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Proc. R. Soc. A (2007) 463, 2415–2427
doi:10.1098/rspa.2007.0003
Published online 10 July 2007
Quantum algebras and
parity-dependent spectra
B Y C. V. S UKUMAR *
AND
A NDREW H ODGES
Wadham College, University of Oxford, Oxford OX1 3PN, UK
We study the structure of a quantum algebra in which a parity-violating term modifies
the standard commutation relation between the creation and annihilation operators of
the simple harmonic oscillator. We discuss several useful applications of the modified
algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case.
We also show a connection with Gaussian and non-Gaussian squeezed states of the
simple harmonic oscillator. Such states have been considered in quantum optics. The
combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate
matrix elements for squeezed states.
Keywords: quantum algebras; parity; Bernoulli numbers
1. Introduction
The SU(1,1) algebra has played an important role in the literature on group theory
and mathematical physics. Various representations of SU(1,1) have been
extensively studied (Barut & Fronsdal 1965; Holman & Biedenharn 1966). In
recent years, the SU(1,1) group and its Schwinger representation (Schwinger 1965)
have been used in quantum optics in connection with the study of parametric
amplifiers (Louisell 1977), interferometers (Yurke et al. 1986; Fearn & Loudon
1989) and squeezed states (Bishop & Vourdas 1986). The generators of SU(1,1)
may be represented in terms of operators obeying certain commutation relations. In
this paper we explore a new way of expressing the generators of SU(1,1).
There is a system of commutation relations which generalizes the standard
commutation relations of the simple harmonic oscillator. The generators of
SU(1,1) may be expressed in terms of the creation and annihilation operators of
the generalized commutation relations. In §2 we study this structure and
interpret it as a parity-violating modification, and we investigate its spectral
properties. In the course of this analysis, we note that in a special case, the
quantum algebra reduces to the algebra introduced by Hodges & Sukumar
(2007), which describes the combinatorics of the Bernoulli and Euler numbers. In
§3 we identify this algebraic structure as relevant to squeezed states such as the
ones considered in quantum optics. In §4 we extend the combinatorial theory
developed earlier for the analysis of Bernoulli and Euler numbers, and apply
these discrete methods to calculate matrix elements for squeezed states. Section 5
comments briefly on possible extensions of these results.
* Author for correspondence ([email protected]).
Received 25 April 2007
Accepted 23 May 2007
2415
This journal is q 2007 The Royal Society
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2416
C. V. Sukumar and A. Hodges
2. Modified commutation relations
Suppose we have operators which satisfy the commutation relations
½R; L Z S;
½R; S Z 2R;
½S; L Z 2L:
ð2:1Þ
These can be considered as the generators of an SU(1,1) algebra. They can
arise from the standard creation and annihilation operators a† and a of the simple
harmonic oscillator when
L Z 12 a†2 ;
R Z 12 a2 ;
S Z 12 ða † a C aa † Þ:
ð2:2Þ
However, there is a more general possibility. Consider a quantum algebra
defined by a non-Hermitian operator A, its adjoint A† and a Hermitian operator
X satisfying
½A; A† Z 1 C X; fA; Xg Z 0; fA† ; Xg Z 0:
ð2:3Þ
The bilinear operators defined by
L Z 12 A†2 ;
R Z 12 A2 ;
S Z 12ðA† A C AA† Þ;
ð2:4Þ
still satisfy the commutation relations (2.1).
The conditions on X are stringent. They imply that the operator X 2 commutes
with A and A†, and so with R, L and S. Thus, X 2 is some multiple of the identity.
Also, X itself commutes with HZA†A. Thus, we can identify the algebra defined
by these commutation relations by studying the simultaneous eigenstates of the
operators X and H. We shall then be able to interpret X in terms of parity.
(a ) Spectral analysis
Let jni be an eigenstate of the operators X and H. The eigenvalue equations
H jni Z ln jni;
Xjni Z an jni;
ð2:5Þ
½A† ; H Z ðX K1ÞA† ;
ð2:6Þ
and the operator relations
½A; H Z ðX C 1ÞA;
imply that
H ðAjniÞ Z ðln C an K1ÞðAjniÞ;
XðAjniÞ ZKan ðAjniÞ
H ðA† jniÞ Z ðln C an C 1ÞðA† jniÞ;
XðA† jniÞ ZKan ðA† jniÞ;
ð2:7Þ
showing that the states (Ajni) and (A†jni) are also simultaneous eigenstates of
the operators H and X. Iteration of the above equations leads to
H ðA2m jniÞ Z ðln K2mÞðA2m jniÞ
H ðA2mK1 jniÞ Z ðln K2m C 1 C an ÞðA2mK1 jniÞ
XðAm jniÞ Z ðK1Þm an ðAm jniÞ
H ðA†2m jniÞ Z ðln C 2mÞðA†2m jniÞ
H ðA†ð2mK1Þ jniÞ Z ðln C 2mK1 C an ÞðA†ð2mK1Þ jniÞ
XðA†m jniÞ Z ðK1Þm an ðA†m jniÞ:
Proc. R. Soc. A (2007)
ð2:8Þ
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Algebras and parity-dependent spectra
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Thus, a sequence of simultaneous eigenstates of H and X may be generated from
the state jni. We first consider the case janj%1. It will be shown later that all other
cases may be related to this case. Then, the above equations show that each
operation with A† on an eigenstate of H produces an eigenstate with a higher
eigenvalue or the same eigenvalue for H and changes the sign of the eigenvalue of
the X operator. Similarly, each operation with A on an eigenstate of H produces
another eigenstate with a lower eigenvalue or the same eigenvalue for H and
changes the sign of the eigenvalue of X. H is a positive semi-definite operator and
so the spectrum of H must be positive semi-definite too. This implies that there
must be a state such that Aj0iZ0 with the lowest eigenvalue l0Z0, so that a state
with an eigenvalue lower than l0 is not possible. This must be the groundstate.
From the groundstate j0i, it is then possible to generate a sequence of eigenstates
jniwA†nj0i. This spectral picture must match the spectrum arising from the
consideration of any eigenstate. For example, if we start from the eigenvalue
equation for j1i, then we must have A2j1iZ0 to stop the spectrum from going
below zero. It may be shown that this is consistent if j1iwA†j0i.
The resulting spectral picture is that there is a groundstate j0i with eigenvalue
0 for H and a0 for X and a set of eigenstates j2ni with l2nZ2n and a2nZa0, for
nZ0, 1, 2, . There is another sequence of eigenstates j2nC1i with eigenvalues
l2nC1Z(2nC1Ca0) and a2nC1ZKa0. The level spacing of the eigenvalues of H
is 2 for both sequences but the odd sequence is displaced from the even sequence
by (1Ca0).
Now that we have identified all the states and the eigenvalues of X in all of
them, we may note that we have shown that
X
X Z a0
ðK1Þn jnihnj:
ð2:9Þ
n
P
That is, X must be a multiple of the parity operator P Z n ðK1Þn jnihnj.
In what follows, we shall write XZaP, where the parameter a may be freely
chosen, so that the eigenvalues of X are given by anZ(K1)na and X 2Za2. By
examining the normalization of the states A†jni and Ajni, it may be shown that
pffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A† j2nK1i Z 2nj2ni
A† j2ni Z 2n C 1 C aj2n C 1i;
ð2:10Þ
pffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Aj2ni Z 2nj2nK1i;
Aj2n C 1i Z 2n C 1 C aj2ni:
It follows that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2n C 1 C aÞð2n C 2Þj2n C 2i
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Lj2nK1i Z A†2 j2nK1i Z ð2n C 1 C aÞð2nÞj2n C 1i
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Rj2n C 2i Z A2 j2n C 2i Z ð2n C 1 C aÞð2n C 2Þj2ni
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Rj2n C 1i Z A2 j2n C 1i Z ð2n C 1 C aÞð2nÞj2nK1i:
2Lj2ni Z A†2 j2ni Z
ð2:11Þ
It is evident from equations (2.11) that R and L do not connect the states jni or
even n with other states of odd n and vice versa. S is diagonal in the space of the
states jni with eigenvalues (2nC1Ca)/2. Thus, the matrices for (R, L, S ) always
split into even and odd parts.
Proc. R. Soc. A (2007)
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C. V. Sukumar and A. Hodges
(b ) Calculating matrix elements
The case XZ0, aZ0 corresponds to the standard harmonic oscillator.
The spectrum of H consists of non-degenerate states A†nj0i with eigenvalues
nZ0, 1, 2, ., and the matrix elements are well known.
In considering the states and matrix elements for non-zero a, it is helpful to study
first the special case aZ1. In this case, the separation between the odd and even
sequences of eigenvalues is 2, thus equalling the spacing of the levels in each sequence.
The two ladders of eigenvalues overlap. There is a non-degenerate groundstate j0i
with eigenvalue 0 for H. All eigenstates of H other than the groundstate are doubly
degenerate with even integers 2, 4, 6, . as eigenvalues. Each degenerate excited
state consists of two states with opposite parities. The even parity states are A†2nj0i
and the corresponding degenerate odd parity states are A†(2nK1)j0i.
The resulting algebra coincides with the (R, L, S ) algebra developed by
Hodges & Sukumar (2007). It was shown in that paper how the matrix elements
naturally yield the tangent and secant numbers (equivalent to the Bernoulli and
Euler numbers). In particular,
E2m Z h0jðR C LÞ2m j0i;
T2mC1 Z h1jðR C LÞ2mC1 j1i;
where E2m and T2mC1 are the secant and tangent numbers, respectively. It was
further shown that the operator X can be regarded as [U †,U ], where U is an anticommuting operator defining a symmetry between the tangent and secant
number structures.
For aZK1, the situation is essentially the same. The difference is only technical.
Equation (2.10) shows that A†j0iZ0 and it is not possible to build a spectral ladder
based on the state j0i. Equation (2.10) also shows that Aj1iZ0 but A†j1is0 and now
j1i is the non-degenerate groundstate with eigenvalue 0 for H1.
The theory for general a may now be developed by analogy with the case aZ1.
Since the matrices for R, L and S split into even and odd parts that do not
connect, we can consider a set of matrices (R e, Le, Se) whose rows and columns
are numbered by integers (0, 1, 2, .) but correspond to the operators in the
space of even values (nZ0, 2, 4, .) of jni, and another set of matrices
(R o, Lo, So) whose rows and columns are numbered by integers (0, 1, 2, .) but
correspond to the operators in the space of odd values (nZ1, 3, 5, .) of jni.
We then write the matrices MeZR eCLe and MoZR oCLo as
1
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0
2ð1 C aÞ
0
0
/
C
B pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C
1B
0
4ð3 C aÞ
0
/C
B 2ð1 C aÞ
ð2:12Þ
Me Z B
C;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C
2B
@
0
4ð3 C aÞ
0
6ð5 C aÞ /A
/
/
/
/
/
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0
2ð3 C aÞ
B pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B
1 B 2ð3 C aÞ
0
Mo Z B
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2B
@
0
4ð5 C aÞ
/
/
Proc. R. Soc. A (2007)
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4ð5 C aÞ
0
/
1
/
C
C
0
/C
C:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C
6ð7 C aÞ /A
/
/
0
ð2:13Þ
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Algebras and parity-dependent spectra
It is clear that the even and odd matrices satisfy the relation Mo(aK2)ZMe(a),
thus justifying our assertion that it is sufficient to consider janj%1. In particular,
the matrices for aZK1 are just relabellings of the matrices for aZ1. In the
ensuing discussion we shall not pay further attention to the case aZK1.
In the case aZ1, the multiplication of these bidiagonal matrices is equivalent
to the use of the recursive triangles set out in §§4–6 of Hodges & Sukumar (2007)
for the calculation of the secant and tangent numbers. We can now see these
triangles as special cases of the more general algebra obtained by considering
these non-standard commutation relations.
For the general case of arbitrary a, the spectrum consists of two paritydependent equispaced ladder of states, which are separated by 1Ca. A search
through the literature in nuclear physics suggests that such parity-dependent
spectra can arise for heavy nuclei with octupole deformations. Semi-classically
nuclei with octupole deformations have a geometrical arrangement similar to
that of the ammonia molecule for which there are two possible mirror symmetric
arrangements of the N atom above and below the plane containing the H atoms.
The evidence for intrinsic reflection asymmetry in heavy atomic nuclei is
discussed by Butler (1998). The level scheme of Ra226
88 shown by Cocks et al.
(1997) suggests that this nucleus may be viewed as a possible candidate for a
physical system exhibiting the type of spectra arising from a non-integer value of a.
We have shown in this section that a simple change in the standard oscillator
algebra leads to a variety of interesting possibilities. We now study some useful
applications of the modified algebra.
3. Application to squeezed states
We consider an application to the study of squeezed states of the simple
harmonic oscillator whose generators obey the same commutation relations as
the operators introduced in the study of Bernoulli and Euler numbers. We first
summarize the results of §§4 and 5 of Hodges & Sukumar (2007) on the algebra of
secant and tangent numbers, putting them into the form of generating functions.
The combinatorial property of the sequence of secant numbers E2n can be
expressed as
h0jexpðbðR C LÞÞj0i Z sec b;
ð3:1Þ
where R and L are just the operators discussed in §2, in the case aZ1. This uses
the ‘even’ matrices Me. In the case of the tangent numbers T2nC1, which involve
the ‘odd’ matrices Mo, we need the additional observation that
ln sec z Z
N
X
T2mK1 z 2m
1
ð2mÞ!
and sec2 z Z
N
X
T2mC1 z 2m
0
ð2mÞ!
;
ð3:2Þ
to give the analogous generating function as
h1jexpðbðR C LÞÞj1i Z sec2 b:
ð3:3Þ
We can also note that h0jðRKLÞ2m j0iZ ðK1Þm h0jðRC LÞ2m j0i and hence
h0jexpðbðRKLÞÞj0i Z sech b;
Proc. R. Soc. A (2007)
h1jexpðbðRKLÞÞj1i Z sech2 b:
ð3:4Þ
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C. V. Sukumar and A. Hodges
We shall now show that these formulae generalize to all a. The key idea is that
the exponential function appearing in these formulae can be reinterpreted as
giving rise to the squeezed states of the simple harmonic oscillator such as the
ones considered in quantum optics.
(a ) Coherent states and squeezed states
In the context of quantum optics, the mechanisms for the generation of
coherent states and Gaussian squeezed states of the electromagnetic field have
been extensively studied (Stoler 1970; Yuen 1976). The general method is to find
a unitary operator that acts on the vacuum to generate a state of the field with
the required properties.
Coherent states are states of the simple harmonic oscillator for which the
variance product DxDp is time independent and has the minimum value allowed
by the Heisenberg uncertainty principle. It is known that the coherent states are
eigenstates of the annihilation operator. Coherent states are Gaussians and may
be generated by considering
Jc Z Uc j0i Z expðba † Kb aÞj0i;
ð3:5Þ
†
where a and a are the annihilation and creation operators of the simple
harmonic oscillator, respectively, obeying the commutation rule [a,a†]Z1. The
operators x and p are Hermitian linear combinations of a and a†, satisfying the
commutation relation ½x; pZ iZ.
Squeezed states are states of the simple harmonic oscillator for which the
variances Dx and Dp are time dependent and the variance product exceeds that
for the coherent states, but the variance of x or p can dip below that for the
coherent states for part of the time during a cycle whose period is determined by
the oscillator frequency. The Gaussian squeezed states, which are eigenstates of a
linear combination of the creation and annihilation operators, may be expressed
in the form
Js Z Us 0i Z exp ba †2 Kb a 2 0i:
ð3:6Þ
It is well known that the coherent states generated by Uc can be brought to the
normal-ordered form, in which all annihilation operators appear to the right and
all the creation operators appear on the left, namely
Jc Z Uc j0i Z expðba † Þexpðbb =2ÞexpðKb aÞj0i;
ð3:7Þ
thus yielding a useful form for calculating expectation values involving Jc. There
is an analogous normal-ordered form for the Gaussian squeezed states, which
arises from the following theorem. If a set of operators satisfy the algebra
½B; B † Z C ; ½B; C Z 2b2 B;
ð3:8Þ
†
with b a complex number, then (B, B , C ) is a closed algebra and the unitary operator
U ðbÞ Z expðbB † Kb BÞ;
may be brought to a normal-ordered form
ð3:9Þ
U ðbÞ Z expðgB † ÞexpðmC ÞexpðKg BÞ
where g Z b
Proc. R. Soc. A (2007)
tanhðjbjbÞ
1
; m Z 2 lnðsechjbjbÞ:
jbjb
b
ð3:10Þ
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Algebras and parity-dependent spectra
2421
This is a new theorem which may be proved by following the same procedure as
that used for establishing the normal-ordered form of the squeeze operator (Truax
1985; Schumaker 1986; Sukumar 1989). In fact, the normal-ordered form of Uc in
equation (3.7) is a special case of this general theorem for CZ1, bZ0, so that
1
jbj2
:
m Z lim 2 lnðsechjbjbÞ ZK
b/0 b
2
A corresponding theorem may be given for the operator
V ðbÞ Z expðbB † C b BÞ:
ð3:11Þ
The normal-ordered form of V has a structure similar to that of U, given by the
replacements tanh/tan, sech/sec. If now we choose
b real; b Z 1; B Z R; B † Z L; C Z S Z A† A C ð1 C aÞ=2;
then these general theorems supply the normal-ordered forms for the expressions
Ua ðbÞ Z expðbLKb RÞ
Va ðbÞ Z expðbL C b RÞ;
ð3:12Þ
in our generalized (R, L, S ) algebra in which a may have any value. These forms
yield the matrix elements
ðtanh bÞn
h2njUa ðbÞj0i Z ðsech bÞð1CaÞ=2
h2njA†2n j0i
2n n!
0
! 11=2
a C1
C
B G nC
C
B
2
ð1CaÞ=2
nB
!C
Z ðsech bÞ
ðtanh bÞ B
C
C
B
@ Gðn C 1ÞG a C 1 A
2
n
ðtanh bÞ
h2n C 1jUa ðbÞj1i Z ðsech bÞð3CaÞ=2
h2n C 1jA†2n j1i
2n n!
0
! 11=2
a C3
C
B G nC
C
B
2
C
ð3CaÞ=2
nB
!
Z ðsech bÞ
ðtanh bÞ B
C ; ð3:13Þ
C
B
a
C
3
A
@ Gðn C 1ÞG
2
and similarly for V, with the obvious replacements. Putting aZ1, nZ0 in these
expressions, we recover the results in equations (3.1)–(3.4).
In the case aZ0, we obtain the expectation values for the standard harmonic
oscillator
h0jU0 ðbÞj0i Z ðsech bÞ1=2 ;
h0jV0 ðbÞj0i Z ðsec bÞ1=2 :
ð3:14Þ
These are well-known results in the literature on Gaussian squeezed states. We
shall derive them by a new combinatorial method in §4.
Proc. R. Soc. A (2007)
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C. V. Sukumar and A. Hodges
(b ) An alternative representation of the tangent and secant number algebra
The algebra of (R e, Le, Se) for aZ1 can also be realized by another algebra
defined in the space of standard harmonic oscillator states (jni, nZ0, 1, 2, .). If
we consider the operators
T1 Z aN 1=2 ; T †1 Z N 1=2 a † ; N Z a† a;
ð3:15Þ
which are defined in terms of the oscillator creation and annihilation operators, then
T1 jni Z njnK1i; T †1 jni Z ðn C 1Þjn C 1i; N jni Z njni:
ð3:16Þ
The resulting algebra
½T1 ; T †1 Z 2N C 1; ½T1 ; 2N C 1 Z 2T1 ;
ð3:17Þ
defined in the space of the oscillator number states that jni is identical to that satisfied
by (R e, Le, Se) as given in equation (2.11) but now restricted to the space of even
states, so that SeZNC1 gives Sej2niZ(2nC1)j2ni. The U and V operators defined
using ðT1 ; T †1 ; 2N C 1Þ will therefore lead to the same results as those discussed
earlier for the (R e, Le, Se) algebra. The operators T1 and T †1 have earlier been used in
a study of squeezed states, which are not Gaussians (Sukumar 1989) but are linear
superpositions of Gaussian forms.
Similarly, the algebra of (R o, Lo, So) can also be realized by another algebra
defined on the harmonic oscillator number states. If we consider
T2 Z aðN C 1Þ1=2 ;
T2 jni Z ðnðn C 1ÞÞ1=2 jnK1i
T †2 Z ðN C 1Þ1=2 a † ; T †2 jni Z ððn C 1Þðn C 2ÞÞ1=2 jn C 1i
ð3:18Þ
the resulting algebra
½T2 ; T †2 Z 2N C 2;
½T2 ; 2N C 2 Z 2T2 ;
ð3:19Þ
defined in the space of the oscillator number states is identical to that satisfied by
(R o, Lo, So) as given in equation (2.11) but now restricted to the space of odd
states, so that SoZNC1 gives Soj2nC1iZ(2nC2)j2nC1i.
We note that operators involving N 1/2 may appear to be unusual, but such
operators have been used to define the phase operator of the electromagnetic field
(Susskind & Glogower 1964; Carruthers & Nieto 1965).
4. Combinatorial evaluation of the standard oscillator matrix elements
We now return to the standard creation and annihilation operators with [a,a†]Z1, i.e.
the case aZ0. We shall consider the matrix elements
h0jða 2 C a†2 Þm j0i;
which, as we have seen, arise in the squeezed states of the simple harmonic oscillator.
These elements can be calculated by the matrices given by the general theory,
and the calculation put in the form of a recursive triangle (figure 1).
Proc. R. Soc. A (2007)
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2423
Algebras and parity-dependent spectra
1
1
1
2
3
1
4
5
2
3
6
7
1
4
5
8
9
2
3
6
7
10
1
4
5
8
2
3
6
1
4
3
2
14
15
28
132
105
556
1500
945
10 668
1112
19 950
212 940
43 784
87 568
1 408 992
5 723 536
2
11 447 072
Figure 1. Recursive triangle for calculating matrix elements.
Here the numbers in the left-hand column give the matrix elements for nZ0,
1, 2, . It also follows from (3.14) that the generating function for this sequence
is given by
N
X
pffiffiffiffiffiffiffiffiffiffiffiffiffi
zn
h0jða2 C a †2 Þn j0i Z sec 2z :
n!
0
ð4:1Þ
But we can also obtain these results by discrete combinatorial methods.
Hodges & Sukumar (2007) developed such methods for the analysis of the
permutation group on n elements, thereby finding an operator formalism for the
calculation of the number of alternating or ‘zig-zag’ permutations. We now apply
a similar analysis to the subgroup of permutations consisting of pure
transpositions, i.e. permutations of 2m elements in which every element lies in
a 2-cycle.
We shall place a transposition in a transposition zig-zag class as follows: the ith
element of the class is C if i is in the 2-cycle (ij ) with jOi, and it is D otherwise.
Thus, the transposition (14)(25)(38)(67) is in class CCCDDCDD.
Proc. R. Soc. A (2007)
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2424
C. V. Sukumar and A. Hodges
12
56
12
56
34
78
34
78
(15) (26) (37) (48)
(15) (28) (36) (47)
Figure 2. Examples of chaining graphs.
We define jXj to be the number of transpositions in a class jXj, and extend the
definition naturally by linearity, so that jaXCbYjZajXjCbjYj. We then find
that jXj can be identified with the matrix element h0jXj0i in the standard
quantum algebra. The proof runs in analogy to the arguments made for the
permutation classes. By simple counting arguments, we may establish such
properties as
jDXj Z 0 Z jXC j for any X;
ð4:2Þ
½C; D Z 1;
ð4:3Þ
If S Z 12 ðCD C DC Þ then jSXj Z 12 jXj for any X;
ð4:4Þ
ð2mÞ!
;
ð4:5Þ
2m m!
where (2m)!/(2mm!) is just the total number of transpositions on 2m elements.
Thus, the algebra of C and D is exactly that of a and a†. The recursive triangle
yielding the matrix elements for h0j(a2Ca†2)mj0i can now be interpreted as using
combinatorial relations to count the transpositions in the classes of j(C 2CD 2)mj.
We can also show by combinatorial means that the generating function for the
resulting sequence 1, 2, 28, 1112, 87 568, . is given by the square root of the
secant. Note first that a typical transposition in the classes of (C 2CD 2)m will
have m pairs of consecutive elements associated with C and the remaining m
pairs with D, with every C-pair member associated with a D-pair member by
the transposition. For example, with mZ2, the class C 2C 2D 2D 2 contains
(15)(26)(37)(48), and it also contains (15)(28)(36)(47).
The key point in what follows is that there is a more refined characterization of
the transpositions, which yields a natural subdivision of the transposition zig-zag
classes. It is a simple connectivity property, which can be illustrated by a
chaining graph. The nodes of the graph are the consecutive pairs, and its edges
are defined by the transposition. For the two transpositions we have considered,
the chaining graphs are as in figure 2 above.
The chaining graph for any transposition will partition the 2m pairs of
consecutive elements into a number of disjoint classes, each with its irreducible
chaining subgraph.
Conversely, given a chaining graph of this form, it defines a class of transpositions
all belonging to some class in (C 2CD 2)2m. It follows that to count the total
number of such transpositions, we can enumerate all the possible chaining graphs,
and then the number of transpositions associated with each such graph.
jðC C DÞ2mC1 j Z 0;
Proc. R. Soc. A (2007)
jðC C DÞ2m j Z
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Algebras and parity-dependent spectra
2425
This enumeration can be readily performed. We first consider the partitions of
the 2m pairs effected by the chaining graphs. A typical partition is given by
2m Z
q
X
2mi ai ;
iZ1
where ai is the multiplicity of 2mi in the partition. There are then
ð2mÞ!
;
ðð2m1 Þ!Þ ðð2m 2 Þ!Þ .ðð2mq Þ!Þaq a1 !a 2 !.aq !
a1
a2
distinct partitions of the pairs.
Given a subset of 2mi pairs, the number of irreducible chaining graphs of that
size is just the same as the number of zig-zag permutations on 2mi elements, viz.
the tangent number
T2miK1 :
The number of transpositions associated with an irreducible chaining graph of
size 2mi is simply
22miK1 :
Putting these facts together, we have
N
X
h0jðC 2 C D 2 Þ2m j0i
mZ0
2m
z
Z1C
ð2mÞ!
P
X 22mK ai ðT2m1K1 Þa1 .ðT2mqK1 Þaq z 2m
ðð2m1 Þ!Þa1 .ðð2mq Þ!Þaq a1 !.aq !
; ð4:6Þ
wherePthe sum on the right-hand side is over all partitions, such that
2mZ qiZ1 2mi ai , for mR1. This can be rewritten as
2m
aq
ð2zÞ2m1 T
a1
ð2zÞ q T2mqK1
2m1K1
2ð2mq Þ!
X
2ð2m1 Þ!
.
;
ð4:7Þ
1C
a1 !
aq !
where now the sum is over all ai , mi , qR1. But, using equation (3.2), this is just
!
N
pffiffiffiffiffiffiffiffiffiffiffiffiffi
X
ð2zÞ2r T2rK1
Z exp 12 ln sec 2z Z sec 2z :
exp
ð4:8Þ
2ð2rÞ!
rZ1
As a generalization, where s is a positive integer,
P
X smK ai ðT2m1K1 Þa1 .ðT2mqK1 Þaq z 2m
pffiffiffi 1=s
ðsec szÞ Z 1 C
;
ð4:9Þ
ðð2m1 Þ!Þa1 .ðð2mq Þ!Þaq a1 !.aq !
P
where the sum is taken over all partitions, such that 2mZ qiZ1 2mi ai , for mR1.
This is readily seen to yield an integer sequence, since
P
(i) mK ai R 0,
(ii) the T2miK1 are all integers,
(iii) ðð2m1 Þ!Þa1 ðð2m 2 Þ!Það2mÞ!
2/ðð2m Þ!Þaq a !a !.a ! is an integer since it counts the number of
q
1 2
q
distinct partitions
Proc. R. Soc. A (2007)
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2426
C. V. Sukumar and A. Hodges
pffiffiffi
More generally ðsec szÞr=s , where r and s are the positive integers, has an
integer sequence. This fact can be interpreted, using equation (3.13), in terms of
matrix elements for the squeezed states for the case aZ2r/sK1. We anticipate
that there are more interesting combinatorial results to be found.
5. Discussion
In this paper we have shown that an alteration of the fundamental commutation
relation between the creation and annihilation operators of the simple harmonic
oscillator by the inclusion of a parity-violating term leads to spectral features
that have several physical applications. We have shown that the algebra
governing the secant and tangent numbers is a special case of a more general
algebra. The expectation values involving Gaussian squeezed states lead to
secant and tangent functions and their higher integer powers. However, the
generalized algebra enables the evaluation of the expectation values involving a
class of non-Gaussian squeezed states that lead to non-integer powers of secant
and tangent functions. The establishment of a connection between the operator
methods used to calculate expectation values in coherent states and squeezed
states and simple triangle schemes arising from a bidiagonal structure of
underlying matrices also suggests that further generalizations are possible.
An algebraic scheme for producing parity-dependent spectra may be a useful
tool in interpreting the rotational and vibrational states of some nuclei, such as
Ra226
88 , and possibly some molecules. There are non-trivial consequences which
follow if a physical system is indeed governed by an algebra such as the ones
considered in this paper. For example, if the even and odd parity rotational
bands built on different vibrational states are described by the same parameter a,
then the separation of the even and odd parity rotational bands must be the same
for different vibrational states of the same nucleus or molecule. The preparation
and study of octupole-deformed nuclei is a difficult subject experimentally and
the published data on nuclei such as Ra226
88 only involve nuclei in their vibrational
groundstate. Further research towards identifying physical systems exhibiting
parity-dependent spectra is in progress.
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