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By comparing with (8.77) it is clear that K plays the role of |t >< t|
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



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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)

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