arXiv:math-ph/0002051v2 16 Mar 2001 - Unicampdeleo/Sissa/s30.pdfarXiv:math-ph/0002051v2 16 Mar 2001...

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Journal Physics A 33, 2971–2995 (2000)math-ph/0002051 ΣδΛ

Right eigenvalue equation in quaternionic quantummechanics

Stefano De Leo1 and Giuseppe Scolarici2

1 Department of Applied Mathematics, University of CampinasPO Box 6065, SP 13083-970, Campinas, [email protected]

2 Department of Physics, University of LeccePO Box 193, I 73100, Lecce, [email protected]

September 10, 1999

Abstract. We study the right eigenvalue equation for quaternionic and complex linearmatrix operators defined in n-dimensional quaternionic vector spaces. For quaternioniclinear operators the eigenvalue spectrum consists of n complex values. For these operatorswe give a necessary and sufficient condition for the diagonalization of their quaternionicmatrix representations. Our discussion is also extended to complex linear operators,whose spectrum is characterized by 2n complex eigenvalues. We show that a consistentanalysis of the eigenvalue problem for complex linear operators requires the choice of acomplex geometry in defining inner products. Finally, we introduce some examples of theleft eigenvalue equations and highlight the main difficulties in their solution.

PACS. 02.10.Tq – 02.30.Tb – 03.65.-w

1 Introduction

In the last decade, after the fundamental works of Finkelstein et al. [1–3] on foundations of quaternionicQuantum Mechanics [qQM] and gauge theories, we have witnessed a renewed interest in algebrizationand geometrization of physical theories by non commutative fields [4, 5]. Among the numerous refer-ences on this subject, we recall the important paper of Horwitz and Biedenharn [6], where the authorsshowed that the assumption of a complex projection of the scalar product, also called complex geom-etry [7], permits the definition of a suitable tensor product [8] between single-particle quaternionicwave functions. We also mention quaternionic applications in special relativity [9], group representa-tions [10–13], non relativistic [14, 15] and relativistic dynamics [16, 17], field theory [18], lagrangianformalism [19], electroweak model [20], grand unification theories [21], preonic model [22]. A clearand detailed discussion of qQM together possible topics for future developments in field theory andparticle physics is found in the recent book of Adler [23].

In writing this paper, the main objective has been to address lack of clarity among mathematicalphysicists on the proper choice of the quaternionic eigenvalue equation within a qQM with complex orquaternionic geometry. In the past, interesting papers have addressed the mathematical discussion ofthe quaternionic eigenvalue equation, and related topics. For example, we find in literature works onquaternionic eigenvalues and characteristic equation [24, 25], diagonalization of matrices [26], Jordanform and q-determinant [27, 28]. More recently, some of these problems have been also discussed forthe octonionic field [29, 30].

Our approach aims to give a practical method in solving the quaternionic right eigenvalue equationin view of rising interest in quaternionic [5, 23, 31] and octonionic [4, 32–36] applications in physics.Given quaternionic and complex linear operators on n-dimensional quaternionic vector spaces, weexplicitly formulate practical rules to obtain the eigenvalues and the corresponding eigenvectors for

http://arXiv.org/abs/math-ph/0002051v2

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2 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

their n-dimensional quaternionic matrix representations. In discussing the right complex eigenvalueproblem in qQM, we find two obstacles. The first one is related to the difficulty in obtaining a suit-able definition of determinant for quaternionic matrices, the second one is represented by the loss, fornon-commutative fields, of the fundamental theorem of the algebra. The lack of these tools, essentialsin solving the eigenvalue problem in the complex world, make the problem over the quaternionic fielda complicated puzzle. We overcome the difficulties in approaching the eigenvalue problem in a quater-nionic world by discussing the eigenvalue equation for 2n-dimensional complex matrices obtained bytranslation from n-dimensional quaternionic matrix operators. We shall show that quaternionic linearoperators, defined on quaternionic Hilbert space with quaternionic geometry, are diagonalizable if andonly if the corresponding complex operators are diagonalizable. The spectral theorems, extended toquaternionic Hilbert spaces [1], are recovered in a more general context. We also study the linearindependence on H of quaternionic eigenvectors by studying their (complex) eigenvalues and discussthe spectrum choice for quaternionic quantum systems. Finally, we construct the hermitian operatorassociated with any anti-hermitian matrix operator and show that a coherent discussion of the eigen-value problem for complex linear operators operators requires a complex geometry. A brief discussionconcerning the possibility to have left eigenvalue equations is also proposed and some examples pre-sented. We point out, see also [29, 36], that left eigenvalues of hermitian quaternionic matrices neednot be real.

This paper is organized as follows: In section 2, we introduce basic notations and mathemat-ical tools. In particular, we discuss similarity transformations, symplectic decompositions and theleft/right action of quaternionic imaginary units. In section 3, we give the basic framework of qQMand translation rules between complex and quaternionic matrices. In section 4, we approach the righteigenvalue problem by discussing the eigenvalue spectrum for 2n-dimensional complex matrices ob-tained by translating n-dimensional quaternionic matrix representations. We give a practical methodto diagonalize n-dimensional quaternionic linear matrix operators and overcome previous problems inthe spectrum choice for quaternionic quantum systems. We also discuss the right eigenvalue problemfor complex linear operators within a qQM with complex geometry. In section 5, we introduce theleft eigenvalue equation and analyze the eigenvalue spectrum for hermitian operators. We explicitlysolve some examples of left/right eigenvalue equations for two-dimensional quaternionic and complexlinear operators. Our conclusions and out-looks are drawn in the final section.

2 Basic notations and mathematical tools

A quaternion, q ∈ H, is expressed by four real quantities [37, 38]

q = a+ ib+ jc+ kd , a, b, c, d ∈ R (1)

and three imaginary units

i2 = j2 = k2 = ijk = −1 .

The quaternion skew-field H is an associative but non-commutative algebra of rank 4 over R, endowedwith an involutory antiautomorphism

q → q = a− ib− jc− kd . (2)

This conjugation implies a reversed order product, namely

pq = q p , p , q ∈ H.

Every nonzero quaternion is invertible, and the unique inverse is given by 1/q = q/|q|2, where thequaternionic norm |q| is defined by

|q|2 = qq = a2 + b2 + c2 + d2 .

• Similarity TransformationTwo quaternions q and p belong to the same eigenclass when the following relation

q = s−1 p s , s ∈ H ,

Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 3

is satisfied. Quaternions of the same eigenclass have the same real part and the same norm,

Re(q) = Re (s−1 p s) = Re(p) , |q| = |s−1 p s| = |p| ,

consequently they have the same absolute value of the imaginary part. The previous equations canbe rewritten in terms of unitary quaternions as follows

q = s−1 p s =s

|s| ps

|s| = u p u , u ∈ H , uu = 1 . (3)

In eq. (3), the unitary quaternion,

u = cos θ2 + h · u sin θ

2 , h ≡ (i, j, k) , u ∈ R3 , |u | = 1 ,

can be expressed in terms of the imaginary parts of q and p. In fact, given two quaternions belong tothe same eigenclass,

q = q0 + h · q , p = p0 + h · p , q0 = p0 , |q | = |p | ,

we find [39], for q 6= ±p,

cos θ =q · p|q| |p| and u =

q × p

|q | |p | sin θ. (4)

The remaining cases q = p and q = −p represent respectively the trivial similarity transforma-tion, i.e. uqu = q, and the similarity transformation between a quaternion q and its conjugate q, i.e.uqu = q. In the first case the unitary quaternion is given by the identity quaternion. In the last casethe similarity transformation is satisfied ∀u = h · u with u · q = 0 and |u | = 1.

• Symplectic DecompositionComplex numbers can be constructed from real numbers by

z = α+ iβ , α, β ∈ R .

In a similar way, we can construct quaternions from complex numbers by

q = z + jw , z, w ∈ C ,

symplectic decomposition of quaternions.• Left/Right ActionDue to the non-commutative nature of quaternions we must distinguish between

qh and hq .

Thus, it is appropriate to consider left and right-actions for our imaginary units i, j and k. Let usdefine the operators

L = (Li, Lj , Lk) , (5)

and

R = (Ri, Rj , Rk) , (6)

which act on quaternionic states in the following way

L : H → H , Lq = hq ∈ H , (7)

and

R : H → H , Rq = q h ∈ H . (8)


The algebra of left/right generators can be concisely expressed by

L2

i = L2

j = L2

k = LiLjLk = R2

i = R2

j = R2

k = RkRjRi = −11 ,

and by the commutation relations

[Li,j,k , Ri,j,k ] = 0 .

From these operators we can construct the following vector space

HL ⊗ HR,

whose generic element will be characterized by left and right actions of quaternionic imaginary unitsi, j, k. In this paper we will work with two sub-spaces of HL ⊗ HR, namely

HL and H

L ⊗ CR ,

whose elements are represented respectively by left actions of i, j, k

a+ b · L ∈ HL , a,b ∈ R , (9)

and by left actions of i, j, k and right action of the only imaginary unit i

a+ b · L + cRi + d · LRi ∈ HL ⊗ C

R , a,b, c,d ∈ R . (10)

3 States and Operators in qQM

The states of qQM will be described by vectors, |ψ〉, of a quaternionic Hilbert space, VH. First ofall, due to the non-commutative nature of quaternionic multiplication, we must specify whether thequaternionic Hilbert space is to be formed by right or left multiplication of quaternionic vectorsby scalars. The two different conventions give isomorphic versions of the theory [40]. We adopt theconvention of right multiplication by scalars.

In quaternionic Hilbert spaces, we can define quaternionic and complex linear operators, whichwill be respectively denoted by OH and OC. They will act on quaternionic vectors, |ψ〉, in the followingway

OH(|ψ〉q) = (OH|ψ〉) q , q ∈ H .

and

OC(|ψ〉λ) = (OC|ψ〉)λ , λ ∈ C .

Such operators are R-linear from the left.As a concrete illustration, let us consider the case of a finite n-dimensional quaternionic Hilbert

space. The ket state |ψ〉 will be represented by a quaternionic n-dimensional column vector

|ψ〉 =

ψ1

...ψn

=

x1 + jy1

...xn + jyn

, x1, y1, ..., xn, yn ∈ C . (11)

Quaternionic linear operators, OH, will be represented by n× n matrices with entries in HL, whereascomplex linear operators, OC, by n× n matrices with entries in HL ⊗ CR.

By using the symplectic complex representation, the n-dimensional quaternionic vector

|ψ〉 = |x〉 + j |y〉 =

x1

...xn

+ j

y1

...yn

,


can be translated in the 2n-dimensional complex column vector

|ψ〉 ↔

x1

y1

...xnyn

. (12)

The matrix representation of Li, Lj and Lk consistent with the above identification is

Li ↔(i 00 -i

)= iσ3 , Lj ↔

(0 -11 0

)= −iσ2 , Lk ↔

(0 -i-i 0

)= −iσ1 . (13)

These translation rules allow to represent quaternionic n-dimensional linear operators by 2n× 2n

complex matrices.The right quaternionic imaginary unit

Ri ↔(i 00 i

), (14)

adds four additional degrees of freedom,

Ri , LiRi , LjRi , LkRi ,

and so, by observing that the Pauli matrices together with the identity matrix form a basis over C,we have a set of rules which allows to translate one-dimensional complex linear operators by 2× 2complex matrices [41]. Consequently, we can construct 2n× 2n complex matrix representations forn-dimensional complex linear operators.

Let us note that the identification in eq. (14) is consistent only for complex inner products (com-plex geometry) [41]. We observe that the right complex imaginary unit, Ri, has not a well definedhermiticity within a qQM with quaternionic inner product (quaternionic geometry),

〈ϕ|ψ〉 = |ϕ〉†|ψ〉 =

n∑

l=1

ϕlψl . (15)

In fact, anti-hermitian operators must satisfy

〈ϕ|Aψ〉 = −〈Aϕ|ψ〉 .

For the right imaginary unit Ri, we have

|Riψ〉 ≡ Ri|ψ〉 = |ψ〉i , 〈Riϕ| ≡ (Ri|ϕ〉)†= −i〈ϕ| ,

and consequently

〈ϕ|ψ〉i = 〈ϕ|Riψ〉 6= −〈Riϕ|ψ〉 = i〈ϕ|ψ〉 .

Nevertheless, by adopting a complex geometry, i.e. a complex projection of the quaternionic innerproduct,

〈ϕ|ψ〉C =〈ϕ|ψ〉 − i〈ϕ|ψ〉i

2, (16)

we recover the anti-hermiticity of the operator Ri,

〈ϕ|Riψ〉C = −〈Riϕ|ψ〉C .


4 The right complex eigenvalue problem in qQM

The right eigenvalue equation for a generic quaternionic linear operator, OH, is written as

OH|Ψ〉 = |Ψ〉q , (17)

where |Ψ〉 ∈ VH and q ∈ H. By adopting quaternionic scalar products in our quaternionic Hilbertspaces, VH, we find states in one to one correspondence with unit rays of the form

|r〉 = {|Ψ〉u} (18)

where |Ψ〉 is a normalized vector and u a quaternionic phase of magnitude unity. The state vector,|Ψ〉u, corresponding to the same physical state |Ψ〉, is an OH-eigenvector with eigenvalue uqu

OH|Ψ〉u = |Ψ〉u (uqu) .

For real values of q, we find only one eigenvalue, otherwise quaternionic linear operators will becharacterized by an infinite eigenvalue spectrum

{q , u1qu1 , ... , ulqul , ...}

with ul unitary quaternions. The related set of eigenvectors

{|Ψ〉 , |Ψ〉u1 , ... , |Ψ〉ul , ...

}

represents a ray. We can characterize our spectrum by choosing a representative ray

|ψ〉 = |Ψ〉uλ ,

so that the corresponding eigenvalue λ = uλquλ is complex. For this state the right eigenvalue equationbecomes

OH|ψ〉 = |ψ〉λ , (19)

with |ψ〉 ∈ VH and λ ∈ C.

We now give a systematic method to determine the complex eigenvalues of quaternionic matrixrepresentations for OH operators.

4.1 Quaternionic linear operators and quaternionic geometry

In n-dimensional quaternionic vector spaces, Hn, quaternionic linear operators, OH, are represented

by n×n quaternionic matrices, Mn(HL), with elements in HL. Such quaternionic matrices admit 2n-

dimensional complex counterparts by the translation rules given in eq. (13). Such complex matricescharacterize a subset of the 2n-dimensional complex matrices

M2n(C) ⊂M2n(C) .

The eigenvalue equation for OH reads

MH|ψ〉 = |ψ〉λ , (20)

where MH ∈ Mn(HL), |ψ〉 ∈ Hn and λ ∈ C.


• The one-dimensional eigenvalue problem

In order to introduce the reader to our general method of quaternionic matrix diagonalization, let usdiscuss one-dimensional right complex eigenvalue equations. In this case eq. (20) becomes

QH|ψ〉 = |ψ〉λ , (21)

where QH = a + b · L ∈ HL, |ψ〉 = |x〉 + j |y〉 ∈ H and λ ∈ C. By using the translation rules,given in section 3, we can generate the quaternionic algebra from the commutative complex algebra(Cayley-Dickson process). The complex counterpart of eq.(21) reads

(z -w∗

w z∗

)(xy

)= λ

(xy

), (22)

z = a+ ib1 , w = b2 − ib3 ∈ C .

Eq. (22) is the eigenvalue equation for a complex matrix whose characteristic equation has real coef-ficients. For this reason, the translated complex operator admits λ and λ∗ as eigenvalues.

Given the eigenvector corresponding to the eigenvalue λ, we can immediately obtain the eigenvectorassociated to the eigenvalue λ∗ by taking the complex conjugate of eq.(22) and then applying asimilarity transformation by the matrix

S =

(0 -11 0

).

In this way, we find

(z -w∗

w z∗

)(-y∗

x∗

)= λ∗

(-y∗

x∗

). (23)

So, for λ 6= λ∗ ∈ C, we obtain the eigenvalue spectrum {λ , λ∗} with eigenvectors

(xy

),

(-y∗

x∗

). (24)

What happens when λ ∈ R ? In this case the eigenvalue spectrum will be determined by two equaleigenvalues λ. To show that, we remark that the eigenvectors (24) associated to the same eigenvaluesλ, are linearly independent on C. In fact,

∥∥∥∥x -y∗

y x∗

∥∥∥∥ = |x|2 + |y|2 = 0 if and only if x = y = 0 .

So in the quaternionic world, by translation, we find two complex eigenvalues respectively λ and λ∗,associated to the following quaternionic eigenvectors

|ψ〉 and |ψ〉j ∈ |r〉 .

The infinite quaternionic eigenvalue spectrum can be characterized by the complex eigenvalue λ andthe ray representative will be |ψ〉. In the next section, by using the same method, we will discusseigenvalue equations in n-dimensional quaternionic vector spaces.

• The n-dimensional eigenvalue problem

Let us formulate two theorems which generalize the previous results for quaternionic n-dimensionaleigenvalue problems. The first theorem [T1] analyzes the eigenvalue spectrum of the 2n-dimensional

complex matrix M , the complex counterpart of the n-dimensional quaternionic matrix MH. In thistheorem we give the matrix S which allows to construct the complex eigenvector |φλ∗〉 from theeigenvector |φλ〉. The explicit construction of |φλ∗〉 enables us to show the linear independence on C


of |φλ〉 and |φλ∗〉 when λ ∈ R and represents the main tool in constructing similarity transformationsfor diagonalizable quaternionic matrices. The second theorem [T2] discusses linear independence onH for MH-eigenvectors.T1 - THEOREM

Let M be the complex counterpart of a generic n × n quaternionic matrix MH. Its eigenvaluesappear in conjugate pairs.

Let

M |φλ〉 = λ |φλ〉 (25)

be the eigenvalue equation for M , where

M ∈M2n(C) , |φλ〉 =

x1

y1

...xnyn

∈ C2n , λ ∈ C .

By taking the complex conjugate of eq. (25),

M∗|φλ〉∗ = λ∗|φλ〉∗ ,

and applying a similarity transformation by the matrix

S = 11n ⊗(

0 -11 0

),

we obtain

SM∗S−1S|φλ〉∗ = λ∗S|φλ〉∗. (26)

From the block structure of the complex matrix M it is easily checked that

SM∗S−1 = M,

and consequently eq. (26) reads

M |φλ∗〉 = λ∗|φλ∗〉 , (27)

where

|φλ∗〉 = S|φλ〉∗ =

-y∗1x∗1...

-y∗n

x∗n

.

Let us show that the eigenvalues appear in conjugate pairs (this implies a double multiplicity for realeigenvalues). To do it, we need to prove that |φλ〉 and |φλ∗〉 are linearly independent on C. In order todemonstrate the linear independence of such eigenvectors, trivial for λ 6= λ∗, we observe that lineardependence, possible in the case λ = λ∗, should require

∥∥∥∥xi -y∗iyi x

∗i

∥∥∥∥ = |xi|2 + |yi|2 = 0 i = 1, ..., n ,

verified only for null eigenvectors. The linear independence of |φλ〉 and |φλ∗〉 ensures an even multi-plicity for real eigenvalues �.


We shall use the results of the first theorem to obtain informations about the MH right complexeigenvalue spectrum. Due to non-commutativity nature of the quaternionic field we cannot give asuitable definition of determinant for quaternionic matrices and consequently we cannot write a char-acteristic polynomial P (λ) for MH. Another difficulty it is represented by the right position of thecomplex eigenvalue λ.T2 - THEOREM

MH admits n linearly independent eigenvectors on H if and only if its complex counterpart Madmits 2n linearly independent eigenvectors on C .

Let{|φλ1

〉 , |φλ∗1 〉 , ... , |φλn〉 , |φλ∗n〉}

(28)

be a set of 2n M -eigenvectors, linearly independent on C, and αl, βl (l = 1, ..., n) be generic complexcoefficients. By definition

n∑

l=1

(αl|φλl〉 + βl|φλ∗

l〉)

= 0 ⇔ αl = βl = 0 . (29)

By translating the complex eigenvector set (28) in quaternionic formalism we find

{|ψλ1

〉 , |ψλ∗1 〉 , ... , |ψλn〉 , |ψλ∗n〉}. (30)

By eliminating the eigenvectors, |ψλ∗l〉 = |ψλl〉j, corresponding for complex eigenvalues to ones with

negative imaginary part, linearly dependent with |ψλl〉 on H, we obtain

{|ψλ1

〉 , ... , |ψλn〉}.

This set is formed by n linearly independent vectors on H. In fact, by taking an arbitrary quaternioniclinear combination of such vectors, we have

n∑

l=1

[|ψλl〉(αl + jβl)

]=

n∑

l=1

(|ψλl〉αl + |ψλ∗

l〉βl)

= 0 ⇔ αl = βl = 0 . (31)

Note that eq. (31) represents the quaternionic counterpart of eq. (29) �.

The MH complex eigenvalue spectrum is thus obtained by taking from the 2n dimensional M -eigenvalues spectrum

{λ1 , λ∗1, ... , λn , λ

∗n},

the reduced n-dimensional spectrum

{λ1 , ... , λn} .

We stress here the fact that, the choice of positive, rather than negative, imaginary part is a sim-ple convention. In fact, from the quaternionic eigenvector set (30), we can extract different sets ofquaternionic linearly independent eigenvectors

{[|ψλ1

〉 or |ψλ∗1 〉], ... ,

[|ψλn〉 or |ψλ∗n〉

]},

and consequently we have a free choice in characterizing the n-dimensional MH-eigenvalue spectrum.A direct consequence of the previous theorems, is the following corollary.T2 - COROLLARY

Two MH quaternionic eigenvectors with complex eigenvalues, λ1 and λ2, with λ2 6= λ1 6= λ∗2, are

linearly independent on H.


Let

|ψλ1〉(α1 + jβ1) + |ψλ2

〉(α2 + jβ2) (32)

be a quaternionic linear combination of such eigenvectors. By taking the complex translation ofeq. (32), we obtain

α1|φλ1〉 + β1|φλ∗1 〉 + α2|φλ2

〉 + β2|φλ∗2 〉 . (33)

The set of M -eigenvectors{|φλ1

〉 , |φλ∗1 〉 , |φλ2〉 , |φλ∗2 〉

}

is linear independent on C. In fact, theorem T1 ensures linear independence between eigenvectorsassociated to conjugate pairs of eigenvalues, and the condition λ2 6= λ1 6= λ∗

2completes the proof by

assuring the linear independence between{|φλ1

〉 , |φλ∗1 〉}

and

{|φλ2

〉 , |φλ∗2 〉}.

Thus the linear combination in eq. (33), complex counterpart of eq. (32), is null if and only if α1,2 =β1,2 = 0, and consequently the quaternionic linear eigenvectors |ψλ1

〉 and |ψλ2〉 are linear independent

on H �.

• A brief discussion about the spectrum choice

What happens to the eigenvalue spectrum when we have two simultaneous diagonalizable quaternioniclinear operators? We show that for complex operators the choice of a common quaternionic eigenvectorset reproduce in qQM the standard results of complex Quantum Mechanics [cQM]. Let

A1 =

(i 00 i

)E and A2 =

~

2

(i 00 -i

)(34)

be anti-hermitian complex operators associated respectively to energy and spin. In cQM, the corre-sponding eigenvalue spectrum is

{iE , iE

}

A1

and

{i~

2, −i~

2

}

A2

, (35)

and physically we can describe a particle with positive energy E and spin 12 . What happens in

qQM with quaternionic geometry? The complex operators in eq. (34) also represent two-dimensionalquaternionic linear operators and so we can translate them in the complex world and then extract theeigenvalue spectrum. By following the method given in this section, we find the following eigenvalues

{iE , − iE , iE , − iE

}

A1

and

{i~

2, −i~

2, i

~

2, −i~

2

}

A2

,

and adopting the positive imaginary part convention we extract{iE , iE

}

A1

and

{i~

2, i

~

2

}

A2

.

It seems that we lose the physical meaning of 12 -spin particle with positive energy. How can we recover

the different sign in the spin eigenvalues? The solution to this apparent puzzle is represented by thechoice of a common quaternionic eigenvector set. In fact, we observe that the previous eigenvaluespectra are related to the following eigenvector sets

{(10

),

(01

)}

A1

and

{(10

),

(0j

)}

A2

.


By fixing a common set of eigenvectors

{(10

),

(01

)}

A1,2

, (36)

we recover the standard results of eq. (35). Obviously,

{(10

),

(01

)}

A1,2

→{

+iE , +iE

}

A1

,

{+i

~

2, −i~

2

}

A2

,

{(10

),

(0j

)}

A1,2

→{

+iE , −iE}

A1

,

{+i

~

2, +i

~

2

}

A2

,

{(j0

),

(01

)}

A1,2

→{−iE , +iE

}

A1

,

{−i~

2, −i~

2

}

A2

,

{(j0

),

(0j

)}

A1,2

→{−iE , −iE

}

A1

,

{−i~

2, +i

~

2

}

A2

,

represent equivalent choices. Thus, in this particular case, the different possibilities in choosing ourquaternionic eigenvector set will give the following outputs

Energy : +E , + E and 12 -spin : ↑ , ↓

+ E , − E and ↑ , ↑− E , + E and ↓ , ↓− E , − E and ↓ , ↑

Thus, we can also describe a 12 -spin particle with positive energy by re-interpretating spin up/down

negative energy as spin down/up positive energy solutions

−E , ↑ (↓) → E , ↓ (↑) .

• From anti-hermitian to hermitian matrix operator

Let us remark an important difference between the structure of an anti-hermitian operator in complexand in quaternionic Quantum Mechanics. In cQM, we can always trivially relate an anti-hermitianoperator, A to an hermitian operator, H, by removing a factor i

A = iH .

In qQM, we must take care. For example,

A =

(-i 3j3j i

), (37)

is an anti-hermitian operator, nevertheless, iA does not represent an hermitian operator. The reasonis simple: Given any independent, over H, set of normalized eigenvectors |vl〉 of A with compleximaginary eigenvalues λl,

A =∑

l

|vl〉 |λl|i 〈vl| ,

the corresponding hermitian operator H is soon obtained by

H =∑

l

|vl〉 |λl| 〈vl| ,

since both the factors are independent of the particular representative |vl〉 chosen. Due to the non-commutative nature of |vl〉, we cannot extract the complex imaginary unit i. Our approach to quater-nionic eigenvalues equations contains a practical method to find eigenvectors |vl〉 and eigenvalues λl


and consequently solves the problem to determine, given a quaternionic anti-hermitian operator, thecorresponding hermitian operator. An easy computation shows that

{i|λ1| , i|λ2|} = {2i , 4i} and {|v1〉 , |v2〉} =

{1√2

(ij

),

1√2

(k1

)}.

So, the hermitian operator corresponding to the anti-hermitian operator of eq. (37) is

H =

(3 k-k 3

). (38)

• A practical rule for diagonalization

We know that 2n-dimensional complex operators, are diagonalizable if and only if they admit 2n

linear independent eigenvectors. It is easy to demonstrate that the diagonalization matrix for M

S M S−1 = M diag ,

is given by

S = Inverse

x(λ1)

1 x(λ∗1 )

1 ... x(λn)1 x

(λ∗n)

1

y(λ1)

1 y(λ∗1)

1 ... y(λn)1

y(λ∗n)

1

......

. . ....

...x(λ1)n

x(λ∗1 )n

... x(λn)n

x(λ∗n)n

y(λ1)n

y(λ∗1)n

... y(λn)n

y(λ∗n)n

. (39)

Such a matrix is in the same subset of M , i.e. S ∈ M2n(C). In fact, by recalling the relationshipbetween |φλ〉 and |φλ∗〉, we can rewrite the previous diagonalization matrix as

S = Inverse

x(λ1)

1 -y∗ (λ1)

1 ... x(λn)1

-y∗ (λn)1

y(λ1)

1 x∗(λ1)

1 ... y(λn)1

x∗ (λn)1

......

. . ....

...x(λ1)n

-y∗ (λ1)n

... x(λn)n

-y∗ (λn)n

y(λ1)n

x∗ (λ1)n

... y(λn)n

x∗ (λn)n

. (40)

The linear independence of the 2n complex eigenvectors of M guarantees the existence of S−1 and theisomorphism between the group of n × n invertible quaternionic matrices GL(n,H) and the complex

counterpart group GL(2n,C) ensures S−1 ∈ M2n(C). So, the quaternionic n-dimensional matrix whichdiagonalizes MH

SH MH S−1

H= Mdiag

H,

can be directly obtained by translating eq. (40) in

SH = Inverse

x(λ1)

1 + jy(λ1)

1 ... x(λn)1 + jy(λn)

1

.... . .

...x(λ1)n

+ jy(λ1)n

... x(λn)n

+ jy(λn)n

. (41)

In translating complex matrices in quaternionic language, we remember that an appropriate mathe-matical notation should require the use of the left/right quaternionic operators Li,j,k and Ri. In thiscase, due to the particular form of our complex matrices,

M , S , S−1 ∈ M2n(C) ,

their quaternionic translation is performed by left operators and so we use the simplified notation i,j, k instead of Li, Lj , Lk.


This diagonalization quaternionic matrix is strictly related to the choice of a particular set ofquaternionic linear independent eigenvectors

{|ψλ1

〉 , ... , |ψλn〉}.

So, the diagonalized quaternionic matrix reads

Mdiag

H= diag {λ1 , ... , λn} .

The choice of a different quaternionic eigenvector set{[

|ψλ1〉 or |ψλ∗1 〉

], ... ,

[|ψλn〉 or |ψλ∗n〉

]},

will give, for not real eigenvalues, a different diagonalization matrix and consequently a differentdiagonalized quaternionic matrix

Mdiag

H= diag {[λ1 or λ∗1 ] , ... , [λn or λ∗

n]} .

In conclusion,

MH diagonalizable ⇔ M diagonalizable ,

and the diagonalization quaternionic matrix can be easily obtained from the quaternionic eigenvectorset.

4.2 Complex linear operators and complex geometry

In this section, we discuss right eigenvalue equation for complex linear operators. In n-dimensionalquaternionic vector spaces, H

n, complex linear operator, OC, are represented by n × n quaternionicmatrices, Mn(H

L⊗CR), with elements in HL⊗CR. Such quaternionic matrices admit 2n-dimensionalcomplex counterparts which recover the full set of 2n-dimensional complex matrices, M2n(C). It isimmediate to check that quaternionic matrices MH ∈ Mn(H

L) are characterized by 4n2 real param-

eters and so a natural translation gets the complex matrix M2n(C) ⊂ M2n(C), whereas a generic2n-dimensional complex matrix M ∈ M2n(C), characterized by 8n2 real parameters needs to doublethe 4n2 real parameters of MH. By allowing right-action for the imaginary units i we recover themissing real parameters. So, the 2n-dimensional complex eigenvalue equation

M |φ〉 = λ|φ〉 , M ∈M2n(C) , |φ〉 ∈ C2n , λ ∈ C , (42)

becomes, in quaternionic formalism,

MC|ψ〉 = |ψ〉λ , MC ∈ Mn(HL ⊗ C

R) , |ψ〉 ∈ Hn , λ ∈ C . (43)

The right position of the complex eigenvalue λ is justified by the translation rule

i112n ↔ Ri11n .

By solving the complex eigenvalue problem of eq. (42), we find 2n eigenvalues and we have no pos-sibilities to classify or characterize such a complex eigenvalue spectrum . Is it possible to extract asuitable quaternionic eigenvectors set? What happens when the complex spectrum is characterizedby 2n different complex eigenvalues? To give satisfactory answers to these questions we must adopt acomplex geometry [6, 7]. In this case

|ψ〉 and |ψ〉jrepresent orthogonal vectors and so we cannot kill the eigenvectors |ψ〉j. So, for n-dimensional quater-nionic matrices MC we must consider the full eigenvalue spectrum

{λ1 , ... , λ2n

}. (44)


The corresponding quaternionic eigenvector set is then given by

{|ψλ1

〉 , ... , |ψλ2n〉}, (45)

which represents the quaternionic translation of the M -eigenvector set

{|φλ1

〉 , ... , |φλ2n〉}. (46)

In conclusion within a qQM with complex geometry [31] we find for quaternionic linear operators,MH, and complex linear operators, MC, a 2n-dimensional complex eigenvalue spectrum and conse-quently 2n quaternionic eigenvectors. Let us now give a practical method to diagonalize complexlinear operators. Complex 2n-dimensional matrices, M , are diagonalizable if and only if admit 2nlinear independent eigenvectors. The diagonalizable matrix can be written in terms of M -eigenvectorsas follows

S = Inverse

x(λ1)

1 x(λ2)

1 ... x(λ2n−1)

1 x(λ2n)

1

y(λ1)

1 y(λ2)

1 ... y(λ2n−1)

1 y(λ2n)

1

......

. . ....

...

x(λ1)n

x(λ2)n

... x(λ2n−1)n x(λ2n)

n

y(λ1)n

y(λ2)n

... y(λ2n−1)n y(λ2n)

n

. (47)

This matrix admits a quaternionic counterpart [41] by complex linear operators

SC = Inverse

q[1,2]1 + p

[1,2]1 Ri ... q

[2n−1,2n]1 + p

[2n−1,2n]1 Ri

.... . .

...q[1,2]n

+ p[1,2]n

Ri ... q[2n−1,2n]n

+ p[2n−1,2n]n

Ri

, (48)

where

q[m,n]l =

x(λm)

l + y(λn) ∗l

2+ j

y(λm)

l − x(λn) ∗l

2,

and

p[m,n]l =

x(λm)

l − y(λn) ∗l

2i+ j

y(λm)

l + x(λn) ∗l

2i.

To simplify the notation we use i j, k instead of Li, Lj , Lk. The right operator Ri indicates the rightaction of the imaginary unit i. The diagonalized quaternionic matrix reproduces the quaternionictranslation of the complex matrix

M diag = diag {λ1 , ... , λ2n}

into

Mdiag

C= diag

{λ1 + λ∗2

2+λ1 − λ∗2

2iRi , ... ,

λ2n−1 + λ∗2n2

+λ2n−1 − λ∗2n

2iRi

}. (49)

5 Quaternionic eigenvalue equation

By working with quaternions we have different possibilities to write eigenvalue equations. In fact,in solving such equations, we could consider quaternionic or complex, left or right eigenvalues. Inthis section, we briefly introduce the problematic inherent to quaternionic eigenvalue equations andemphasize the main difficulties present in such an approach.


5.1 Right quaternionic eigenvalue equation for complex linear operators

As seen in the previous sections, the right eigenvalue equation for quaternionic linear operators, OH,reads

MH|ψ〉 = |ψ〉q , q ∈ H .

Such an equation can be converted into a right complex eigenvalue equation by re-phasing the quater-nionic eigenvalues, q,

MH|ψ〉u = |ψ〉u uqu = |ψ〉λ , λ ∈ C .

This trick fails for complex linear operators. In fact, by discussing right quaternionic eigenvalueequations for complex linear operators,

MC|ψ〉 = |ψ〉q , (50)

due to the presence of the right imaginary unit i in MC, we cannot apply quaternionic similaritytransformations,

(MC|ψ〉

)u 6= MC

(|ψ〉u

), u ∈ H .

Within a qQM with complex geometry [20, 31, 41], a generic anti-hermitian operator must satisfy

〈φ|ACψ〉C = −〈ACφ|ψ〉C . (51)

We can immediately find a constraint on our AC-eigenvalues by putting in the previous equation|φ〉 = |ψ〉,

〈ψ|ψqψ〉C = −〈ψqψ|ψ〉C ⇒ qψ = iαψ + jwψ , (52)

αψ ∈ R , wψ ∈ C .

Thus, complex linear anti-hermitian operators, AC, will be characterized by purely imaginary quater-nions. An important property must be satisfied for complex linear anti-hermitian operators, namelyeigenvectors |φ〉 and |ψ〉 associated to different eigenvalues, qφ 6= qψ, have to be orthogonal in C. Bycombining Eqs. (51,52), we find

〈φ|ψqψ〉C = 〈qφφ|ψ〉C .

To guarantee the complex orthogonality of the eigenvectors |φ〉 and |ψ〉, namely 〈φ|ψ〉C = 0, we mustrequire a complex projection for the eigenvalues, (q)C,

qψ,φ −→ λψ,φ ∈ C .

In conclusion, a consistent discussion of right eigenvalue equations within a qQM with complex ge-ometry requires complex eigenvalues.

5.2 Left quaternionic eigenvalue equation

What happens for left quaternionic eigenvalue equations? In solving such equations for quaternionicand complex linear operators,

{MH|ψ〉 = q |ψ〉 ,MC|ψ〉 = q |ψ〉 , q ∈ H ,

we have not a systematic way to approach the problem. In this case, due to the presence of leftquaternionic eigenvalues (translated in complex formalism by two-dimensional matrices), the transla-tion trick does not apply and so we must solve directly the problem in the quaternionic world.


In discussing left quaternionic eigenvalue equations, we underline the difficulty hidden in diagonal-izing such operators. Let us suppose that the matrix representations of our operators be digonalizedby a matrix SH/C

SH MH S−1

H= Mdiag

Hand SC MC S−1

C= Mdiag

C.

The eigenvalue equation will be modified in

{Mdiag

HSH|ψ〉 = SHqS−1

HSH |ψ〉 ,

Mdiag

CSC|ψ〉 = SCqS−1

CSC |ψ〉 ,

and now, due to the non-commutative nature of q,

SH/C q S−1

H/C6= q .

So, we can have operators with the same left quaternionic eigenvalues spectrum but no similaritytransformation relating them. This is explicitly show in appendix B, were we discuss examples oftwo-dimensional quaternionic linear operators. Let us now analyse other difficulties in solving leftquaternionic eigenvalue equation. Hermitian quaternionic linear operators satisfy

〈φ|HHψ〉 = 〈HHφ|ψ〉 .

By putting |φ〉 = |ψ〉 in the previous equation we find constraints on the quaternionic eigenvalues q

〈ψ| q ψ〉 = 〈q ψ|ψ〉 .

From this equation we cannot extract the conclusion that q must be real, q = q †. In fact

〈ψ|(q − q †)|ψ〉 = 0

could admit quaternionic solutions for q (see the example in appendix B). So, the first complicationis represented by the possibility to find hermitian operators with quaternionic eigenvalues. Withina qQM, we can overcome this problem by choosing anti-hermitian operators to represent observablequantities. In fact,

〈φ|AHψ〉 = −〈AHφ|ψ〉 ,

will imply, for |φ〉 = |ψ〉,

〈ψ|(q + q †)|ψ〉 = 0 . (53)

In this case, the real quantity, q + q †, commutes with |ψ〉, and so eq. (53) gives the constraint

q = iα+ jw .

We could work with anti-hermitian operators and choose |q| as observable output. Examples ofleft/right eigenvalue equation for two-dimensional anti-hermitian operators will be discussed in ap-pendix B. In this appendix, we explicitly show an important difference between left and right eigen-value equation for anti-hermitian operators: Left and right eigenvalues can have different absolutevalues and so cannot represent the same physical quantity.

6 Conclusions

The study undertaken in this paper demonstrates the possibility to construct a practical method todiagonalize quaternionic and complex linear operators on quaternionic vector spaces. Quaternioniceigenvalue equations have to be right eigenvalue equations. As shown in our paper, the choice of aright position for quaternionic eigenvalues is fundamental in searching for a diagonalization method.


A left position of quaternionic eigenvalues gives unwished surprises. For example, we find operatorswith the same eigenvalues which are not related by similarity transformation, hermitian operatorswith quaternionic eigenvalues, etc.

Quaternionic linear operators in n-dimensional vector spaces take infinite spectra of quaternioniceigenvalues. Nevertheless, the complex translation trick ensures that such spectra are related by sim-ilarity transformations and this gives the possibility to choose n representative complex eigenvaluesto perform calculations. The complete set of quaternionic eigenvalues spectra can be generated fromthe complex eigenvalue spectrum,

{λ1 , ... , λn

},

by quaternionic similarity transformations,

{u1λ1u1 , ... , unλnun

}

Such a symmetry is broken when we consider a set of diagonalizable operators. In this case thefreedom in constructing the eigenvalue spectrum for the first operator, and consequently the freechoice in determining an eigenvectors basis, will fix the eigenvalue spectrum for the other operators.

The power of the complex translation trick gives the possibility to study general properties forquaternionic and complex linear operators. Complex linear operators play an important role withina qQM with complex geometry by reproducing the standard complex results in reduced quaternionicvector spaces [31]. The method of diagonalization becomes very useful in the resolution of quaternionicdifferential equations [42]. Consequently an immediate application is found in solving the Schrodingerequation with quaternionic potentials [43].

Mathematical topics to be developed are represented by the discussion of the eigenvalue equationfor real linear operators, OR, and by a detailed study of the left eigenvalue equation. Real linearoperators are characterized by left and right actions of the quaternionic imaginary units i, j, k. Thetranslation trick now needs to be applied in the real world and so, for a coherent discussion, it willrequire the adoption of a real geometry.

Acknowledgements

The authors wish to express their gratitude to Nir Cohen for several helpful comments concerningquaternionic matrix theory and Jordan form. They are also indebted to Gisele Ducati for suggestionson possible applications of the diagonalization method to quaternionic differential equations and formany stimulating conversations. G.S. gratefully acknowledges the Department of Applied Mathe-matics, IMECC-Unicamp, for the invitation and hospitality. This work was partially supported bya fellowship of the Department of Physics, Lecce University, (G.S.) and by a research grant of theFapesp, Sao Paulo State, (S.d.L.). Finally, the authors wish to thank the referees for drawing attentionto interesting references and for their remarks which helped to clarify the notation and improve thediscussion presented in this paper.


Appendix A

Two dimensional right complex eigenvalue equations

In this appendix we explicitly solve the right eigenvalues equations for quaternionic, OH, and complexOC, linear operators, in two-dimensional quaternionic vector spaces.

• Quaternionic linear operators

Let

MH =

(i jk i

)(54)

be the quaternionic matrix representation associated to a quaternionic linear operator in a two-dimensional quaternionic vector space. Its complex counterpart reads

M =

i 0 0 -10 -i 1 00 -i i 0-i 0 0 -i

.

In order to solve the right eigenvalue problem

MH|ψ〉 = |ψ〉λ , λ ∈ C ,

let us determine the M -eigenvalue spectrum. From the constraint

det[M − λ114

]= 0 ,

we find for the M -eigenvalues the following solutions

{λ1 , λ

∗1, λ2 , λ

∗2

}

M

=

{2

14 ei 3

8π , 214 e−i 3

8π , −214 e−i 3

8 π , −214 ei 3

8π

}

M

.

The M -eigenvector set is given by

−1 + iλ1

001

,

0−1 − iλ∗

1

−10

,

01 − iλ∗

1

10

,

−1 − iλ1

001

M

.

The MH-eigenvalue spectrum is soon obtained from that one of M . For example by adopting thepositive imaginary part convention we find

{λ1 , λ2

}

MH

=

{2

14 ei 3

8π , − 214 e−i 3

8π

}

MH

, (55)

and the corresponding quaternionic eigenvector set, defined up to a right complex phase, reads

{(−1 + iλ1

j

),

(j(1 − iλ∗

1)

1

)}

MH

. (56)

The quaternionic matrix which diagonalizes MH is

SH = Inverse

[(−1 + iλ1 j(1 − iλ∗1 )

j 1

)]= − 1

2 |λ1|2(iλ∗1 j

[iλ1 + |λ1|2

]

kλ∗1 iλ1 − |λ1|2). (57)


As remarked in the paper, we have infinite possibilities of diagonalization

{u1λ1u1 , u2λ2u2

}.

Equivalent diagonalized matrices can be obtained from

Mdiag

H= diag

{λ1 , λ2

}

by performing a similarity transformation

U−1Mdiag

HU = U†Mdiag

HU ,

and

U = diag

{u1 , u2

}.

The diagonalization matrix given in eq. (57) becomes

SH → U†SH .

• Complex linear operators

Let

MC =

(−iRi + j −kRi + 1−kRi − 1 iRi + j

)(58)

be the quaternionic matrix representation associated to a complex linear operator in a two-dimensionalquaternionic vector space. Its complex counterpart is

M =

1 -1 1 -11 -1 -1 1

-1 -1 -1 -1-1 -1 1 1

.

The right complex eigenvalue problem

MC|ψ〉 = |ψ〉λ , λ ∈ C ,

can be solved by determining the M -eigenvalue spectrum

{λ1 , λ2 , λ3 , λ4

}

M/MC

=

{2 , −2 , 2i , −2i

}

M/MC

. (59)

Such eigenvalues also determine the MC-eigenvalues spectrum. The MC-eigenvector set is obtainedby translating the complex M -eigenvector set

100-1

,

0110

,

1-ii1

,

-i1-1-i

M

,

in quaternionic formalism

{(1-j

),

(j1

),

(1 + ki+ j

),

(j − ik − 1

)}

MC

. (60)


The quaternionic matrix which diagonalizes MC is

SC = Inverse

[(1 1 + k-j i+ j

)]=

1

2

(1 j

1−k2 - i+j

2

), (61)

and the diagonalized matrix is given by

Mdiag

C= 2

(-iRi 00 i

). (62)

This matrix can be directly obtained from the M/MC eigenvalue spectrum by translating, in quater-nionic formalism, the matrix

M diag =

2 0 0 00 -2 0 00 0 2i 00 0 0 -2i

.

It is interesting to note that equivalent diagonalized matrices can be obtained from Mdiag

Cin eq. (62)

by the similarity transformation

U† Mdiag

CU .

For example by choosing

U =

(-j 00 1+k√

2

),

one find

Mdiag

C→ 2

(iRi 00 j

). (63)


Appendix B

Two dimensional left quaternionic eigenvalue equations

Let us now examine left quaternionic eigenvalue equations for quaternionic linear operators.

• Hermitian operators

Let

HH =

(0 k-k 0

)

be the quaternionic matrix representation associated to an hermitian quaternionic linear operator.We consider its left quaternionic eigenvalue equation

HH|ψ〉 = q |ψ〉 , (64)

where

|ψ〉 =

(ψ1

ψ2

)∈ H

2 , q ∈ H .

Eq. (64) can be rewritten by the following quaternionic system

{k ψ2 = q ψ1 ,

−k ψ1 = q ψ2 .

The solution is{q

}

HH

=

{z + jβ

}

HH

,

where

z ∈ C , β ∈ R , |z|2 + β2 = 1 .

The HH-eigenvector set is given by

{(ψ1

-k(z + jβ)ψ1

)}

HH

.

It is easy to verify that in this case

< ψ|(q − q†)|ψ〉 = 0

is verified for quaternionic eigenvalues q 6= q†.

• Anti-hermitian operators

Let

AH =

(j ii k

)

be the quaternionic matrix representation associated to an anti-hermitian quaternionic linear operator.Its right complex spectrum is given by

{λ1 , λ2

}

HH

=

{i

√2 −

√2 , i

√2 +

√2

}

HH

.


We now consider the left quaternionic eigenvalue equation

AH|ψ〉 = q |ψ〉 , (65)

where

|ψ〉 =

(ψ1

ψ2

)∈ H

2 , q ∈ H .

By solving the following quaternionic system

{j ψ1 + i ψ2 = q ψ1 ,

i ψ1 + k ψ2 = q ψ2 ,

we find{q1 , q2

}

AH

=

{i√2

+j + k

2,−i√

2+j + k

2

}

AH

,

and{(

ψ1

( 1√2

+ j+k2 )ψ1

),

(ψ1

(−1√2

+ j+k2 )ψ1

)}

HH

.

We observe that{|u1λ1u1| =

√2 −

√2 , |u2λ2u2 =

√2 +

√2

}

and{|q1| = 1 , |q2| = 1

}.

Thus, left and right eigenvalues cannot associated to the same physical quantity.

• A new possibility

In order to complete our discussion let us discuss for the quaternionic linear operator given in eq. (54)its left quaternionic eigenvalue equation

MH|ψ〉 = q |ψ〉 , (66)

where

|ψ〉 =

(ψ1

ψ2

)∈ H

2 , q ∈ H .

Eq. (66) can be rewritten by the following quaternionic system

{i ψ1 + j ψ2 = q ψ1 ,

k ψ1 + i ψ2 = q ψ2 .

The solution gives for the quaternionic eigenvalue spectrum

{q1 , q2

}

MH

=

{i+

j + k√2

, i− j + k√2

}

MH

, (67)


and for the eigenvector set

{(1

1−i√2

),

(1

i−1√2

)}

MH

. (68)

Let us now consider the following quaternionic linear operator

NH =

(i+ j+k√

20

0 i− j+k√2

). (69)

This operator represents a diagonal operator and has the same left quaternionic eigenvalue spectrum ofMH, notwithstanding such an operator is not equivalent to Mdiag

H. In fact, the NH-complex counterpart

is characterized by the following eigenvalue spectrum

{i√

2 , −i√

2 , i√

2 , −i√

2

}

N

,

different from the eigenvalue spectrum of the M -complex counterpart of MH. Thus, there is nosimilarity transformation which relates these two operators in the complex world and consequently bytranslation there is no a quaternionic matrix which relates NH to MH. So, in the quaternionic world,we can have quaternionic linear operators which have a same left quaternionic eigenvalue spectrumbut not related by a similarity transformation.

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