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Multi-matrix vector coherent states K. Thirulogasanthara,*, G. Honnouvoa, A. Krzy_zakb a

Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal, Que., Canada H4B 1R6 b Department of Computer Science, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Que., Canada H3G 1M8 Received 2 June 2004; accepted 22 July 2004 Available online 17 September 2004

Abstract A class of vector coherent states is derived with multiple of matrices as vectors in a Hilbert space, where the Hilbert space is taken to be the tensor product of several other Hilbert spaces. As examples vector coherent states with multiple of quaternions and octonions are given. The resulting generalized oscillator algebra is brieﬂy discussed. Further, vector coherent states for a tensored Hamiltonian system are obtained by the same method. As particular cases, coherent states are obtained for tensored Jaynes–Cummings type Hamiltonians and for a two-level two-mode generalization of the Jaynes–Cummings model. 2004 Elsevier Inc. All rights reserved. PACS: 81R30 Keywords: Coherent states; Vector coherent states; Oscillator algebra

1. Introduction The deﬁnition of the canonical coherent states, CS for short, of the harmonic oscillator has been generalized in a number of ways and successfully applied in quantum physics for a long time. For various generalizations of this concept and their *

Corresponding author. Fax: +1 514 848 2830. E-mail addresses: [email protected] (K. (G. Honnouvo), [email protected] (A. Krzy_zak).

Thirulogasanthar),

0003-4916/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2004.07.006

[email protected]

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properties one may consult [1–7]. Hilbert spaces are the underlying mathematical structure of several physical phenomena. In general CS form an overcomplete family of vectors in the separable Hilbert space of the physical problem. 1 Let H be a separable Hilbert space with an orthonormal basis f/m gm¼0 and C be the complex plane. For z 2 D, an open subset of C, the states 1 X zm 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ /m jzi ¼ N ðjzjÞ 2 ð1:1Þ qðmÞ m¼0 are said to form a set of CS if (a) The states |zæ are normalized, (b) The states |zæ satisfy a resolution of the identity, that is Z

jzihzj dl ¼ I;

ð1:2Þ

D 1

where N ðjzjÞ is the normalization factor, fqðmÞgm¼0 is a sequence of non-zero positive real numbers, dl is an appropriately chosen measure on D, and I is the identity operator on H. In a recent paper [6], the labeling parameter z of (1.1) was replaced by an n · n matrix valued function of the form Z = A (r)eifH (k), where A (r) and H (k) are n · n matrices and thereby a class of vector coherent states (VCS for short) was generated as n component vectors in a Hilbert space H ¼ Cn H 1 , where H 1 is an abstract separable Hilbert space. In order to get the normalization and resolution of the identity following conditions were imposed on the matrices. y

HðkÞ ¼ HðkÞ ;

½AðrÞ; HðkÞ ¼ 0;

y

½AðrÞ; AðrÞ ¼ 0;

ð1:3Þ

where stands for the conjugate transpose of a matrix. The square bracket is equal to zero if the matrices commute. A further generalization of the same concept was given in [7], where the VCS were realized as inﬁnite component vectors in a suitable Hilbert space and as special cases VCS were also given as ﬁnite component vectors. It is shown in [6,7] that the VCS obtained with the complex representation of quaternions are very eﬀective in describing two-level atomic systems placed in a single mode electromagnetic ﬁeld. In particular [7] argues that it is a good ﬁt for the Jaynes–Cummings model. Gazeau and Klauder [4] proposed a class of temporally stable CS for Hamiltonians with non-degenerate discrete spectrum by introducing a generalization of the canonical coherent states. At present this class of CS is known as the Gazeau–Klauder CS. Following their method CS were derived for several quantum systems, see, for example [8,9]. As a further generalization of this concept multidimensional CS were introduced in [5] and as a special case of this generalization CS were obtained as a tensor product of two canonical CS in the following form: 1 X 1 X 2 2 zn11 zn22 pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ jn1 ; n2 i: jz1 ; z2 i ¼ eðjz1 j þjz2 j Þ=2 ð1:4Þ n1 ! n2 ! n1 ¼0 n2 ¼0

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121

The result of [5] can be considered as an extension to the results of [4] in a sense that it allows one to construct CS for a system with several degrees of freedom. For example, for a system with two degrees of freedom it can be written as follows: jJ 1 ; J 2 ; c1 ; c2 i ¼ N 1 ðJ 1 ; J 2 Þ

1

X n1

X n2

n =2

J 11 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ eic1 En1 N 2 ðJ 2 ; n1 Þ1 q1 ðn1 Þ

n =2 J 22

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ eic2 En2 jn1 ; n2 i; q2 ðn1 ; n2 Þ

ð1:5Þ

where the labeling parameters and the Hilbert space were chosen appropriately for the system and q (n1), q (n1, n2), En1 , En2 are associated with the spectrum En1 ;n2 of the system. The dependence of the ﬁrst sum on the other severely restricts one to consider the states of Eq. (1.5) as a tensor product of two CS. The detailed explanation can be found in [5]. This procedure is also used in [10] to obtain CS for a free magnetic Schro¨dinger operator. It should also be mentioned that in [11] Klauder has obtained CS for the hydrogen atom using a similar method of [5]. Motivated by these recent developments we generalize the results of [5–7] by introducing multi-matrix VCS in the form 1 X 1 1 X X An11 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ jA1 ; A2 ; . . . ; As ; ji ¼ N 2 q1 ðn1 Þ n1 ¼0 n2 ¼0 ns ¼0 An22 Ans s ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vj /n1 /n2 /ns q2 ðn2 Þ qs ðns Þ

ð1:6Þ

on a Hilbert space H ¼ Cn ½sk¼1 H k , where AkÕs are n · n matrix valued functions 1 on appropriate domains, f/nj gnj ¼0 is an orthonormal basis of the separable Hilbert space H j ; j ¼ 1; 2; . . . ; s and fqj ðnj Þg1 nj ¼0 ; j ¼ 1; 2; . . . ; s are positive sequences of real numbers. A similar result with complex numbers z1, z2, . . ., zs may not be hard to obtain because the complex numbers commute with each other. For matrices, the non-commutativity restricts the construction severely. Furthermore, notice that jA1 ðr1 Þ; A2 ðr2 Þ; . . . ; As ðrs Þ; ji 6¼ jA1 ðr1 Þ; ji jA2 ðr2 Þ; ji jAs ðrs Þ; ji: Next we discuss the physical motivation of the generalization (1.6). It is well known that the most elementary model describing the interaction of two-level atoms with a single quantized ﬁeld is the Jaynes–Cummings model. This model is exactly solvable in the rotating wave approximation, see [12–14] and the references listed therein. Suppose we have a diagonalizable Hamiltonian H for a two-level atom in a single mode cavity ﬁeld with non-degenerate energies Ekm and wavefunctions wkm ; k ¼ 1; 2; m ¼ 0; 1; ...; 1 1. Let xkm ¼ Ekm Ek0 ; k ¼ 1; 2; m ¼ 0; 1; . . . 1. Let RðmÞ ¼ ½diagðx1m !; x2m !Þ 2 and Z = diag (z1, z2) be diagonal matrices, where xkm ! ¼ xk1 xkm is the generalized factorial. Assume that the following vectors are normalized and satisfy a resolution of the identity, 1 X 1 RðmÞZ m vk /m 2 C2 H; k ¼ 1; 2; ð1:7Þ jZ; ki ¼ N ðZÞ 2 m¼0

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where the wavefunctions wkm ; k ¼ 1; 2 are identiﬁed to the basis of C2 H as /km :¼ vk /m ; k ¼ 1; 2; m ¼ 0; 1; 2; . . . The collection of vectors jZi ¼

2 X

ck jZ; ki

with

2

2

jc1 j þ jc2 j ¼ 1

k¼1

forms a set of CS for the diagonalized Hamiltonian HD. Such an argument for a twolevel system leading to the quaternionic VCS of [6] was given in [7]. If one neglects losses, multi-mode multi-level generalizations of the Jaynes– Cummings model can be solved exactly using the exact solvability of the Jaynes– Cummings model [13,15]. By assuming the existence of a solvable multi-mode multi-level system, we also construct VCS as CS of a multi-mode multi-level quantum model as an application to the multi-matrix VCS. In fact, an extension of (1.7) to multi-level multi-mode systems requires multi-matrix VCS. For example, let us consider the following model. In the rotating wave approximation the Hamiltonian of a two-level atom in the two-mode ﬁeld is given by [15] qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 X 1 1 H¼ xi ayi ai þ ay1 a1 ay2 a2 ay1 ay2 r þ rþ a1 a2 ay1 a1 ay2 a2 ; þ x0 r3 þ g 2 2 i¼1 ð1:8Þ ði ¼ 1; 2Þ are the annihilation and creation operators of the ith where h ¼ 1, mode of the ﬁeld, xi (i = 1, 2), x0 = x1 + x2 (i.e, the two-photon resonance process) are the frequencies of the ith mode of the radiation ﬁeld and the atomic transition, r3, r, and r+ are the pseudospin operators of the atom, and g is the coupling constant. By diagonalizing H one can obtain the spectrum of H (see [15] and the references listed there): ai ; ayi

En;m ¼ x1 ðn þ 1Þ þ x2 ðm þ 1Þ gðn þ 1Þðm þ 1Þ:

ð1:9Þ

For the zero coupling (g = 0) it can be easily seen that if x1/x2 is an irrational number then the spectrum is non-degenerate. By appropriately identifying the wavefunctions of (1.8) as the basis of separable abstract Hilbert spaces, as an example of the general model, when (g = 0) we shall introduce CS for the Hamiltonian (1.8) in the following form: 1 1 X X 1 1 1 jZ; Z; ki ¼ N ðZ; ZÞ 2 RðnÞ 2 Z n Rðn; mÞ 2 Zm vk /n wm ; k ¼ 1; 2; n¼0

m¼0

ð1:10Þ where Z, Z, R(n), and R(n, m) are 2 · 2 diagonal matrices. In addition to this example, we shall also present VCS for tensored Jaynes–Cummings type Hamiltonians. In order to avoid technicalities, we carry out our general construction with two n · n matrices A and B. Later using the real matrix representation of quaternions and octonions we demonstrate it for several matrices. We also generate VCS in the above form with matrix valued qj (nj)Õs. Further, we discuss VCS of type (1.5). We shall brieﬂy discuss generalized oscillator algebras associated to the VCS. Finally, we introduce VCS for multi-level multi-mode quantum systems.

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123

2. General construction Let H 1 ; H 2 be two separable Hilbert spaces and C be the complex plane. Let 1 1 n f/m gm¼0 ; fwl gl¼0 , and fvj gj¼0 be orthonormal bases of the Hilbert spaces in the respecb ¼ Cn H 1 H 2 , where stands for the tensor prodtive order and of Cn . Denote H b. uct. Then the set of vectors {vj /m wl}j, m, l serve as an orthonormal basis of H 0 0 Let K, K be measure spaces with probability measures dK and dK , respectively, and R, S be a second pair of measure spaces with measures dR and dS, respectively. For (r, k, f) 2 R · K · [0, 2p) and (s, k 0 , g) 2 S · K 0 · [0, 2p), let A ¼ AðrÞeifHðkÞ

and

0

B ¼ BðsÞeigKðk Þ ;

ð2:1Þ

where A (r), H (k), B (s), and K (k 0 ) are measurable n · n matrix valued functions with the following properties (assumed to hold almost everywhere with respect to the corresponding measure(s)): y

HðkÞ ¼ HðkÞ ; y

y

½AðrÞ; AðrÞ ¼ 0; y

Kðk 0 Þ ¼ Kðk 0 Þ ;

½BðsÞ; BðsÞ ¼ 0;

½BðsÞ; AðrÞ ¼ 0;

½Kðk 0 Þ; AðrÞ ¼ 0;

½AðrÞ; HðkÞ ¼ 0;

ð2:2Þ

½BðsÞ; Kðk 0 Þ ¼ 0;

ð2:3Þ

y

½BðsÞ; AðrÞ ¼ 0;

½HðkÞ; BðsÞ ¼ 0: ð2:4Þ

Let us recall that ½BðsÞ; AðrÞ ¼ 0; ½Kðk 0 Þ; AðrÞ ¼ 0; ½BðsÞ; AðrÞy ¼ 0; ½HðkÞ; BðsÞ ¼ 0 ) ½BðsÞy ; AðrÞy ¼ 0; ½Kðk 0 Þ; AðrÞy ¼ 0; ½BðsÞy ; AðrÞ ¼ 0; ½HðkÞ; BðsÞy ¼ 0: ð2:5Þ Let D ¼ R S K K 0 ½0; 2pÞ ½0; 2pÞ and deﬁne the measure dl (r, s, k, k 0 , f, g) = dRdSdKdK 0 dfdg on it. For each pair of matrices A, B we deﬁne multi-matrix vector coherent states (MVCS) as follows: jA; B; ji ¼ N ðr; sÞ

12

1 X 1 X m¼0 l¼0

Am Bl pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ vj /m wl ; qðmÞ kðlÞ

ð2:6Þ

where the normalization factor N ðr; sÞ and the positive sequences q (m), k (l) have to be chosen so that the MVCS are normalized and give a resolution of identity. In the 1 sequel we use the following notation: For an n · n matrix A, we denote jAj ¼ ½AAy 2 . In order to normalize the states (2.6) and to obtain a resolution of the identity we impose the following assumptions on the matrices A and B: Let A and B be matrices of the form (2.1) satisfying conditions (2.2)–(2.4). Suppose that H (k) and K (k 0 ) have non-zero integer spectrum. With these conditions the vectors (2.6) are normalized to unity as n X hA; B; jjA; B; ji ¼ 1 ð2:7Þ j¼1

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with the normalization factor N ðr; sÞ ¼

m l 2 1 X 1 X TrjAðrÞ BðsÞ j : qðmÞkðlÞ m¼0 l¼0

ð2:8Þ

In order to see this let us start from (2.7) and establish (2.8) in the following way: n n X 1 X 1 X 1 X 1 X X 1 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ hA; B; jjA; B; ji ¼ N ðr; sÞ qðmÞqðmÞkðlÞkðsÞ j¼1 j¼1 m¼0 l¼0 m¼0 s¼0 hAm Bl vj /m wl jAm Bs vj /m ws ib H

¼ N ðr; sÞ

n X 1 X 1 X 1 X 1 X 1 m¼0

j¼1 m¼0 l¼0 m

m

l j

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qðmÞqðmÞkðlÞkðsÞ

s¼0

s j

hA B v jA B v iCn h/m j/m iH 1 hwl jws iH 2 ¼ N ðr; sÞ

1

n X 1 X 1 X j¼1 m¼0 l¼0

¼ N ðr; sÞ

1

¼ N ðr; sÞ

1

1 hAm Bl vj jAm Bl vj iCn qðmÞkðlÞ

1 X 1 X TrjAm Bl j2 qðmÞkðlÞ m¼0 l¼0 m l 2 1 X 1 X TrjAðrÞ BðsÞ j ; qðmÞkðlÞ m¼0 l¼0

because hAm Bl vj jAm Bl vj iCn ¼ hBly Amy Am Bl vj jvj iCn ¼ hBly AðrÞmy AðrÞm Bl vj jvj iCn ly

my

m

½by (2.2)

l

¼ hBðsÞ AðrÞ AðrÞ BðsÞ vj jvj iCn

½by (2.3) and (2.4):

Therefore, we have (2.8). For the resolution of the identity, we have n Z X jA; B; jihA; B; jjdlðr; s; k; k 0 ; f; gÞ j¼1

¼

D n X 1 X 1 X 1 X 1 X j¼1 m¼0 l¼0

m¼0

s¼0

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qðmÞqðmÞkðlÞkðsÞ

l ilgKðk 0 Þ j

m imfHðkÞ

m imfHðkÞ

jAðrÞ e BðsÞ e v ihAðrÞ e h/m j jwl ihws jdlðr; s; k; k 0 ; f; gÞ ¼

1 X 1 X 1 X 1 X m¼0 l¼0 m¼0 s¼0 m imfHðkÞ

AðrÞ e

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qðmÞqðmÞkðlÞkðsÞ l ilgKðk 0 Þ

BðsÞ e 0

hws jdlðr; s; k; k ; f; gÞ

because

R

S

R

n X j¼1

Z K0

K

2p

0

Z 0

2p

1 N ðr; sÞ

s isgKðk 0 Þ j

v j j/m i

BðsÞ e

Z Z Z Z

sy isgKðk 0 Þ

BðsÞ e "

Z Z Z Z

S

K

Z K0

my imfHðkÞ

AðrÞ e j

j

2p

0

Z 0

2p

1 N ðr; sÞ

j/m ih/m j jwl i

#

jv ihv j ¼ In ; n n identity matrix

K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

¼

1 X 1 X m¼0 l¼0

4p2 qðmÞkðlÞ

Z Z Z Z R

S

K

K0

125

1 AðrÞm BðsÞl BðsÞly AðrÞmy j/m i N ðr; sÞ

h/m j jwl ihwl jdR dS dK dK 0 Z 2p 0 if m 6¼ m by (2.2)--(2.4) and eiðmmÞfHðkÞ df ¼ 2p if m ¼ m 0 Z Z 2m 2l 1 1 2 X X 4p jAðrÞj jBðsÞj ¼ j/m ih/m j jwl ihwl jdR dS qðmÞkðlÞ R S N ðr; sÞ m¼0 l¼0 ¼ In I H 1 I H 2 ½because dK; dK 0 are probability measures; and by (2.4); provided that the measures dR and dS satisfy Z Z jAðrÞj2m jBðsÞj2l 2 dR dS ¼ qðmÞkðlÞIn : 4p N ðr; sÞ R S

ð2:9Þ

Thereby, the situation can be summarized as follows: The set of states (2.6) forms a set of MVCS if N ðr; sÞ of (2.8) is ﬁnite and the condition (2.9) is satisﬁed. In this case the states are normalized as (2.7) and a resolution of the identity is obtained as n Z X jA; B; jihA; B; jjdlðr; s; k; k 0 ; f; gÞ ¼ In I H 1 I H 2 : j¼1

D

In the following let us see some illustrative examples of the main construction. 2.1. Multi-quaternionic VCS with complex representation As an example of our general construction, in this section we derive MVCS using the complex representation of quaternions by 2 · 2 matrices. Using the basis matrices, 1 0 0 i 0 1 i 0 r0 ¼ ; ir1 ¼ ; ir2 ¼ ; ir3 ¼ ; 0 1 i 0 1 0 0 i where r1, r2, and r3 are the usual Pauli matrices, a general quaternion is written as q ¼ x0 r0 þ i x r with x0 2 R; x ¼ ðx1 ; x2 ; x3 Þ 2 R3 , and r ¼ ðr1 ; r2 ; r3 Þ. Thus, x0 þ ix3 x2 þ ix1 q¼ : x2 þ ix1 x0 ix3

ð2:10Þ

It is convenient to introduce the polar coordinates: x0 ¼ r cos h;

x1 ¼ r sin h sin / cos w;

x2 ¼ r sin h sin / sin w;

x3 ¼ r sin h cos /; where r 2 [0, 1), / 2 [0, p], and h, w 2 [0, 2p). In terms of these, q ¼ AðrÞeihrðbnÞ ;

ð2:11Þ

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K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

where AðrÞ ¼ rr0 ;

rðb nÞ ¼

cos /

sin /eiw

sin /eiw

cos /

2

rðb n Þ ¼ r0 :

and

;

ð2:12Þ

We denote the ﬁeld of quaternions by H. Now let us take two quaternions q1 ; q2 as follows: q1 ¼ AðrÞeih1 rðbn 1 Þ ;

q2 ¼ BðsÞeih2 rðbn 2 Þ ;

where

rðb n1Þ ¼

AðrÞ ¼ rr0 ; rðb n2Þ ¼

cos /2 sin /2 eiw2

ð2:13Þ

cos /1

sin /1 eiw1

sin /1 eiw1 sin /2 eiw2 : cos /2

cos /1

;

BðsÞ ¼ sr0 ;

and

n 2 Þ satisfy the conditions (2.2)–(2.4) and The matrices A (r), B (s), rðb n 1 Þ, and rðb 1 1 the eigenvalues of rðb n j Þ; j ¼ 1; 2 are 1 and 1. Let fUm gm¼0 ; fWl gl¼0 be ortho1 2 normal bases of abstract Hilbert spaces H 1 ; H 2 , respectively, and v , v be an orthonormal basis of C2 . Let D ¼ ½0; 1Þ ½0; pÞ ½0; 2pÞ ½0; 2pÞ. Let us take the measure to be dlðr;s;/1 ;w1 ;h1 ;/2 ;w2 ;h2 Þ ¼

W ðr;sÞ rdrsdsðsin/1 Þd/1 dw1 dh1 ðsin/2 Þd/2 dw2 dh2 16p2

on D D, where W (r, s) is a density function to be determined. With the above setup we have a set of MVCS, 1 X 1 X qm1 ql2 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ ð2:14Þ jq1 ; q2 ; ji ¼ N ðr; sÞ 2 vj Um Wl ; j ¼ 1; 2: qðmÞ kðlÞ m¼0 l¼0 From (2.12), (2.13), and (2.8) the normalization factor can be obtained as 1 X 1 X r2m s2l : N ðr; sÞ ¼ 2 qðmÞkðlÞ m¼0 l¼0

ð2:15Þ

Since Z

2p 0

Z

2p 0

Z 0

p

eiðmmÞh1 rðbn 1 Þ sin /1 d/1 dh1 dw1 ¼

2p I2

if m ¼ m

0

if m 6¼ m

and a similar integral holds when the index 1 changes to 2, from (2.12), (2.13), and (2.9), following the argument of Section 2, when W (r, s) satisﬁes Z 1Z 1 W ðr; sÞr2mþ1 s2lþ1 dr ds ¼ qðmÞkðlÞ: ð2:16Þ 4p2 N ðr; sÞ 0 0 We can readily obtain the resolution of the identity Z jq1 ; q2 ; ji W ðr; sÞ hq1 ; q2 ; jj dl ¼ I2 I H 1 I H 2 : DD

ð2:17Þ

K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

127

In the particular case q (m) = C (m + 1) and k (l) = C (l + 1) we have 2

2

N ðr; sÞ ¼ 2er es ¼ 2er

2 þs2

and W ðr; sÞ ¼ p22 satisﬁes (2.16), which can be seen with the integral representation of the gamma function. Remark 2.1. One could solve the moment problem (2.16) for various choices of q (m) and k (l) using the methods demonstrated in [16] and thereby create several classes of multiquaternionic MVCS. While keeping conditions (2.2) and (2.3), if we replace conditions (2.4) by y

y

AðrÞAðrÞ ¼ AðrÞ AðrÞ ¼ f ðrÞIn

and

y

y

BðsÞBðsÞ ¼ BðsÞ BðsÞ ¼ gðsÞIn ;

ð2:18Þ

where f, g are functions of r and s, we can carry out the construction in a somewhat similar way. We demonstrate it through the following section. 2.2. An extension of the quaternion case In the case of quaternions we had a more convenient form for the matrix A (r) in which the matrix A (r) is simply r times the identity. Here we will establish MVCS with a little more complicated A (r). This example is, in fact, a generalization of the quaternion case. Let rI2 sI2 n 2 Þ cos h rðb n 1 Þ sin h irðb Aðr; sÞ ¼ n 2 ; hÞ ¼ and H ¼ Hðb n1; b ; irðb n 2 Þ cos h rðb sI2 rI2 n 1 Þ sin h n 2 ¼ ðn21 ; n22 ; n23 Þ are unit perpendicular vectors and where b n 1 ¼ ðn11 ; n12 ; n13 Þ and b rðb n j Þ ¼ ðnj1 ; nj2 ; nj3 Þ ðr1 ; r2 ; r2 Þ. Let B1 ¼ Aðr1 ; s1 Þeif1 H1 ð1Þ

and

B2 ¼ Aðr2 ; s2 Þeif2 H2 ;

ð1Þ

ð2Þ

ð2Þ

where H1 ¼ H1 ðb n1 ; b n 2 ; h1 Þ; H2 ¼ H2 ðb n1 ; b n 2 ; h2 Þ. Through straightforward calculations we can see that A (r1, s1), A (r2, s2), H1, and H2 satisfy conditions (2.2), (2.3), ð1Þ ð1Þ ð2Þ ð2Þ and (2.18). Let r1, s1, r2, s2 2 [0, 1), f1, h1, f2, h2 2 [0, 2p), and b n1 ; b n2 ; b n1 ; b n 2 2 X, 3 where X is an open subset of R . For the measure let us take dl = dm dj dX, where dm = r1s1r2s2 dr1 ds1 dr2 ds2, dj ¼ 4p12 df1 df2 , and dX is the probability measure on X · X · [0, 2p) · [0, 2p). Let 4

D ¼ ½Rþ ½0; 2pÞ ½0; 2pÞ X X ½0; 2pÞ ½0; 2pÞ: With the above choices and with the usual notations we form a set of MVCS: 1

jB1 ; B2 ;ji ¼ N ðr1 ;s1 ;r2 ;s2 Þ2

1 X 1 X m¼0 l¼0

Bm1 Bl2 j pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ v /m wl ; j ¼ 1; 2;3;4: qðmÞ kðlÞ ð2:19Þ

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Since m 2 2 m Bmy 1 B 1 ¼ ðr 1 þ s1 Þ I4 ;

l 2 2 l Bly 2 B 2 ¼ ðr2 þ s2 Þ I4 ;

and

and my m l j j 2 2 m 2 2 l hBm1 Bl2 vj jBm1 Bl2 vj i ¼ hBly 2 ; B 1 B 1 B 2 v jv i ¼ 4ðr1 þ s1 Þ ðr2 þ s2 Þ ;

we have 4 X

hB1 ; B2 ; jjB1 ; B2 ; ji ¼ 4N ðr1 ; s1 ; r2 ; s2 Þ

1

j¼1

1 X 1 X ðr21 þ s21 Þm ðr22 þ s22 Þl : qðmÞ kðlÞ m¼0 l¼0

ð2:20Þ Thereby the normalization factor is given by N ðr1 ; s1 ; r2 ; s2 Þ ¼ 4

1 X 1 X ðr21 þ s21 Þm ðr22 þ s22 Þl : qðmÞ kðlÞ m¼0 l¼0

Further, since 4 X

jBm1 Bl2 vj ihBm1 Bl2 vj j

¼

4 X

Bm1 Bl2

j¼1

! y

m

l

jvj ihvj j ðBm1 Bl2 Þ ¼ ðr21 þ s21 Þ ðr22 þ s22 Þ I4 ;

j¼1

we have 4 Z X j¼1

¼

D

W ðr1 ;s1 ;r2 ;s2 ÞjB1 ; B2 ; jihB1 ; B2 ;jjdl

1 X 1 X m¼0 l¼0

1 qðmÞkðlÞ

Z

1

Z

0

1

0

Z

1

Z

0

0

1

m

l

W ðr1 ;s1 ;r2 ; s2 Þðr21 þ s21 Þ ðr22 þ s22 Þ dmI4 N ðr1 ;s1 ;r2 ;s2 Þ

j/m ih/m j jwl ihwl j: Thus a resolution of the identity is satisﬁed provided that there exists a density W (r1, s1, r2, s2) satisfying Z 1Z 1Z 1Z 1 m l W ðr1 ; s1 ; r2 ; s2 Þðr21 þ s21 Þ ðr22 þ s22 Þ dm ¼ qðmÞkðlÞ: N ðr1 ; s1 ; r2 ; s2 Þ 0 0 0 0 For the particular case, q (m) = m! and k (l) = l! the normalization factor can be easily deduced from (2.20). 2

2

2

2

N ðr1 ; s1 ; r2 ; s2 Þ ¼ 4eðr1 þs1 þr2 þs2 Þ and a resolution of the identity is obtained with W ðr1 ; s1 ; r2 ; s2 Þ ¼ W ðr1 ; s1 ÞW ðr2 ; s2 Þ; where W ðr1 ; s1 Þ ¼

16 ; r21 þ s21

W ðr2 ; s2 Þ ¼

16 ; r22 þ s22

K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

129

which can be seen in the following way: Since in general we have Z 1Z 1 m wðr; sÞðr2 þ s2 Þ sr ds dr N ðBÞ 0 0 Z 1Z 1 16ðr2 þ s2 Þm ¼ sr ds dr 4ðr2 þ s2 Þer2 þs2 0 0 Z 1Z 1 4ðr2 þ s2 Þm1 ¼ sr ds dr e r 2 e s2 0 0 Z 1 Z 1 m1 X m1 m1j 2 j r2 s2 ¼ 4rsðr2 Þ ðs Þ e e dr ds j 0 0 j¼0 Z 1 Z 1 m1 X m1 2 2 ðr2 Þm1j er 2r dr ðs2 Þj es 2s ds ¼ j 0 0 j¼0 m1 X m1 ¼ Cðm jÞCðj þ 1Þ ¼ m!; j j¼0 we obtain 4 Z X j¼1

D

W ðr1 ; s1 ; r2 ; s2 ÞjB1 ; B2 ; jihB1 ; B2 ; jjdl ¼ I4 I H 1 I H 1 :

Remark 2.2. For r, s 2 [0, 1) and h, g 2 [0, 2p), if we take our matrices A and B as, A ¼ eih AðrÞ

and

B ¼ eig BðsÞ

ð2:21Þ

with conditions, y

y

AðrÞAðrÞ ¼ AðrÞ AðrÞ ¼ f ðrÞIn

and

y

y

BðsÞBðsÞ ¼ BðsÞ BðsÞ ¼ gðsÞIn ;

ð2:22Þ

we can carry out the above construction with conditions (2.2)–(2.4) replaced by the single condition (2.22). The condition (2.22) is satisﬁed by any real Cliﬀord type matrix [7,17]. Following the construction given in Section 2 one can construct MVCS with any n · n Clifford matrix. However, since an explicit real matrix representation is known only for quaternions and octonions [17], in the following sections we demonstrate it with the real matrix representation of quaternions and octonions.

2.3. Multi-quaternionic VCS with real representation Here we present quaternionic MVCS with the real matrix representation of quaternions without imposing any conditions on the matrices. Let H ¼ fq0 ¼ a0 þ a1 i þ a2 j þ a3 kji2 ¼ j2 ¼ k 2 ¼ 1; ijk ¼ 1; a0 ; a1 ; a2 ; a3 2 Rg

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be the real quaternion division algebra. It is known that H is algebraically isomorphic to the real matrix algebra 9 8 0 1 a0 a1 a2 a3 > > > > > > = < Ba C a a a 1 0 3 2 B C 0 M¼ q ¼B C j a0 ; a1 ; a2 ; a3 2 R : > > @ a2 a3 a0 a1 A > > > > ; : a3 a2 a1 a0 For a detailed explanation see [17] and the references therein. For z = eih 2 S1 and q0 2 M, let qðzÞ ¼ zq0 , then we have y

y

2

2

qðzÞqðzÞ ¼ qðzÞ qðzÞ ¼ ða20 þ a21 þ a22 þ a23 ÞI4 ¼ jqj I4 ¼ jq0 j I4 ; where jq0 j is the norm of the quaternion q0 and I4 is the 4 · 4 identity matrix. Let q0 ðaÞ denotes the matrix representation of q0 with matrix elements ais. Let us consider the Hilbert space C4 H 1 H 2 . The quaternion norm can be considered as a continuous function, j j : H ! Rþ ; q 7! t: Thus, we make the following identiﬁcation: jq1 j ¼ t, jq2 j ¼ s. Since z1 ¼ eih1 and z2 ¼ eih2 , set the measure dl (t, s, h1, h2) = dh1 dh2 t dt s ds on D ¼ ½0; 2pÞ ½0; 2pÞ Rþ Rþ . With the notations of the previous section let us deﬁne a set of multi-matrix VCS as follows: For j = 1, 2, 3, 4 the following set of vectors forms a set of MVCS. 1 X 1 X qm1 ql2 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ jq1 ðz1 ; aÞ; q2 ðz2 ; bÞ; ji ¼ N ðjq1 j; jq2 jÞ 2 vj /m /l : qðmÞ kðlÞ m¼0 l¼0 Let us ﬁnd the normalization factor. Since 4 X hq1 ðz1 ; aÞ; q2 ðz2 ; bÞ; jjq1 ðz1 ; aÞ; q2 ðz2 ; bÞ; ji

ð2:23Þ

j¼1

¼ N ðjq1 j; jq2 jÞ1

4 X 1 X 1 X j¼1 m¼0 l¼0

1 hqm ql vj jqm1 ql2 vj i qðmÞkðlÞ 1 2

2m 2l 1 X 1 X jq1 j jq2 j ; qðmÞkðlÞ m¼0 l¼0 the normalization factor is given by 2m 2l 1 X 1 X jq1 j jq2 j : N ðjq1 j; jq2 jÞ ¼ 4 qðmÞkðlÞ m¼0 l¼0

¼ 4N ðjq1 j; jq2 jÞ1

ð2:24Þ

Next we obtain a resolution of the identity. Since 4 Z X jq1 ðz1 ; aÞ; q2 ðz2 ; bÞ; jihq1 ðz1 ; aÞ; q2 ðz2 ; bÞ; jjdlðt; s; h1 ; h2 Þ j¼1

¼

D

1 X 1 X m¼0 l¼0

4p2 qðmÞkðlÞ

¼ I4 I H 1 I H 2 ;

Z 0

1

Z 0

1

W ðt; sÞt2mþ1 s2lþ1 dt dsI4 j/m ih/m j jwl ihwl j N ðt; sÞ

K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

131

provided that there exits a density function W (t, s) such that 4p2

Z

1

0

Z

1

0

W ðt; sÞt2mþ1 s2lþ1 dt ds ¼ qðmÞkðlÞ: N ðt; sÞ

ð2:25Þ

Remark 2.3. For the choice q (m) = m! and k (l) = l! from (2.24) we have N ðjq1 j; jq2 jÞ ¼ 4et and W ðs; tÞ ¼ p42 satisﬁes (2.25). One could easily notice that the whole procedure depends on the following:

y

2m

2l

hqm1 ql2 vj jqm1 ql2 vj i ¼ hðqm1 ql2 Þ qm1 ql2 vj jvj i ¼ jq1 j jq2 j and 4 X

jqm1 ql2 vj ihqm1 ql2 vj j

¼

qm1 ql2

j¼1

4 X

2 þs2

ð2:26Þ

! y

2m

2l

my jv ihv j ðqm1 ql2 Þ ¼ qm1 ql2 qly 2 q1 ¼ jq1 j jq2 j I4 : j

j

j¼1

ð2:27Þ The identities (2.26) and (2.27) holds for any number of quaternions, that is, q1 q2 can be replaced by q1 qs , therefore the procedure can be extended to have MVCS on the Hilbert space Cn ½sr¼1 H r , where H r ; r ¼ 1; . . . ; s are separable Hilbert spaces. 2.4. Multi-octonionic VCS Let O denote the octonion algebra over the real number ﬁeld R. In [17] it was shown that any a 2 O has a left matrix representation x (a) and a right matrix representation m (a) and these representations were given, respectively, by 0

a0

B a1 B B B a2 B Ba B 3 xðaÞ ¼ B B a4 B Ba B 5 B @ a6 a7 and

a1

a2

a3

a4

a5

a6

a0 a3

a3 a0

a2 a1

a5 a6

a4 a7

a7 a4

a2 a5

a1 a6

a0 a7

a7 a0

a6 a1

a5 a2

a4

a7

a6

a1

a0

a3

a7 a6

a4 a5

a5 a4

a2 a3

a3 a2

a0 a1

a7

1

a6 C C C a5 C C a4 C C C a3 C C a2 C C C a1 A a0

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0

1

a0

a1

a2

a3

a4

a5

a6

a7

B a1 B B B a2 B Ba B 3 mðaÞ ¼ B B a4 B Ba B 5 B @ a6 a7

a0 a3

a3 a0

a2 a1

a5 a6

a4 a7

a7 a4

a2 a5

a1 a6

a0 a7

a7 a0

a6 a1

a5 a2

a4

a7

a6

a1

a0

a3

a7 a6

a4 a5

a5 a4

a2 a3

a3 a2

a0 a1

a6 C C C a5 C C a4 C C C: a3 C C a2 C C C a1 A

ih

a0

1

Let z = e 2 S and xða; zÞ ¼ zxðaÞ and

mða; zÞ ¼ zmðaÞ:

Then y

y

y

y

xða; zÞxða; zÞ ¼ xða; zÞ xða; zÞ ¼ mða; zÞmða; zÞ ¼ mða; zÞ mða; zÞ ¼ a20 þ a21 þ a22 þ a23 þ a24 þ a25 þ a26 þ a27 ¼ kak2 I8 : Thus, identities similar to (2.26) and (2.27) can be obtained for x (a, z) and m (a, z). Now, with previous notations we can have MVCS as jxða1 ; z1 Þ; xða2 ; z2 Þ; . . . ; xðas ; zs Þ; ji; jmða1 ; z1 Þ; mða2 ; z2 Þ; . . . ; mðas ; zs Þ; ji; or jmða1 ; z1 Þ; xða2 ; z2 Þ; mða3 ; z3 Þ; xða4 ; z4 Þ; . . . ; mðas ; zs Þ; ji: In the last set of states we can mix x (a, z) and m (a, z) in any order.

3. Multi-matrix VCS with matrix q (m) Here we consider a class of MVCS in the following form: 1 1 X X 1 jZ 1 ; Z 2 ; . . . ; Z s ; ji ¼ N 2 R1 ðn1 ÞZ n11 Rs ðns ÞZ ns s vj /n1 /ns ; n1 ¼0

ns ¼0

ð3:1Þ where Rk (nk) and Zk are n · n matrices for all k = 1, . . ., s. In order to get a normalization and a resolution of identity, we need to take the Zks in a certain way and we have to impose conditions on all the matrices. For example, for k = 1, . . ., s let Ck be a real n · n Cliﬀord matrix, Zk = zkCk with zk 2 S1 and Rk (nk) be a n · n matrix satisfying y

y

Rk ðnk ÞRk ðnk Þ ¼ Rk ðnk Þ Rk ðnk Þ ¼ fk ðnk ÞIn ;

ð3:2Þ

where fk (nk) is a real valued function of nk. Then following the argument of Section 2 one can construct MVCS. For simplicity, we work it out in detail with the real matrix representation of quaternions. Let us take qk to be as in Section 2.3 for all k = 1, 2, . . ., s and Z k ¼ qk ðhk Þ ¼ qk eihk :

ð3:3Þ

K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

133

Further, let us take Rk (nk) such that Rk ðnk ÞRk ðnk Þy ¼ Rk ðnk Þy Rk ðnk Þ ¼ fk ðnk ÞI4 ¼

1 I4 ; nk !

k ¼ 1; 2; . . . ; s:

ð3:4Þ

For example, one such possible Rk (nk) is cosðxk ÞI2 sinðxk ÞI2 1 ﬃﬃﬃﬃﬃ ﬃ p Rk ðnk Þ ¼ ; nk ! sinðxk ÞI2 cosðxk ÞI2 where x1, . . ., xs are ﬁxed. For k = 1, . . ., s let Zk be as in (3.3) and Rk (nk) be as in (3.4) then the vectors (3.1) form a set of MVCS. Since hR1 ðn1 Þq1 ðh1 Þn1 Rs ðns Þqs ðhs Þns vj jR1 ðn1 Þq1 ðh1 Þn1 Rs ðns Þqs ðhs Þns vj i n y

y

n y

y

n

n

¼ hqs ðhs Þ s Rs ðns Þ q1 ðh1 Þ 1 R1 ðn1 Þ R1 ðn1 Þq1 ðh1 Þ 1 Rs ðns Þqs ðhs Þ s vj jvj i 2n

jq1 j 1 jqs j n1 ! ns !

¼

2ns

hvj jvj i

and 4 X

n

n

l

l

jR1 ðn1 Þq1 ðh1 Þ 1 Rs ðns Þqs ðhs Þ s vj ihR1 ðl1 Þq1 ðh1 Þ 1 Rs ðls Þqs ðhs Þ s vj j

j¼1

¼e

iðn1 l1 Þh1

¼e

iðn1 l1 Þh1

n X

! jv ihv j qsls y Rs ðls Þy q1l1 y R1 ðl1 Þy

e

iðns ls Þhs

R1 ðn1 Þqn11

Rs ðns Þqns s

e

iðns ls Þhs

R1 ðn1 Þqn11

y y Rs ðns Þqns s qsls y Rs ðls Þ q1l1 y R1 ðl1 Þ ;

j

j

j¼1

we have the normalization factor 4 X 1 1 X X 2 2 jq1 j2n1 jqs j2ns j j hv jv i ¼ 4ejq1 j þþjqs j : N ¼ n ! n ! 1 s j¼1 n ¼0 ns ¼0

ð3:5Þ

1

Set dl ¼ ðpÞ4 s jq1 j jqs jdjq1 j djqs jdh1 dhs : With this measure a resolution of the identity is obtained as follows: Z 1 Z 2p Z 2p 4 Z 1 X jq1 ðh1 Þ; . . . ; qs ðhs Þ; jihq1 ðh1 Þ; . . . ; qs ðhs Þjjdl j¼1

¼

0

0

1 X

1 X

n1 ¼0

Z

1

0

ls ¼0

Z 0

1

Z

2p

0

Z 0

2p

1 iðn1 l1 Þh1 e eiðns ls Þhs R1 ðn1 Þ N

Rs ðns Þqns s vj qsls y Rs ðls Þy ql11 y R1 ðl1 Þy dl j/1 ih/1 j j/s ih/s j Z 1 Z 1 1 X jq1 j2n1 jqs j2ns s

n1 ¼0

0 1 X

ns ¼0 l1 ¼0

qn11 1 X

¼

0 1 X

2

ns ¼0 2

0 2

0

n1 ! ns !

ejq1 j ejqs j jq1 j jqs jdjq1 j djqs j j/1 ih/1 j j/s ih/s j ¼ I4 I H 1 I H s :

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Remark 3.1. Usually the q (m) of (1.1) is a positive sequence of real numbers and in getting a resolution of the identity we end up at solving a moment problem with positive moments. When we replace the q (m) by a matrix, one could expect to replace it by a positive deﬁnite matrix. The matrices we have used are not positive deﬁnite. But one can observe that, at the end of the calculations, we end up with a moment problem with positive moments. In this sense, it really does not matter with which matrix we start with but what does matter is the ﬁnal moment problem.

4. Summations depending on one another So far we have demonstrated several classes of MVCS as a generalization of (1.4). In this section we present MVCS as a generalization to the states (1.5), where the summations depend one another. First we give a general construction and then we discuss an example using the complex representation of quaternions. Let A, B be matrices as in Section 2 with all the assumptions of Section 2 on them. Then the following set of vectors forms a set of MVCS. jA; B; ji ¼ N 1 ðr; sÞ

12

1 1 X 1 X Af ðmÞ BgðlÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N 2 ðs; mÞ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vj /m wl ; q1 ðmÞ q2 ðm; lÞ m¼0 l¼0

ð4:1Þ where f, g are some functions of m and l, respectively. Following the steps of Section 2 we can get the normalization factor N 1 ðr; sÞ ¼

f ðmÞ gðlÞ 2 1 X 1 X TrjAðrÞ BðsÞ j : N 2 ðs; mÞq1 ðmÞq2 ðm; lÞ m¼0 l¼0

ð4:2Þ

Again by following the steps of Section 2 we can have the resolution of the identity n Z X jA; B; jihA; B; jjdlðr; s; k; k 0 ; f; gÞ j¼1

¼

D 1 X 1 X m¼0 l¼0

4p2 q1 ðmÞq2 ðm; lÞ

Z Z R

2f ðmÞ

S

2gðlÞ

jAj jBj j/m ih/m j jwl i N 1 ðr; sÞN 2 ðs; mÞ

hwl jk1 ðrÞk2 ðr; sÞ dR dS ¼ In I H 1 I H 2 provided that there are densities k1 (r) and k2 (s, m) such that Z Z jAj2f ðmÞ jBj2gðlÞ 4p2 k1 ðrÞk2 ðs; mÞ dR dS ¼ q1 ðmÞq2 ðm; lÞIn ; R S N 1 ðr; sÞN 2 ðs; mÞ

ð4:3Þ

where R, S and dR, dS are as in Section 2. Now let us see an illustrative example with the complex representation of quaternions. Take the quaternions as in (2.11). Let G1 ¼ fq ¼ AðrÞeifrð nb1 Þ : r 2 ½0; 1Þg;

G2 ¼ fp ¼ AðsÞeigrð nb2 Þ : s 2 ½0; 1Þg:

K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

135

The variable s is restricted on the set [0, 1), the rest of the variables obey the conditions of Section 2.1. For q 2 G1 ; p 2 G2 ; f ðmÞ ¼ m2 ; gðlÞ ¼ 2l with 1 mþl ; q1 ðmÞ ¼ m! and j ¼ 1; 2 A ¼ q; B ¼ p; q2 ðm; lÞ ¼ l let us consider the states (4.1). From (4.2) and the properties of quaternions we obtain 1 X 1 X m þ l N 2 ðs; mÞ1 rm sm N 1 ðr; sÞ ¼ 2 ; ð4:4Þ m! l m¼0 l¼0 where 1 X mþl m ðmþ1Þ : s ¼ ð1 sÞ N 2 ðs; mÞ ¼ l l¼0

ð4:5Þ

Therefore we get N 1 ðr; sÞ ¼ 2

ðmþ1Þ m 1 X ð1 sÞ r ¼ ð1 sÞerð1sÞ : m! m¼0

For a resolution of the identity (4.3) reduces to 1 Z 1Z 1 m l m mþl r s ð1 sÞ 2 k1 ðrÞk2 ðr; s; mÞ dr ds ¼ m! : 4p erð1sÞ l 0 0

ð4:6Þ

ð4:7Þ

Since Z

1 l

s ð1 sÞ

m

m1

ds ¼

mþl

1

l

0

ð4:8Þ

we take k2 ðr; s; mÞ ¼

m ers : 2pð1 sÞ

Thereby from (4.7) we only have to solve Z 1 rm er k1 ðrÞ dr ¼ m!: 2p

ð4:9Þ

0 1 By the deﬁnition of the Gamma function, we obtain (4.9) when k1 ðrÞ ¼ 2p . Thus we have a resolution of the identity.

5. Generalized oscillator algebra For the states (1.1), a general way of deﬁning an associated oscillator algebra is as follows [1]: Let xm ¼

qðmÞ 8m P 1 qðm 1Þ

and

x0 ! ¼ 1

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then q (m) = xm!. A set of operators, namely annihilation, creation, and number operators on the basis vectors {/m}, is deﬁned as pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ a/m ¼ xm /m1 ; ay /m ¼ xmþ1 /mþ1 ; N /m ¼ xm /m : The states (1.1) satisfy the relation a|zæ = z|zæ, that is, the CS are eigenvectors of the operator a. These three operators together with the identity operator, under the commutator bracket, generate a Lie algebra, which is the so-called generalized oscillator algebra. In analogy with the above case, for the states (1.6) let us take xnj ¼

qj ðnj Þ ; x0 ! ¼ 1; qj ðnj 1Þ

8j ¼ 1; . . . ; s

Let us deﬁne a similar set of operators for the basis vectors fvj /n1 /ns g as follows: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Avj /n1 /ns ¼ xn1 xn2 xns vj /n1 1 /ns 1 ; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð5:1Þ Ay vj /n1 /ns ¼ xn1 þ1 xn2 þ1 xns þ1 vj /n1 þ1 /ns þ1 ; N vj /n1 /ns ¼ xn1 xn2 xns vj /n1 /ns : If xnj xnj 1 ¼ c; a constant 8j ¼ 1; . . . ; s 8nj

ð5:2Þ

we get ½A; Ay ¼ cI;

½N ; A ¼ cA;

½N ; Ay ¼ cAy

and the algebra is isomorphic to the Wyel–Heisenberg algebra. Since AjÕs do not commute, in general, we have AjA1 ; A2 ; . . . ; As ; ji 6¼ A1 A2 As jA1 ; A2 ; . . . ; As ; ji: Observe that if the matrices A1, A2, . . ., As are replaced by complex variables z1, . . ., zs and the basis is replaced by /n1 /ns then the states (1.6) can be considered as multi-boson states with no interaction between bosons. In such a case, when (5.2) is satisﬁed, if we deﬁne a set of operators similar to the above case we can have Ajz1 ; . . . ; zs ; ji ¼ z1 zs jz1 ; . . . ; zs ; ji:

ð5:3Þ

For the states (1.6), if we assume that the matrices satisfy [Aj, Ak] = 0 for all pairs j „ k we can have a relation similar to (5.3). Such an assumption will be very strong. In order to have an annihilation operator so that the MVCS |A1, A2, . . .,As, jæ as an eigenvector of it, it has to be deﬁned in such a way that the action of the operator A aﬀects only the ﬁrst component of the vector, that is, pﬃﬃﬃﬃﬃﬃ Avj /n1 /ns ¼ xn1 vj /n1 1 /n2 /ns : ð5:4Þ In this case the other two operators take the form pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ay vj /n1 /ns ¼ xn1 þ1 vj /n1 þ1 /n2 /ns ;

ð5:5Þ

N vj /n1 /ns ¼ xn1 vj /n1 /n2 /ns :

ð5:6Þ

K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

137

Under the deﬁnition (5.4) we have AjA1 ; A2 ; . . . ; As ; ji ¼ A1 jA1 ; A2 ; . . . As ; ji

ð5:7Þ

and if we assume (5.2) we again get an algebra isomorphic to the Wyel–Heisenberg algebra. For the states of Section 4 one can deﬁne the operators by the same relations of (5.4)– (5.6) and get a relation similar to (5.7). If we attempt to deﬁne operators as (5.1) we end at a situation similar to multi-boson case with interaction between bosons. 6. A physical model In the following we adapt the above methods to construct a class of CS for a Hamiltonian, which is a tensor product of r multi-level quantum Hamiltonians. In the sequel we also derive CS for tensored Jaynes–Cummings models and for a two-mode two-level generalization of the Jaynes–Cummings model. For each j = 1, 2, . . ., r, let Hj be a Hamiltonian for an n-level atomic system. Let Ejk;mj and /jk;mj ; k ¼ 1; . . . ; n; mj ¼ 1; 2; . . . ; 1 be the eigenenergies and eigensolutions of the Hamiltonian Hj for j = 1, . . ., r. Suppose that there is no degeneracy and the energies are ordered as follows. 0 < Ejk;0 < Ejk;1 < Ejk;2 < ;

k ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; r:

For each j = 1, 2, . . ., r suppose also that the eigensolutions /jk;mj ; k ¼ 1; . . . ; n; mj ¼ 1; 2; . . . ; 1 form an orthonormal basis of the quantum Hilbert space H jQM of the Hamiltonian Hj. Let us rearrange the energies as follows: ejk;mj ¼ Ejk;mj Ejk;0 ;

k ¼ 1; . . . ; n; mj ¼ 0; 1; . . . ; 1; j ¼ 1; 2; . . . ; r:

Thereby we have 0 ¼ ejk;0 < ejk;1 < ejk;2 < ;

k ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; r:

Let H 1 ; . . . ; H r be abstract separable Hilbert spaces with orthonormal bases 1 n fwjmj gmj ¼0 ; j ¼ 1; 2; . . . ; r and fvk gk¼1 be the natural basis of Cn . Let r

H ¼ H j; j¼1

r

H QM ¼ H jQM ; j¼1

and

r

H ¼ Hj: j¼1

Then H is a Hamiltonian describing n-level tensored systems with the state Hilbert space H QM . The eigenenergies and the eigensolutions of H can be written as r Y r Ejk;mj ; /k;m1 ;...;mr ¼ /jk;mj and Ek;m1 ;...;mr ¼ j¼1

j¼1

where k = 1, . . ., n. Observe that the energies Ek;m1 ;...;mr of the Hamiltonian H can be degenerate. However, it will not aﬀect the CS construction. Since we sum over all the indices of the energies and the energies of the individual Hamiltonians Hj, j = 1, . . ., r are non-degenerate, the degeneracy of Ek;m1 ;...;mr will not cause any problem in obtaining a resolution of the identity. Deﬁne

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V : H QM ! Cn H

by

V ¼

n X 1 X k¼1 m1 ¼0

1 X

jvk wm1 ;...;mr ih/k;m1 ;...;mr j;

mr ¼0

where r

wm1 ;...;mr ¼ wjmj : j¼1

Then V is a unitary operator with /k;m1 ;...;mr 7! vk wm1 ;...;mr for all k = 1, . . ., n and mj = 0, 1, 2, . . .; j = 1, . . ., n. With the above setup we have VHV 1 ¼ H D ¼ diagðH D1 ; H D2 ; . . . ; H Dn Þ a diagonal n · n matrix, where H Dk ; k ¼ 1; 2; . . . ; n are self-adjoint operators on H with H Dk wm1 ;...;mr ¼ Ek;m1 ;...;mr wm1 ;...;mr ; Let

zjk

2 C and

Zj ¼

zjk

¼

j rjk ehk ;

k ¼ 1; 2; . . . ; n; mj ¼ 0; 1; 2; . . .

j ¼ 1; 2; . . . ; r; k ¼ 1; 2; . . . ; n. Let

diagðzj1 ; zj2 ; . . . ; zjn Þ

and

qj ðmj Þ ¼ diagðej1;mj !; ej2;mj !; . . . ; ejn;mj !Þ;

where ejk;mj ! ¼ ejk;1 ejk;2 ejk;mj . For each j and k, suppose that the series 1 X ðrj Þ2mj qkﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ mj ¼0 ejk;mj !

ð6:1Þ

is convergent and the radius of convergence is Ljk . Let Dj ¼ fðzj1 ; zj2 ; . . . ; zjn Þ 2 C C j jzjk j < Ljk g: Suppose that there are densities kjk ðrjk Þ such that Z Lj k ðrjk Þ2m kjk ðrjk Þ drjk ¼ ejk;mj !; k ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; r:

ð6:2Þ

0

Set dlðZ 1 ; . . . ; Z r Þ ¼

r Y n 1 Y kj ðrj Þ drjk dhjk rn ð2pÞ j¼1 k¼1 k k

on D1 · · Dr. With this setup the set of vectors 1

jZ 1 ; Z 2 ; . . . ; Z r ; ki ¼ N ðZ 1 ; . . . ; Z r Þ2 " # 1 X 1 1 r Y X X 12 mj qj ðmj Þ Z j vk w1m1 w2m2 wrmr m1 ¼0 m2 ¼0

mr ¼0

j¼1

forms a set of MVCS, when the states are normalized as n X hZ 1 ; Z 2 ; . . . ; Z r ; kjZ 1 ; Z 2 ; . . . ; Z r ; ki ¼ 1 k¼1

ð6:3Þ

K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

139

and the normalization factor is given by N ðZ 1 ; . . . ; Z r Þ ¼

1 X 1 X

m1 ¼0 m2 ¼0

1 X

Tr

mr ¼0

r Y

! 1

2mj

qj ðmj Þ jZ j j

:

j¼1

With the argument of Section 2 and with (6.1) it can be seen that N ðZ 1 ; . . . ; Z r Þ < 1: Again with the argument of Section 2 and with (6.2) one can prove the resolution of the identity in a straightforward way as follows: n Z X jZ 1 ; Z 2 ; . . . ; Z r ; kihZ 1 ; Z 2 ; . . . ; Z r ; kj k¼1

D1 D2 Dr

N ðZ 1 ; . . . ; Z r ÞdlðZ 1 ; . . . ; Z r Þ ¼ In I H 1 I H r :

ð6:4Þ

Remark 6.1. Using the technique of [7] the diagonal nature of the VCS (6.3) can be removed. For this let dm be the normalized invariant measure of SU (n) and let U 2 SU (n). In (6.3) let us replace r Y j¼1

1

m

qj ðmj Þ 2 Z j j

by

U

r Y

1

m

qj ðmj Þ 2 Z j j U y

j¼1

then a resolution of the identity can be obtained with the measure dlðZ 1 ; . . . ; Z n ÞdmðU Þ and the normalization factor remains the same. For a detail explanation we refer [7].

Remark 6.2. Let q1 and q2 be two quaternions as in Section 2.3. Then it is straightforward to seepthat the eigenvalues of q1 are z1 = a0 + ib and z1 each with multiplicity 2, ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ð11Þ where b ¼ a21 þ a22 þ a23 . Furthermore, there are orthonormal eigenvectors vj and ð12Þ vj ; j ¼ 1; 2. Let U1 be unitary matrix such that q1 ¼ U 1 diagðz1 ; z1 ; z1 ; z1 Þ U y1 ¼ U 1 Z 1 U y1 . Let the corresponding counterparts of q2 be z2 and U2. In (6.3) one could take k = 4, r = 2, Z 1 ¼ diagðz1 ; z1 ; z1 ; z1 Þ; Z 2 ¼ diagðz2 ; z2 ; z2 ; z2 Þ; q1 ðm1 Þ ¼ qðmÞI4 , and q2 ðm2 Þ ¼ kðlÞI4 . In particular, if Z1 and Z2 are replaced, respectively, by eih U 1 Z 1 U y1 and eig U 2 Z 2 U y2 we have the results of Eq. (2.23). The special case q (m) = m! and k (l) = l! may also be of interest. For an explicit demonstration, in the following example of the physical model we construct MVCS for a Hamiltonian, which is taken as the tensor product of two simple cases of the Jaynes–Cummings Hamiltonian (JC). As a second example, we construct VCS for the Hamiltonian (1.8) with the spectrum (1.9) shifted in a suitable manner.

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K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

6.1. Tensored Jaynes–Cummings model Let us construct MVCS for the Hamiltonian H = Hw Hc, where Hw is the JC in the weak coupling limit and Hc is the JC with the exact resonance and zero coupling (as described below). Information regarding the JC model is extracted from [12]. We outline it according to our needs. For a detailed analysis please see [12]. The JC model describing a spin-1/2 system in interaction with a one-mode magnetic ﬁeld, in the rotating wave approximation, can be given by 1 x0 þ H JC ¼ x a a þ r0 þ r3 þ jðaþ r þ a rþ Þ; ð6:5Þ 2 2 where x is the ﬁeld mode frequency, x0 the atomic frequency, j a coupling constant, a+ and a are the usual creation and annihilation operators for the radiation ﬁeld, r± and r3 are associated with the usual Pauli matrices, and r0 is the identity matrix. The Hamiltonian Hw: It is known that the Hamiltonian HJC can be diagonalized as 0 H DðþÞ y O H JC O ¼ H D ¼ ; 0 H DðÞ where O is the diagonalizing operator. From the diagonalized form the energy eigenvalues can be obtained as and þ n ¼ xn þ jrðnÞ n ¼ xðn þ 1Þ jrðn þ 1Þ; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 D where rðnÞ ¼ d þ n; d ¼ ð2j Þ , and Dp= x x0, the detuning with D > 0. Since ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D > 0 we have nþ1 > n . If 0 6 j=x 6 2 d þ 1, the energies þ n are strictly increasing and non-degenerate. Let x ðjÞ ¼

x j2 D

and

E n ¼ n 0 :

In the weak coupling limit case we expand E n and by keeping at most terms of order 2 in j we get E n ðjnÞ ¼ x ðjÞn: Let w n be the corresponding normalized energy states. In this case we have n

q ðnÞ ¼ E 1 E 2 E n ¼ ½x ðjÞ Cðn þ 1Þ:

The Hamiltonian Hc: In the absence of the oscillating component of the magnetic ﬁeld (j = 0) and for the exact resonance (D = x x0 = 0) we take HJC := Hc. In this case it is straightforward to see that b ¼ xn E n

and

b q ðnÞ ¼ xn Cðn þ 1Þ:

Let / states. n be the corresponding normalized energy j For j = 1, 2, let zj1 ; zj2 2 C and zjk ¼ rjk eihk ; k ¼ 1; 2; j ¼ 1; 2: Let Z 1 ¼ diagðz11 ; z12 Þ; Z 2 ¼ diagðz21 ; z22 Þ; q1 ðm1 Þ ¼ diagðqþ ðm1 Þ; q ðm1 ÞÞ; q2 ðm2 Þ ¼ diagðb q þ ðm2 Þ; b q ðm2 ÞÞ; and q1 ð0Þ ¼ q2 ð0Þ ¼ I2 :

K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

141

Since, for k = 1, 2, 1 X m1 ¼0

2m

ðr1k Þ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ q ðm1 Þ

and

1 X m2 ¼0

2m

ðr2k Þ 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b q ðm2 Þ

converge everywhere we have L1k ¼ L2k ¼ 1: From the inverse Mellin transform [18] Z 1 eax xs1 dx ¼ as CðsÞ 0

and (6.2) we observe that

! ! 1 2 1 2 ðr Þ ðr Þ k11 ðr11 Þ ¼ 2r11 exp 1 ; k12 ðr12 Þ ¼ 2r12 exp 2 ; xþ ðjÞ x ðjÞ ! ! 2 2 ðr21 Þ ðr22 Þ 2 2 2 2 2 2 k1 ðr1 Þ ¼ 2r1 exp and k2 ðr2 Þ ¼ 2r2 exp : x x

In this case a set of MVCS for H can be written as follows: jZ 1 ; Z 2 ; ki ¼ N ðZ 1 ; Z 2 Þ

12

1 X 1 X

1

1

q1 ðm1 Þ 2 Z m1 1 q2 ðm2 Þ 2 Z m2 2 vk Um1 Um2 ;

m1 ¼0 m2 ¼0

ð6:6Þ k = 1, 2, where fUm1 g and fUm2 g are bases for two abstract separable Hilbert spaces H 1 and H 2 , ! 2 2 2 2 ðr11 Þ ðr12 Þ ðr21 Þ ðr22 Þ þ þ þ N ðZ 1 ; Z 2 Þ ¼ exp xþ ðjÞ x ðjÞ x x and a resolution of the identity is satisﬁed as 2 Z 1 Z 1 Z 1 Z 1 Z 2p Z 2p Z 2p Z 2p X k¼1

0

0

0

0

0

0

0

jZ 1 ; Z 2 ; kihZ 1 ; Z 2 ; kjdlðZ 1 ; Z 2 Þ

0

¼ I2 I H 1 I H 2 ; where dlðZ 1 ; Z 2 Þ ¼

1

2 Y 2 Y

xþ x x2 p4

j¼1 k¼1

rjk drjk dhjk :

As a last illustrative example of the construction let us introduce VCS for the Hamiltonian (1.8) with the spectrum (1.9). 6.2. The Hamiltonian (1.8) Consider the Hamiltonian H of (1.8) with g = 0 and denote it by H0. In this case, the spectrum En;m of (1.9) reads

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K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144 En;m ¼ x1 ðn þ 1Þ þ x2 ðm þ 1Þ ¼ en þ em :

Let w n /m be the corresponding normalized eigensolutions. Set n;m m Rðn; mÞ ¼ diagðEn;m þ !; E !Þ ¼ x2 ðaÞm I2 ;

where a = 1 + [x1 (n + 1) + x2]/x2 and (a)m = C (m + a)/C (a) is the Pochhammer symbol. Let n Z ¼ diagðz1 ; z2 Þ and Z ¼ diagðv; vÞ; RðnÞ ¼ diag eþ n !; en ! ¼ Cðn þ 2Þx1 I2 where z1 ¼ reih1 , z2 ¼ r2 eih2 , v = seig, v ¼ seig , and h1, h2, g 2 [0, 2p), where the matrix 1 Z is chosen in a convenient way (see below). As before, let f/n g1 n¼0 , fwm gm¼0 be orthonormal bases of two separable abstract Hilbert spaces H 1 and H 2 , respectively, and {v1, v2} be the natural basis of C2 . Identify the eigensolutions of H0 to the basis of the abstract Hilbert spaces as follows: þ 0 wn wþ m :¼ v2 /n wm : :¼ v1 /n wm w 0 n wm With the above setup we write a set of VCS for the Hamiltonian H0 as follows: jZ; Z; ki ¼ N ðZ; ZÞ

12

1 X n¼0

1

RðnÞ 2 Z n N ðZ; nÞ

12

1 X

1

Rðn; mÞ 2 Zn vk /n wm ;

m¼0

k = 1, 2, where N ðZ; nÞ normalizes the second sum as introduced in [5] (see Eq. (1.5) or [5,10]). The normalization requirement 2 X

hZ; Z; kjZ; Z; ki ¼ 1

k¼1

yields that N ðZ; ZÞ ¼

" 1 X n¼0

# 1 2n 2m X r2n r s 1 1 2 þ ; N ðZ; nÞ xn1 Cðn þ 2Þ xn1 Cðn þ 2Þ xm2 ðaÞm m¼0

where s2m s2 ¼1 F 1 1; a; N ðZ; nÞ ¼ xm2 ðaÞm x2 m¼0

2 converges everywhere. Since 1 F 1 1; a; xs 2 P 1 for all s > 0, we have 1 X 1 r2n r2n 1 2

þ N ðZ; ZÞ ¼ s2 xn1 Cðn þ 2Þ xn1 Cðn þ 2Þ n¼0 1 F 1 1; a; x 2 1 X r2n r2n x1 2 x1 2 1 2 þ 6 ¼ 2 ðer1 =x1 Þ þ 2 ðer2 =x1 Þ; n n x Cðn þ 2Þ x Cðn þ 2Þ r r2 1 1 1 n¼0 1 X

K. Thirulogasanthar et al. / Annals of Physics 314 (2004) 119–144

143

which converges everywhere. Let kðrÞ ¼

2r2 r2 =x2 1; e x21

2s2a1 s2 =x2 b e kðs; nÞ ¼ a : x2 CðaÞ

Then we have Z 1 r2n kðrÞ dr ¼ xn1 Cðn þ 2Þ; 0

or

Z

Z

1

kðs; nÞ ds ¼ xm2 ðaÞm s2m b

0

1

r2n kðrÞ 0

Z

1

kðs; nÞ ds dr ¼ xn1 Cðn þ 2Þxm2 ðaÞm : s2m b

0

Let D = [0, 1) · [0, 2p) then by setting n = 0 and m = 0 in the above integrals one can see that Z 1 2 kðr1 Þkðr2 Þb kðs; nÞ dr1 dh1 dr2 dh2 ds dg ds dg ¼ 1: 4 ð2pÞ DDDD Thereby we have a resolution of the identity 2 Z X jZ; Z; kihZ; Z; kjdlðZ; ZÞ ¼ I2 I H 1 I H 2 ; k¼1

DDDD

where dlðZ; ZÞ ¼

2 Y

1 ð2pÞ

4

N ðZ; ZÞkðrj ÞN ðZ; nÞb kðs; nÞ drj dhj ds ds dg dg: 2

j¼1

Here again Remark 6.1 is applicable. For the Hamiltonian H0 we can have another set of CS similar to the CS (6.6) by appropriately shifting the spectrum. We describe it in the following remark. Remark 6.3. Let us shift the spectrum as follows: n;m n;0 n;m n;m n;0 en;m þ ¼ E þ E þ ¼ e ¼ E E ¼ x2 m; n;m 0;n n;m n;m 0;n n;m þ ¼ E þ E þ ¼ ¼ E E ¼ x1 n:

Let

n;m m Rðn; mÞ ¼ diag en;m þ !; e ! ¼ m!x2 I2 ; Z ¼ diagðz1 ; z2 Þ;

and

RðnÞ ¼ diag nþ !; n ! ¼ n!xn1 I2 ;

Z ¼ diagðv1 ; v2 Þ;

where z1 ¼ reih1 ; z2 ¼ r2 eih2 ; v1 ¼ s1 eig1 ; v2 ¼ s2 eig2 , and h1, h2, g1, g2 2 [0, 2p). With the above setup we write a set of VCS for the Hamiltonian H0 as follows: 1 1 X X 1 1 1 RðnÞ 2 Z n Rðn; mÞ 2 Zn vk /n wm ; k ¼ 1; 2: jZ; Z; ki ¼ N ðZ; ZÞ 2 n¼0

m¼0

By following the details of the Section 6.1 we can obtain the normalization factor and the resolution of the identity.

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7. Conclusion The results of [6] have been generalized with multiple matrices. The results presented in this paper serve as a generalization of [5] since they can be employed for the purpose of obtaining CS for tensored solvable matrix Hamiltonians as well as to obtain CS for multi-level multi-mode solvable systems. Further the results of Section 6 may also be considered as a generalization of the physical model presented in [7]. In order to demonstrate the physical model, in Example 6.1 we have considered the tensor product of two simple cases of the Jaynes–Cummings model. The model can also be applied to other tensored systems, where to solve the moment problem one may adapt the results or the methods of [16]. Furthermore, the methods presented here may facilitate the derivation of CS for super Hamiltonians with multiple degrees of freedom. The complex and real representation of quaternions and the real representation of octonions are used in the construction of multi-matrix VCS in several ways, namely the variables Jjs of (1.5) are replaced by matrices, the qj (nj)s are replaced by matrices, within the multiple sums one sum is allowed to depend on the other. In the same spirit, with the aid of [7], it is possible to extend the construction to the matrix representation of any Cliﬀord algebra and to any Cartesian product of classical matrix domains. Acknowledgment Partial ﬁnancial support of this work from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged (A.K. and K.T.). References [1] S.T. Ali, J.-P. Antoine, J.-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer, New York, 2000. [2] J.R. Klauder, B.S. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics, World Scientiﬁc, Singapore, 1985. [3] A.M. Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag, Berlin, 1986. [4] J.-P. Gazeau, J.R. Klauder, J. Phys. A: Math. Gen. 32 (1999) 123–132. [5] M. Novaes, J.-P. Gazeau, J. Phys. A: Math. Gen. 36 (2003) 199–212. [6] K. Thirulogasanthar, S.T. Ali, J. Math. Phys. 44 (2003) 5070–5083. [7] S.T. Ali, M. Englis, J.-P. Gazeau, J. Phys. A: Math. Gen. 37 (2004) 6067–6089. [8] J.-P. Antoine, J.-P. Gazeau, P. Monceau, J.R. Klauder, K.A. Penson, J. Math. Phys. 42 (2001) 2349– 2387. [9] H. Fakhri, Phys. Lett. A 313 (2003) 243–251. [10] K. Thirulogasanthar, N. Saad, A.B. von Keviczky, J. Math. Phys. 45 (2004) 2694–2717. [11] J.R. Klauder, J. Phys. A: Math. Gen. 29 (1996) L293–L298. [12] M. Daoud, V. Hussin, J. Phys. A: Math. Gen. 35 (2002) 7381–7402. [13] M.W. Janowicz, J.M.A. Ashbourn, Phys. Rev. A 55 (1997) 2348–2359. [14] M.M. Ashraf, Phys. Rev. A 50 (1994) 5116–5121. [15] Y.F. Gao, J. Feng, S.R. Shi, Int. J. Theoret. Phys. 41 (2002) 867–875. [16] J.R. Klauder, K. Penson, J.-M. Sixdeniers, Phys. Rev. A 64 (2001) 013817. [17] Yongge Tian, Adv. Appl. Cliﬀord Algebras 10 (2000) 61–90. [18] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Table of Integral Transforms, New York, McGraw-Hill, 1953.

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