Right Eigenvalue Equation in Quaternionic Quantum Mechanics

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their n-dimensional quaternionic matrix representations. In discussing the right complex eigenvalue problem in qQM, we find two obstacles. The first one is related to the difficulty in obtaining a suitable definition of determinant 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 over the quaternionic field a complicated puzzle. We overcome the difficulties in approaching the eigenvalue problem in a quaternionic world by discussing the eigenvalue equation for 2n-dimensional complex matrices obtained by translation from n-dimensional quaternionic matrix operators. We shall show that quaternionic linear operators, defined on quaternionic Hilbert space with quaternionic geometry, are diagonalizable if and only if the corresponding complex operators are diagonalizable. The spectral theorems, extended to quaternionic Hilbert spaces [1], are recovered in a more general context. We also study the linear independence on H of quaternionic eigenvectors by studying their (complex) eigenvalues and discuss the spectrum choice for quaternionic quantum systems. Finally, we construct the hermitian operator associated with any anti-hermitian matrix operator and show that a coherent discussion of the eigenvalue problem for complex linear operators operators requires a complex geometry. A brief discussion concerning the possibility to have left eigenvalue equations 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 mathematical tools. In particular, we discuss similarity transformations, symplectic decompositions and the left/right action of quaternionic imaginary units. In section 3, we give the basic framework of qQM and translation rules between complex and quaternionic matrices. In section 4, we approach the right eigenvalue problem by discussing the eigenvalue spectrum for 2n-dimensional complex matrices obtained by translating n-dimensional quaternionic matrix representations. We give a practical method to diagonalize n-dimensional quaternionic linear matrix operators and overcome previous problems in the spectrum choice for quaternionic quantum systems. We also discuss the right eigenvalue problem for complex linear operators within a qQM with complex geometry. In section 5, we introduce the left eigenvalue equation and analyze the eigenvalue spectrum for hermitian operators. We explicitly solve some examples of left/right eigenvalue equations for two-dimensional quaternionic and complex linear 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, 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, where the quaternionic norm q is deﬁned by | | | | q 2 = qq = a2 + b2 + c2 + d2 . | | Similarity Transformation Two• quaternions q and p belong to the same eigenclass when the following relation

q = s−1 ps, 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 ps) = Re(p) , q = s−1 ps = p , | | | | | | consequently they have the same absolute value of the imaginary part. The previous equations can be rewritten in terms of unitary quaternions as follows s s q = s−1 ps = p = upu, u H , uu =1 . (3) s s ∈ | | | | In eq. (3), the unitary quaternion,

u = cos θ + h u sin θ , h (i, j, k) , u R3 , u =1 , 2 · 2 ≡ ∈ | | can be expressed in terms of the imaginary parts of q and p. In fact, given two quaternions belong to the same eigenclass,

q = q + h q , p = p + h p , q = p , q = p , 0 · 0 · 0 0 | | | | we find [39], for q = p, 6 ± q p q p cos θ = · and u = × . (4) q p q p sin θ | | | | | | | | The remaining cases q = p and q = p represent respectively the trivial similarity transformation, i.e. uqu¯ = q, and the similarity transformation− between a quaternion q and its conjugateq ¯, i.e. uqu¯ =q ¯. In the first case the unitary quaternion is given by the identity quaternion. In the last case the similarity transformation is satisfied u = h u with u q = 0 and u = 1. ∀ · · | | Symplectic Decomposition Complex• 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 Action Due• 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 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) → ∈ 4 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

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

L2 = L2 = L2 = L L L = R2 = R2 = R2 = R R R = 1 , i j k i j k i j k k j i − 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 units i, j, k. In this paper we will work with two sub-spaces of HL HR, namely ⊗ HL and HL 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 + cR + d LR HL CR , a, b, c, d R . (10) · i · i ∈ ⊗ ∈

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| i multiplication, we must specify whether the quaternionic Hilbert space is to be formed by right or left multiplication of quaternionic vectors by scalars. The two diﬀerent conventions give isomorphic versions of the 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 H and C. They will act on quaternionic vectors, ψ , in the following way O O | i

H( ψ q) = ( H ψ ) q , q H . O | i O | i ∈ and

C( ψ λ) = ( C ψ ) λ , λ C . O | i O | i ∈ 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 | i

ψ1 x1 + jy1 . . ψ = . = . , x ,y , ..., xn,yn C . (11) | i  .   .  1 1 ∈ ψn xn + jyn         Quaternionic linear operators, H, will be represented by n × n matrices with entries in HL, whereas O L R complex linear operators, C, by n × n matrices with entries in H C . By using the symplecticOcomplex representation, the n-dimensional⊗ quaternionic vector

x1 y1 ψ = x + j y = . + j . , | i | i | i  .   .  xn yn         Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 5 can be translated in the 2n-dimensional complex column vector

y1 ψ  .  . (12) | i ↔ .    xn     yn    The matrix representation of Li, Lj and Lk consistent with the above identiﬁcation is

i 0 0 -1 0 -i L = iσ ,L = iσ ,L = iσ . (13) i ↔ 0-i 3 j ↔ 1 0 − 2 k ↔ -i 0 − 1 These translation rules allow to represent quaternionic n-dimensional linear operators by 2n × 2n complex matrices. The right quaternionic imaginary unit

i 0 R , (14) i ↔ 0 i adds four additional degrees of freedom,

Ri ,LiRi ,Lj Ri ,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 × 2 complex matrices [41]. Consequently, we can construct 2n × 2n complex matrix 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 well deﬁned hermiticity within a qQM with quaternionic inner product (quaternionic geometry),

n † ϕ ψ = ϕ ψ = ϕ¯lψl . (15) h | i | i | i Xl=1 In fact, anti-hermitian operators must satisfy

ϕ ψ = ϕ ψ . h |A i −hA | i

For the right imaginary unit Ri, we have

† Riψ Ri ψ = ψ i , Riϕ (Ri ϕ ) = i ϕ , | i≡ | i | i h |≡ | i − h | and consequently

ϕ ψ i = ϕ Riψ = Riϕ ψ = i ϕ ψ . h | i h | i 6 −h | i h | i Nevertheless, by adopting a complex geometry, i.e. a complex projection of the quaternionic inner product,

ϕ ψ i ϕ ψ i ϕ ψ C = h | i− h | i , (16) h | i 2 we recover the anti-hermiticity of the operator Ri,

ϕ Riψ C = Riϕ ψ C . h | i −h | i 6 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

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) | i | i

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

r = Ψ u (18) | i {| i } where Ψ is a normalized vector and u a quaternionic phase of magnitude unity. The state vector, | i Ψ u, corresponding to the same physical state Ψ , is an OH-eigenvector with eigenvalue uqu | i | i

OH Ψ u = Ψ u (uqu) . | i | i For real values of q, we ﬁnd only one eigenvalue, otherwise quaternionic linear operators will be characterized by an inﬁnite eigenvalue spectrum

q , u qu , ... , ulqul , ... { 1 1 }

with ul unitary quaternions. The related set of eigenvectors

Ψ , Ψ u , ... , Ψ ul , ... | i | i 1 | i represents a ray. We can characterize our spectrum by choosing a representative ray

ψ = Ψ uλ , | i | i

so that the corresponding eigenvalue λ =u ¯λquλ is complex. For this state the right eigenvalue equation becomes

OH ψ = ψ λ , (19) | i | i

with ψ VH and λ C. | i ∈ ∈ 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 represented L L by n n quaternionic matrices, n(H ), with elements in H . Such quaternionic matrices admit 2n- dimensional× complex counterpartsM by the translation rules given in eq. (13). Such complex matrices characterize a subset of the 2n-dimensional complex matrices

M n(C) M n(C) . 2 ⊂ 2 The eigenvalue equation for OH reads f

H ψ = ψ λ , (20) M | i | i where H (HL), ψ H n and λ C. M ∈Mn | i ∈ ∈ Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 7

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) | i | i 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| i the| quaternionici | i ∈ algebra from∈ the commutative complex algebra (Cayley-Dickson process). The complex counterpart of eq.(21) reads

z -w x x ∗ = λ , (22) w z∗ y y

z = a + ib , w = b ib C . 1 2 − 3 ∈ 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 eigenvector associated to the eigenvalue λ∗ by taking the complex conjugate of eq.(22) and then applying a similarity transformation by the matrix

0 -1 S = . 1 0 In this way, we ﬁnd

z -w∗ -y∗ -y∗ = λ∗ . (23) w z∗ x∗ x∗

So, for λ = λ∗ C, we obtain the eigenvalue spectrum λ , λ∗ with eigenvectors 6 ∈ { } x -y , ∗ . (24) y x∗ What happens when λ R ? In this case the eigenvalue spectrum will be determined by two equal eigenvalues λ. To show that,∈ we remark that the eigenvectors (24) associated to the same eigenvalues λ, are linearly independent on C. In fact,

x -y ∗ = x 2 + y 2 = 0 if and only if x = y =0 . y x∗ | | | |

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 . | i | i ∈ | i The inﬁnite quaternionic eigenvalue spectrum can be characterized by the complex eigenvalue λ and the ray representative will be ψ . In the next section, by using the same method, we will discuss eigenvalue equations in n-dimensional| i 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-dimensional quaternionic matrix H. In this M theorem we give the matrix S which allows to construct the complex eigenvector φλ∗ from the | i C eigenvector φλ .f The explicit construction of φλ∗ enables us to show the linear independence on | i | i 8 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

of φλ and φλ∗ when λ R and represents the main tool in constructing similarity transformations for| diagonalizablei | i quaternionic∈ matrices. The second theorem [T2] discusses linear independence on H for H-eigenvectors. T1 - THEOREMM Let M be the complex counterpart of a generic n n quaternionic matrix H. Its eigenvalues appear in conjugate pairs. × M Let f

M φλ = λ φλ (25) | i | i be the eigenvalue equation for M, where f

f x1 y1  .  2n M M n(C) , φλ = . C , λ C . ∈ 2 | i . ∈ ∈   xn f   yn    By taking the complex conjugate of eq. (25),

M ∗ φλ ∗ = λ∗ φλ ∗ , | i | i and applying a similarity transformationf by the matrix 0 -1 S =11n , ⊗ 1 0 we obtain

−1 SM ∗S S φλ ∗ = λ∗S φλ ∗. (26) | i | i From the block structure of the complexf matrix M it is easily checked that

−1 SM ∗Sf = M, and consequently eq. (26) reads f f

M φλ∗ = λ∗ φλ∗ , (27) | i | i where f

-y1∗

x1∗  .  φλ∗ = S φλ ∗ = . . | i | i . -y   n∗  x   n∗    Let us show that the eigenvalues appear in conjugate pairs (this implies a double multiplicity for real eigenvalues). To do it, we need to prove that φλ and φλ∗ are linearly independent on C. In order to | i | i demonstrate the linear independence of such eigenvectors, trivial for λ = λ∗, we observe that linear 6 dependence, possible in the case λ = λ∗, should require

xi -yi∗ 2 2 = xi + yi =0 i =1, ..., n , yi x∗ | | | | i

veriﬁed only for null eigenvectors. The linear independence of φλ and φλ∗ ensures an even multiplicity for real eigenvalues . | i | i Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 9

We shall use the results of the first theorem to obtain informations about the H right complex eigenvalue spectrum. Due to non-commutativity nature of the quaternionic fieldM we cannot give a suitable definition of determinant for quaternionic matrices and consequently we cannot write a characteristic polynomial P (λ) for H. Another difficulty it is represented by the right position of the complex eigenvalue λ. M T2 - THEOREM H admits n linearly independent eigenvectors on H if and only if its complex counterpart M admitsM 2n linearly independent eigenvectors on C . Let f

φλ , φλ∗ , ... , φλ , φλ∗ (28) | 1 i | 1 i | n i | n i be a set of 2n M-eigenvectors, linearly independent on C, and αl, βl (l =1, ..., n) be generic complex coeﬃcients. By deﬁnition

f n

αl φλ + βl φλ∗ =0 αl = βl =0 . (29) | l i | l i ⇔ Xl=1 By translating the complex eigenvector set (28) in quaternionic formalism we ﬁnd

ψλ , ψλ∗ , ... , ψλ , ψλ∗ . (30) | 1 i | 1 i | n i | n i

By eliminating the eigenvectors, ψλ∗ = ψλ j, corresponding for complex eigenvalues to ones with | l i | l i negative imaginary part, linearly dependent with ψλ on H, we obtain | l i

ψλ , ... , ψλ . | 1 i | n i This set is formed by n linearly independent vectors on H. In fact, by taking an arbitrary quaternionic linear combination of such vectors, we have

n n

ψλ (αl + jβl) = ψλ αl + ψλ∗ βl =0 αl = βl =0 . (31) | l i | l i | l i ⇔ Xl=1 Xl=1 Note that eq. (31) represents the quaternionic counterpart of eq. (29) . The H complex eigenvalue spectrum is thus obtained by taking from the 2n dimensional M- eigenvaluesM spectrum f λ , λ∗ , ... , λn , λ∗ , { 1 1 n} the reduced n-dimensional spectrum

λ , ... , λn . { 1 } We stress here the fact that, the choice of positive, rather than negative, imaginary part is a simple convention. In fact, from the quaternionic eigenvector set (30), we can extract diﬀerent sets of quaternionic linearly independent eigenvectors

ψλ or ψλ∗ , ... , ψλ or ψλ∗ , | 1 i | 1 i | n i | n i and consequently we have a free choice in characterizing the n-dimensional H-eigenvalue spectrum. A direct consequence of the previous theorems, is the following corollary. M T2 - COROLLARY

Two H quaternionic eigenvectors with complex eigenvalues, λ1 and λ2, with λ2 = λ1 = λ2∗, are linearly independentM on H. 6 6 10 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

Let

ψλ (α + jβ )+ ψλ (α + jβ ) (32) | 1 i 1 1 | 2 i 2 2 be a quaternionic linear combination of such eigenvectors. By taking the complex translation of eq. (32), we obtain

α φλ + β φλ∗ + α φλ + β φλ∗ . (33) 1| 1 i 1| 1 i 2| 2 i 2| 2 i The set of M-eigenvectors

f φλ , φλ∗ , φλ , φλ∗ | 1 i | 1 i | 2 i | 2 i is linear independent on C. In fact, theorem T1 ensures linear independence between eigenvectors associated to conjugate pairs of eigenvalues, and the condition λ2 = λ1 = λ2∗ completes the proof by assuring the linear independence between 6 6

φλ , φλ∗ and φλ , φλ∗ . | 1 i | 1 i | 2 i | 2 i

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 . | i | i

A brief discussion about the spectrum choice • What happens to the eigenvalue spectrum when we have two simultaneous diagonalizable quaternionic linear operators? We show that for complex operators the choice of a common quaternionic eigenvector set reproduce in qQM the standard results of complex Quantum Mechanics [cQM]. Let

i 0 ~ i 0 = E and = (34) A1 0 i A2 2 0-i be anti-hermitian complex operators associated respectively to energy and spin. In cQM, the corresponding eigenvalue spectrum is ~ ~ iE , iE and i , i , (35) 2 − 2 A1 A2 1 and physically we can describe a particle with positive energy E and spin 2 . 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 world and then extract the eigenvalue spectrum. By following the method given in this section, we ﬁnd the following eigenvalues ~ ~ ~ ~ iE , iE , iE , iE and i , i , i , i , − − 2 − 2 2 − 2 A1 A2 and adopting the positive imaginary part convention we extract ~ ~ iE , iE and i , i . 2 2 A1 A2 1 It seems that we lose the physical meaning of 2 -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 the choice of a common quaternionic eigenvector set. In fact, we observe that the previous eigenvalue spectra are related to the following eigenvector sets 1 0 1 0 , and , . 0 1 0 j A1 A2 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 11

By ﬁxing a common set of eigenvectors

1 0 , , (36) 0 1 , A1 2 we recover the standard results of eq. (35). Obviously,

1 0 ~ ~ , +iE , +iE , +i , i , 0 1 2 2 , → − A1 2 A1 A2 1 0 ~ ~ , +iE , iE , +i , +i , 0 j 2 2 , → − A1 2 A1 A2 j 0 ~ ~ , iE , +iE , i , i , 0 1 2 2 , → − − − A1 2 A1 A2 j 0 ~ ~ , iE , iE , i , +i , 0 j 2 2 , → − − − A1 2 A1 A2 represent equivalent choices. Thus, in this particular case, the diﬀerent possibilities in choosing our quaternionic eigenvector set will give the following outputs 1 Energy: + E, + E and 2 -spin : , + E, E and ↑ , ↓ E, −+ E and ↑ , ↑ − E, E and ↓ , ↓ − − ↓ ↑ 1 Thus, we can also describe a 2 -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 complex and in quaternionic Quantum Mechanics. In cQM, we can always trivially relate an anti-hermitian operator, to an hermitian operator, , by removing a factor i A H = i . A H In qQM, we must take care. For example,

-i 3j = , (37) A 3j i is an anti-hermitian operator, nevertheless, i does not represent an hermitian operator. The reason A is simple: Given any independent, over H, set of normalized eigenvectors vl of with complex | i A imaginary eigenvalues λl,

= vl λl i vl , A | i | | h | Xl the corresponding hermitian operator is soon obtained by H

= vl λl vl , H | i | | h | Xl since both the factors are independent of the particular representative vl chosen. Due to the non- | i commutative nature of vl , we cannot extract the complex imaginary unit i. Our approach to quater- | i nionic eigenvalues equations contains a practical method to ﬁnd eigenvectors vl and eigenvalues λl | i 12 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

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

1 i 1 k i λ , i λ = 2i , 4i and v , v = , . { | 1| | 2|} { } {| 1i | 2i} √ j √ 1 2 2 So, the hermitian operator corresponding to the anti-hermitian operator of eq. (37) is

3 k = . (38) H -k 3

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

−1 diag S M S = M , f is given by e f e f

(λ∗) (λ∗ ) (λ1 ) 1 (λn) n x1 x1 ... x1 x1 (λ∗) (λ∗ ) y(λ1) y 1 ... y(λn) y n  1 1 1 1  S = Inverse ...... (39) ......  λ λ∗ λ λ∗  ( 1 ) ( 1 ) ( n) ( n) xn xn ... xn xn   ∗ ∗  e y(λ1) y(λ1 ) ... y(λn) y(λn)   n n n n   

Such a matrix is in the same subset of M, i.e. S M n(C). In fact, by recalling the relationship ∈ 2 between φλ and φλ∗ , we can rewrite the previous diagonalization matrix as | i | i f e f (λ1) (λ1) (λn) (λn) x1 -y1∗ ... x1 -y1∗ (λ ) (λ ) y 1 x∗ 1 ... y(λn) x (λn)  1 1 1 1∗  Inverse . . . . . S = ...... (40)  (λ1) (λ1) (λn) (λn) xn -yn∗ ... xn -yn∗  e  (λ1) (λ1) (λn) (λn)  yn x∗n ... yn xn∗      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 ×−1 counterpart group GL(2n, C) ensures S M n(C). So, thef quaternionic n-dimensional matrixe which ∈ 2 diagonalizes H M f e f −1 diag H H = , S M SH MH can be directly obtained by translating eq. (40) in

(λ1) (λ1) (λn) (λn) x1 + jy1 ... x1 + jy1 . . H = Inverse . .. . . (41) S  . . .  x(λ1) + jy(λ1) ... x(λn) + jy(λn)  n n n n    In translating complex matrices in quaternionic language, we remember that an appropriate mathematical 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,

−1 M, S, S M n(C) , ∈ 2 their quaternionic translation is performedf e by lefte operatorsf and so we use the simpliﬁed notation i, j, k instead of Li, Lj , Lk. Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 13

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

ψλ , ... , ψλ . | 1 i | n i So, the diagonalized quaternionic matrix reads

diag = diag λ , ... , λn . MH { 1 } The choice of a diﬀerent quaternionic eigenvector set

ψλ or ψλ∗ , ... , ψλ or ψλ∗ , | 1 i | 1 i | n i | n i will give, for not real eigenvalues, a diﬀerent diagonalization matrix and consequently a diﬀerent diagonalized quaternionic matrix

diag = diag [λ or λ∗] , ... , [λn or λ∗ ] . MH { 1 1 n } In conclusion,

H diagonalizable M diagonalizable , M ⇔ and the diagonalization quaternionic matrix can be easily obtained from the quaternionic eigenvector f set.

4.2 Complex linear operators and complex geometry

In this section, we discuss right eigenvalue equation for complex linear operators. In n-dimensional n quaternionic vector spaces, H , complex linear operator, OC, are represented by n n quaternionic L R L R × matrices, n(H C ), with elements in H C . Such quaternionic matrices admit 2n-dimensional M ⊗ ⊗ C complex counterparts which recover the full set of 2n-dimensional complex matrices, M2n( ). It is L 2 immediate to check that quaternionic matrices H n(H ) are characterized by 4n real param- M ∈M C C eters and so a natural translation gets the complex matrix M2n( ) M2n( ), whereas a generic C 2 ⊂ 2n-dimensional complex matrix M M2n( ), characterized by 8n real parameters needs to double 2 ∈ the 4n real parameters of H. By allowing right-action forf the imaginary units i we recover the missing real parameters. So,M the 2n-dimensional complex eigenvalue equation

2n M φ = λ φ ,M M n(C) , φ C , λ C , (42) | i | i ∈ 2 | i ∈ ∈ becomes, in quaternionic formalism,

L R n C ψ = ψ λ , C n(H C ) , ψ H , λ C . (43) M | i | i M ∈M ⊗ | i ∈ ∈ The right position of the complex eigenvalue λ is justiﬁed by the translation rule

i1 n R 1 n . 2 ↔ i By solving the complex eigenvalue problem of eq. (42), we ﬁnd 2n eigenvalues and we have no possibilities to classify or characterize such a complex eigenvalue spectrum . Is it possible to extract a suitable quaternionic eigenvectors set? What happens when the complex spectrum is characterized by 2n diﬀerent complex eigenvalues? To give satisfactory answers to these questions we must adopt a complex geometry [6, 7]. In this case ψ and ψ j | i | i represent orthogonal vectors and so we cannot kill the eigenvectors ψ j. So, for n-dimensional quater- | i nionic matrices C we must consider the full eigenvalue spectrum M

λ1 , ... , λ2n . (44) 14 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

The corresponding quaternionic eigenvector set is then given by

ψλ , ... , ψλ , (45) | 1 i | 2n i which represents the quaternionic translation of the M-eigenvector set

φλ , ... , φλ . (46) | 1 i | 2n i In conclusion within a qQM with complex geometry [31] we ﬁnd for quaternionic linear operators, MH, and complex linear operators, MC,a2n-dimensional complex eigenvalue spectrum and consequently 2n quaternionic eigenvectors. Let us now give a practical method to diagonalize complex linear operators. Complex 2n-dimensional matrices, M, are diagonalizable if and only if admit 2n linear independent eigenvectors. The diagonalizable matrix can be written in terms of M-eigenvectors as follows

λ (λ1) (λ2 ) ( 2n−1) (λ2n) x1 x1 ... x1 x1 λ y(λ1) y(λ2) ... y( 2n−1) y(λ2n)  1 1 1 1  Inverse . . . . . S = ...... (47)  (λ )  (λ1) (λ2 ) 2n−1 (λ2n) xn xn ... xn xn   λ  (λ ) (λ ) ( 2n−1) (λ ) y 1 y 2 ... yn y 2n   n n n    This matrix admits a quaternionic counterpart [41] by complex linear operators

[1,2] [1,2] [2n−1,2n] [2n−1,2n] q1 + p1 Ri ... q1 + p1 Ri C = Inverse . .. . , (48) S  . . .  q[1,2] + p[1,2]R ... q[2n−1,2n] + p[2n−1,2n]R  n n i n n i   where

(λm) (λn) (λm) (λn) [m,n] xl + yl ∗ yl xl ∗ ql = + j − , 2 2 and

(λm) (λn) (λm) (λn) [m,n] xl yl ∗ yl + xl ∗ pl = − + j . 2i 2i

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

diag M = diag λ , ... , λ n { 1 2 } into

diag λ1 + λ2∗ λ1 λ2∗ λ2n−1 + λ2∗n λ2n−1 λ2∗n C = diag + − R , ... , + − R . (49) M 2 2i i 2 2i i

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. In this section, we briefly introduce the problematic inherent to quaternionic eigenvalue equations and emphasize the main difficulties present in such an approach. Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 15

5.1 Right quaternionic eigenvalue equation for complex linear operators

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

H ψ˜ = ψ˜ q , q H . M | i | i ∈ Such an equation can be converted into a right complex eigenvalue equation by re-phasing the quaternionic eigenvalues, q,

H ψ˜ u = ψ˜ u uqu = ψ˜ λ , λ C . M | i | i | i ∈ This trick fails for complex linear operators. In fact, by discussing right quaternionic eigenvalue equations for complex linear operators,

C ψ = ψ q , (50) M | i | i due to the presence of the right imaginary unit i in C, we cannot apply quaternionic similarity transformations, M

C ψ˜ u = C ψ˜ u , u H . M | i 6 M | i ∈ Within a qQM with complex geometry [20, 31, 41], a generic anti-hermitian operator must satisfy

φ Cψ C = Cφ ψ C . (51) h |A i −hA | i We can immediately ﬁnd a constraint on our C-eigenvalues by putting in the previous equation φ = ψ , A | i | i

ψ ψqψ C = ψqψ ψ C qψ = iαψ + jwψ , (52) h | i −h | i ⇒

αψ R , wψ C . ∈ ∈ Thus, complex linear anti-hermitian operators, C, will be characterized by purely imaginary quaternions. An important property must be satisfiedA for complex linear anti-hermitian operators, namely eigenvectors φ and ψ associated to different eigenvalues, qφ = qψ , have to be orthogonal in C. By combining Eqs.| i (51,52),| i we find 6

φ ψqψ C = qφφ ψ C . h | i h | i To guarantee the complex orthogonality of the eigenvectors φ and ψ , namely φ ψ C = 0, we must | i | i h | i 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 complex geometry requires complex eigenvalues.

5.2 Left quaternionic eigenvalue equation

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

H ψ˜ =q ˜ ψ˜ , M | i | i C ψ˜ =q ˜ ψ˜ , q˜ H , (M | i | i ∈ we have not a systematic way to approach the problem. In this case, due to the presence of left quaternionic eigenvalues (translated in complex formalism by two-dimensional matrices), the translation trick does not apply and so we must solve directly the problem in the quaternionic world. 16 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

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 digonalized by a matrix H C S / −1 diag −1 diag H H = and C C = . S M SH MH S M SC MC The eigenvalue equation will be modiﬁed in

diag −1 H H ψ˜ = Hq˜ H H ψ˜ , Mdiag S | i S S−1 S | i C ψ˜ = Cq˜ C ψ˜ , (MC S | i S SC S | i and now, due to the non-commutative nature of q,

−1 H C q = q . S / SeH/C 6 So, we can have operators with the same lefte quaternionice eigenvalues spectrum but no similarity transformation relating them. This is explicitly show in appendix B, were we discuss examples of two-dimensional quaternionic linear operators. Let us now analyse other diﬃculties in solving left quaternionic eigenvalue equation. Hermitian quaternionic linear operators satisfy

φ Hψ = Hφ ψ . h |H i hH | i By putting φ = ψ in the previous equatione e we find constraintse e on the quaternionic eigenvalues q | i | i e e ψ q ψ = q ψ ψ . h | i h | i e From this equation we cannot extract the conclusion thatq ˜ must be real, q = q †. In fact e e e e e e ψ (q q †) ψ =0 h | − | i e e could admit quaternionic solutions forq ˜ (seee the examplee in appendix B). So, the first complication is represented by the possibility to find hermitiane e operators with quaternionic eigenvalues. Within a qQM, we can overcome this problem by choosing anti-hermitian operators to represent observable quantities. In fact,

φ Hψ = Hφ ψ , h |A i −hA | i will imply, for φ = ψ , e e e e | i | i e e ψ (q + q †) ψ =0 . (53) h | | i In this case, the real quantity, q + q †, commutese withe ψ , and so eq. (53) gives the constraint e e | i q = iα + jw . e e e We could work with anti-hermitian operators and choose q as observable output. Examples of e left/right eigenvalue equation for two-dimensional anti-hermitian| | operators will be discussed in appendix B. In this appendix, we explicitly show an important diﬀerencee between left and right eigenvalue equation for anti-hermitian operators: Left and right eigenvalues can have diﬀerent absolute values and so cannot represent the same physical quantity.

6 Conclusions

The study undertaken in this paper demonstrates the possibility to construct a practical method to diagonalize quaternionic and complex linear operators on quaternionic vector spaces. Quaternionic eigenvalue equations have to be right eigenvalue equations. As shown in our paper, the choice of a right position for quaternionic eigenvalues is fundamental in searching for a diagonalization method. Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 17

A left position of quaternionic eigenvalues gives unwished surprises. For example, we ﬁnd operators with the same eigenvalues which are not related by similarity transformation, hermitian operators with quaternionic eigenvalues, etc. Quaternionic linear operators in n-dimensional vector spaces take inﬁnite spectra of quaternionic eigenvalues. Nevertheless, the complex translation trick ensures that such spectra are related by similarity transformations and this gives the possibility to choose n representative complex eigenvalues to perform calculations. The complete set of quaternionic eigenvalues 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 case the freedom in constructing the eigenvalue spectrum for the first operator, and consequently the free choice 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 for quaternionic and complex linear operators. Complex linear operators play an important role within a qQM with complex geometry by reproducing the standard complex results in reduced quaternionic vector spaces [31]. The method of diagonalization becomes very useful in the resolution of quaternionic differential equations [42]. Consequently an immediate application is found in solving the Schrödinger equation with quaternionic potentials [43]. Mathematical topics to be developed are represented by the discussion of the eigenvalue equation for real linear operators, R, and by a detailed study of the left eigenvalue equation. Real linear operators are characterizedO by left and right actions of the quaternionic imaginary units i, j, k. The translation trick now needs to be applied in the real world and so, for a coherent 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 concerning quaternionic matrix theory and Jordan form. They are also indebted to Gisele Ducati for suggestions on possible applications of the diagonalization method to quaternionic diﬀerential equations and for many stimulating conversations. G.S. gratefully acknowledges the Department of Applied Mathe- matics, IMECC-Unicamp, for the invitation and hospitality. This work was partially supported by a fellowship of the Department of Physics, Lecce University, (G.S.) and by a research grant of the Fapesp, S˜ao Paulo State, (S.d.L.). Finally, the authors wish to thank the referees for drawing attention to interesting references and for their remarks which helped to clarify the notation and improve the discussion presented in this paper. 18 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

Appendix A Two dimensional right complex eigenvalue equations

In this appendix we explicitly solve the right eigenvalues equations for quaternionic, H, and complex O C, linear operators, in two-dimensional quaternionic vector spaces. O

Quaternionic linear operators • Let i j H = (54) M k i be the quaternionic matrix representation associated to a quaternionic linear operator in a two- dimensional quaternionic vector space. Its complex counterpart reads

i 0 0-1 0 -i 1 0 M = . 0 -i i 0  -i 00-i   f   In order to solve the right eigenvalue problem

H ψ = ψ λ , λ C , M | i | i ∈ let us determine the M-eigenvalue spectrum. From the constraint

f det M λ1 =0 , − 4 h i we ﬁnd for the M-eigenvalues the following solutionsf

1 i 3 π 1 −i 3 π 1 −i 3 π 1 i 3 π λf , λ∗ , λ , λ∗ = 2 4 e 8 , 2 4 e 8 , 2 4 e 8 , 2 4 e 8 . 1 1 2 2 − − M M The M-eigenvector set is given byf f

f 1+ iλ1 0 0 1 iλ1 − 0 1 iλ 1 iλ − −0 , 1∗ , 1∗ , .  0  − −1   −1   0   1 −0 0 1          M          f The H-eigenvalue spectrum is soon obtained from that one of M. For example by adopting the positiveM imaginary part convention we ﬁnd f 1 3 1 3 − 4 i 8 π 4 i 8 π λ1 , λ2 = 2 e , 2 e , (55) H − H M M and the corresponding quaternionic eigenvector set, deﬁned up to a right complex phase, reads

1+ iλ j(1 iλ ) 1 , 1∗ . (56) − j −1 H M The quaternionic matrix which diagonalizes H is M 2 1+ iλ1 j(1 iλ∗) 1 iλ∗ j iλ1 + λ1 H Inverse 1 1 = − − = 2 | 2| . (57) S j 1 −2 λ1 kλ1∗ iλ1 λ1 | | − | | Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 19

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

u1λ1u1 , u2λ2u2 . Equivalent diagonalized matrices can be obtained from

diag H = diag λ1 , λ2 M by performing a similarity transformation

−1 diag = † diag , U MH U U MH U and

= diag u , u . U 1 2 The diagonalization matrix given in eq. (57) becomes

† H H . S → U S

Complex linear operators • Let

iRi + j kRi +1 C = − − (58) M kRi 1 iRi + j − − be the quaternionic matrix representation associated to a complex linear operator in a two-dimensional quaternionic vector space. Its complex counterpart is

1-1 1-1 1-1-1 1 M = .  -1 -1 -1 -1  -1-1 1 1     The right complex eigenvalue problem

C ψ = ψ λ , λ C , M | i | i ∈ can be solved by determining the M-eigenvalue spectrum

λ1 , λ2 , λ3 , λ4 = 2 , 2 , 2i , 2i . (59) M/ C − − M/ C M M Such eigenvalues also determine the C-eigenvalues spectrum. The C-eigenvector set is obtained by translating the complex M-eigenvectorM set M

1 0 1 -i 0 1 -i 1 , , , ,  0  1  i  -1  -1 0 1 -i          M         in quaternionic formalism  

1 j 1+ k j i , , , . (60) -j 1 i + j k − 1 C − M 20 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

The quaternionic matrix which diagonalizes C is M Inverse 1 1+ k 1 1 j C = = 1 k i+j , (61) S -j i + j 2 − - 2 2 and the diagonalized matrix is given by

diag -iRi 0 C =2 . (62) M 0 i This matrix can be directly obtained from the M/ C eigenvalue spectrum by translating, in quaternionic formalism, the matrix M

200 0 0-2 0 0 M diag = .  0 02i 0  0 0 0-2i     diag It is interesting to note that equivalent diagonalized matrices can be obtained from C in eq. (62) by the similarity transformation M

† diag . U MC U For example by choosing

-j 0 = 1+k , U 0 √2 one ﬁnd

diag iRi 0 2 . (63) MC → 0 j Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 21

Appendix B Two dimensional left quaternionic eigenvalue equations

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

Hermitian operators • Let 0 k H = H -k 0 be the quaternionic matrix representation associated to an hermitian quaternionic linear operator. We consider its left quaternionic eigenvalue equation

H ψ = q ψ , (64) H | i | i where e e e ψ ψ = 1 H2 , q˜ H . | i ψ ∈ ∈ 2 e e Eq. (64) can be rewritten by the following quaternionice system

k ψ2 =q ˜ψ1 ,

( k ψ1 =q ˜ψ2 . −e e The solution is e e

q˜ = z + jβ , H H H H where z C , β R , z 2 + β2 =1 . ∈ ∈ | | The H-eigenvector set is given by H ψ 1 . -k(z + jβ)ψ 1 H e H It is easy to verify that in this case e < ψ (˜q q˜†) ψ =0 | − | i is veriﬁed for quaternionic eigenvaluesq ˜ =q ˜†. 6 e e

Anti-hermitian operators • Let j i H = A i k be the quaternionic matrix representation associated to an anti-hermitian quaternionic linear operator. Its right complex spectrum is given by

λ1 , λ2 = i 2 √2 , i 2+ √2 . H − H H q q H 22 Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics

We now consider the left quaternionic eigenvalue equation

H ψ = q ψ , (65) A | i | i where e e e ψ ψ = 1 H 2 , q˜ H . | i ψ ∈ ∈ 2 e e By solving the following quaternionic systeme

j ψ1 + i ψ2 =q ˜ψ1 ,

(i ψ1 + k ψ2 =q ˜ψ2 , e e e we ﬁnd e e e i j + k i j + k q˜1 , q˜2 = + , − + , H √2 2 √2 2 H A A and

ψ1 ψ1 1 j+k , 1 j+k . ( + 2 )ψ1 ( − + 2 )ψ1 ( √2 ! √2 !) H e e H We observe that e e

u λ u = 2 √2 , u λ u = 2+ √2 | 1 1 1| − | 2 2 2 q q and

q˜ =1 , q˜ =1 . | 1| | 2| 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

H ψ =q ˜ ψ , (66) M | i | i where e e ψ ψ = 1 H 2 , q˜ H . | i ψ ∈ ∈ 2 e e Eq. (66) can be rewritten by the following quaternionice system

i ψ1 + j ψ2 =q ˜ψ1 ,

(k ψ1 + i ψ2 =q ˜ψ2 . e e e The solution gives for the quaternionic eigenvaluee e spectrume j + k j + k q˜1 , q˜2 = i + , i , (67) H √2 − √2 H M M Stefano De Leo, Giuseppe Scolarici: Right eigenvalue equation in quaternionic quantum mechanics 23 and for the eigenvector set

1 1 1 i , i 1 . (68) − − √2 √2 H M Let us now consider the following quaternionic linear operator

j+k i + √ 0 H = 2 . (69) N 0 i j+k − √2 ! This operator represents a diagonal operator and has the same left quaternionic eigenvalue spectrum of diag H, notwithstanding such an operator is not equivalent to H . In fact, the H-complex counterpart Mis characterized by the following eigenvalue spectrum M N

i√2 , i√2 , i√2 , i√2 , − − N diﬀerent from the eigenvalue spectrum of the M-complex counterpart of H. Thus, there is no similarity transformation which relates these two operators in the complex worldM and consequently by translation there is no a quaternionic matrix which relates H to H. So, in the quaternionic world, we can have quaternionic linear operators which have a sameN leftM quaternionic eigenvalue spectrum but not related by a similarity transformation.

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