Nuclear Physics B245 (1984) 17-44 c North-Holland Pubhshmg Company

S U P E R C O N F O R M A L Q U A N T U M M E C H A N I C S

s FUBINI

CfiRN, Geneua, 5wltzerland

E RABINOVICI*

Rajah Instttute of Ph)~t~ ~ Jerusalem, l~rael

Received 5 April 1984

N = 1 and N = 2 superconformal quantum mechanics are formulated and sol~ed Sponta- neous symmetry breaking occurs A de Sitter subalgebra is used to define new supersymmetnc evolution operators The modified system has a discrete spectrum exhibiting ne,~ supersymmetraes The N= 2 extended de Slner superalgebra is spontaneously broken to N = 1 The relation to the behawour of flmte field theones and Poancare ln~anance 1~ discussed

1. Introduct ion

The discovery of finite supersymmetrlc field theories [1] is of great interest In

such field theories, one can dispose of the infinite renormahzat lons with which only

a cold detente has existed dur ing the years If these theories have also non-per tu rba-

tlvely a zero beta function, one can calculate exactly all Green funct ions The finite

theories have an addi t ional property, they are conformal ly symmetric In the ca~e of

renormahzable theories, which were conformally lnvar lant at the classical level, the

conformal symmetry was sacrificed in favour of Lorentz invarlance in the process of

renormahza t lon For finite theories, it may seem that one can have the best of both

worlds Could this indeed be the case 9 Whale th~s theory is well defined m the

ultra-violet, it may suffer from mfra-red problems which would manifest themselves

in the properties of the ground state

Fayet [2] has studied the consequences of a spontaneous breaking of the 0(6)

symmetry of the N = 4 supersymmetric model In that case the colour slnglet sector

massless sector of the system would consist, between others, of s~x spin-zero particles

(a dl la ton and the five Golds tone bosons of O(6)/O(5)) , four Majorana fermlons

and one gauge boson 0(6) and SU(N¢) are broken, supersymmetry and Polncar6

invar iance are main ta ined Using a large-N expansion, Bardeen, Bander and Moshe

* Work supported in part by the Amen~.an-Israeh Bi-National ~c~ence foundation

17

18 Y Fuhtm, L Rahmot{tt ~,upet(,m/otmalquap~tumm~dlam<~

[3] (BBM) have studied q5 ~ in three dimensions Perturbatl~el~, the theory has a zero

beta function They have shown that ~hlle the vacuum seems to be ill defined

perturbatlvely for large enough couplings, one could stabxhze the ~acuum b) having

the beta function non-zero above a certain coupling The conformal symmetr) ~s

broken beyond that couphng In the supersymmetrlc ~ , the) have obtained results along the lines envisaged by Fayet

A rather different approach was proposed by one of us [4] Since the Pomcard

group is just one of several subgroups of the conformal O(D,2) group, where D is

the number of space-time dimensions, ~t was suggested that there ma? be circum-

stances under which it would become advantageous to pursue the description of the

physics of the system in terms of other subgroups of O( D, 2). such as O( D - 1,2) or

O(D, 1) This cautious description results from some difficulty m using the current

terminology to describe the breaking of Lorentz lnvarlance

Let us explain the ground state wave functlonals o| the various theories are

generally unknown It may be that a normahzable ~acuum does not exist m such theories This has been shown by de Alfaro-Fublm-Fur lan [5] (DFF) to be the case

m conformal quantum mechanics and has been conjectured to occur in general

conformal lnvarlant models as well as in the tyro-dimensional Liouvmlle model [6]

In the absence of a vacuum, one may wish to mvesugate the consequences of

adding an additional prescription to the theory, such that the new theor) has a

well-defined vacuum That addmonal rule .sacrifices th~s nine the Pomcare lnvarl- ance in favour of maintaining a well-defined vacuum and a de Smer subgroup The

details of the construction appear in refs [4,5] We shall recall some of these

construct ions later

It is clear that the procedure breaks scale and Polncard lnvarlance, but ho~ should

we characterize this breaking If we concentrate on the properties of the breaking, ~t was shown that these theories have an algebraic structure which results m low-energ~ theorems, not unlike the low-energy theorems of current algebra In th~s sense, the

resulting symmetry breaking in the theory is spontaneous, however, ~t ~s not

spontaneous in the sense of having a theory m which it is necessary to choose

ab lnltlO between one of many ~acua In any case, the procedure will also have an impact on the supersymmetr> of the

system, this has been studied by de Alfaro-Furlan [7] (DF) m the days when supersymmetr) was originally proposed We wish now to return to this ~ssue in view

of the emergence of quantum conformall5, lnvarlant field theories In order to learn the possible emerging structure, we study a well-defined soluble model, which

possesses both conformal and supersymmetrlc lnvariances Such a model is confor- real supersymmetnc quantum mechanics, which is the bupersymmetrlc extension of the conformal quantum mechanics which was studied by D F F [5]

The results in quan tum mechanics are interesting in their own right, we will also try to suggest what interpretation should be given m a field theoretic context The

structure of the paper is the following m sect 2 we describe the conformal

S Fubml. E Rabtnot,t~t / Superconyorrnalquantum rne~ham(~ 19

supersymmetrlc structure of the quantum mechamcal model In sect 3 we discuss

the N = 1 case m the presence of an internal quantum number and an particular the

case D = 2 m which a larger group of supersymmetry is present Under both circumstances, the original supersymmetry of the model is spontaneously broken

We will describe how the resulting level structure may also have a new supersymmet- nc interpretation

2. Quantum mechanical conformal supersymmetry

In this section, we wish to discuss the fusion of conformal lnvarlance and

supersymmetry as ~t occurs m quantum mechanics Conformal quantum mechanics

has been studied m great detail from the point of interest of this work by DFF [5]

We next review the essentml features of the model The lagrangmn describing the

system is

where g ~s the d~menslonless couphng constant The action of this system ~s

lnvarlant under the conformal group O(2, 1) which is spanned by three generators,

H the hamlltonmn, D the d~latatlon generator and K the conformal generator

These generators form together the algebra

[H, D] = tH, [K, D] = - t K , [H, K] = 2ID (2 2)

The harmltoman describes the motion of a particle m a repulsive potential and is given by

The hamdtonlan can be dxagonahzed exactly The elgenspectrum includes all E > 0

values for each of which there exists a plane wave normahzable state The spectrum

does not have an endpolnt or ground state as the state E = 0 as not even plane-wave

normahzable We do not know of a principle which requires that a system always

needs to have a ground state. However, our understanding of field theory was greatly

enhanced by studying the symmetry properties of the vacuum and excitations around it It is possible to add a new rule [4] that would enable the choice of a ground state, that rule, however, results in the breakdown of translauonal lnvariance

in the time t direction The rule chosen by DFF [5] was to notice that the quantum evolution of the system can be controlled by a compact operator Such an operator would have normahzable eIgenstates in general, and a ground state in particular Using the three symmetry operators, the compact operator can be chosen from any

2O

linear combmauon, G, ol the generators

G= uH + ~,D + )~A,

as long as

'} Fubtm, L Rahtnoltct / bupe:(¢m[otmalquantum mcdlamt~

(24)

r o = ~ ( l + v g + ~ } (29)

The mtroducuon of the linear comblnauon (26) between H and K guarantees that

all elgenfunctlons are indeed normahzable We are interested m studying the supersymmetrlc version of contormal quantum

mechanics To achieve this we shall make use of the general construcuon by Wltten [8,9], wl-nch enables one to convert any quantum mechanical model into its super-

symmetric counterpart The supersymmemc algebra ~s given b_~

~{Q,Q+}=H,

{ Q , Q } = { Q ' , Q ' } = O (2 lo)

A minimal reahzatlon of this algebra is given by a point particle in one dimension

where r o is given bTy

.~ = ~'-' - 4uw < 0 (25)

Any such choice would break translational and scale mvarIance DFF hax e lound it convenient to define

G = R = ~ K+aH), (26)

where a is a constant (with the dimensions of length) whose introduction breaks

scale lnvanance For simplicity, we shall from now on keep a = 1 Making use of the algebra

L =

[ R , L

[ L , , L ] = - 2 R , (2 7)

it has been shown that the mgenvalues of R are given by a discrete series

r,,=lo+n, n = 0 , l ,2 (2 8)

S Fubmt E Rahmot ut / ,5uper(on]otnla[ qttantum me(hamts 21

which has two degrees of freedom

dW

( dW) 0 + = 4 tp+~-~ . (2 11)

where p = - t ( O / O x ) , the superpotentml W(x) is any function of a and the spmor operators 4 and 4 + obey the antlcommutatlon relations

{ 4 ~ , 4 } = 1 (2 12)

We define

B = ½ 1 4 + , 4 ] (213)

B is the generator of the U(1) transformation 4-~e"~4, 4+-- ,e '"4 + and has etgenvalues + ~ and - ~ We see that the algebra of 4 +, 4 and B is isomorphic to that ot the Pauh spin mamces

4 - ~ }( o~ - ; o 2 ) ,

B --* ~o, (2 14)

The hamlltoman governing the t~me behavlour of the system is

dW _ 2Bd-W 1 (215) H=!{Q,Q+}=½ p2+ ~ d.t3 I

The conformal supersymmetnc quantum mechamcs is g~ven by the superpotentJal

W(x) = ½flogx 2, (2 16)

where f is the &menslonless coupling constant In th~s case, the system has a richer algebraic structure, ~t is the superconformal algebra of Haag, Lopuszanskx and Sohnms [10] This leads to the introduction of an extra pair of spmor operators, S and S + which are, in some way, the "square roots" of the operator K [11] We define

S=~,+x,

S += 4x (2.17)

22 S Fuhrer, E Rahmot,t(t £upet,.(n, tlotmal qua~,ttum mcHtant,.',

The enlarged algebra is now given by

~{Q,Q*}=H, 4 { S , S ' } = K ,

4{q ,s ~ } : ½ ) ' - ~ B + tD,

! { Q ~ , S } = ~ J - } B - t D

All other anucommuta to r s hke { Q, Q }, { Q, s }, etc , ~ amsh

H= ~( p2 + f2 +2fB

K=+~-,:, D : - ~ { , p},

(2 181

Let us now examine m whmh way the system reahze~ the supersymmemc algebra

Because of the relation H = } { Q, Q~ }, supersymmetry is unbroken, i e Q and Q

anmhllate the vacuum if and only if the hamll toman annlhdates the vacuum as well

In other words, supersymmetry is broken if the system does not have a normahzable

zero energy state We already know that m conformal quantum mechamcs, the zero

energy state ~s not even plane-wave normahzable Endowing the system v~th

supersymmetry does not mtroduce a taming scale We would thus suspect that the zero energy state will remam non-renormahzable This is indeed checked by an exact

calculatmn Wltten [8] has shown that a zero energy state always exists and is given

by solwng the first-order differential equahons

which can be expressed as

Ol,/,> : Q+ I,/,) = O,

d W + 2tBp )] q"} = 0 da

The general solution of eq (2 20') is

I ' / ') = exp( -2BW(a ))1 ) ,

where t

We thus ha~e the a dependence

(2 2 l )

) is any .x independent two-component state vector In the conformal case

I'/'(-~)) = x :/~1

for B = !

' / ' - x / for B = -

(2 20)

(2 2o')

S Fuhml E Rahmot,t¢t / 5uper~onfotmalquantum meUmm(~ 23

Neither choice of B can result in a normahzable state the wave function will blow

up for either large or small values of x Supersymmetry ~s broken for superconformal

quantum mechanics

In fact, superconformal quantum mechamcs can be solved exactly m the same wa)

as conformal quantum mechanics For a posmve f , one looks at the B = ~ sector, m which the mgenvalue equation reduces to that of the conformal quantum mechanical

case with g = f 2 + f The B = ½ sector has a positive-definite spectrum, every E > 0 is an elgenvalue with a corresponding plane-wave normallzable Bessel v~ave tunctlon

~¢1 ~2j/+l ,2(A 2V/2V/~) AS H commutes with Q, each B = ~ state has a B = - { state

w~th the same energy as the supersymmetrlc partner At th~s stage, the dlagonahza-

non of the hamfitonlan has been completed Once again one is faced w~th an exactly

solvable system whose energy spectrum is E > 0, the ground state at E = 0 being

non-normalIzable

We follow here the point of view of DDF that it is possible to study the nine

evolunon of the system by means of a compact operator that has a discrete set of

normahzable mgenfunctlon It is first mstrucnve to dlagonahze the operator R

defined by eq (2 6).

R = ½( aH + I K )

where a 1s a scale factor required by dimensional analysis which i8 set to be unity As already mentioned, eq (2 8), the R spectrum in conformal quantum mechanics

is equally spaced This equal spacing is the quantum mechanical analogue of the

low-energy theorems related to the breakdown of a symmetry, in this case the

dilatation lnvarlance [4]. In the case of conformal supersymmetnc quantum mechan- ics, the appearance of a zero energy fermlonlc excitation (the quantum mechanical

"Goldstone fermlon") was already apparent m the spectrum of H Once R is

dlagonahzed, an exphclt difference between the energy necessary to excite bosons

and fermlons is expected

Indeed looking at the form of the hamlltonlan given in eq (2 19), we see that it

reduces to the standard form (2 3) with g = f ( f + 1) for B = 4 and g =J(J - 1) One can thus apply eq (2 8) to both cases The lowest level, given by eq (2 9), is given by

r ° = ½(~ + f ) for B = ~ and r ° = ~(} + f ) for B = - ½ respectively We see that two such series corresponding to the two different fermionlc numbers are shifted by This semi-integer shift can be understood algebramally in the follov~lng manner

Dehne the fermlon lowering and raising operators by

M = Q - S , M + = Q + - S + ,

N = Q + + S ~ , N + = Q + S, (2 22)

24 s Fuhrer, k. Rahmot I~l , Supcr~on/ormal quantum maham¢

whach obey the antmommutator algebra

¼ { M , M ~ } = R + ~ B - ½ f = T 1.

¼ { N , N ~ } = R - ~ B + ~ j = T,,

¼ { M ~ , N +}=L+

The exphc~t form of our operators as

dm

dn ),

to be compared to

where

dW

M + = ~ ( tp +-~),dm

( N ~= ~ ~ - tp + ~-~ ,

dW

r e ( x ) = _:e),

n ( x ) = ~ ( f l o g x 2 + x2),

(2 23a)

(2 23b)

(2 23c)

(2 23d)

(2 24)

~4{ N,N+ }= T~

These relations follov~ from a de Sitter subalgebra present m the superconformal algebra [11] (eq (2 18))

The doublet structure of the mgenstates of H follows from the fact that H commutes w~th Q and Q + In a similar wa~, a doublet ~tructure for T1 follows from

¼{ M , M ~ }= TI,

¼ { Q , Q ' } =½H,

We are now ready to study the level structure of our system, m particular the doublet structure present m standard supersymmetry Let us first nonce a striking algebraic analogy between the operators defined m eq (2 24) and the ongmal operators of standard supersymmetry Th~s ts embodied m the formulae

W( "~ ) = ½f log :~ 2 (2 25 )

S Fuhrer, E Rabmovl¢t / Superconformalquantum mecham~;

and a doublet structure for T 2 follows from

[T2,N]= [T2. N+] =O

25

It is, however, important to notice that when we look at the expllclt expressions (2 24) and (2 25) of the three sets of operators, we find them to have a different type of spectrum. This is due to the three different forms of the superpotentlals W(a), m(x) and n(x) . Indeed when we search for the lowest elgenstates of the operators H, T 1 and T 2, we have to consider wave functions whose space dependence should be e -+ ~, e + " , e + n respectively We see that out of the six possible choices only

e m ( ~ ) = x / e - ,:'-/2

gives rise to a normallzable wave function We thus conclude that out of the three operators, only T 1 has a normahzable

lowest elgenstate [ ' / ' ) obeying

M[ q ' ) = M + [g ') = 0, (2 26)

whose solution is

]g ' )=e 2m~)B[ ) (2 27)

In order to get a normahzable state, we chose I ) = [4 ) where ]+) is the no-fermlon state corresponding to B = - ~ We thus have

I~)=em¢~)[ $ )= xfe 'L '215) (2 28)

The level spectrum of the operator T 1 can easily be obtained by considering the two operators N and N + which play the role of lowering and raising operators respectively, since

[T1,N]= - N . [T1,N +] = N * (229)

It ~s easy to verify that

N I ~ ) = 0 (2 30)

The spectrum 1s thus given by

T 1 = 0 ,1 .2 ,3 ( B = - ½ )

= 1 . 2 . 3 ( B = ½ )

A similar role Js taken by the operators L and L+ with commutators

[T 1.L ] = - L , [T 1 , L + ] = L + (231)

26 5 Fuhtm, F Rabtnm'tu / ,Mq~et(,m/ortnalquantum metham(s

Nonce that L+ raises T t by keeping B constant whereas N- does it by changing the value of B The operators N +, N can be v~ewed as creanon and destrucUon operators of a Goldstone ferm~on whereas L+ and L as creatton and destructton operators of a Goldstone boson

It xs useful to describe the spectrum m terms of the original operators R used in DDF We use the relation (2 23a) between /'1 and R and recall that a/~ and 84 decrease R by ~ whereas N +, M + increase R by

J R , N ] = - ~ N , [R, M ] = - ~ M ,

+

The lowest mgenstate I'/') gtxen by eq (2 28) corresponds to

The higher levels are obtained by repeated appllcanons of the M +, N'

We thus get

B= - : , J + ~ ) + n ,

B = l , R:~(/+~)+~+,z

(2 32)

(2 33)

operators

(2 34}

The level structure of the operator R Js depicted in fig 1 Having solved the model

i x

I

I

I

- 1 / 2

f I I L1 I I r

i _ _ - 4

I ro i

1 /2 - - B

Fig 1 The s p e c t r u m of R for ,'~ = I as a l u n c n o n of B The sp t .c t rum ha' , no degcne rac 5 The. i n l n l m u m xalue, rl~of R ts r o = i l l _ : + ) ' ) The le'~el spac ing for a fixed B (bosonu. ex~.ltatlon) is ore. u m t The lcxel

spac ing for k B =~ 0 ( f e rmlomc ex~.ltatlon) is ha l l a u m t

S Fubmt, E Rabmot, ttt / Superconformalquantum tne~hantc~ 27

exactly, we summanze the results displayed in fig 1, by viewing it from different

angles

In conformal supersymmetrlcal quantum mechanics, supersymmetry ~s sponta-

neously broken, the hamlltonlan has a continuous E > 0 spectrum Under such

circumstances, it is useful to observe the time evolution of the system using as a

complete set the elgenfunctlons of a compact operator In general, the diagonah-

zatlon of any compact operator will do, however, under particular circumstances, it may turn out that using different operators may emphasize different symmetry

structures of the model Such circumstances arise in the case at hand as we v~ew the

system through elgenfunctlons of the operators B, T 1 and T 2

The first way is described in fig 1 itself, the spectrum of R consists of two B

sectors In each B sector the bosonlc excitations are equally spaced by one umt The

"Goldstone boson" operators L+, L relating the states Fermlomc excitations are one-half of a unit and are carried by the "Goldstone fermion" operators N, N +

resulting from Q supersymmetry breaking

The second way is to assign to T I the role of the observation operator In that

case, whale dilatation invarlance is still broken, the supersymmetry generated b2¢

M, M + is unbroken The states will exhibit a supersymrnetrlcal structure This new supersymmetric operator mixing Q and S is responsible for the twin degeneracy of

all non-zero T~ states There exists a single state of zero T x Bosonlc and fermlomc excitations from the ground state are'equal and non-zero The equal level spacing is

again unity The equal bosonac spacing is the analogue of the low-energy theorems

related to the breakdown of dilatatlonal invariance The ferrmonlc equal spacing is

the analogue of a fernuomc companion of a Goldstone bosom such companions

appear when supersymmetry is unbroken

The third way to view the system is through T 2 In that case, the system is equally

spaced by one unit, however, there is no normahzable zero T~ state The supersym- merry apparent m that form, N, is spontaneously broken The ground state is to be

chosen between the lowest T~ states having B = - ~ and 1 _ 2, once the choice ~s made,

a zero "T 2 energy" fermlomc excitation connecting the ground state and ~ts degener- ate partners, exists This corresponds to a "massless'" Goldstone fermlon of N

supersymmetry breaking The equal bosomc spacing has already been discussed The different points of view are shown in fig 2

The basic features uncovered seem to be rather general, in order to be certain we

shall study conformal supersymmetrlcal examples in which the particle may mo~e in more than one dimension, we shall also study an N = 2 system in which one expects to encounter difficulties when attempting to break supersymmetry A more technical reason for extending the study is related to the fact that the varmble x can acl~eve all positive as well as all negative values We w~sh to be certain that some regularlzatlon prescription at the ongin will not invahdate the qualitative results

W~th a larger number of dimensions, this problem would not occur as the radial vanable ~s never negative

28 S Fuhtnt L Rablnm,t~t ,/ ,Mtpetum/ormalquantum mcthattlt

R

I

) I

t.J' I

TI= I I i

[ - ~ l

Tl= 0 ~ /

-1/2

T2= const

I

Tz= const

I

I a

1/2

Fig 2 Lines of hxed T l and T 2 as def ined m eq (2 27) Fhe opt . ra tors ,V and ,", ) connec t twain, ol a

f ixed T 2 The degene racy of the lowest T, s tate reflects the b r e a k i n g ol ~ ~uper~vmmetr 'v a n d th~

ex is tence of a " G o l d s t o n e f e r m m n ' (a massless f e r m m m c excatat~on) The Watten index of T~ Ls z~ro The

o p e r a t o r s M a n d ~t* connec t twins of a fixed T 1 The lowest T l s ta te is not degene ra t e Ot ha~ T 1 = 0)

Th i s ref lects the u n b r o k e n M s u p e r s ) m m e t r l c Exci ted T I a n d T, s ta tes a p p e a r in twm~ ThL Wl t t en

index of T t t s one L + o p e r a t o r s ~.reate bosom~ ex~_~tat~ons

Let us end this section with an amusing technical remark related to the Wltten

index [8] The Wlt ten index is dehned to be the number of bosomc zero energy

solut ions minus the number of fermlonlc zero energy soluuons When supersymme-

try ~s broken, there exists no zero energy state and thus the Wxtten index ~b zero If

the Wlt ten index ~s non-zero supersymmetD ~s unbroken The Wltten index may be

zero also m a supersymmetncal case Assume that f were n e g a m e then it would

appear that the state 1 0 ) = x I t l exp ( - , c ~) ts sUll normahzable a~ long a~ Ill < ~.

thus the model would haze a supersymmemc t r ansmon at Ifl = ~,. the Wltten index

[8] is zero m the broken supersymmetrlc phase and ~s non-zero in this case in the

s u p e r s y m m e m c phase It would thus seem that by changing f (and not at f = 0) one

can change discont inuously the W~tten index Hov~ever. one shall note that when the

state {0) ]s included in the Hdber t space, for 0 > J > - ~. the hamdtoman ceases to

be h e r m m a n (and physical) as long as there ~s no source at the origin and thus the

index theorem need not hold for it

3. Superconformal quantum mechanics in many dimensions

3 1 T H E O E N E R A L C A S E

The supersymmetnc quan tum mechanics of one particle in many &mens[ons (or

al ternatively one particle ~ l th many degrees of freedom in one dimension) is pobed

S Fubmt, E Rahtnocl~l / Super~onformalquantum me~hum~ 29

In a similar manner to the supersymmetric quantum mechamcs of one particle in one dimension [8, 9, 12] The algebra as stated in eq (2 10) is unchanged The operators reahzmg the algebra are generalized Eqs (2 11)-(2 13) are generalized to be

Q= ~.. 4,+ - t p .+ , a = l

°( t oc=1

H=~(Q'Q+}=~EP~+,~ ~ -ZB~"oB O:,~Ox#" (31)

where 4"s obey the antlcommutauon relanons

{ 4,+'4,B} = 6,~B' { 4,,~, 4,/~ } = ( 4,+. 4,~ } = 0 . (3 2)

and B~# is defined by

Bo~ = ~[4,; , 4,~] (3 3)

We chose the superpotentlal W(a,~) to depend only on x z= ,%a. In th~s case, the theory will be lnvarmnt with respect to simultaneous rotations of x., p,,. ~p~. 4,+° The theory will also be anvarmnt w~th respect to inversions, ~e under transformattons hke

'c i - - ' - - 'q , P l ---' - P ~ , 4 , i --" - 4 1 , 4 , { - - ' - 4 ;

We now add the fundamental reqmrement of conformal mvarlance The superpoten- tlal is then umquely defined as

W(a,~) = ½f loga -~ (3 4)

In this case we can introduce the extended algebra of eqs. (2 17)-(2 19) where the relevant operator will now have the form

s=E4,2xo, s+=E4,.xo,

= {p2 f2 2x,,xt~- 26 } H ~ +- -+2 fB~# x ~ ,-~- ~0 " (3 5)

where

B = EBoo (3 5')

30 S Fubmt, L Rablnot,t(t / Super(onfornmlquantunl me¢llant(~

We w~sh to show that. as in the previous case of sect 2. the hamfltonlan has only non-normahzable zero energy states Let us discuss, following Affleck [12]. the general method of obtaining solutmns to the ground state equatlonb

QI,/') = Q~ I,/') = 0 (3 6)

We concentrate on O(D) mvanant states, we also d~stlngmsh between even ( + ) and odd ( - ) states which decouple because of the panty conservation We start by' defining the empty and the totally full states as

+.105 = O, +; ID} = 0 (3 7)

We also mlroduce the two e~en scalar states

and the odd doublet

15 + > = r la t"zl0 > [ - % = " "

I1" + > = r ,o l),2(.~,~+.)10)/ ix[ (3 8)

d ( 2 h ~

where b is an operator defmed by

dw+ +- -~- r ) l ' / ' ~ )=O, (311)

bid>= -~1~5,

and the effecuve potenuals w ~ are given by

w~= + ½ ( D - 1) logr+ W(r t

We thus get the general soluUons

I'P+> = exp ( -2b~ +)l + >,

I'/" > = e x p ( - 2 b w ) 1 - 5 .

h l T ) = 4 l $ ) , (3 12)

(3 13)

(3 14)

1.1. -- ) = r ,D 1) 2 ( . ~ . , ~ . ) [ D ) "

[ 1" - ) = r {D 1) 2 [ D ) (3 9)

We can now introduce the most general even and odd states I'P~ / v)) and I'/' ( x )) defined as hnear combmaUons of [ 1" > and I $ }

['/'+(x)> =so. , ( , )l ~ + > +SO~T(t)IT +

['/" (x)}=SO , ( r ) I $ - } + S O T ( r ) l l " - (310)

We transform eq ( 3 6 ) into exphclt equahons for the So funchons

S Fubmt, E Rahmovtu / Super(onformalquantum rne(ham¢~ 31

where [ + ) and [ - ) are r independent combinations of I1'_), [$ +) and I $ ), I $ +) respectively

If we apply the general formulation to our conformal choice (3 4), we get for the effective potential

w+= ( f + ½ ( D - 1)) logr ,

w = ( f - l ( D - - 1))logr. (3 15)

This shows that the lowest state wave functions are in general non-normahzable and exhibit the famlhar power behavlour

I'kIt+ ) = r-2h(f+ t~(D-1)) I q- )

[,/, ) = r - ~ h ( / ~(D l))[__) (3 16)

Supersymmetry is spontaneously broken. In conclusion, as in sect 2, we are led to study the evolution of the system in

terms of a compact operator The fundamental operators M and N will now have the form

where

and we have again

Om " - E <

ol

Om M + = ~ p , ~ +tp,~ +--Ox,,

m(x)= l ( f l ogx2 -x2 ) ,

n ( x ) = ½(f logx z + x2) ,

(3 17)

(3 18)

¼ { M , M + } = R + ½ B - ½ f = T ~ (3 19)

32 S Fubmt, E Rabtnotl~t / 5uper~onp.mal quantum mechant{

We apply Affleck's procedure to obtain the zero 1] state solutmns of T~lg" > = 0, ~ e

MI '/'> = M ~ I'/'> = 0 3 20)

We finally get

where

1'/'+> = exp ( -2bm+) l + >,

J~/" > = e x p ( - 2 b m ) ] - ) (3 21)

m . = { f + ½ ( D - 1 ) ) l o g r - ~,r 2 3 22)

Again, m both cases the normahzatmn reqmrement forces b = - ~ so that we finally get

i , / ,+>=r(l+' ,~o ~% / 21+ + ) ,

]q, > = r l / ':l~ 1~1 e r: 2 1 ~ _ > , (3 23)

l e

I ' / ' +>=r l e r'~ Zl0>

I'P > = r / tD 11(+~,¢G) e ; 2[D > (3 24)

It is important to notice that there is a substantial difference between the even and the odd zero eigenstates of T 1 Indeed, from eq (3 24), we ~ee that whereas I'/" ) is normahzable for all posmve values of f , I'I" ) starts being normahzable only for f > D - 1 (recall that the radial normalization integral is f'/" ~'pr lJ 1dr) M super- symmetry is unbroken

The two lowest levels I'P÷ > and I'/" ) are, of course, also e~genstates of R Using eq (3 13) and the fact that

BI'P~ > = - ~DI'P~ >,

B['/" > = ½ ( D - 2 ) I ' / " 5, (325)

we obtain

R] ' / '+) = ~ ( f +½D)lg'÷ >,

RI'/" > = ~ ( f - ~ ( D - 2 ) ) [ ' / " > (326)

S Fubml. E Rabmocul / Supereonformalq~mntum me, bantus 33

Star t ing f rom eq (3 26), using the same procedure of last section, we can cons t ruc t

the spec t rum of scalar mgenstates of R.

B=-½D. even,

B = - ½ ( D - 2 ) . even,

B = ½ ( D - 2 ) , o d d ,

B = ½D, o d d ,

32 THE D = 2 MODEL

In the case of D - 2, we have

B = - I , even,

B = 0, even,

B = 0, o d d ,

r,,=4(f+{D)+n,

r,,=½(f+½D)-}+n,

r,.=½(f-½D+ l)+n,

r,,=½(f-½D+2)+n (3 27)

rn=~(f+l)+n,

r~=~f+n.

r. = ½ f + n ,

B = I , odd r.=4(f+l)+n ( 3 2 8 )

We see that for D = 2, the mgenvalues of R co r respond ing to B = - 1 and B = 1 and

the two B = 0 states are the same, it would be easy to check that the elgenstates have

also a Slrmlar form This suggests that m the case of D = 2, we are in presence of a

new kind of symmet ry re la t ing even to odd states We want to show that th~s new

p rope r ty amount s to an ex tended N = 2 supe r symmet ry

Let us first ident i fy an internal SU(2) group, not unhke quas>sp ln [13] m nuclear

physics, whose genera tors are

~- = q' lq 'e, Y + = - + ; + J ,

B = 1 ( [ 1 ~ , I,~L,1] q- [ ~/J2 , 1~/2] } , (3 29)

whmh obey the SU(2) a lgebra

[B,Y+I=2Y +, [B,Y]=-2Y,

W e also need the commuta to r s

[r +, r ] = ~ (3 30)

[ r , q,2] = - ~oBq',, = -~7~, (3 31)

3 4 S Fubtnt. L Rabtnot't~t ,~)upet~onJorma[ qlta~ttl#Tt me~ hantt s

~here r./~ ~s the 2 × 2 antIsymmetrlc tensor

[ Y. B.~] = Y6.l~,

[ w , eo~] = - ~ "~or, 13 32)

It is now important to introduce the unitary operator U which performs the

t ransformauon

t __ - - 4 U I~,~U = Eo~[3~[ ~ = l . ~ a , ~_, l},~ U = e,,/~//~ - q ; . . (3 33 )

we also have

U 1 y u = - }'~ , U 1y~ U = - }', U 1BU = - B

If we introduce the isomorphism with angular momentum operator~

B ~ 2 o ~ , t ' ~ o ~ . Y ~ o

eqs (3 34) read

(3 34)

obtains

[B,H]=0,

[ ¥ - ' 1 = - 2 r ~ ,

[r +,H]=~-~ ~D~ (3 36)

In the generalized conformal case, one has

D - 2 D W = / - - (3 37)

A 2

In particular, in the case of D = 2 one has ~14"= 0 1he same result would be correct

for any analytic function W Thus, for analytic functions the quas~-spm ~s an extra

U ~ o l U = o x U ~ o . U = o ~ . U t o , U = - o ~

The umtary t ransformation b' represents a 180 ° rotation around the (2) axis

We finally have the fundamental t ransformanon property

U ~B,q~U = -~,,z~B + B./~ (3 35)

Let ub now look at the effect of our t ran~lormauon on the hamlltonian One

S Fublm. E Rabmo~t~t / Superconformulquantum me~ham~ 35

mvarlance group of the system. This can also be seen directly by

U-IHU= H + 2B E3W (3.38)

Again for E3W = 0, one obtains

U IHU=H (3 39)

Our extended supersymmetry is now obtained by combining the usual operator Q, Q+, S and S + with the generators of our internal SU(2) group The N = 2 extended supersymmetry group will be obtained by assocmtlng to each even supersymmetry operator A and odd partner A defined as

.,T= U 1.4 U (3 40)

In this way standard N = 2 supersymmetry will contain besides Q and Q+

Ow

Q=Y~'~['P'~+\ Ox,~OW)=~'e'~fl~P~( tp'~+ ~OW) (3 41,

One can also cast the operators Q in a form more readily related to an N = 2 model m two dimensions, from which N = 2 quantum mechanics can be obtained by dimensional reducnon [14-17] That is

OW '~x2 (~1+~_) Ql=Q++t-Q=t(pl +lp2)(~Pl-t~b2) + ~x 1-

( ow aw ) Qa=Q+-'-Q=I(P~-~P2)(~ +'4':)+ ~ + ' - ~ 2 (4'1-'~2), (342)

where W is an analytic function Let us now discuss the extended supersymmetry algebra containing the operators

Q, Q ~, Q, Q+ The only non-vamshlng antlcommutators are

¢(Q,Q+}=H, ~(Q,Q+}=H (343)

From eqs (3 42), we learn that Q~ Q+, Q, Q+, all commute with H We thus e,~pect for a general N = 2 model [14-17] a quartet structure, each quartet containing two even and two odd levels The zero energy states

HI ' / ' ) = 0 (3 44)

36 ') tZuhmt E Rabmovtcl ~uperconjortnalquamum medumtcs

have to obey the equations

Qt ' / ' ) = Q+ I'/ ') = Q[q'> = Q ~ ]q'> = O, (3 45)

and are, in general, I $ ) and 15 ~ ) type states Let us now make the usual choice

W = {j' log a :

In this case, the theory wdl be mvariant with respect to 0(2) rotations of x,, p,, ~,, ,~/ The generator of this group will be the angular momentum operator

J = P l X 2 - P 2 ~ l +/(~b~ ~ 2 - ~-~1) (3 46)

We shall then obtain N = 2 supersymmetr? which contains the extra operators S, and S, S + given by

g = ~ t ~ , % , ~-~ = ~+~: ,~ (3 47)

We already know from our general analysis of the N = 1 model that m the case ol

superconformal quantum mechanics, supersymmetry is broken and there ~s no

normalizable ground state This result Is unaltered by the discovery that for D - 2,

the model has extra symmetries Note that, in general, tt is xery difficuh to break

N = 2 supersymmetry [15] We are faced again with a model which has a spectrum which is bounded from below but has no ground state Supersymmetr? is broken

spontaneously We are thus led again to the algebra of the M, N operators together with their odd partners M, N We shall discuss the general antlcommutator algebra

in terms ol those operators It ~s convement to classff3 them into two lamllles (1) M M - N N ~

~ { M , M ' } =R + ~ B - !]= T~,

4~{N,N + } = R + ! B + { l= 7",= Tt+ j ,

~{M , •}= , J (3 4s)

(2) MM ~ NN ÷

~ { M , M + ) = R + ~ B - ~J= T~,

~4{N,N +} =R + t ,B+~]= T~= T l + j .

1 ~ + 1 - - + ( 3 4 9 ) ?{N, } = - ~ ( M , N } = - t J

The non-vanishing anticommutators between members of the tyro different famahes

Fllblnl, E Rabmoct~t / Super{,m/otnta/ quantum mc~hant{ ~ 37

are

½ { M , M + } = - ½ { N , N + } = Y ,

~ { M + , M } = - ½ { N . N + } = Y ~

- ¼ { M , N } = - ~ { M , N } = L ,

- ~ { M + , N + } = - ¼ { M + . N + } = L +

Let us now proceed to discuss the level spectrum of our problem with the supersymmetrlc version of our model (eqs (3 44) and (3 45)) suggests that we shall have again a quartet structure Indeed, ff we consider the operator

T 1 = R + 4 B - ½ f ,

we do have a quartet structure since it commutes with the operators M, M + N, A/4, two of the levels are even, two are odd However, the analogy w~th the hamdtoman is not complete since, whereas

I { M , M + ) = r~,

we have the analogous anticommutator

] { N , N + } = T a + f

Th~s means that the lowest level condmon

T~ [ 'P) =0,

does only imply

(3 50)

The analogy

M I ~ ) = M+ I,/*) = O,

and does not involve at all the operators N, N + Th~s weaker condition w~ll be satisfmd by a pair of states (one even and one odd) We will later offer an

interpretation of th~s result The search for those states has already been made for the general case at the beginning of this section, and we shall .lust adapt eqs (3 24) to our case.

The odd state which we shall define as IF 0) will satisfy the equation

BIF0) = 0, RrF0) = 1fiFo), (3 51)

and the exphclt form of IF0) is

]Fo) =r I/ 11(~2~)e "-/- '[2), (3 52)

~8 Fill)lilt ~ l~¢lhl?lOUlc l / ~l lpcr( otl[ot t~la[ qllatllllttl D1c( llcllllt S

where 12) is the fully occupied state (corresponding to B = 1) The new group theoretical feature is that the two states belong to a doublet created b~ the N and N+ operators so that

~IF,,> = 0, (3 53)

and the even state of the doublet ]Co) can be obtained by

so that

NIF,,> = I < , ) . (3 54)

Bit, ,> = - I t , , ) ,

RICo) = (½f+ ~)[V,> (3 55)

One can make an analogous classification of states using

T1 = R - ! B - ~J (3 56)

Again we have quartets of equal T: states generated by M, M +, N, N ~ The zero TI

condition will again amount to

MIq'> = M" I"t"), (3 57)

and will be composed by a doublet of odd states interconnected by N and N ~ The B = 0 level of the doublet will be again the celebrated [k~>) state which will have the

added property

XtF,,> = 0 (3 58)

The B = 1 level [ ~ ) will be given by

NTIFo) = I(~,), (3 59)

and will correspond to R = : : f+ ½ ICo) and [(7o) will be connected by the transformation

Y~ ICo> = I ( , ) . r i c o ) = It;,> (3 60)

In order to construct a complete table of levels, it ~s also useful to introduce exphcltly the lowest even B = 0 level LEo} defined by

l eo) = M ~ I(,,> = M + IC,,),

RIEo) = ( ~ f + 1)IE,, > (3 61)

The spectrum of R is shown In fig 3

S Fubtm, E Rahtno~'t(l / Super(onfotmalquantum me(ham(~ 39

TI= 1

TI= 0

F'I x

F0

3.

-1

x F2

~ . ~ TI=O

Tl= o

f / 2 t ]

1

Fag 3 The spec t rum of R for N = 2 as a func tmn of B B = 1 is an empt~ state B = 0 ~s a

one - f e rm l on state (U can be usually eather a s y m m e m c or a n t l s 3 m m e t n c c o m b l n a t m n of the tyro

fe rmlon lc species) B = 1 con tams two different fermaons The lowest R state ~s r o = ~2] and B = 0 (the

a n t a s y m m e t n c combina t ion) at as non-degenera te The spacing in R as half a u mt The excited states are m 1 turn B = + ~ states and degenerate B = 0 states The daspla'ved spec t rum is an the O12) smglet sector

The states IF,,). I C . ) , I C , ) , I E . ) are jus t higher recurrences ob ta ined by app ly ing

( L + ) " to the co r respond ing lowest levels J F o), ]Co). [C0). I E0)

I t is c o n v e m e n t to use as the reference state the lowest level [Fo) Since it belongs

at the same t ime to the lowest T 1 and the lowest T 1 double t , it obeys the equat ions

M]Fo) = M + [Fo) = M I F o ) = M + [Fo) = 0 (3 62)

If we take the state [F0) as " v a c u u m " , we see that the symmet ry reduced by the

M ope ra to r s is preserved whereas that reduced by the N opera tors is spontaneous ly

b roken . In o rde r to have a perspecnve of our full results shown m figs 3 and 4, we

wish to use di f ferent observa t ion opera tors in the same manner and with s imilar

p u r p o s e to wha t we d id in sect 2 Using R as the obse rvauon opera tor , we f ind the

spec t rum of fig 3 As in the s tmple N = 1, D = 1 superconformal system, the

sp l i t t ing in R rela ted to bosomc exci tat ions l A B = 0) is t~ lce that associa ted with

the fer rmonlc exci ta t ion l A B = _+ 1) The fer rmomc and bosomc sph t tmgs are equal

all over the spec t rum A R = ~ for a fe rmlomc exci ta t ion and J R = 1 for bosomc

exc i ta t ions (with a B = 0)

This equal spacing is an analogue, in a system evolving with R, for a Go lds tone

b o s o n and a G o l d s t o n e fern'non Note, however, that all exci ted states are two-fold

degenera te , for R -- ~ f + ½(2n + 1), these are degenera te bosons, for R = I f + ~2n

40 5 Fuhlm, 1= Rahmo~,t(t / M~per{on/ormal quantum medlattl(

X . . . . . X X . . . . X

I I r

I

I ~ - - - x x - - - 7

I

-I 0 I - - B

. . . . . . . x

1 I

. . . . . . X

I

i

-1

I

I

I

I I I i

x . . . . . . I I

1 1

TI= R- B / 2 - f / 2

Fig 4 (a) T,,plcal N - 2 ~upers)mmetncal spectrum of a hamlltonlan 11 The glound qatc has a non-zero negat~,,e W]tten index It consists of one or more one-ferm~on states ( B - ()) The Lxc]ted states appear m quartet~ (b) The spectrum ol T 1 = R ~2 B ~2! as a ~unctlon of B The excited ~tates dlsplax an % = 2 supers'emmetQ structure The xacuum state > degenerate one vacuum ~tate is a lermlon ~tate I B = 0) and the other a boson ( B = 1, a two-Fermx state) The W]tten index 1% zero \ = 2 de Sitter c-,tended supers)mmetr). ~s broken to ~/= 1 The energ,¢ to excite a boson ~ fimte, but the ¢nergx to excite a fermton ~s zero This reflects the Goldqone ferm]on resulting Dom the breaking ol one

supers} mmetr~

t h e s e a re d e g e n e r a t e f e r m i o n s T h e d o u b l i n g fo l lows f r o m the fac t t h a t w h e n N = 2

s u p e r s y m m e t r y is b r o k e n t u o G o l d s t o n e - h k e f e r m l o n s s h o u l d a p p e a r , a n e v e n

l e r m l o n r e l a t e d to M, M " a n d a n o d d f e r m l o n r e l a t e d to M, M ~ W h e n T:, T1

( T - ~,B _+ !j' ) a re c h o s e n as o b s e r v a t i o n o p e r a t o r s , t he f e r m l o m c o p e r a t o r s w h i c h

c o m m u t e w i t h R - ~,B are c a n d i d a t e s for s u p e r s y m m e t r l c o p e r a t o r s T h e lowes t

m g e n v a l u e of T1 is ze ro a n d it c o r r e s p o n d s to a n o r m a h z a b l e e l g e n f u n c t l o n T h e

e q u a l s p a c i n g b e t w e e n b o s o m c levels re f lec ts the b r e a k d o w n of t i m e t r a n s l a t i o n a l

m v a r t a n c e It ~s t e m p t i n g to say t h a t the q u a r t e t d e g e n e r a c y of the e,~clted s t a t e s

r e f l ec t s a r e d e f m e d N = 2 e x t e n d e d s u p e r s y m m e t r y N o t e , h o w e v e r , t h a t on ly

M, M e a re s u p e r s y m m e t n c o p e r a t o r s f u l f i l h n g a n a l g e b r a in w h i c h H is r e p l a c e d by

T1 T h i s is m o r e h k e an N = 1 a lgeb ra , the q u a r t e t d e g e n e r a c y resu l t s f r o m the fact

t h a t a lso N a n d N + c o m m u t e w i t h T> e v e n t h o u g h t he a n t l c o m m u t a t o r { N, ,¥ ~ }

d i f f e r s f r o m Tt (eq (3 49)) Th~s is a r a t h e r p e c u h a r s i t u a t i o n T h e v a c u u m s ta t e ts

t w o - f o l d d e g e n e r a t e b u t c o n t a i n s a lso a b o s o n i c ( B - 1) s t a t e a n d t h u s ha s a ze ro

W l t t e n i n d e x T h i s is n o t p o s s i b l e for a m o d e l w i t h u n b r o k e n N - 2 s u p e r s y m m e t r y

In th i s e x a m p l e , a s y s t e m wi th a n N = 1 s u p e r s y m m e t r y & s p l a y b an exc i t ed N = 2

s p e c t r u m

In t he c a s e o n e c h o o s e s T 2 as an e v o l u t i o n o p e r a t o r , o n e h n d s t h a t i ts lowes t

m g e n v a l u e is f > 0 T h e a n t l c o m m u t a t o r of N, N + is

{ N , N + } = T , ( 3 6 3 )

N h a s a ze ro e l g e n v a l u e n o n - n o r m a h z a b l e s t a t e Le t us n o w take a shgh t l y d i f f e r e n t

v i ew of t he s i t u a t i o n A s s u m e t h a t b o t h N , N + a n d M, M are the s u p e r s y m m e m c

S Fubml, E Rabmovl~t / Superconformalquantum me, ham,s 41

generators of an N = 2 de Sitter supersymmetry which contains central charges

(operators which commute with the supersymmetry operators) We would now view

the pairs of anucommuta to r s of M, M + and N, N + as

{ N , N + } = U + ~ f , { N , M } = { N + , M + } = O ,

- - m

{ M , M + } = U - ½ f , { N , M + } = - { N + , M } = - 2 t J , ( 3 6 4 )

where I f is the central charge (which 1s non-zero m this case of all states in the

spectrum) U = R - ½ B is chosen to be the evolutlon operator (one could use

U = R - ½B - ½f just as well) In this case, we interpret the results as follows, the

N = 2 supersymmetry was broken down to an N = 1 supersymmetry N is the

broken opera tor while M is the surviving operator The breakdown of the supersym-

met ry associated with N is responsible for the degenerate ground states B = 0 and

B = 1 The "Go lds tone ferrmon" connects them The Wltten index is thus zero The excited spectrum, as m any supersymmetnc theory, reflects the symmetry of the

harlul toman, which is the full N = 2 supersymmetry, that explains the existence of

the quartets Note that the lowest elgenvalue of U = R - ~B is I f > 0 (choosing U = R - ~B - ½)' ) would set it zero but would not change the Wltten index The

spectrum is depicted m fig 4b The fact that N = 2 was not totally broken, ~s rather _ l 1 surprising since one usually concludes [8] that since H - ~,~Q~Q~ (fixed t) once

there exists an l 0 such that Q'°[0) ~ 0, the same is true for all t This results in a

total b reakdown of the supersymmetry However, in the presence of a central charge,

we have shown that there can be a gradual breakdown of an extended supersymme-

try In this case, however, the central charge does not seem to have any direct

topological meaning The same &scusslon is vahd for the pair of operators R + ~ B +

~b and their corresponding supersymmetry generators M, M +, N, N ~

Our last effort will be to discuss the excited states corresponding to elgenvalues o! j 2 different f rom zero We notice that to any j 2 = j 2 ( j integer) correspond the

levels [ + j ) and [ - j ) which transform into each other under inversion In order to get the level spectrum for j v~ 0, we go back to our starting equations

(3 48) and (3 49) We immediately see that we shall stdl have a quartet structure

since T i commutes with M, M +, N, N + However, the prewous procedure for

determining the lowest elgenstate has to be modified Indeed, suppose that T 1 has a

I k/'o) zero elgenstate subject to the equations

M[,/ '0) = M+ 1,/'0) = 0 , (3 65)

using the third of eqs ( 3 4 8 ) , we shall have

½t(q'o[ ( M, N+ }[ '/'o) = (q 'olJ[ '/'o) = O, (3 66)

42 g Fuhtnt L Rahtnot,t~t / ,Supetcon/?tmulquuntum ntedlant~

in contrast with our assumption of j ~ 0 (a d @ 0 state would break a bosomc 0(2) symmetry) We are thus led to look for another pair of operators say ~ and v such that the mixed anticommutators involving ~ and v vanish Let us define

tg 0 = j / ] .

sln~0=-~[1-/(j2+]'-) ,,~]l,~ (3 67)

, = c o s ( ½ 0 ) M + t sln(½0)N.

v =- - ~ sm(~0 ) M + cos( ~0 ) ~ , (3 6s)

~t is easy to show that

1 _ 2 = T ( ), ~ { , + , , } = R + ~ z l ~v'S +J~- s

~ { 1 , + . 1 , } = R + { B + ~7'f2 + j ~ = T( j ) + V'72 + j 2 ,

( g + , v } = {/~v + } = 0 (369)

The form (3 68) of g a n d v tells us that only the states I+o) obeying

t~lq',,> = t , ~ Iq'o> (3 70)

are acceptable They correspond to

v ( i ) l ' P o ) = (R + } n - ~,,'/2 + s_ ~ )1 '/ 'o) = 0 (3 71)

In particular, the lowest level IF0(j)) , corresponding to B = 0, will correspond to the lowest elgenvalue of R, i e

R [ F o ( J ) ) = ~V')2 + j Z l F 0 ( j ) ) (3 72)

It is easy to apply the same procedure to the analogous anticommutators of eq (3 49), this will lead to similar results

In conclusion, the level structure will be the same as that of fig 4 The only

difference is that now the lowest eigenvalue of R ts ~t,,f2 + j 2 We nonce that the energy gap between the d = 1 and j = 0, B = 0 state ~s smaller than the energy gap ( = ½) between the closest equal-j B = 0 and B = + 1 states This concludes our discussion of N = 2 extended conformal supersymmetry

S Fubmt. E Rabmovtct / Superconformalquantum mecham~s 43

4. Conclusions

The solut ion of superconformal q u a n t u m mechanics is in teres t ing in its own right

I t was shown that for N = 1, in one or more &mens lons , supe r symmet ry as b roken

Supe r symmet ry is also b roken for a mode l with an ex tended N = 2 supe r symmet ry

It was a l ready known that conformal quan tum mechanics has no no rmahzab le zero

energy solution, that seemed to indicate that supe r symmet ry would be b roken m the

supe rconfo rma l verslon If f imte theories would suffer f rom the absence of an E = 0

state, s lrmlar consequences as to the b reak ing of the supe r symmet ry may well fol low

In the absence of a g round state, it was prescr ibed that compac t opera to r s should

p l ay the role of evolut ion and observa t ion opera to r s of the system The spec t rum of

such opera to r s was found to be discrete. A par t i cu la r symmetr ic choice for the

modi f i ed evolut ion ope ra to r is to d lagonahze the compac t bosonlc opera to rs appea r -

mg in the de Sit ter suba lgebra of the conformal a lgebra Wi th such a choice, new

conserved supe r symmet ry opera to r s can be def ined These opera to rs are app rop r i a t e

com b i na t i ons of supe r symmet ry and fermlonlc conformal opera tors

V~ewmg the system in that way, one found that for N = 1 and a system consis t ing

of one part icle , the new supe r symmet ry is unbroken F o r the N = 2 case, a non-com-

mon par tml b r e a k d o w n to N = 1 is observed These features, while demons t r a t ed

only in the e l emen ta ry case of q u a n t u m mechanics , may also occur in finite quan tum

field theories

The consequence of using compac t evolut ion opera to rs is the b r e a k d o w n of t ime

t rans la t iona l lnvar lance The equal spacing in the spec t rum is an ind ica t ion of that

p r o p e r t y In view of the necessi ty of ident i fy ing mechanisms for d imens iona l

reduct ion , tlus l iabi l i ty may turn into an asset We believe that these obse rvanons

mer i t fur ther inves t igat ions of f inite field theories m h~gher d imens ions

E R wishes to thank M Jacob for the hosp i t ah ty ex tended to him dur ing his stay

at the Theore t ica l Physics Divis ion at C E R N E R wishes to thank C Bernard and

B Lau t rup for discussions

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