A New Algebraic Structure of Finite Quantum
Systems and the Modified Bessel Functions
Kazuyuki FUJII ∗
Department of Mathematical Sciences
Yokohama City University
Yokohama, 236-0027
Japan
Abstract
In this paper we present a new algebraic structure (a super hyperbolic system in
our terminology) for finite quantum systems, which is a generalization of the usual
one in the two–level system.
It fits into the so–called generalized Pauli matrices, so they play an important
role in the theory. Some deep relation to the modified Bessel functions of integer
order is pointed out.
By taking a skillful limit finite quantum systems become quantum mechanics on
the circle developed by Ohnuki and Kitakado.
∗E-mail address : fujii@yokohama-cu.ac.jp
http://arxiv.org/abs/0704.1844v2
Quantum Computation is usually based on two–level system of atoms (qubit theory).
In the realistic construction of quantum logic gates we must solve some Schrödinger equa-
tions. Then the Pauli matrices {σ1, σ3} is essentially used and not only the periodic
functions {cos(x), sin(x)} but also the hyperbolic functions {cosh(x), sinh(x)} play an
important role.
On the other hand, they are deeply related to the modified Bessel functions of integer
order {In(x) | n ∈ Z}. The functions are in general given by the generating function.
Atom has usually many (finite or infinite) energy levels. However, to treat infinitely
many ones at the same time is not realistic, so we treat an atom with finite (for example
n) energy levels. We call this a finite quantum system and for this system the so–called
generalized Pauli matrices {Σ1,Σ3} play a crucial role, see for example [1], [2] and [3].
In this system we have a natural question on what functions corresponding to the hy-
perbolic functions are. In the paper we present such a system {c0(x), c1(x), · · · , cn−1(x)} (a
super hyperbolic system in our terminology) as a “natural” generalization of {cosh(x), sinh(x)}.
Moreover, we define a generating matrix based on the generalized Pauli matrices as
a “natural” generalization of the generating function and obtain interesting results by
taking some traces.
Lastly, we want to take a limit of finite quantum systems, which is of course impossible.
However, there is a bypass. That is, by taking a skillful limit finite quantum systems
become quantum mechanics on the circle developed by Ohnuki and Kitakado [4].
Through this paper we have a clear and unified picture of quantum systems.
First of all we make some mathematical preliminaries on the 2–level system. Let
{σ1, σ2, σ3} be Pauli matrices and 12 the unit matrix :
 0 1
 , σ2 =
 0 −i
 , σ3 =
 1 0
 , 12 =
 1 0
 . (1)
List the well–known properties of σ1 and σ3 :
σ21 = σ
3 = 12, σ
1 = σ1, σ
3 = σ3, σ3σ1 = −σ1σ3 = eπiσ1σ3. (2)
Let W be the Walsh–Hadamard matrix
 1 1
 = W−1 , (3)
then we can diagonalize σ1 as σ1 = Wσ3W
−1 by making use of W .
The modified Bessel functions of integer order {Ik(x) | k ∈ Z} are given by the
generating function
(w+ 1
Ik(x)w
k. (4)
Now let us list some (well–known) important properties (see for example [5]) :
1 = I0(x) + 2
(−1)kI2k(x),
ex = I0(x) + 2
Ik(x), e
−x = I0(x) + 2
(−1)kIk(x)
cosh(x) = I0(x) + 2
I2k(x), sinh(x) = 2
I2k−1(x).
In the following we set
c0(x) ≡ cosh(x) =
(2k)!
, c1(x) ≡ sinh(x) =
x2k+1
(2k + 1)!
for simplicity. The fundamental equation
c20(x)− c21(x) = 1 (6)
is interpreted as a simple relation
Sσ3S = σ3 ⇐⇒ σ3Sσ3S = 12 ⇐⇒ (σ3S)2 = 12
for S defined by
 c0(x) c1(x)
c1(x) c0(x)
 = c0(x)12 + c1(x)σ1 = exσ1 . (7)
Next we would like to extend the 2–level system to general n–level one. To make our
purpose clearer we treat the 3–level case in detail. Let σ be exp(2πi
), then we have
σ3 = 1, σ̄ = σ2, 1 + σ + σ2 = 0. (8)
Let Σ1 and Σ3 be generators of generalized Pauli matrices in the case of n = 3, namely
, Σ3 =
. (9)
Then it is easy to see
Σ31 = Σ
3 = 13, Σ
1 = Σ
3 = Σ
3, Σ3Σ1 = σΣ1Σ3. (10)
Now we can show that Σ1 can be diagonalized by making use of the matrix
1 1 1
1 σ2 σ
1 σ σ2
∈ U(3) (11)
Σ1 = WΣ3W
† = WΣ3W
−1. (12)
In fact
1 1 1
1 σ2 σ
1 σ σ2
1 1 1
1 σ σ2
1 σ2 σ
0 0 3
3 0 0
0 3 0
= Σ1,
where we have used the relations in (8).
From (5) we set
c0(x) =
(3k)!
, c1(x) =
x3k+1
(3k + 1)!
, c2(x) =
x3k+2
(3k + 2)!
. (13)
Then it is easy to check
c0(x) =
ex + eσx + eσ
, c1(x) =
ex + σ2eσx + σeσ
, c2(x) =
ex + σeσx + σ2eσ
by use of σ in (8) or reversely
ex = c0(x)+c1(x)+c2(x), e
σx = c0(x)+σc1(x)+σ
2c2(x), e
σ2x = c0(x)+σ
2c1(x)+σc2(x).
Now, our question is as follows : What is the fundamental equation that {c0(x), c1(x), c2(x)}
satisfy ?
The answer is given by the equation
(c0(x) + c1(x) + c2(x))(c0(x) + σc1(x) + σ
2c2(x))(c0(x) + σ
2c1(x) + σc2(x))
= exeσxeσ
2x = e(1+σ+σ
2)x = e0 = 1.
By expanding the left-hand side and using the relations (8) we obtain
c30(x) + c
1(x) + c
2(x)− 3c0(x)c1(x)c2(x) = 1. (15)
Next let us consider the addition formulas. By expanding
eσxeσy = eσ(x+y) ⇐= eσt = c0(t) + σc1(t) + σ2c2(t)
we have
c0(x)c0(y) + c1(x)c2(y) + c2(x)c1(y) = c0(x+ y),
c0(x)c1(y) + c1(x)c0(y) + c2(x)c2(y) = c1(x+ y),
c0(x)c2(y) + c1(x)c1(y) + c2(x)c0(y) = c2(x+ y). (16)
From here let us give a unified approach by use of the generalized Pauli matrices
{Σ1,Σ3} above. We consider the matrix
exΣ1 = c0(x)13 + c1(x)Σ1 + c2(x)Σ
c0(x) c2(x) c1(x)
c1(x) c0(x) c2(x)
c2(x) c1(x) c0(x)
. (17)
Then by Σ1 = WΣ3W
† in (12)
c0(x) c2(x) c1(x)
c1(x) c0(x) c2(x)
c2(x) c1(x) c0(x)
= exΣ1 = W exΣ3W † = W
so taking the determinant leads to
∣∣∣∣∣∣∣∣∣
c0(x) c2(x) c1(x)
c1(x) c0(x) c2(x)
c2(x) c1(x) c0(x)
∣∣∣∣∣∣∣∣∣
= e(1+σ+σ
2)x = 1.
Namely, we recovered (15).
On the other hand, by use of (11) it is straightforward to show
W † =
1 1 1
1 σ2 σ
1 σ σ2
1 1 1
1 σ σ2
1 σ2 σ
ex + eσx + eσ
ex + σ2eσx + σeσ
ex + σeσx + σ2eσ
so we recovered (14).
The matrix form is very convenient. Moreover, we can give new relations. For that
we consider the simple equation
exΣ1eyΣ
1 = exΣ1+yΣ
1 . (18)
The left hand side is
exΣ1eyΣ
1 = (c0(x)13 + c1(x)Σ1 + c2(x)Σ
1)(c0(y)13 + c1(y)Σ
1 + c2(y)Σ1)
= (c0(x)c0(y) + c1(x)c1(y) + c2(x)c2(y))13
+ (c0(x)c2(y) + c1(x)c0(y) + c2(x)c1(y))Σ1
+ (c0(x)c1(y) + c1(x)c2(y) + c2(x)c0(y))Σ
because Σ
1 = Σ
1. The right hand side is
exΣ1+yΣ
1 = eW (xΣ3+yΣ
)W † = W exΣ3+yΣ
3W † = W
exσ+yσ
ex+y + exσ+yσ
+ exσ
ex+y + σ2exσ+yσ
+ σexσ
ex+y + σexσ+yσ
+ σ2exσ
so we obtain
c0(x)c0(y) + c1(x)c1(y) + c2(x)c2(y) =
ex+y + exσ+yσ
+ exσ
c0(x)c2(y) + c1(x)c0(y) + c2(x)c1(y) =
ex+y + σ2exσ+yσ
+ σexσ
c0(x)c1(y) + c1(x)c2(y) + c2(x)c0(y) =
ex+y + σexσ+yσ
+ σ2exσ
. (19)
Next, let us consider the matrix exΣ1+yΣ
1 . If we set y = 1/x, then the matrix
exΣ1+(1/x)Σ
1 is similar to (4) the generating function of modified Bessel functions of integer
order. Therefore from (4) it is reasonable to consider
(wΣ1+ 1
1) = e
(wΣ1+ 1
1 ) =
Ik(x)w
kΣk1 . (20)
In the following we call this the generating matrix of modified Bessel functions of
integer order. Let us look for some typical properties. The result is
(wΣ1+ 1
(w+ 1
) + e
(wσ+ 1
σ2) + e
(wσ2+ 1
I3k(x)w
(wΣ1+ 1
(w+ 1
) + σe
(wσ+ 1
σ2) + σ2e
(wσ2+ 1
I3k−1(x)w
3k−1,
(wΣ1+ 1
1)Σ21
(w+ 1
) + σ2e
(wσ+ 1
σ2) + σe
(wσ2+ 1
I3k−2(x)w
where σ−1 = σ2 and σ−2 = σ.
A comment is in order. In the case of n = 2 the generating matrix is
(wσ1+ 1
1) = e
(w+ 1
)σ1 = cosh
12 + sinh
because σ1 is hermitian, so the situation becomes much easier.
From the lesson for the case of n = 3, let us set up the general case. Let {Σ1,Σ3} be
generalized Pauli matrices

. . .
. . .

, Σ3 =

. . .

where σ is a primitive element σ = exp(2πi
) which satisfies
σn = 1, σ̄ = σn−1, 1 + σ + · · ·+ σn−1 = 0. (23)
Then it is easy to see
Σn1 = Σ
3 = 1n, Σ
1 = Σ
1 , Σ
3 = Σ
3 , Σ3Σ1 = σΣ1Σ3. (24)
If we define a Vandermonde matrix W based on σ as

1 1 1 · · · 1 1
1 σn−1 σ2(n−1) · · · σ(n−2)(n−1) σ(n−1)2
1 σn−2 σ2(n−2) · · · σ(n−2)2 σ(n−1)(n−2)
1 σ2 σ4 · · · σ2(n−2) σ2(n−1)
1 σ σ2 · · · σn−2 σn−1

, (25)
then it is not difficult to see
Σ1 = WΣ3W
† = WΣ3W
−1. (26)
That is, Σ1 can be diagonalized by making use of W .
We set
cj(x) =
xkn+j
(kn+ j)!
for 0 ≤ j ≤ n− 1. It is of course
cj(x)
and easy to see
exΣ1 = c0(x)1n + c1(x)Σ1 + c2(x)Σ
1 + · · ·+ cn−2(x)Σn−21 + cn−1(x)Σn−11

c0(x) cn−1(x) · · · c2(x) c1(x)
c1(x) c0(x) cn−1(x) · · · c2(x)
c1(x) c0(x) cn−1(x)
. . .
. . .
. . .
cn−2(x) · · · c1(x) c0(x) cn−1(x)
cn−1(x) cn−2(x) · · · c2(x) c1(x) c0(x)

. (28)
Let us look for the fundamental equation that {c0(x), c1(x), · · · , cn−2(x), cn−1(x)} sat-
isfy. By use of (26)
exΣ1 = exWΣ3W
= W exΣ3W †
we have
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
c0(x) cn−1(x) · · · c2(x) c1(x)
c1(x) c0(x) cn−1(x) · · · c2(x)
c1(x) c0(x) cn−1(x)
. . .
. . .
. . .
cn−2(x) · · · c1(x) c0(x) cn−1(x)
cn−1(x) cn−2(x) · · · c2(x) c1(x) c0(x)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
. . .
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
= ex(1+σ+σ
2+···+σn−2+σn−1) = e0 = 1 (29)
because W is unitary (|W | = 1). For example
n = 2 c20(x)− c21(x) = 1 (⇐= (6))
n = 3 c30(x) + c
1(x) + c
2(x)− 3c0(x)c1(x)c2(x) = 1 (⇐= (15))
n = 4 c40(x)− c41(x) + c42(x)− c43(x)− 2c20(x)c22(x) + 2c21(x)c23(x)
−4c20(x)c1(x)c3(x) + 4c0(x)c21(x)c2(x)− 4c1(x)c22(x)c3(x) + 4c0(x)c2(x)c23(x) = 1.
We call {c0(x), c1(x), · · · , cn−1(x)} the super hyperbolic system.
The addition formulas are given by the simple equation
exΣ1eyΣ1 = e(x+y)Σ1
and become
cj(x+ y) =
k+l=j (mod n)
ck(x)cl(y) for 0 ≤ j ≤ n− 1. (31)
More explicitly,
cj(x+y) = c0(x)cj(y)+c1(x)cj−1(y)+· · ·+cj(x)c0(y)+cj+1(x)cn−1(y)+· · ·+cn−1(x)cj+1(y).
The new relations are given by the simple equation
exΣ1eyΣ
1 = exΣ1+yΣ
and become
ck(x)cn−j+k(y) +
ck(x)ck−j(y) =
σk(n−j)exσ
k+yσn−k (32)
for 0 ≤ j ≤ n− 1.
The generating matrix of modified Bessel functions of integer order is given by
(wΣ1+ 1
Ik(x)w
kΣk1 (33)
and from this we have
(wΣ1+ 1
(wσl+ 1
σ−l) =
Ink−j(x)w
nk−j (34)
for 0 ≤ j ≤ n− 1.
The result in the case of j = 0 is known in [6] and [7].
We want to take a (formal) limit n −→ ∞. That is, what is Σ1 −→ ?, Σ3 −→ ? It is
of course impossible to take a limit with this form. For that let us make a small change.
We set n = 2N + 1 and
Σ̃1 = Σ1, Σ̃3 =

. . .
. . .

where σ = exp( 2πi
). Here we rewrite Σ̃3 as Σ̃3 = exp(
G̃) where

. . .
. . .

. (36)
The commutator [G̃, Σ̃1] becomes
[G̃, Σ̃1] =

0 −2N
. . .
. . .


. . .
. . .

(mod 2N + 1).
That is, we have the relation
[G̃, Σ̃1] = Σ̃1 (mod 2N + 1). (37)
In this stage, it may be better to write the (finite dimensional) Hilbert space as
2N+1 = VectC{|−N〉, · · · , |−1〉, |0〉, |1〉, · · · , |N〉}
because G̃|n〉 = n|n〉.
Now, if we take a formal limit N −→ ∞ then we have the fundamental relation
[G,W ] = W (38)
where

. . .
. . .

, W =

. . .
. . . 0
. . .
. . .

where the notations {G,W} in [4] were used. Note that G is a hermitian operator and
W a unitary operator on the Hilbert space
L2(Z) =
cn|n〉 |
|cn|2 < ∞
; W |n〉 = |n+ 1〉, G|n〉 = n|n〉. (40)
The relation (38) is just the fundamental one in quantum mechanice on the circle devel-
oped by Ohnuki and Kitakado [4].
A comment is in order. There is some freedom on the choice of G. That is, if we
choose G like
G −→ G+ α1, 0 ≤ α < 1
the relation (38) still holds. α is interpreted as a kind of abelian gauge induced in quantum
mechanice on the circle.
Therefore it may be better to write the generators {Gα ≡ G + α1,W} in place of
{G,W} in [4]. We don’t repeat the contents, so see [4] and its references.
Readers may find many interesting problems from the paper. For example, we can
consider the generating operator
(wW+ 1
W †).
We leave some calculations to readers.
In this paper we developed the super hyperbolic structure for (all) finite quantum
systems, and defined the generating matrix for the modified Bessel functions of integer
order and obtained some interesting results. We also gave a connection to quantum
mechanics on the circle by Ohnuki and Kitakado by taking a skillful limit.
Our motivation is to apply the development in the paper to qudit theory based on
finite quantum systems, which will be reported in another paper.
Acknowledgment.
The author wishes to thank K. Funahashi for helpful comments and suggestions.
References
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http://arxiv.org/abs/quant-ph/0207002
[2] K. Fujii : How to Treat an N-Level System : A Proposal, quant-ph/0302050.
[3] K. Fujii, K. Funahashi and T. Kobayashi : Jarlskog’s Parametrization of Uni-
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quant-ph/0508006.
[4] Y. Ohnuki and S. Kitakado : Fundamental Algebra for Quantum Mechanics on SD
and Gauge Potentials, J. Math. Phys. 34(1993), 2827.
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Int. J. Mod. Phys. 20(2006), 593, arXiv : 0704.2898 (math-ph).
http://arxiv.org/abs/quant-ph/0302050
http://arxiv.org/abs/quant-ph/0508006
http://arxiv.org/abs/math-ph/0703010

  In this paper we present a new algebraic structure (a super hyperbolic system
in our terminology) for finite quantum systems, which is a generalization of
the usual one in the two-level system.
  It fits into the so-called generalized Pauli matrices, so they play an
important role in the theory. Some deep relation to the modified Bessel
functions of integer order is pointed out.
  By taking a skillful limit finite quantum systems become quantum mechanics on
the circle developed by Ohnuki and Kitakado.

A New Algebraic Structure of Finite Quantum
Systems and the Modified Bessel Functions
Kazuyuki FUJII ∗
Department of Mathematical Sciences
Yokohama City University
Yokohama, 236-0027
Japan
Abstract
In this paper we present a new algebraic structure (a super hyperbolic system in
our terminology) for finite quantum systems, which is a generalization of the usual
one in the two–level system.
It fits into the so–called generalized Pauli matrices, so they play an important
role in the theory. Some deep relation to the modified Bessel functions of integer
order is pointed out.
By taking a skillful limit finite quantum systems become quantum mechanics on
the circle developed by Ohnuki and Kitakado.
∗E-mail address : fujii@yokohama-cu.ac.jp
http://arxiv.org/abs/0704.1844v2
Quantum Computation is usually based on two–level system of atoms (qubit theory).
In the realistic construction of quantum logic gates we must solve some Schrödinger equa-
tions. Then the Pauli matrices {σ1, σ3} is essentially used and not only the periodic
functions {cos(x), sin(x)} but also the hyperbolic functions {cosh(x), sinh(x)} play an
important role.
On the other hand, they are deeply related to the modified Bessel functions of integer
order {In(x) | n ∈ Z}. The functions are in general given by the generating function.
Atom has usually many (finite or infinite) energy levels. However, to treat infinitely
many ones at the same time is not realistic, so we treat an atom with finite (for example
n) energy levels. We call this a finite quantum system and for this system the so–called
generalized Pauli matrices {Σ1,Σ3} play a crucial role, see for example [1], [2] and [3].
In this system we have a natural question on what functions corresponding to the hy-
perbolic functions are. In the paper we present such a system {c0(x), c1(x), · · · , cn−1(x)} (a
super hyperbolic system in our terminology) as a “natural” generalization of {cosh(x), sinh(x)}.
Moreover, we define a generating matrix based on the generalized Pauli matrices as
a “natural” generalization of the generating function and obtain interesting results by
taking some traces.
Lastly, we want to take a limit of finite quantum systems, which is of course impossible.
However, there is a bypass. That is, by taking a skillful limit finite quantum systems
become quantum mechanics on the circle developed by Ohnuki and Kitakado [4].
Through this paper we have a clear and unified picture of quantum systems.
First of all we make some mathematical preliminaries on the 2–level system. Let
{σ1, σ2, σ3} be Pauli matrices and 12 the unit matrix :
 0 1
 , σ2 =
 0 −i
 , σ3 =
 1 0
 , 12 =
 1 0
 . (1)
List the well–known properties of σ1 and σ3 :
σ21 = σ
3 = 12, σ
1 = σ1, σ
3 = σ3, σ3σ1 = −σ1σ3 = eπiσ1σ3. (2)
Let W be the Walsh–Hadamard matrix
 1 1
 = W−1 , (3)
then we can diagonalize σ1 as σ1 = Wσ3W
−1 by making use of W .
The modified Bessel functions of integer order {Ik(x) | k ∈ Z} are given by the
generating function
(w+ 1
Ik(x)w
k. (4)
Now let us list some (well–known) important properties (see for example [5]) :
1 = I0(x) + 2
(−1)kI2k(x),
ex = I0(x) + 2
Ik(x), e
−x = I0(x) + 2
(−1)kIk(x)
cosh(x) = I0(x) + 2
I2k(x), sinh(x) = 2
I2k−1(x).
In the following we set
c0(x) ≡ cosh(x) =
(2k)!
, c1(x) ≡ sinh(x) =
x2k+1
(2k + 1)!
for simplicity. The fundamental equation
c20(x)− c21(x) = 1 (6)
is interpreted as a simple relation
Sσ3S = σ3 ⇐⇒ σ3Sσ3S = 12 ⇐⇒ (σ3S)2 = 12
for S defined by
 c0(x) c1(x)
c1(x) c0(x)
 = c0(x)12 + c1(x)σ1 = exσ1 . (7)
Next we would like to extend the 2–level system to general n–level one. To make our
purpose clearer we treat the 3–level case in detail. Let σ be exp(2πi
), then we have
σ3 = 1, σ̄ = σ2, 1 + σ + σ2 = 0. (8)
Let Σ1 and Σ3 be generators of generalized Pauli matrices in the case of n = 3, namely
, Σ3 =
. (9)
Then it is easy to see
Σ31 = Σ
3 = 13, Σ
1 = Σ
3 = Σ
3, Σ3Σ1 = σΣ1Σ3. (10)
Now we can show that Σ1 can be diagonalized by making use of the matrix
1 1 1
1 σ2 σ
1 σ σ2
∈ U(3) (11)
Σ1 = WΣ3W
† = WΣ3W
−1. (12)
In fact
1 1 1
1 σ2 σ
1 σ σ2
1 1 1
1 σ σ2
1 σ2 σ
0 0 3
3 0 0
0 3 0
= Σ1,
where we have used the relations in (8).
From (5) we set
c0(x) =
(3k)!
, c1(x) =
x3k+1
(3k + 1)!
, c2(x) =
x3k+2
(3k + 2)!
. (13)
Then it is easy to check
c0(x) =
ex + eσx + eσ
, c1(x) =
ex + σ2eσx + σeσ
, c2(x) =
ex + σeσx + σ2eσ
by use of σ in (8) or reversely
ex = c0(x)+c1(x)+c2(x), e
σx = c0(x)+σc1(x)+σ
2c2(x), e
σ2x = c0(x)+σ
2c1(x)+σc2(x).
Now, our question is as follows : What is the fundamental equation that {c0(x), c1(x), c2(x)}
satisfy ?
The answer is given by the equation
(c0(x) + c1(x) + c2(x))(c0(x) + σc1(x) + σ
2c2(x))(c0(x) + σ
2c1(x) + σc2(x))
= exeσxeσ
2x = e(1+σ+σ
2)x = e0 = 1.
By expanding the left-hand side and using the relations (8) we obtain
c30(x) + c
1(x) + c
2(x)− 3c0(x)c1(x)c2(x) = 1. (15)
Next let us consider the addition formulas. By expanding
eσxeσy = eσ(x+y) ⇐= eσt = c0(t) + σc1(t) + σ2c2(t)
we have
c0(x)c0(y) + c1(x)c2(y) + c2(x)c1(y) = c0(x+ y),
c0(x)c1(y) + c1(x)c0(y) + c2(x)c2(y) = c1(x+ y),
c0(x)c2(y) + c1(x)c1(y) + c2(x)c0(y) = c2(x+ y). (16)
From here let us give a unified approach by use of the generalized Pauli matrices
{Σ1,Σ3} above. We consider the matrix
exΣ1 = c0(x)13 + c1(x)Σ1 + c2(x)Σ
c0(x) c2(x) c1(x)
c1(x) c0(x) c2(x)
c2(x) c1(x) c0(x)
. (17)
Then by Σ1 = WΣ3W
† in (12)
c0(x) c2(x) c1(x)
c1(x) c0(x) c2(x)
c2(x) c1(x) c0(x)
= exΣ1 = W exΣ3W † = W
so taking the determinant leads to
∣∣∣∣∣∣∣∣∣
c0(x) c2(x) c1(x)
c1(x) c0(x) c2(x)
c2(x) c1(x) c0(x)
∣∣∣∣∣∣∣∣∣
= e(1+σ+σ
2)x = 1.
Namely, we recovered (15).
On the other hand, by use of (11) it is straightforward to show
W † =
1 1 1
1 σ2 σ
1 σ σ2
1 1 1
1 σ σ2
1 σ2 σ
ex + eσx + eσ
ex + σ2eσx + σeσ
ex + σeσx + σ2eσ
so we recovered (14).
The matrix form is very convenient. Moreover, we can give new relations. For that
we consider the simple equation
exΣ1eyΣ
1 = exΣ1+yΣ
1 . (18)
The left hand side is
exΣ1eyΣ
1 = (c0(x)13 + c1(x)Σ1 + c2(x)Σ
1)(c0(y)13 + c1(y)Σ
1 + c2(y)Σ1)
= (c0(x)c0(y) + c1(x)c1(y) + c2(x)c2(y))13
+ (c0(x)c2(y) + c1(x)c0(y) + c2(x)c1(y))Σ1
+ (c0(x)c1(y) + c1(x)c2(y) + c2(x)c0(y))Σ
because Σ
1 = Σ
1. The right hand side is
exΣ1+yΣ
1 = eW (xΣ3+yΣ
)W † = W exΣ3+yΣ
3W † = W
exσ+yσ
ex+y + exσ+yσ
+ exσ
ex+y + σ2exσ+yσ
+ σexσ
ex+y + σexσ+yσ
+ σ2exσ
so we obtain
c0(x)c0(y) + c1(x)c1(y) + c2(x)c2(y) =
ex+y + exσ+yσ
+ exσ
c0(x)c2(y) + c1(x)c0(y) + c2(x)c1(y) =
ex+y + σ2exσ+yσ
+ σexσ
c0(x)c1(y) + c1(x)c2(y) + c2(x)c0(y) =
ex+y + σexσ+yσ
+ σ2exσ
. (19)
Next, let us consider the matrix exΣ1+yΣ
1 . If we set y = 1/x, then the matrix
exΣ1+(1/x)Σ
1 is similar to (4) the generating function of modified Bessel functions of integer
order. Therefore from (4) it is reasonable to consider
(wΣ1+ 1
1) = e
(wΣ1+ 1
1 ) =
Ik(x)w
kΣk1 . (20)
In the following we call this the generating matrix of modified Bessel functions of
integer order. Let us look for some typical properties. The result is
(wΣ1+ 1
(w+ 1
) + e
(wσ+ 1
σ2) + e
(wσ2+ 1
I3k(x)w
(wΣ1+ 1
(w+ 1
) + σe
(wσ+ 1
σ2) + σ2e
(wσ2+ 1
I3k−1(x)w
3k−1,
(wΣ1+ 1
1)Σ21
(w+ 1
) + σ2e
(wσ+ 1
σ2) + σe
(wσ2+ 1
I3k−2(x)w
where σ−1 = σ2 and σ−2 = σ.
A comment is in order. In the case of n = 2 the generating matrix is
(wσ1+ 1
1) = e
(w+ 1
)σ1 = cosh
12 + sinh
because σ1 is hermitian, so the situation becomes much easier.
From the lesson for the case of n = 3, let us set up the general case. Let {Σ1,Σ3} be
generalized Pauli matrices

. . .
. . .

, Σ3 =

. . .

where σ is a primitive element σ = exp(2πi
) which satisfies
σn = 1, σ̄ = σn−1, 1 + σ + · · ·+ σn−1 = 0. (23)
Then it is easy to see
Σn1 = Σ
3 = 1n, Σ
1 = Σ
1 , Σ
3 = Σ
3 , Σ3Σ1 = σΣ1Σ3. (24)
If we define a Vandermonde matrix W based on σ as

1 1 1 · · · 1 1
1 σn−1 σ2(n−1) · · · σ(n−2)(n−1) σ(n−1)2
1 σn−2 σ2(n−2) · · · σ(n−2)2 σ(n−1)(n−2)
1 σ2 σ4 · · · σ2(n−2) σ2(n−1)
1 σ σ2 · · · σn−2 σn−1

, (25)
then it is not difficult to see
Σ1 = WΣ3W
† = WΣ3W
−1. (26)
That is, Σ1 can be diagonalized by making use of W .
We set
cj(x) =
xkn+j
(kn+ j)!
for 0 ≤ j ≤ n− 1. It is of course
cj(x)
and easy to see
exΣ1 = c0(x)1n + c1(x)Σ1 + c2(x)Σ
1 + · · ·+ cn−2(x)Σn−21 + cn−1(x)Σn−11

c0(x) cn−1(x) · · · c2(x) c1(x)
c1(x) c0(x) cn−1(x) · · · c2(x)
c1(x) c0(x) cn−1(x)
. . .
. . .
. . .
cn−2(x) · · · c1(x) c0(x) cn−1(x)
cn−1(x) cn−2(x) · · · c2(x) c1(x) c0(x)

. (28)
Let us look for the fundamental equation that {c0(x), c1(x), · · · , cn−2(x), cn−1(x)} sat-
isfy. By use of (26)
exΣ1 = exWΣ3W
= W exΣ3W †
we have
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
c0(x) cn−1(x) · · · c2(x) c1(x)
c1(x) c0(x) cn−1(x) · · · c2(x)
c1(x) c0(x) cn−1(x)
. . .
. . .
. . .
cn−2(x) · · · c1(x) c0(x) cn−1(x)
cn−1(x) cn−2(x) · · · c2(x) c1(x) c0(x)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
. . .
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
= ex(1+σ+σ
2+···+σn−2+σn−1) = e0 = 1 (29)
because W is unitary (|W | = 1). For example
n = 2 c20(x)− c21(x) = 1 (⇐= (6))
n = 3 c30(x) + c
1(x) + c
2(x)− 3c0(x)c1(x)c2(x) = 1 (⇐= (15))
n = 4 c40(x)− c41(x) + c42(x)− c43(x)− 2c20(x)c22(x) + 2c21(x)c23(x)
−4c20(x)c1(x)c3(x) + 4c0(x)c21(x)c2(x)− 4c1(x)c22(x)c3(x) + 4c0(x)c2(x)c23(x) = 1.
We call {c0(x), c1(x), · · · , cn−1(x)} the super hyperbolic system.
The addition formulas are given by the simple equation
exΣ1eyΣ1 = e(x+y)Σ1
and become
cj(x+ y) =
k+l=j (mod n)
ck(x)cl(y) for 0 ≤ j ≤ n− 1. (31)
More explicitly,
cj(x+y) = c0(x)cj(y)+c1(x)cj−1(y)+· · ·+cj(x)c0(y)+cj+1(x)cn−1(y)+· · ·+cn−1(x)cj+1(y).
The new relations are given by the simple equation
exΣ1eyΣ
1 = exΣ1+yΣ
and become
ck(x)cn−j+k(y) +
ck(x)ck−j(y) =
σk(n−j)exσ
k+yσn−k (32)
for 0 ≤ j ≤ n− 1.
The generating matrix of modified Bessel functions of integer order is given by
(wΣ1+ 1
Ik(x)w
kΣk1 (33)
and from this we have
(wΣ1+ 1
(wσl+ 1
σ−l) =
Ink−j(x)w
nk−j (34)
for 0 ≤ j ≤ n− 1.
The result in the case of j = 0 is known in [6] and [7].
We want to take a (formal) limit n −→ ∞. That is, what is Σ1 −→ ?, Σ3 −→ ? It is
of course impossible to take a limit with this form. For that let us make a small change.
We set n = 2N + 1 and
Σ̃1 = Σ1, Σ̃3 =

. . .
. . .

where σ = exp( 2πi
). Here we rewrite Σ̃3 as Σ̃3 = exp(
G̃) where

. . .
. . .

. (36)
The commutator [G̃, Σ̃1] becomes
[G̃, Σ̃1] =

0 −2N
. . .
. . .


. . .
. . .

(mod 2N + 1).
That is, we have the relation
[G̃, Σ̃1] = Σ̃1 (mod 2N + 1). (37)
In this stage, it may be better to write the (finite dimensional) Hilbert space as
2N+1 = VectC{|−N〉, · · · , |−1〉, |0〉, |1〉, · · · , |N〉}
because G̃|n〉 = n|n〉.
Now, if we take a formal limit N −→ ∞ then we have the fundamental relation
[G,W ] = W (38)
where

. . .
. . .

, W =

. . .
. . . 0
. . .
. . .

where the notations {G,W} in [4] were used. Note that G is a hermitian operator and
W a unitary operator on the Hilbert space
L2(Z) =
cn|n〉 |
|cn|2 < ∞
; W |n〉 = |n+ 1〉, G|n〉 = n|n〉. (40)
The relation (38) is just the fundamental one in quantum mechanice on the circle devel-
oped by Ohnuki and Kitakado [4].
A comment is in order. There is some freedom on the choice of G. That is, if we
choose G like
G −→ G+ α1, 0 ≤ α < 1
the relation (38) still holds. α is interpreted as a kind of abelian gauge induced in quantum
mechanice on the circle.
Therefore it may be better to write the generators {Gα ≡ G + α1,W} in place of
{G,W} in [4]. We don’t repeat the contents, so see [4] and its references.
Readers may find many interesting problems from the paper. For example, we can
consider the generating operator
(wW+ 1
W †).
We leave some calculations to readers.
In this paper we developed the super hyperbolic structure for (all) finite quantum
systems, and defined the generating matrix for the modified Bessel functions of integer
order and obtained some interesting results. We also gave a connection to quantum
mechanics on the circle by Ohnuki and Kitakado by taking a skillful limit.
Our motivation is to apply the development in the paper to qudit theory based on
finite quantum systems, which will be reported in another paper.
Acknowledgment.
The author wishes to thank K. Funahashi for helpful comments and suggestions.
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