Dynamical parasuperalgebras and harmonic oscillators

Volume 146, number 6

PHYSICS LETTERS A

4 June 1990

DYNAMICAL PARASUPERALGEBRAS AND HARMONIC OSCILLATORS * Stéphane DURAND Laboratoire dePhysique Nucléaire, Université de Montréal, C.P. 6128, Succ. A, Montreal, Quebec, Canada H3C 3J7

and Luc VINET’ Department of Physics, University ofCalifornia, 405 HilgardAvenue, Los Angeles, CA 90024, USA Received 9 March 1990; accepted for publication 26 March 1990 Communicated by D.D. Hoim

The dynamical algebra of the n-dimensional harmonic oscillator with one second-order parafermionic degree of freedom is exhibited. It provides an example and an explicit realization of a second-order parasuperalgebra. These constructs generalize the more familiar superalgebra structures. They possess a Z 2 grading and involve a symmetric trilinear product among their odd elements. Particular attention is givento the one- and two-dimensional cases. It is pointed out that the one-dimensional oscillator Hamiltonian is parasupersymmetric, that QtQ2 it belongs to the of Hamiltonians for which one has aHamiltonians realization ofconthe 2Qt+i.e., QQtQ+ = 4QH. It isclass further noted that theHparasupersymmetric structed by Rubakov Spiridonov always possess two additional conserved parasupercharges. It is finally explained how the algebra [H, QJ= [H, and Qt J=0, Q two-dimensional oscillator parasuperalgebra provides a spectrum-generating algebra for the one-dimensional parasupersymmetric Morse Hamiltonian.

Loosely speaking a quantum-mechanical system is said to possess a dynamical algebra if its states form representation spaces for this algebra. The simplest situation where this occurs is when the Hamilton operator can itself be expressed as an algebra element in a realization of some algebra [1]. There are also many interesting cases where the Hamiltonian can be written as a bilinear combination of algebra generators [2]. In both instances, there exist various techniques that allow for a group theoretic resolution of the dynamics. We shall here be mostly concerned with examples of the first type. When systems involving both bosonic and fermionic canonical degrees of freedom are considered it is appreciated [31 that superalgebras can arise as * This work is supported in part through funds provided by the

UCLA Department of Physics, the Natural Sciences and Engineenng Research Council (NSERC) of Canada and the Fonds FCAR of the Quebec Ministry of Education. On sabbatical leave from the Laboratoire de Physique Nucléaire, Université de Montréal, Montreal, Canada.

dynamical algebras. This is so, when there are constants of motion that generate supersymmetry transformations, that is transformations mixing the bosonic and the fermionic variables. It has been known [4] for a long time that the canonical quantization rules are not the only relations that can be imposed on the basic dynamical vanables in order for the Heisenberg equations to be compatible with the “classical” equations ofmotion. There is in factan infinity of consistent schemes and quantities satisfying the corresponding generalized quantization rules are said to be parabosonic or parafermionic [5]. As a matter of fact, there exist systems which involve dynamical variables both of the ordinary Bose type and of the para-Fermi type. In such a context, we may expect symmetry operations transforming the bosonic variables into the parafermionic ones .

(and vice versa) to occur. These operations which generalize the familiar supersymmetry transformations have been called parasupersymmetries [61.

0375-9601/90/s 03.50 © Elsevier Science Publishers B.V. (North-Holland)

299

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4 June 1990

Their generators realize new algebraic structures, parasuperalgebras, of which superalgebras are a special case. The main feature of parasuperalgebras lies in the fact that they involve a multi-linear product rule for their fermionic elements. The purpose of this contribution is to give an example where this new type of dynamical algebra is encountered, Let (4 and a,, i= 1, n, respectively stand for ordinary bosonic creation and annihilation operators satisfying ~ —(5 —1 ‘1

dimensional oscillator. We shall now present its dynamical algebra which, as we shall see, is a fairly large second-order parasuperalgebra. (This explains our interest in this otherwise rather trivial system.) Aspects and applications that are specific to the oneand two-dimensional cases will be discussed in the conclusions. In particular, we shall indicate what is meant by a parasupersymmetric Hamiltonian and shall explain how the invariance parasuperalgebra of the two-dimensional oscillator provides a spectrumgenerating algebra for the parasupersymmetric Morse

1a,, a~ t~j— fl “ ~‘ To these variables let us add one single set of secondorder parafermionic creation and annihilation op-

Hamiltonian. Let us introduce the following constants ofmotion,

...,

—

~

~

erators a and a. By definition these quantities obey the trilinear relations [5] 3=0, aata=2a, a2at+ata2=2a. (2) a We shall take the following Hamiltonian to govern the time evolution of these dynamical variables:

r—

,

P~=,~iwe”a~,P~=~jwe”a~,t=l,...,n, T~

/

ae

iwi

= (a~e

1~~)= (~io~T” )~,, H= ~ lw{4, a,}+iw[at, a]

(3)

.

1=1

It is well(1) known that the ifcanonical lations are realized one uses commutation n coordinatesrex, and takes a,=

(p,—iwx,), a,

(p,+icox,)

=

(4) 9x,. As for the parafermionic vanwith ables,p~=—iô/ it is not difficult to see that a and at are irreducibly represented by the following 3 x 3 matrices: /0 0

0\

1 0 \0 1

01,

/0

1

0\

8

with

2_a2at)

b= ~ (ata

/i =

1,

2,

(9)

(10)

. .

Their conservation is easily verified by checking that they belong to the kernel of d/dt = ~9/ôt + i [H, ]. Let us also identify as follows all bilinear monomials in P,, P~,T~and i~that can be constructed: A,~=P 1P~,H~P’=~{P~,P3}, A~,=P~Pj, (lIb) (ha) L =iTtcr T a=l 2 3 a

4

a

Y= ~Tt T, ~

(11 c) Q 1~=P~T~,

at=~/~(0 0 \0 0

0/

1). 0/

Substitution of (4) and (5) in H gives 2x~)+wE n H= ~ ~(p~+w 3,

(5)

(6)

with

fl I3=~[at,a]=I 0

0 0 0

0\

01,

(lid)

(In eq. (hlb), o~,,a=l, 2, 3 stands for the standard Pauli matrices.) This exhausts the list of the independent bilinears; note that -

-

-

-

~TtaaT=~TtaaT=~TtJaT=La,

(l2a)

~TtT=~TtT=4I_3Y,

(l2b)

~~t~=y (7)

—

making clear that this Hamiltonian describes an n300

S.~,_—P~TM,~

(l2c) .

As the structure relations that we are about to give will show, T, Q and S (and T~ and gas well) transform as two-spinors under the SU (2) generated by

Q

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PHYSICS LETTERS A

the La’S. We claim that the operators defined in (8), (9) and (11) form a closed algebraic set V, if one includes the identity operator I. Note that V possesses a natural Z25~ as the bosonic basisP1elements grading with (P1, A ii, At~ if , a, Li, H~°~ L Y, I)eV~ and (Tn, T~,Q,~,

a,,, ~ ~)eV~”~) as the parafermionic ones. We thus have the decomposition

4 June 1990

(l5d)

[La, Lb]ja&Lc.

The transformation properties of the parafermionic charges under the bosonic symmetries are given by ii

[H~°1, Q~,j=oJöikQf~, [A,~, Qk~,]= 0, [A,

[H~°~

3,Skp]

=

~‘

‘—

— (~0ôIkQJ~— WôfkQ1~,

V=V(B) ~ The structure relations of this algebra are of the following form, 8~ , [V(B), y(PF)] ~ [V(B), v(B)] V~

[A1~Qk~I = (DôikSf,~+ wôfkSI,~~[A ~ Sk~i= 0, [P 1,Q~]=O, [P1, S~]= —wô0T~,

{V~),y(PF), V~)}

[P1,

with [ , finedby

(13)

y(B)V(PF) ,

] the usual commutator and {

} de-

, ,

Q~]=wô0T,,, [P1,S~]=0 [I,Q~~]=0, [I,S1~]=0, [Y,Q,,j=~11,,

{X,,X2,X3}X1(X2X3+X3X2)

+cyclic permutations of (1, 2, 3)

[La,Qj~]__j(aa),,vQjv,

.

(14)

This fully symmetric trilinear product arises because we have chosen the parafermionic variables a and t of second order. (A p-linear symmetric product would need to be used ifthe parafermionic variables were taken to be of order p.) It is straightforward to determine the Various products of the basis elements (in our realization), they are given below, The subspace V~is itself a Lie algebra. It is identified asSp (n) -Oh ( n) ~ SU (2) ~U (1), i.e. the semidirect sum of the rank n symplectic algebra Sp (n) (generated by ~ A0 and 4) and of the Heisenberg algebra, h(n), in n dimensions [7] (generated by P,, P1 and I) to which an SU(2) (generated by La) a U(l) (generated by Y) added as direct and summancis. In our basis, thearenon-vanishing commutators are [H,~,°1,Ak!] = Wö,kAfl + wôIlAJk, =

[Y,S11,]=S~~,

Wt~JkA1,a4,A1k,

[At, Ak!] =wôikHJl°~+wö 11H~,~” +o.4kH~°~ +wo~1H~,

[La, 5,,,] = —

~(t7a),,vSiv,

(l6a)

[4,

~

T,,] = [At,, T,,] = T~]= [P,, —~ T~]= [I, T~]= 0, — [Y, T~J=1’,,, [La, T~]= — ~(aa)pvTp.

~—

~—‘

~

{Q,,,, Q~,Q~}=0, {S,,,, S~,,,,SAp}=O, {T~,,T~,T~}=0, (l7a) 2ox, 2w~ôJkQI,,, {Q,,,, QJ~,S,~}= 4ö~kQJ~+ ~ S~,,,,Q~} = — 2W 20)(vpôjkSip, 14,ÔIkSJP — {Q,,~,Q~,T,,, } = 0, { Ti,, T~,Q,,} = 0,

{S~,,,S~,,,,T~}=0, {T,,, T~,S~}=0,

(l7b)

{Q,,,, S~,T,,} =2w~~o0T~, {Q,,,, Q~,~,}=4,~,,[AuQk]v+

(l7c) (~t~-~ ,~‘),

(ISa)

{S11,, S~,Si~}=4,~,[A~Sk]~+ (~z~— v)

[P1,P3]coö0J,

(l5b)

{T,,,T~,T~}=4,,~[IT]~+(p.—’v),

[At,, Pk]wc5akPf+wôfkPl,

2,~[(HJj~+~WôkiLa(Ya)Qj+(l~~J)]v

[Ht,P),P~]=_wôJ~P~, =

—aöIkPJ—WöJkPI,

(17d)

{Q,,,, Q~1,S,,~}

~k] =(mSIkPf,

[A0, Pfl

(l6b)

The commutation relations involving Q, Si and Dare obtained from the above formulas by effecting the substitutions Q Q, S S~T 7’, Finally, the tnlinear products of the parafermionic basis elements read as follows with 12 = — 21 = 1 and ~t, v~p = 1, 2:

[H~°~, H~?~] =2w(511H~?)—2wôJkH~°~ ,

~

Ti,]

+(lz’—”v)—co,,,,[ök1~,~—(i~-”j)]

(15c) 301

Volume 146, number 6

s,,,, S

~k 1~

PHYSICS LETTERS A

}

= 2pp[(Hjj~ — ~wöjkLaaa)S,

+ (p

v) —we~~[

~

öjk

+~

with more than one parafermionic degree of freedom; second, one could allow for parafermionic variables of degree greater than two (see refs. [5,6]).

~]p

— (i ~

{ Q,,~,Q~,,,,T~} = 4e,~,[P, Q~]v + T~,T~,Q,,, } = 4,~. [P, T],. + (p

~

These situations remain to be analysed. We shall now close this paper by discussing some aspects of the one-dimensional case and by explain-

v)

(~i ~

ii)

{S,~,Si,,, T~} =4~[P1S] + (ii

ing how the invariance parasuperalgebra of the twodimensional oscillator provides a spectrum generating algebra for the parasupersymmetric Morse

,

v)

{T,1, T~,S~p}=4e~,[P1T]~+(p*-+v) ,

{Q111,

(l7e)

~ Q~}

(p

=~w({at,a}+[at,a])

+—~

v) —we~~ök,S~~,

These constants commute with the Hamiltonian in

{Q,,,, T~,~~}=2e,~[A,1T+P1Q1]~+(p+-+ v) ,

accordance with eqs. (16):

{Q,,,, T~,~ v) {S1~,T~,S~ } = 2e~,[At, T+ P~Si],, + (p v)

[H,

,

~

~

,

(l7f)

0~~}

{Q, Q, Qt}~HQ, {Q, Qt, Qt}zzr8HQt,

1 ~

Q,~,S~}

= 2cR,, [P5Qi

+ (Hj 0~+ ~ W(5iiLatla) T] v 1 v)+a~i~,,(5 3.~7. (l7g) The relations of the form {A, B, C} or {A, B, C} can be obtained from the above using +(pi—*

_______

{A,B,C}=—{A,B,C} and

(l8a)

{A, II, C} = {A, B, C}. (1 8b) Investigating the symmetries of oscillators with a single parafermionic degree of freedom has thus revealed a remarkable algebraic structure. Two obvious generalizations would lead to even richer parasuperalgebras. First, one could consider oscillators —

(21)

Their trilinear products are obtained by specializing formulas (17) and using the fact that

One finds Q3z=0, Qt3=0,

{S,,,,2pp[PjSj T~, + (H~ 0~ — ~Wô,jLaaa)T],.

302

Q]= [H, Qt]=0.

(LaaaT)M=3(L3t73T),~,.

{Q,,~,,SJ,,,~

{T,,,

(19)

Let us single out the following two odd elements from its dynamical parasuperalgebra basis: Q_=Q11 _\/~aa, Qt=Sii2=.,J~waat. (20)

‘~

2e,,p[AJt,(Q1+ (H~ +~WôkiLaaa)Sj]p

+(u

.

~

Q114’ ~ + (p

Hamiltonian [8]. In one dimension the Hamiltonian (6) reduces to H=H~?~ +2wL3

2,,p[AkASj(H5~~WöjkLarJa)Qi]v +

4 June 1990

(22) (23a) (23b) (23c)

This parasuperalgebra is the second-order generalization of the superalgebra 2_Qt20 {Q, Qt}2H, Q [H, Q] = [H, Qt] =0. (24) It was first introduced by Rubakov and Spinidonov [6]. A Hamiltonian H is said to be parasupersymmetnc (of order 2) if there exist two parasupercharges Q and Qt such that the parasuperalgebra specified by the relations (21) and (23) is realized. The one-dimensional harmonic oscillator Hamiltonian (19) is therefore parasupersymmetric. As shown in ref. [6], a one-dimensional parasupersymmetric Hamiltonian can be constructed quite

Volume 146, number 6

PHYSICS LETTERS A

generally in terms of two functions W1 (x) and W2(x), that satisfy W~(x)—W~(x)+W’2(x)+W’1(x)=k,

(25)

with k a real constant and the prime standing for differentiation. (The “potentials” W, and W2 are otherwise arbitrary.) Given two such functions, one can check that the relations (21) and (23) are realized for

{~,~, ~} = 8H~, {Q, Q, ~t} = — 2kQ, Q Qt}= —2k~, {Q, Q Qt}=4J-JQkQ, {~,Q, Qt}4H~kQ, ~ ~ = {Q, ~, Q} = ~Q,~, =

a}

(plus Hermitian conjugated relations)

+ W’2ata_Whiaat]

(26a)

and 2at] . (26b) (p+iW,)ata2+ (p+iW )a 2 In our oscillator special case W, = W2 = wx and k= 2w. We would here like to point out that the above Hamiltonians always possess two further bosonic constants, 13 and Y, and two additional conserved parasupercharges, ~ and Qt ~‘. Indeed, let ~ be the parasupercharge which is obtained by commuting Q with Y=~(aat+ata),i.e. let

.

(29)

In our oscillator example,

O—Q1~

~

4 June 1990

(30)

~

One may verify that the trilinear relations (17) among Q11, Si,2, ~ and S,2 correspondingly reduce to those given in (29) with k=2w. The last that webetween would like make is to indicate the point connection thetoparasupersymmetric one-dimensional Morse Hamiltonian and the two-dimensional oscillator, hence showing that the dynamical parasuperalgebra of the latter system is also a spectrum-generating algebra for the former [8]. The parasupersymmetric Morse Hamiltonian is obtained by substituting W1 = —~he_x+ ~(h— 1) , W2= W1 + 1 (31) in eq. (26a). One finds 2(e~_l)2+he_xls+~ (32) 2HM=— ~ +~h The eigenstates of the Hamiltonian .

O=[Y,Q]

[(p+iW,)btb2+(p+iW

=

2bt] 2)b

[(p+iWi)ata2_(p+iW

=

2at].

(27)

H=H~?~ +H~°~ +13

(33) can be obtained by constructing the unitary repre-

2)a One finds that [H,

[13,

~] =

sentations of its dynamical parasuperalgebra; more precisely, by determining a basis for the correspond-

[H,

Q] =

[13, Qt] [Y,~]=Q,

[y,Qt]~...Qt,

0t] = [H, 13] —Q, [1~, = ~

Qt

,

= [H,

Y]

ing representation space. (We have set w= 1.) A nat-

=0,

ural set of quantum numbers is provided by the following eigenvalue equations: 0~,(5, m (HI?~+H~)Iht 2>=h~°~I ~ (5, m~>

_.~

[13,~t] ~7t

[y,Qt]~Qt,

(28)

The trilinear products involving Q, Qt, Q and ~ are straightforwardly evaluated and in addition to those given in (23) one has

~ 5=1, 0, — I

(34a)

,

~12Ih,ö,m2>m2I12~,ö,m2> ~ ~‘

These results were obtained in collaboration with M. Mayrand.

(34b)

(34c)

with

303

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PHYSICS LETTERS A

M2 = ~i(a~a1 —ala2) The range of values for m

(35)

,

2 is determined by observing that M, =~(a1a2+a~a,), M2 and M3= ~(a~a,— a ~a2) form an SU (2) algebra with ~ Now let 1+213. (36) R=HI? +H~

Instead of h ~ we can equivalently use as quantum number the eigenvalue h = h ~ + 2(5 ofI?, identify the basis states as I h, (5, m 2> and replace (34a) by

4 June 1990

2=h e_x in (41) and mulvariable.this Indeed, setting tiplying equation onrboth sides by r2 yields HM!PhOm2(X) 2+l)I~Phom =[—~m~+~(h

2(x),

(43)

with HM as given in (32). This paper is based in part on results that we presented at the 1989 Annual Seminar of the Canadian Mathematical Society. We would like to thank the organizers of this meeting for their interest.

RIh,ö,m2>=hlh,ô,m2> , h=—l,0, 1 (37) References In terms of the polar coordinates x, = r cos 0~ x2 = r sin 0~

(38)

[1] A.O. Barut, A. Bohm and Y. Ne’eman, Dynamical groups

M2= — ~i~9/8Ø. Eq. (34c) is then immediately integrated and one gets

and spectrum generating algebras (World Scientific, Singapore, 1989). [21N. Kamran and P. Olver, J. Math. Anal. AppI. 145 (1990); M.A. Shifman, I. J. Mod. Phys. A 4 (1989) 2897. [3] E. D’Hoker, V.A. Kostelecky and L. Vinet, Spectrum

2im2i~Wh,o,m

=e Separating the variables in
ØIRIh, (5, m2> =h

one finds

~1f — Id r d — ~

= h ~PhOm2

~

(39)

2(r)

,

(40)

2

+ —j~ 4m + r2) + 2(5] ~~hOm2 ( r)

( r)

(41)

The solutions to this equation can be obtained either directly or by applying ladder operators on the ground state wave functions. They are given by ~1h,8,m2

( r) = C~r2’~ e — r/2

2) , (42) 2 — ~(h—2ö— 1), 2m2 + 1; r with C,, normalization constants and F the confluent hypergeometric function. The eigenfunctions ofthe Morse Hamiltonian can now be obtained from (42) by a mere change of x

304

F(m

generating superalgebras, in: Dynamical groups and spectrum generating algebras, eds. A.O. Barut, A. Bohm and Y. Ne’eman (World Scientific, Singapore, 1989). [4] E.P. Wigner, Phys. Rev. 77 (1950) 711. [5] Y. Ohnuki and S. Kamefuchi, Quantum field theory and parastatistics (Springer, Berlin, 1982). [6] V.A. Rubakov and V.P. Spindinov, Mod. Phys. Lett. A 3 (1988) 1337; S. Durand and L. Vinet, Conformal parasupersymmetry in quantum mechanics, preprint UdeM-LPN-THO8, to appear in Mod. Phys. Lett. A (1989); S. Durand and L. Vinet, Dynamical parasupersymmetries in quantum mechanics, preprint UdeM-LPN-THO7, to appear in: Pr,oc. V J.A. Swieca Summer School on Particles andfields (World Scientific, Singapore, 1989). [7] S. Durand, Supersymétries des systémes mécaniques nonrelativistes en une et deux dimensions, M.Sc. Thesis, UniversitédeMontréal (1985); J. Beckers, D. Dehin and V. Hussin, J. Phys. A 21 (1988) 651. [81S. Durand and L. Vinet, Dynamical parasuperalgebras of parasupersymmetnc harmonic oscillator, cyclotron motion and Morse Hamiltonians, preprint UCLA-89-TEP-62/ UdeM-LPN-TH14, submitted to J. Phys. A (1989).