid:16) ¯˜ Q a (cid:17) j , ( S a ) i := (cid:16) ˜ S a (cid:17) i (cid:0) ¯ T a (cid:1) / , (cid:0) ¯ S a (cid:1) j = ( T a ) / (cid:16) ¯˜ S a (cid:17) j , (Π a ) i := (cid:16) ˜ Q a (cid:17) i (cid:0) ¯ T a (cid:1) − / , (cid:0) ¯Π a (cid:1) j := ( T a ) − / (cid:16) ¯˜ Q a (cid:17) j , (Ξ a ) i := (cid:16) ˜ S a (cid:17) i ( T a ) − / , (cid:0) ¯Ξ a (cid:1) j := (cid:0) ¯ T a (cid:1) − / (cid:16) ¯˜ S a (cid:17) j . (A.2)Using the su (2 | ,
1) (anti)commutation relations (3.19), it is straightforward to check that all thesefermionic operators commute with the Hamiltonian H (conf) a . Note that the generators T a , ¯ T a inthe present model start with non-zero numerical constants (as follows from the definitions (3.6)and (3.3), (3.4)), so the operators invert to them, equally as their square roots, are well defined.In fact, not all of these odd integrals of motion are independent. We can choose among them thebasis ( Q a ) i , (cid:0) ¯ Q a (cid:1) j and represent the remaining ones, taking into account the relations (3.19), as( S a ) i = 2 m h ( Q a ) i − ( T a ) / (cid:0) ¯ T a (cid:1) / ( Q a ) i ( T a ) − / (cid:0) ¯ T a (cid:1) − / i ( T a ) / (cid:0) ¯ T a (cid:1) / , (cid:0) ¯ S a (cid:1) j = 2 m ( T a ) / (cid:0) ¯ T a (cid:1) / h(cid:0) ¯ Q a (cid:1) j − ( T a ) − / (cid:0) ¯ T a (cid:1) − / (cid:0) ¯ Q a (cid:1) j ( T a ) / (cid:0) ¯ T a (cid:1) / i , (Π a ) i = ( Q a ) i ( T a ) − / (cid:0) ¯ T a (cid:1) − / , (cid:0) ¯Π a (cid:1) j = ( T a ) − / (cid:0) ¯ T a (cid:1) − / (cid:0) ¯ Q a (cid:1) j , (Ξ a ) i = 2 m h ( Q a ) i − ( T a ) / (cid:0) ¯ T a (cid:1) / ( Q a ) i ( T a ) − / (cid:0) ¯ T a (cid:1) − / i , (cid:0) ¯Ξ a (cid:1) j = 2 m h(cid:0) ¯ Q a (cid:1) j − ( T a ) − / (cid:0) ¯ T a (cid:1) − / (cid:0) ¯ Q a (cid:1) j ( T a ) / (cid:0) ¯ T a (cid:1) / i . (A.3)The integrals of motion ( T a ) ± / (cid:0) ¯ T a (cid:1) ± / can be argued to functionally depend on M a ¯ a and H (conf) a .Hence, our system has 2 N commuting bosonic integrals H (conf) a , L a and 4 N Hermitian (or 2 N complex) fermionic ones ( Q a ) i , (cid:0) ¯ Q a (cid:1) j , i.e. it is integrable in the sense of supergeneralization ofLiouville theorem . B Commutators of super Uhlenbeck tensor
The commutator of off-diagonal elements ( a = b, b = c, c = a ) of the hidden symmetry generators˜ I ab introduced in (4.25) is given by h ˜ I ab , ˜ I bc i = ˜ T abc + ˜ Z b (cid:16) ˜ M a ¯ c − ˜ M c ¯ a (cid:17) , (B.1) The Hamiltonian system on (2 k | n )-dimensional symplectic supermanifold is integrable if it possesses n function-ally independent fermionic odd integrals and k functionally independent commuting bosonic integrals, see [24]. T abc is defined as˜ T abc = 112 ( I a ) ij h ( I b ) jk ( I c ) ki − ( I b ) ki ( I c ) jk i + H (conf) a m (cid:0) ¯ T b T c − T b ¯ T c (cid:1) − ˜ Z a m (cid:0) ¯ T b T c − T b ¯ T c (cid:1) + ˜ Z a m (cid:26)(cid:20)(cid:16) ˜ S b (cid:17) k (cid:16) ¯˜ S c (cid:17) k + (cid:16) ¯˜ S b (cid:17) k (cid:16) ˜ S c (cid:17) k (cid:21) + (cid:20)(cid:16) ˜ Q b (cid:17) k (cid:16) ¯˜ Q c (cid:17) k + (cid:16) ¯˜ Q b (cid:17) k (cid:16) ˜ Q c (cid:17) k (cid:21)(cid:27) − H (conf) a m (cid:26)(cid:20)(cid:16) ˜ S b (cid:17) k (cid:16) ¯˜ S c (cid:17) k + (cid:16) ¯˜ S b (cid:17) k (cid:16) ˜ S c (cid:17) k (cid:21) + (cid:20)(cid:16) ˜ Q b (cid:17) k (cid:16) ¯˜ Q c (cid:17) k + (cid:16) ¯˜ Q b (cid:17) k (cid:16) ˜ Q c (cid:17) k (cid:21)(cid:27) + ( I a ) ji m (cid:26)(cid:20)(cid:16) ˜ S b (cid:17) i (cid:16) ¯˜ S c (cid:17) j + (cid:16) ¯˜ S b (cid:17) j (cid:16) ˜ S c (cid:17) i (cid:21) − (cid:20)(cid:16) ˜ Q b (cid:17) i (cid:16) ¯˜ Q c (cid:17) j + (cid:16) ¯˜ Q b (cid:17) j (cid:16) ˜ Q c (cid:17) i (cid:21)(cid:27) + T a m (cid:20)(cid:16) ˜ Q b (cid:17) k (cid:16) ¯˜ S c (cid:17) k + (cid:16) ¯˜ S b (cid:17) k (cid:16) ˜ Q c (cid:17) k (cid:21) + ¯ T a m (cid:20)(cid:16) ˜ S b (cid:17) k (cid:16) ¯˜ Q c (cid:17) k + (cid:16) ¯˜ Q b (cid:17) k (cid:16) ˜ S c (cid:17) k (cid:21) + (cyclic permutation of abc ) . (B.2)The rest of the non-vanishing commutators of the algebra is given by h ˜ I aa , ˜ I ab i = ˜ Z a (cid:16) ˜ M a ¯ b − ˜ M b ¯ a (cid:17) . (B.3) C Conformal symmetry in the bosonic sector
The C N S.-W. quantum system can be conveniently considered in the framework of conformalquantum mechanics. This peculiar feature was not noticed in [9].Let us concentrate on the C case. The Hamiltonian in the presence of magnetic field B reads H SW = − ∂ ¯ z ∂ z + B L + m z ¯ z g z ¯ z , L = z∂ z − ¯ z∂ ¯ z . (C.1)The spectrum of H SW is given by H SW Φ ( n,l ) = " m n + ˜ l ! + B l Φ ( n,l ) , L Φ ( n,l ) = l Φ ( n,l ) , (C.2)where Φ ( n,l ) = n ! ( z ¯ z ) ˜ l ( z/ ¯ z ) l e − mz ¯ z L (˜ l ) n ( mz ¯ z ) , ˜ l = p l + g . (C.3)One sees that the presence of magnetic field B affects the energy spectrum.On the other hand, we can represent the Hamiltonian (C.1) in the form H SW = H conf + B L , (C.4)where H conf is a conformal Hamiltonian of the trigonometric type, H conf = − ∂ ¯ z ∂ z + m z ¯ z g z ¯ z . (C.5)One can check that L commutes with the conformal generators T , ¯ T and H conf satisfying the algebra(3.7), where T = H conf − m z ¯ z m z∂ z + ¯ z∂ ¯ z + 1) , ¯ T = H conf − m z ¯ z − m z∂ z + ¯ z∂ ¯ z + 1) . (C.6)The introduction of magnetic field according to (C.4) modifies the conformal algebra so (2 ,
1) writtenin terms of the original Hamiltonian H SW as (cid:2) ¯ T , T (cid:3) = 2 m H SW − B L , (cid:2) H SW , ¯ T (cid:3) = − m ¯ T , [ H SW , T ] = m T, (C.7)23ith L playing the role of a central charge. The conformal algebra has the standard form (3.7) justin the basis with the conformal Hamiltonian, with L becoming an external generator commutingwith all conformal generators.The generators T , ¯ T and H conf act on the wave functions Φ ( n,l ) , as T Φ ( n,l ) = m Φ ( n +1 ,l ) , ¯ T Φ ( n,l ) = n (cid:16) n + ˜ l (cid:17) m Φ ( n − ,l ) , H conf Φ ( n,l ) = m n + ˜ l ! Φ ( n,l ) . (C.8)We observe that the explicit dependence on the external magnetic field disappears in the spectrumof the conformal Hamiltonian H conf .The relevant Casimir operator of so (2 ,
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