in the FHS. Note that N is identi¬ed with the integer (t + 1). As

emphasized by Pegg and Barnett [32,33], when we deal with phase

states, the traceless relation (8.84) is to be used in place of [b, b+ ] = 1.

The truncated relation (8.77) looks deceptively similar to the

generalized quantum condition (5.70) considered in Chapter 5. To

avoid confusion of notations, let us rewrite the latter as

∼ ∼+

[ b , b ] = 1 + 2νL (8.85)

where ν ∈ R and L is an idempotent operator (L2 = 1) that com-

∼+

∼

mutes with b and b . We now remark that (8.85) cannot be trans-

formed to the form (8.77) and hence is not a representative of a trun-

cated scheme. Indeed if we deform (8.77) in terms of the parameters

», µ ∈ R as

[B, B + ] = » ’ µN K (8.86)

and compare with (8.85), we ¬nd for the representations (8.79), and

choosing L = σ3 , the solutions for » and µ turn out to be

µ = » ’ 1 = 2ν (8.87)

So » cannot be put equal to unity since µ and ν will simultaneously

vanish.

The root cause of this di¬culty is related to the fact that the

truncated oscillator is distinct from normal harmonic oscillator pos-

sessing in¬nite dimensional representations [30]. While a generalized

quantum condition such as (8.85) speaks for parabosonic oscillators,

there are remarkable similarities between the rules obeyed by B, B +

and those of the parafermionic operators c, c+ which will be consid-

ered now.

© 2001 by Chapman & Hall/CRC

(b) Construction of a PSUSY model

Let us consider a truncation at the (p + 1)th level (p > 0, an

integer). Then B, B + are represented by (p + 1) — (p + 1) matrices

and (8.77) acquires the form

[B, B + ] = 1 ’ (p + 1)K (8.88)

where 1 stands for a (p + 1) — (p + 1) unit matrix. The irreducible

representations of (8.88) are the same as those for the scheme [37]

described by a set of operators d and d+

[d, d+ d] = d

dp+1 = 0

dj = , j <p+1

0 (8.89)

Using the decompositions of B and B given in (8.81) we ¬nd

p

r

(B) = p(p ’ 1)(p ’ 2) . . . (p ’ r + 1)

r=0

|k ’ r >< k|

p

+r

(B ) = p(p ’ 1)(p ’ 2) . . . (p ’ r + 1)

r=0

|k >< k ’ r| (8.90)

Thus

(B)r = (B + )r = 0 for r > p (8.91)

Further, the following nontrivial multilinear relation between B

and B + hold [31,39]

p(p + 1) p’1

B p B + + B p’1 B + B + . . . + B + B p = B (8.92)

2

along with its hermitean conjugated expression. Of course, these

coincide for p = 1 and we have the familiar fermionic condition

BB + + B + B = 1. For p = 2 we ¬nd from (8.91) and (8.92) the

following set

(B)3 = 0 = (B + )3 (8.93a)

B 2 B + + BB + B + B + B 2 = 3B (8.93b)

© 2001 by Chapman & Hall/CRC

(B + )2 B + B + BB + + B(B + )2 = 3B + (8.93c)

We notice that, except for the coe¬cients of B and B + in the

right-hand-sides of (8.93b) and (8.93c), the above equations resemble

remarkabley the trilinear relations which the parafermionic operators

c and c+ obey, namely [25,26]

c3 = 0 = (c+ )3

c2 c+ + cc+ c + c+ c2 = 4c

(c+ )2 c + c+ cc+ + c(c+ )2 = 4c+ (8.94)

The generalization of (8.93) to p = 3 and higher values are straight-

forward.

We may thus interpret B and B + to be the annihilation and

creation operators of exotic para Fermi oscillators (p ≥ 2) governed

by the algebraic relation (8.90) - (8.92). Certainly such states here

have ¬nite-dimensional representations.

One is therefore motivated to construct a kind of PSUSY scheme

of order p in which there will be a symmetry between normal bosons

and truncated bosons of order p. So we de¬ne a new PSUSY Hamil-

tonian Hp [which is distinct from either the Rubakov-Spiridonov or

the Beckers-Debergh type] generated by parasupercharges Q and Q+

de¬ned by

β A+ δ±,β+1

(Q)±β = b — B + = β

√

(Q+ )±β = b+ — B = ± A’ δ±+1,β (8.95)

±

where A± have been de¬ned by (8.55) and (8.56).

±

One then ¬nds the underlying Hamiltonian Hp to be

(H)±β = H± δ±β

where

1 d2 1 2

Hr =’ + Wr + Wr

2

2 dx 2

1

+ cr , r = 1, 2, . . . , p (8.96)

2

1 d2 1 1

2

Hp+1 =’ + Wp ’ Wp + cp (8.97)

2 dx2 2 2

© 2001 by Chapman & Hall/CRC

with the constraint

2 2

Ws’1 ’ Ws’1 + cs’1 = Ws + Ws + cs , s = 2, 3, . . . p (8.98)

and the parameters c1 , c2 , . . . , cp (which have the dimension of en-

ergy) obeying

c1 + 2c2 + . . . + pcp = 0 (8.99)

Note that (8.99) is of a di¬erent character as compared to (8.61).

However, the Hamiltonian and the relationships between the super-

potentials as given by (8.96)-(8.98) are similar to the case described

in (8.58)-(8.60). As a result the consequences from the two di¬erent

schemes of PSUSY of order p are identical. To summarize, we have

for both the cases the following features to hold

(i) The spectrum is not necessarily positive semide¬nite which is

the case with SUSYQM.

(ii) The spectrum is (p+1) full degenerate at least above the ¬rst p

levels while the ground state could be 1, 2, . . . , p fold degenerate

depending upon the form of superpotentials.

(iii) One can associate p ordinary SUSYQM Hamiltonians. This

is easily checked by writing in place of (8.24) the combination

p

Q= jQj . Qj ™s then turn out to be supercharges with

j=1

Q2 = 0.

j

Finally, we answer the question as to why two seemingly di¬erent

PSUSY schemes have the same consequences. The answer is that in

the case of PSUSY of order p, one has p independent PSUSY charges.

In the two schemes of order p that we have addressed, we have merely

used two of the p independent forms of Q. It thus transpires that

one can very well construct p di¬erent PSUSY schemes of order p

but all of them will yield almost identical consequences.

8.5 Multidimensional Parasuperalgebras

In this section we explore the PSUSY algebra of order p = 2 to

study a noninteracting three-level system [40,41] and two bosonic

© 2001 by Chapman & Hall/CRC

modes possessing di¬erent frequencies. Such a system is described

by the Hamiltonian

12 1

ωk bk , b+ + V

H= (8.100)

k

2 k=1 2

where bk and b+ are bosonic annihilation and creation operators of

k

the type (2.2)

1 d

√

b1 = + ω1 x

2ω1 dx

1 d

b+ = √ ’ + ω1 x

1

dx

2ω1

1 d

√

b2 = + ω2 x

2ω2 dx

1 d

b+ = √ ’ + ω2 x (8.101)

2

dx

2ω2

In the above, the frequencies ωk are distinct (ω1 = 2 ) and V has a

ω

diagonal form to be speci¬ed shortly.

In a three-level system there are three possible schemes of con¬g-

urations of levels, namely the Ξ type, V type, and § type. Note that

a three-level atom can sense correlations between electromagnetic

¬eld modes with which it interacts [42].

Let us consider the case when the frequencies ω1 and ω2 of the

bosonic modes are equal to the splitting between various energy lev-

els. Accordingly, we have the following possibilities [40]

Ξ type : ω1 = E1 ’ E2 , ω2 = E2 ’ E3

V type : ω1 = E1 ’ E2 , ω2 = E3 ’ E2

§ type : ω1 = E2 ’ E1 , ω2 = E2 ’ E3 (8.102)

For the Ξ type the Hamiltonian may be considered in terms of

the bosonic operators, namely,

12

ωj bj , b+ + diag(E1 , E2 , E3 )

HΞ = (8.103)

j

2 j=1

The transition operators between levels 1 and 2 which are denoted

as t± and those between levels 2 and 3 which are denoted as t± are

1 2

© 2001 by Chapman & Hall/CRC

explicitly given by

«

0 x 0

1

t+ = 0

√ 0 0

1

20 0 0

«

0 0 0

1

t+ = y

√ 0 0 (8.104)

2

20 0 0

along with their conjugated representations. In (8.104), x and y are

nonzero real quantities.

The above forms for t+ and t+ induce charges Q+ and Q+ given

1 2 1 2

by

«

0 b1 0

1

Q+ = 0 0 0

√

1

ω1

0 00

«

0 00

1

Q+ = 0 0 b2 (8.105)