arX

iv:m

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0020

51v1

28

Feb

2000

Right Eigenvalue Equation

inQuaternionic Quantum Mechanics

Stefano De Leo(a) and Giuseppe Scolarici(b)

(a)Department of Applied Mathematics, IMECC, Campinas University

CP 6065, SP 13081-970, Campinas, Brazil

deleo@ime.unicamp.br(b)Department of Physics, Lecce University

CP 193, I 73100, Lecce, Italy

scolarici@le.infn.it

[ September, 1999 - Last Revised Version: February, 2000 ]

Abstract

We study the right eigenvalue equation for quaternionic and complex linear matrixoperators defined in n-dimensional quaternionic vector spaces. For quaternionic linearoperators 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 ofthe left eigenvalue equations and highlight the main difficulties in their solution.

1 Introduction

In the last decade, after the fundamental works of Finkelstein et al. [1, 2, 3] on founda-tions of quaternionic Quantum Mechanics [qQM] and gauge theories, we have witnessed arenewed interest in algebrization and geometrization of physical theories by non commutativefields [4, 5]. Among the numerous references on this subject, we recall the important paperof Horwitz and Biedenharn [6], where the authors showed that the assumption of a complexprojection of the scalar product, also called complex geometry [7], permits the definition of asuitable tensor product [8] between single-particle quaternionic wave functions. We also men-tion quaternionic applications in special relativity [9], group representations [10, 11, 12, 13],non relativistic [14, 15] and relativistic dynamics [16, 17], field theory [18], lagrangian for-malism [19], electroweak model [20], grand unification theories [21], preonic model [22]. Aclear and detailed discussion of qQM together possible topics for future developments in fieldtheory and particle physics is found in the recent book of Adler [23].

1

In writing this paper, the main objective has been to address lack of clarity among math-ematical physicists on the proper choice of the quaternionic eigenvalue equation within aqQM with complex or quaternionic geometry. In the past, interesting papers have addressedthe mathematical discussion of the quaternionic eigenvalue equation, and related topics. Forexample, we find in literature works on quaternionic eigenvalues and characteristic equa-tion [24, 25], diagonalization of matrices [26], Jordan form and q-determinant [27, 28]. Morerecently, some of these problems have been also discussed for the octonionic field [29, 30].

Our approach aims to give a practical method in solving the quaternionic right eigen-value equation in view of rising interest in quaternionic [5, 23, 31] and octonionic [4, 32,33, 34, 35, 36] applications in physics. Given quaternionic and complex linear operators onn-dimensional quaternionic vector spaces, we explicitly formulate practical rules to obtainthe eigenvalues and the corresponding eigenvectors for their n-dimensional quaternionic ma-trix representations. In discussing the right complex eigenvalue problem in qQM, we findtwo obstacles. The first one is related to the difficulty in obtaining a suitable definition ofdeterminant for quaternionic matrices, the second one is represented by the loss, for non-commutative fields, of the fundamental theorem of the algebra. The lack of these tools,essentials in solving the eigenvalue problem in the complex world, make the problem overthe quaternionic field a complicated puzzle. We overcome the difficulties in approachingthe eigenvalue problem in a quaternionic world by discussing the eigenvalue equation for2n-dimensional complex matrices obtained by translation from n-dimensional quaternionicmatrix operators. We shall show that quaternionic linear operators, defined on quaternionicHilbert space with quaternionic geometry, are diagonalizable if and only if the correspond-ing complex operators are diagonalizable. The spectral theorems, extended to quaternionicHilbert spaces [1], are recovered in a more general context. We also study the linear in-dependence on H of quaternionic eigenvectors by studying their (complex) eigenvalues anddiscuss the spectrum choice for quaternionic quantum systems. Finally, we construct thehermitian operator associated with any anti-hermitian matrix operator and show that a co-herent discussion of the eigenvalue problem for complex linear operators operators requiresa complex geometry. A brief discussion concerning the possibility to have left eigenvalueequations is also proposed and some examples presented. We point out, see also [29, 36],that left eigenvalues of hermitian quaternionic matrices need not be real.

This paper is organized as follows: In section 2, we introduce basic notations and math-ematical tools. In particular, we discuss similarity transformations, symplectic decomposi-tions and the left/right action of quaternionic imaginary units. In section 3, we give the basicframework of qQM and translation rules between complex and quaternionic matrices. In sec-tion 4, we approach the right eigenvalue problem by discussing the eigenvalue spectrum for2n-dimensional complex matrices obtained by translating n-dimensional quaternionic matrixrepresentations. We give a practical method to diagonalize n-dimensional quaternionic linearmatrix operators and overcome previous problems in the spectrum choice for quaternionicquantum systems. We also discuss the right eigenvalue problem for complex linear operatorswithin a qQM with complex geometry. In section 5, we introduce the left eigenvalue equa-tion and analyze the eigenvalue spectrum for hermitian operators. We explicitly solve someexamples of left/right eigenvalue equations for two-dimensional quaternionic and complexlinear operators. Our conclusions and out-looks are drawn in the final section.

2

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 overR, endowed with 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, wherethe quaternionic 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 ,

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 equationscan be 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 θ2 ,

~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 quaternionsbelong to the 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)

3

The remaining cases ~q = ~p and ~q = − ~p represent respectively the trivial similarity trans-formation, i.e. uqu = q, and the similarity transformation between a quaternion q and itsconjugate q, i.e. uqu = q. In the first case the unitary quaternion is given by the identityquaternion. In the last case the similarity transformation is satisfied ∀u = ~h ·~u with ~u · ~q = 0and |~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

q~h and ~hq .

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

L2i = L2

j = L2k = LiLjLk = R2

i = R2j = 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 imaginaryunits i, 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)

4

3 States and Operators in qQM

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

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

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 columnvector

|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,

whereas complex linear operators, OC, by n× n matrices with entries in HL ⊗ C

R.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 by2n× 2n complex matrices.

5

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 basisover C, we have a set of rules which allows to translate one-dimensional complex linearoperators by 2× 2 complex matrices [41]. Consequently, we can construct 2n× 2n complexmatrix representations for n-dimensional complex linear operators.

Let us note that the identification in eq. (14) is consistent only for complex inner products(complex geometry) [41]. We observe that the right complex imaginary unit, Ri, has not a welldefined hermiticity 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 quaternionicinner product,

〈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 quaternionicHilbert spaces, VH, we find states in one to one correspondence with unit rays of the form

|r〉 = {|Ψ〉u} (18)

6

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

OH|Ψ〉u = |Ψ〉u (uqu) .

For real values of q, we find only one eigenvalue, otherwise quaternionic linear operators willbe characterized 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 eigenvalueequation becomes

OH|ψ〉 = |ψ〉λ , (19)

with |ψ〉 ∈ VH and λ ∈ C.We now give a systematic method to determine the complex eigenvalues of quaternionic

matrix representations for OH operators.

4.1 Quaternionic linear operators and quaternionic geometry

In n-dimensional quaternionic vector spaces, H n, quaternionic linear operators, OH, are repre-sented by n×n quaternionic matrices, Mn(H

L), with elements in HL. Such quaternionic ma-

trices admit 2n-dimensional complex counterparts by the translation rules given in eq. (13).Such complex matrices characterize 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〉 ∈ H

n and λ ∈ C.

• The one-dimensional eigenvalue problem

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

QH|ψ〉 = |ψ〉λ , (21)

7

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 complexalgebra (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 realcoefficients. 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 theeigenvector associated to the eigenvalue λ∗ by taking the complex conjugate of eq.(22) andthen applying a similarity 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 bytwo equal eigenvalues λ. To show that, we remark that the eigenvectors (24) associated tothe same eigenvalues λ, 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 λand λ∗, associated to the following quaternionic eigenvectors

|ψ〉 and |ψ〉j ∈ |r〉 .

The infinite quaternionic eigenvalue spectrum can be characterized by the complex eigenvalueλ and the ray representative will be |ψ〉. In the next section, by using the same method, wewill discuss eigenvalue 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-dimensional eigenvalue problems. The first theorem [T1] analyzes the eigenvalue spectrum

of the 2n-dimensional complex matrix M , the complex counterpart of the n-dimensionalquaternionic matrix MH. In this theorem we give the matrix S which allows to construct thecomplex eigenvector |φλ∗〉 from the eigenvector |φλ〉. The explicit construction of |φλ∗〉 enables

8

us to show the linear independence on C of |φλ〉 and |φλ∗〉 when λ ∈ R and represents themain tool in constructing similarity transformations for diagonalizable quaternionic matrices.The second theorem [T2] discusses linear independence on H for MH-eigenvectors.T1 - THEOREM

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

eigenvalues appear 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∗nx∗n

.

Let us show that the eigenvalues appear in conjugate pairs (this implies a double multiplicityfor real eigenvalues). To do it, we need to prove that |φλ〉 and |φλ∗〉 are linearly independent

9

on C. In order to demonstrate the linear independence of such eigenvectors, trivial for λ 6= λ∗,we observe that linear dependence, 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 evenmultiplicity for real eigenvalues �.

We shall use the results of the first theorem to obtain informations about the MH rightcomplex eigenvalue spectrum. Due to non-commutativity nature of the quaternionic field wecannot give a suitable definition of determinant for quaternionic matrices and consequentlywe cannot write a characteristic polynomial P (λ) for MH. Another difficulty it is representedby the right position of the complex eigenvalue λ.T2 - THEOREM

MH admits n linearly independent eigenvectors on H if and only if its complex counterpart

M admits 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 genericcomplex coefficients. 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 arbitraryquaternionic linear 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) �.TheMH 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} .

10

We stress here the fact that, the choice of positive, rather than negative, imaginary partis a simple convention. In fact, from the quaternionic eigenvector set (30), we can extractdifferent sets of quaternionic linearly independent eigenvectors

{[|ψλ1〉 or |ψλ∗

1〉], ... ,

[|ψλn〉 or |ψλ∗

n〉]}

,

and consequently we have a free choice in characterizing the n-dimensional MH-eigenvaluespectrum. 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 translationof eq. (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 eigen-vectors associated to conjugate pairs of eigenvalues, and the condition λ2 6= λ1 6= λ∗2 completesthe 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 onlyif α1,2 = β1,2 = 0, and consequently the quaternionic linear eigenvectors |ψλ1〉 and |ψλ2〉 arelinear independent on H �.

• A brief discussion about the spectrum choice

What happens to the eigenvalue spectrum when we have two simultaneous diagonalizablequaternionic linear operators? We show that for complex operators the choice of a commonquaternionic eigenvector set reproduce in qQM the standard results of complex QuantumMechanics [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, thecorresponding eigenvalue spectrum is

{iE , iE

}

A1

and

{i~

2, −i~

2

}

A2

, (35)

11

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-dimensional quaternionic linear operators and so we can translate them in the complex worldand then extract the eigenvalue spectrum. By following the method given in this section, wefind 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 isrepresented by the choice of a common quaternionic eigenvector set. In fact, we observe thatthe previous eigenvalue spectra 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 choos-ing our quaternionic 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 , ↓ (↑) .

12

• From anti-hermitian to hermitian matrix operator

Let us remark an important difference between the structure of an anti-hermitian operatorin complex and in quaternionic Quantum Mechanics. In cQM, we can always trivially relatean anti-hermitian operator, 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 reason is simple: Given any independent, over H, set of normalized eigenvectors |vl〉 ofA with complex imaginary 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 thenon-commutative nature of |vl〉, we cannot extract the complex imaginary unit i. Our ap-proach to quaternionic eigenvalues equations contains a practical method to find eigenvectors|vl〉 and eigenvalues λl and consequently solves the problem to determine, given a quater-nionic anti-hermitian operator, the corresponding hermitian operator. An easy computationshows 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 admit2n linear independent eigenvectors. It is easy to demonstrate that the diagonalization matrixfor 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)

nx(λ∗

n)n

y(λ1)n y

(λ∗

1)n ... y(λn)

n y(λ∗

n)n

. (39)

13

Such a matrix is in the same subset of M , i.e. S ∈ M2n(C). In fact, by recalling therelationship between |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 the isomorphism 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 quaternionicn-dimensional matrix which diagonalizes MH

SHMH 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 appropriatemathematical notation should require the use of the left/right quaternionic operators Li,j,k

and Ri. In this case, 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 simplifiednotation i, j, k instead of Li, Lj , Lk.

This diagonalization quaternionic matrix is strictly related to the choice of a particularset of quaternionic 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 adifferent diagonalized 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 quaternioniceigenvector set.

14

4.2 Complex linear operators and complex geometry

In this section, we discuss right eigenvalue equation for complex linear operators. In n-dimensional quaternionic vector spaces, Hn, complex linear operator, OC, are representedby n × n quaternionic matrices, Mn(H

L ⊗ CR), with elements in H

L ⊗ CR. Such quater-

nionic matrices admit 2n-dimensional complex counterparts which recover the full set of2n-dimensional complex matrices, M2n(C). It is immediate to check that quaternionic matri-ces MH ∈ Mn(H

L) are characterized by 4n2 real parameters and so a natural translation gets

the complex matrix M2n(C) ⊂ M2n(C), whereas a generic 2n-dimensional complex matrixM ∈ M2n(C), characterized by 8n2 real parameters needs to double the 4n2 real parame-ters of MH. By allowing right-action for the imaginary units i we recover the missing realparameters. 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 nopossibilities to classify or characterize such a complex eigenvalue spectrum . Is it possible toextract a suitable quaternionic eigenvectors set? What happens when the complex spectrumis characterized by 2n different complex eigenvalues? To give satisfactory answers to thesequestions we must adopt a complex geometry [6, 7]. In this case

|ψ〉 and |ψ〉j

represent orthogonal vectors and so we cannot kill the eigenvectors |ψ〉j. So, for n-dimensionalquaternionic 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 lin-ear operators, MH, and complex linear operators, MC, a 2n-dimensional complex eigenvaluespectrum and consequently 2n quaternionic eigenvectors. Let us now give a practical method

15

to diagonalize complex linear operators. Complex 2n-dimensional matrices, M , are diago-nalizable if and only if admit 2n linear independent eigenvectors. The diagonalizable matrixcan be written in terms of M -eigenvectors as 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 indicatesthe right action of the imaginary unit i. The diagonalized quaternionic matrix reproducesthe quaternionic translation 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 righteigenvalues. In this section, we briefly introduce the problematic inherent to quaternioniceigenvalue equations and emphasize 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 opera-tors, OH, reads

MH|ψ〉 = |ψ〉q , q ∈ H .

16

Such an equation can be converted into a right complex eigenvalue equation by re-phasingthe quaternionic eigenvalues, q,

MH|ψ〉u = |ψ〉uuqu = |ψ〉λ , λ ∈ C .

This trick fails for complex linear operators. In fact, by discussing right quaternionic eigen-value equations for complex linear operators,

MC|ψ〉 = |ψ〉q , (50)

due to the presence of the right imaginary unit i in MC, we cannot apply quaternionicsimilarity transformations,

(MC|ψ〉

)u 6= MC

(|ψ〉u

), u ∈ H .

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

〈φ|ACψ〉C = −〈ACφ|ψ〉C . (51)

We can immediately find a constraint on our AC-eigenvalues by putting in the previousequation |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 imaginaryquaternions. An important property must be satisfied for complex linear anti-hermitianoperators, namely eigenvectors |φ〉 and |ψ〉 associated to different eigenvalues, qφ 6= qψ, haveto be orthogonal in C. By combining 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 must require 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 com-plex geometry requires complex eigenvalues.

5.2 Left quaternionic eigenvalue equation

What happens for left quaternionic eigenvalue equations? In solving such equations forquaternionic and 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

17

translation trick does not apply and so we must solve directly the problem in the quaternionicworld.

In discussing left quaternionic eigenvalue equations, we underline the difficulty hidden indiagonalizing such operators. Let us suppose that the matrix representations of our operatorsbe digonalized by a matrix SH/C

SHMH 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/C 6= q .

So, we can have operators with the same left quaternionic eigenvalues spectrum but nosimilarity transformation relating them. This is explicitly show in appendix B, were wediscuss examples of two-dimensional quaternionic linear operators. Let us now analyse otherdifficulties in solving left quaternionic eigenvalue equation. Hermitian quaternionic linearoperators satisfy

〈φ|HHψ〉 = 〈HHφ|ψ〉 .

By putting |φ〉 = |ψ〉 in the previous equation we find constraints on the quaternionic eigen-values 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 com-plication is represented by the possibility to find hermitian operators with quaternionic eigen-values. Within a qQM, we can overcome this problem by choosing anti-hermitian operatorsto represent observable quantities. 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. Examplesof left/right eigenvalue equation for two-dimensional anti-hermitian operators will be dis-cussed in appendix B. In this appendix, we explicitly show an important difference betweenleft and right eigenvalue equation for anti-hermitian operators: Left and right eigenvaluescan have different absolute values and so cannot represent the same physical quantity.

18

6 Conclusions

The study undertaken in this paper demonstrates the possibility to construct a practicalmethod to diagonalize quaternionic and complex linear operators on quaternionic vectorspaces. Quaternionic eigenvalue equations have to be right eigenvalue equations. As shownin our paper, the choice of a right position for quaternionic eigenvalues is fundamental insearching for a diagonalization method. A left position of quaternionic eigenvalues givesunwished surprises. For example, we find operators with the same eigenvalues which are notrelated by similarity transformation, hermitian operators with quaternionic eigenvalues, etc.

Quaternionic linear operators in n-dimensional vector spaces take infinite spectra ofquaternionic eigenvalues. Nevertheless, the complex translation trick ensures that such spec-tra are related by similarity transformations and this gives the possibility to choose n rep-resentative complex eigenvalues to perform calculations. The complete set of quaternioniceigenvalues spectra can be generated from the 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 casethe freedom in constructing the eigenvalue spectrum for the first operator, and consequentlythe free choice in determining an eigenvectors basis, will fix the eigenvalue spectrum for theother operators.

The power of the complex translation trick gives the possibility to study general propertiesfor quaternionic and complex linear operators. Complex linear operators play an importantrole within a qQM with complex geometry by reproducing the standard complex results inreduced quaternionic vector spaces [31]. The method of diagonalization becomes very usefulin the resolution of quaternionic differential equations [42]. Consequently an immediateapplication is found in solving the Schrodinger equation with quaternionic potentials [43].

Mathematical topics to be developed are represented by the discussion of the eigenvalueequation for real linear operators, OR, and by a detailed study of the left eigenvalue equation.Real linear operators are characterized by left and right actions of the quaternionic imaginaryunits i, j, k. The translation trick now needs to be applied in the real world and so, for acoherent discussion, it will require the adoption of a real geometry.

Acknowledgements

The authors wish to express their gratitude to Nir Cohen for several helpful comments con-cerning quaternionic matrix theory and Jordan form. They are also indebted to Gisele Ducatifor suggestions on possible applications of the diagonalization method to quaternionic differ-ential equations and for many stimulating conversations. G.S. gratefully acknowledges theDepartment of Applied Mathematics, IMECC-Unicamp, for the invitation and hospitality.This work was partially supported by a fellowship of the Department of Physics, Lecce Uni-versity, (G.S.) and by a research grant of the Fapesp, Sao Paulo State, (S.d.L.). Finally,the authors wish to thank the referees for drawing attention to interesting references and for

19

their remarks which helped to clarify the notation and improve the discussion presented inthis paper.

20

Appendix A

Two dimensional right complex eigenvalue equations

In this appendix we explicitly solve the right eigenvalues equations for quaternionic, OH, andcomplex OC, 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 atwo-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

38π , 2

14 e−i 3

8π , −2

14 e−i 3

8π , −2

14 ei

38π

}

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 adoptingthe positive imaginary part convention we find

{λ1 , λ2

}

MH

=

{2

14 ei

38π , − 2

14 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)

21

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-dimensional quaternionic 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)

22

Such eigenvalues also determine the MC-eigenvalues spectrum. The MC-eigenvector set isobtained by 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 quaternionic 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 0

0 1+k√2

),

one find

Mdiag

C→ 2

(iRi 00 j

). (63)

23

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 linearoperator. 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†.

24

• Anti-hermitian operators

Let

AH =

(j ii k

)

be the quaternionic matrix representation associated to an anti-hermitian quaternionic linearoperator. 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+k

2 )ψ1

),

(ψ1

(−1√2+ j+k

2 )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.

25

• A new possibility

In order to complete our discussion let us discuss for the quaternionic linear operator givenin 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 eigenvaluespectrum of MH, 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, thereis no similarity transformation which relates these two operators in the complex world andconsequently by translation 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 leftquaternionic eigenvalue spectrum but not related by a similarity transformation.

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