Acta Polytechnica Vol. 51 No. 4/2011

Coquaternionic Quantum Dynamics for Two-level Systems

D. C. Brody, E. M. Graefe

Abstract

Thedynamical aspects of a spin-1
2
particle inHermitian coquaternionic quantumtheory are investigated. It is shown that

the time evolution exhibits three different characteristics, depending on the values of the parameters of theHamiltonian.
When energy eigenvalues are real, the evolution is either isomorphic to that of a complexHermitian theory on a spherical
state space, or else it remains unitary along an open orbit on a hyperbolic state space. When energy eigenvalues form a
complex conjugate pair, the orbit of the time evolution closes again even though the state space is hyperbolic.

Keywords: complexified mechanics, PT symmetry, hyperbolic geometry.

Over the last decade or so there has been con-
siderable interest in the study of complexified dy-
namical systems; both classically [1–6] and quantum
mechanically [7–17]. For a classical system, its com-
plex extension typically involves the use of complex
phase-space variables: (x, p) → (x0 + ix1, p0 + ip1).
Hence the dimensionality of the phase space, i.e. the
dynamical degrees of freedom, is doubled, and the
Hamiltonian H(x, p) in general also becomes com-
plex. For a quantum system, on the other hand,
its complex extension typically involves the use of a
Hamiltonian that is not Hermitian, whereas the dy-
namical degrees of freedom associated with the space
of states — the quantum phase space variables — are
kept real. However, a fully complexified quantum dy-
namics, analogous to its classical counterpart, can be
formulated, where state space variables are also com-
plexified [18, 19].

The present authors recently observed that there
are two natural ways in which quantum dynamics can
be extended into a fully complex domain [19], where
both the Hamiltonian and the state space are com-
plexified. In short, one way is to let the state space
variables and the Hamiltonian be quaternion valued;
the other is to let them be coquaternion valued. The
former is related to quaternionic quantum mechanics
of Finkelstein and others [20, 21], whereas the lat-
ter possesses spectral structures similar to those of
PT-symmetric quantum theory of Bender and oth-
ers [7–10]. The purpose of this paper is to work out in
some detail the dynamics of an elementary quantum
system of a spin-1

2 particle under a coquaternionic ex-
tension, in a manner analogous to the quaternionic
case investigated elsewhere [22].

As illustrated in [19], a coquaternionic dynamical
system arises from the extension of the real and the
imaginary parts of the state vector in the complex-j
direction, where j is the second coquaternionic ‘imag-
inary’ unit (described below). The general dynamics

is governed by a coquaternionic Hermitian Hamilto-
nian, whose eigenvalues are either real or else ap-
pear as complex conjugate pairs. Here we examine
the evolution of the expectation values of the five
Pauli matrices generated by a generic 2 × 2 coquater-
nionic Hermitian Hamiltonian. We shall find that,
depending on the values of the parameters appearing
in the Hamiltonian, the dynamics can be classified
into three cases: (a) the eigenvalues of H are real
and the dynamics is strongly unitary in the sense that
the ‘real part’ of the dynamics on the reduced state
space is indistinguishable from that generated by a
standard complex Hermitian Hamiltonian; (b) the
eigenvalues of H are real and the states evolve uni-
tarily into infinity without forming closed orbits; and
(c) the eigenvalues of H form a complex-conjugate
pair but the dynamics remains weakly unitary in the
sense that the real part of the dynamics, although
generating closed orbits, no longer lies on the state
space of a standard complex Hermitian system. In-
terestingly, properties (b) and (c) are in some sense
interchanged in a typical PT-symmetric Hamiltonian
where the orbits of a spin-12 system are closed when
eigenvalues are real and open otherwise. These char-
acteristics are related to the three cases investigated
recently by Kisil [23] in a more general context of
Heisenberg algebra, based on the use of: (i) spherical
imaginary unit i2 = −1; (ii) parabolic imaginary unit
i2 = 0; and (iii) hyperbolic imaginary unit i2 = 1.
The use of coquaternionic Hermitian Hamiltonians
thus provides a concise way of visualising these dif-
ferent aspects of generalised quantum theory.

Before we analyse the dynamics, let us begin by
briefly reviewing some properties of coquaternions
that are relevant to the ensuing discussion. Co-
quaternions [24], perhaps more commonly known as
split quaternions, satisfy the algebraic relation

i2 = −1, j2 = k2 = ijk = +1 (1)

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Acta Polytechnica Vol. 51 No. 4/2011

and the skew-cyclic relation

ij = −ji = k, jk = −kj = −i, ki = −ik = j. (2)

The conjugate of a coquaternion q = q0 + iq1 + jq2 +
kq3 is q̄ = q0 − iq1 − jq2 − kq3. It follows that
the squared modulus of a coquaternion is indefinite:
q̄q = q20 + q

2
1 − q22 − q23. Unlike quaternions, a co-

quaternion need not have an inverse q−1 = q̄/(q̄q) if
it is null, i.e. if q̄q = 0. The polar decomposition
of a coquaternion is thus more intricate than that of
a quaternion. If a coquaternion q has the property
that q̄q > 0 and that its imaginary part also has a
positive norm so that q21 − q22 − q23 > 0, then q can be
written in the form

q = |q|eiq θq = |q|(cos θq + iq sin θq ), (3)

where

iq =
iq1 + jq2 + kq3√

q21 − q22 − q23
and

θq = tan
−1
(√

q21 − q22 − q23
q0

)
. (4)

That a coquaternion with a ‘time-like’ imaginary part
admits the representation (3) leads to the strong uni-
tary dynamics generated by a coquaternionic Hermi-
tian Hamiltonian. On the other hand, if q̄q > 0 but
q21 − q22 − q23 < 0, i.e. if the imaginary part of q is
‘space-like’, then

q = |q|eiq θq = |q|(cosh θq + iq sinh θq ), (5)

where

iq =
iq1 + jq2 + kq3√
−q21 + q22 + q23

and

θq = tanh
−1
(√

−q21 + q22 + q23
|q0|

)
. (6)

If q̄q > 0 and q21 − q22 − q23 = 0, then q = q0(1 + iq ),
where iq = q

−1
0 (iq1 + jq2 + kq3) is the null pure-

imaginary coquaternion. Finally, if q̄q < 0, then we
have

q = |q|eiq θq = |q|(sinh θq + iq cosh θq ), (7)

where

iq =
iq1 + jq2 + kq3√
−q21 + q22 + q23

and

θq = tanh
−1
(√

−q21 + q22 + q23
q0

)
. (8)

As indicated above, the fact that the polar decompo-
sition of a coquaternion is represented either in terms
of trigonometric functions or in terms of hyperbolic
functions manifests itself in the intricate mixture of
spherical and hyperbolic geometries associated with
the state space of a spin-12 system, as we shall de-
scribe in what follows.

In the case of a coquaternionic matrix Ĥ, its Her-
mitian conjugate Ĥ† is defined in a manner identical
to a complex matrix, i.e. Ĥ† is the coquaternionic
conjugate of the transpose of Ĥ. Therefore, for a
coquaternionic two-level system, a generic Hermitian
Hamiltonian satisfying Ĥ† = Ĥ can be expressed in
the form

Ĥ = u0� +
5∑

l=1

ulσ̂l, (9)

where {ul}l=0..5 ∈ �, and

σ̂1 =

(
0 1

1 0

)
, σ̂2 =

(
0 −i
i 0

)
,

σ̂3 =

(
1 0

0 −1

)
, (10)

σ̂4 =

(
0 −j
j 0

)
, σ̂5 =

(
0 −k
k 0

)

are the coquaternionic Pauli matrices. The eigenval-
ues of the Hamiltonian (9) are given by

E± = u0 ±
√

u21 + u
2
2 + u

2
3 − u24 − u25. (11)

Thus, they are both real if u21+u
2
2+u

2
3 > u

2
4+u

2
5; oth-

erwise they form a complex conjugate pair. This, of
course, is a characteristic feature of a PT-symmetric
Hamiltonian.

A unitary time evolution in a coquaternionic
quantum theory is generated by a one-parameter
family of unitary operators e−Ât, where  is skew-
Hermitian: † = −Â. As in the case of complex
quantum theory, we would like to let the Hamilto-
nian Ĥ be the generator of the dynamics. For this
purpose, let us write

i =
1
ν

(iu2 + ju4 + ku5), (12)

where ν =
√

u22 − u24 − u25 if u24 + u25 < u22, and
ν =

√
u24 + u

2
5 − u22 if u22 < u24 + u25. Then we set

 = iĤ and the Schrödinger equation in units h̄ = 1
is thus given by (cf. [22])

|ψ̇〉 = −iĤ|ψ〉. (13)
It is worth remarking that when u24+u

2
5 < u

2
2 we have

i2 = −1, whereas when u22 < u24+u25 we have i2 = +1.

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Acta Polytechnica Vol. 51 No. 4/2011

In either case iĤ is a skew-Hermitian operator satis-
fying (iĤ)† = −iĤ; thus e−iĤt formally generates
a unitary time evolution that preserves the norm
〈ψ|ψ〉 = ψ̄1ψ1 + ψ̄2ψ2, where ψ̄ is the coquaternionic
conjugate of ψ so that 〈ψ| represents the Hermitian
conjugate of |ψ〉. The conservation of the norm can
be checked directly by use of the explicit form of the
Schrödinger equation in terms of the components of
the state vector:(

ψ̇1

ψ̇2

)
=

(
−(u0 + u3)iψ1 − u1iψ2 − νψ2
−(u0 − u3)iψ2 − u1iψ1 + νψ1

)
. (14)

Here we have assumed u24 + u
2
5 < u

2
2 so that ν =√

u22 − u24 − u25; if u22 < u24 + u25, we have ν =√
u24 + u

2
5 − u22 and the sign of ν in (14) changes.

To investigate properties of the unitary dynamics
generated by the Hamiltonian (9) we shall derive the
evolution equation satisfied by what one might call a
‘coquaternionic Bloch vector’ �σ, whose components
are given by

σl =
〈ψ|σ̂l|ψ〉
〈ψ|ψ〉 , l = 1, . . . , 5. (15)

By differentiating σl in t for each l and using the
dynamical equation (14), we deduce, after rearrange-
ments of terms, the following set of generalised Bloch
equations:

1
2
σ̇1 = νσ3 −

u3
ν

(u2σ2 + u4σ4 + u5σ5)

1
2
σ̇2 =

1
ν

(u2u3σ1 − u1u2σ3 + u0u5σ4 − u0u4σ5)
1
2
σ̇3 = −νσ1 +

u1
ν

(u2σ2 + u4σ4 + u5σ5) (16)

1
2
σ̇4 =

1
ν

(−u3u4σ1 + u0u5σ2 + u1u4σ3 + u0u2σ5)
1
2
σ̇5 =

1
ν

(−u3u5σ1 − u0u4σ2 + u1u5σ3 − u0u2σ4),

where we have assumed u24 + u
2
5 < u

2
2 so that ν =√

u22 − u24 − u25. This is the region in the parame-
ter space where the coquaternion appearing in the
Hamiltonian has a time-like imaginary part. Note
that these evolution equations preserve the condition:

σ21 + σ
2
2 + σ

2
3 − σ24 − σ25 = 1, (17)

which can be viewed as the defining equation for the
hyperbolic state space of a coquaternionic two-level
system.

Let us now show how the dynamics can be re-
duced to three-dimensions so as to provide a more
intuitive understanding. For this purpose, we define
the three reduced spin variables

σx = σ1, σy =
1
ν

(u2σ2 + u4σ4 + u5σ5),

σz = σ3. (18)

We can think of the space spanned by these reduced
spin variables as representing the ‘real part’ of the
state space (17). Then a short calculation making
use of (16) shows that

1
2
σ̇x = νσz − u3σy

1
2
σ̇y = u3σx − u1σz (19)

1
2
σ̇z = u1σy − νσx,

or, more concisely, �̇σ = 2 �B ×�σ where �B = (u1, ν, u3).
Hence although the state space of a coquaternionic
spin-12 system is a hyperboloid (17), remarkably in
the region u24 + u

2
5 < u

2
2 the reduced spin variables

σx, σy, σz defined by (18) obey the standard Bloch
equations (19). In particular, the reduced motions
are confined to the two sphere S2:

σ2x + σ
2
y + σ

2
z = const., (20)

where the value of the right side of (20) depends on
the initial condition (but is positive and is time in-
dependent). To put the matter differently, in the
parameter region u24 + u

2
5 < u

2
2, the dynamics on the

reduced state space S2 induced by a coquaternionic
Hermitian Hamiltonian is indistinguishable from the
conventional unitary dynamics generated by a com-
plex Hermitian Hamiltonian. This corresponds to
the situation in a PT-symmetric quantum theory
whereby in some regions of the parameter space the
Hamiltonian is complex Hermitian (e.g., a harmonic
oscillator in the Bender-Boettcher Hamiltonian fam-
ily H = p2 + x2(ix)� [7], or the six-parameter 2 × 2
matrix family in [25]). Some examples of dynamical
trajectories are sketched in Figure 1.

The evolution of the other dynamical variables
σ2, σ4, σ5 can be analysed as follows. Recall that the
dynamics (19) preserves the relation (20). Thus, by
subtracting (20) from (17) and rearranging the terms
we deduce that

−(u2σ4 + u4σ2)2 + (u4σ5 − u5σ4)2
−(u5σ2 + u2σ5)2 = const.

(21)

This shows that the evolution of the vector
(σ2, σ4, σ5) is confined to a hyperbolic cylinder. It
turns out that the time evolution of these ‘hidden’
dynamical variables σ2, σ4, σ5 can also be represented
in a form similar to Bloch equations if we trans-
form the variables according to σy1 = u4σ5 − u5σ4,
σy2 = u5σ2 + u2σ5, and σy3 = u2σ4 + u4σ2. In terms
of these auxiliary variables we have

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Acta Polytechnica Vol. 51 No. 4/2011

Fig. 1: (colour online) Dynamical trajectories on the reduced state spaces. In the parameter region u22 > u
2
4 + u

2
5

the reduced state space is just a two-sphere, upon which the dynamical equations (19) generate Rabi oscillations (left
figure). In theparameter region u22 < u

2
4+u

2
5 the reduced state space is a two-dimensional hyperboloid, and thedynamical

equations (26) generate open trajectories on this hyperbolic state space, if the energy eigenvalues are real (right figure).
If the eigenvalues are complex, the open trajectories are rotated to form hyperbolic Rabi oscillations

1
2
σ̇y1 = −

u0
ν

(u5σy2 + u4σy3)

1
2
σ̇y2 = −

u0
ν

(u2σy3 + u5σy1) (22)

1
2
σ̇y3 = −

u0
ν

(u4σy1 − u2σy2).

It should be evident that these dynamics are confined
to a hyperboloid:

−σ2y1 + σ2y2 + σ2y3 = const. (23)

Note, however, that when u0 = 0 we have σ̇y1 =
σ̇y2 = σ̇y3 = 0 from (22), while σ2, σ4, σ5 are in gen-
eral evolving in time. Hence in transforming the vari-
ables into σy1, σy2, σy3, part of the information con-
cerning the dynamics is lost.

We see from (20) and (21) that on the ‘imaginary
part’ of the state space the dynamics is endowed with
hyperbolic characteristics, which nevertheless is not
visible on the reduced state space, or the ‘real part’
of the state space S2 spanned by σx, σy, σz .

When u22 = u
2
4 + u

2
5 so that the imaginary part

of the coquaternion appearing in the Hamiltonian is
null, a calculation shows that the reduced spin vari-
ables obey the following dynamical equations:

1
2
σ̇x = −u3σy

1
2
σ̇y = −u3σx + u1σz (24)

1
2
σ̇z = u1σy,

and preserve σ2x − σ2y + σ2z .
When u22 < u

2
4 + u

2
5 so that the imaginary part of

the coquaternion in the Hamiltonian is space-like, the

structure of the state space, as well as the dynamics,
change, and they exhibit an interesting and nontriv-
ial behaviour. The five-dimensional spin variables in
this case evolve according to

1
2
σ̇1 = −νσ3 −

u3
ν

(u2σ2 + u4σ4 + u5σ5)

1
2
σ̇2 =

1
ν

(u2u3σ1 − u1u2σ3 + u0u5σ4 − u0u4σ5)
1
2
σ̇3 = νσ1 +

u1
ν

(u2σ2 + u4σ4 + u5σ5) (25)

1
2
σ̇4 =

1
ν

(−u3u4σ1 + u0u5σ2 + u1u4σ3 + u0u2σ5)
1
2
σ̇5 =

1
ν

(−u3u5σ1 − u0u4σ2 + u1u5σ3 − u0u2σ4),

where ν =
√

u24 + u
2
5 − u22. These evolution equa-

tions preserve the normalisation (17). However, in
the region u22 < u

2
4 + u

2
5 the reduced spin variables

σx, σy, σz defined by (18) no longer obey the standard
Bloch equations (19); instead, they satisfy

1
2
σ̇x = −νσz − u3σy

1
2
σ̇y = −u3σx + u1σz (26)

1
2
σ̇z = u1σy + νσx,

and preserve the relation

σ2x − σ2y + σ2z = const. (27)

We thus see that at the level of reduced spin vari-
ables in three dimensions, the state space changes
from a two-sphere (20) to a hyperboloid (27), as the
parameters u2, u4, u5 appearing in the Hamiltonian

17



Acta Polytechnica Vol. 51 No. 4/2011

Fig. 2: (colour online) Conic sections and PT phase transition: changes of orbit structures. A projection of the orbits on
the hyperboloid, for parameters just above the transition to complex energy eigenvalues, is shown on the left side. The
orbits form circular sections. On the right side we plot orbits of hyperbolic Rabi oscillations further into complex energy
eigenvalues. The energy eigenvalues determine the angle between the axis of rotation and the axis of the hyperboloid.
When eigenvalues are complex, the axis of rotation is within the hyperboloid, leading to closed orbits on the state space
generated by circular sections. When the imaginary part of the coquaternion appearing in the Hamiltonian is null, we
have parabolic sections of the hyperboloid; whereas when the energy eigenvalues are real, the angle of the two axes is
larger than π/4, and open orbits are generated by hyperbolic sections

change. This transition corresponds to the tran-
sition from a complex Hermitian Hamiltonian into
a PT-symmetric non-Hermitian Hamiltonian. Since
the energy eigenvalues can still be real even when
u22 < u

2
4 + u

2
5, we expect the dynamics to exhibit two

distinct characteristics depending on whether the re-
ality condition u21 + u

2
2 + u

2
3 > u

2
4 + u

2
5 is satisfied.

Indeed, we find that on a hyperbolic state space,
orbits of the unitary dynamics associated with real
energies are the ones that are open and run off to
infinities. Conversely, when the reality condition is
violated, these open orbits are in effect Wick rotated
to generate closed orbits. These features can be iden-
tified by a closer inspection on the structure of the
underlying state space, upon which the dynamical
orbits lie. In particular, (26) shows that the dynam-
ics generates a rotation around the axis (u1, ν, u3);
whereas the state space (27) is a hyperboloid about
the axis (0, 1, 0). We have sketched in Figure 2 dy-
namical orbits resulting from (26), indicating that
there indeed is a transition from open to closed or-
bits as real eigenvalues turn into complex conjugate
pairs.

Intuitively, one might have expected an opposite
transition since in a PT-symmetric model of a spin-12
system the renormalised Bloch vectors on a spheri-
cal state space follow closed orbits when eigenvalues
are real, whereas sinks and sources are created when
eigenvalues become complex [15]. The apparent op-
posite behaviour seen here is presumably to do with
the facts that the underlying state space is hyper-
bolic, not spherical, and that no renormalisation is
performed here. In Figure 2 we have sketched some

dynamical trajectories when energy eigenvalues are
complex. A projection of the dynamical orbits from
the σz axis (for the choice of parameters used in these
plots) shows in which way the topology of the orbits
are affected by the reality of the energy eigenvalues.

The evolutions of the other dynamical variables
σ2, σ4, σ5 are confined to the space characterised by
the relation

(u2σ4 + u4σ2)
2 − (u4σ5 − u5σ4)2

+(u5σ2 + u2σ5)
2 = const.,

(28)

instead of the relation (21) of the previous case. How-
ever, if we define, as before, three auxiliary vari-
ables σy1 = u4σ5 − u5σ4, σy2 = u5σ2 + u2σ5, and
σy3 = u2σ4 + u4σ2, then the dynamical equations
satisfied by these variables are identical to those in
(22), except, of course, that the definition of ν is dif-
ferent.

It is interesting to remark that when the imag-
inary part of the coquaternion appearing in the
Hamiltonian is space-like, the imaginary unit i has
the characteristic of a ‘double number’ or a ‘Study
number’ introduced by Clifford [26], that is, i2 = 1.
Quantum theories generated by such a number field
(instead of the field of complex numbers) and other
hyperbolic generalisations, as well as various issues
that might arise from such generalisations, have
been discussed by various authors (e.g., [27, 28]; see
also [23] and references cited therein). The use of
coquaternionic Hermitian Hamiltonian thus captures
dynamical behaviours of different generalisations of
quantum mechanics in a simple unified scheme.

18



Acta Polytechnica Vol. 51 No. 4/2011

Acknowledgement

We thank the participants of the International
Conference on Analytic and Algebraic Methods in
Physics VII, Prague, 2011, for stimulating discus-
sions. EMG is supported by an Imperial College Ju-
nior Research Fellowship.

References

[1] Bender, C. M., Boettcher, S., Meisinger, P. N.:
PT-symmetric quantum mechanics, J. Math.
Phys. 40, 1999, 2201.

[2] Kaushal, R. S., Korsch, H. J.: Some remarks
on complex Hamiltonian systems, Phys. Lett. A
276, 2000, 47.

[3] Mostafazadeh, A.: Real description of classical
Hamiltonian dynamics generated by a complex
potential, Phys. Lett. A 357, 2006, 177.

[4] Bender, C. M., Holm, D. D., Hook, D. W.:
Complex trajectories of a simple pendulum,
J. Phys. A 40, 2007, F81.

[5] Bender, C. M., Hook, D. W., Kooner, K. S.:
Classical particle in a complex elliptic pendu-
lum, J. Phys. A 43, 2010, 165 201.

[6] Cavaglia, A., Fring, A., Bagchi, B.: PT-symmet-
ry breaking in complex nonlinear wave equations
and their deformations. J. Phys. A 44, 2011,
325201, arXiv:1103.1832

[7] Bender, C. M., Boettcher S.: Real spectra in
non-Hermitian Hamiltonians having PT symme-
try, Phys. Rev. Lett. 80, 1998, 5 243.

[8] Lévai, G., Znojil, M.: Systematic search for PT-
symmetric potentials with real energy spectra.
J. Phys. A 33, 2000, 7 165.

[9] Bender, C. M., Brody, D. C., Jones, H. F.: Com-
plex extension of quantum mechanics, Phys.
Rev. Lett. 89, 2002, 270 401.

[10] Mostafazadeh, A.: Pseudo-Hermiticity ver-
sus PT-symmetry III: Equivalence of pseudo-
Hermiticity and the presence of antilinear sym-
metries, J. Math. Phys. 43, 2002, 3 944.

[11] Oko	lowicz, J., P	loszajczak, M., Rotter, I.: Dy-
namics of quantum systems embedded in a con-
tinuum, Phys. Rep. 374, 2003, 271.

[12] Jones, H. F., Rivers, R. J.: Which Green func-
tions does the path integral for quasi-Hermitian
Hamiltonians represent? Phys. Lett. A 373,
2009, 3 304.

[13] Dorey, P., Dunning, C., Lishman, A., Tateo, R.:
PT symmetry breaking and exceptional points
for a class of inhomogeneous complex potentials,
J. Phys. A 42, 2009, 465 302.

[14] Witten, E.: A new look at the path integral
of quantum mechanics. In Surveys in Differen-
tial Geometry XV. Mrowka, T., Yau, S.-T. (eds.)
Boston : International Press, 2011.

[15] Graefe, E. M., Korsch, H. J., Niederle, A. E.:
Quantum-classical correspondence for a non-
Hermitian Bose-Hubbard dimer, Phys. Rev. A
82, 2010, 013 629.

[16] Günther, U., Kuzhel, S.: PT-symmetry, Cartan
decompositions, Lie triple systems and Krein
space-related Clifford algebras. J. Phys. A 43,
2010, 39 002.

[17] Moiseyev, N.: Non-Hermitian Quantum Me-
chanics, Cambridge : Cambridge University
Press, 2011.

[18] Nesterov, A. I.: Non-Hermitian quantum sys-
tems and time-optimal quantum evolution.
SIGMA, 5, 2009, 069.

[19] Brody, D. C., Graefe, E. M.: On complexified
mechanics and coquaternions, J. Phys. A 44,
2011, 072 001.

[20] Finkelstein, D., Jaueh, J. M., Schiminovieh, S.,
Speiser, D.: Foundations of quaternion quantum
mechanics, J. Math. Phys. 3, 1962, 207.

[21] Adler, S. L.: Quaternionic Quantum Mechanics
and Quantum Fields. Oxford : Oxford Univer-
sity Press, 1995.

[22] Brody, D. C., Graefe, E. M.: Six-dimensional
space-time from quaternionic quantum mechan-
ics. 2011, arXiv:1105.3604.

[23] Kisil, V. V.: Erlangen programme at large 3.1:
Hypercomplex representations of the Heisenberg
group and mechanics. 2010, arXiv:1005.5057v2

[24] Cockle, J.: On systems of algebra involving more
than one imaginary; and on equations of the fifth
degree, Phil. Magazine 35, 1849, 434.

[25] Wang, Q., Chia, S., Zhang, J.: PT symmetry as
a generalisation of Hermiticity. J. Phys. A 43,
2010, 295 301.

[26] Clifford, W. K.: Applications of Grassmann’s ex-
tensive algebra. Am. J. Math. 1, 1878, 350.

[27] Hudson, R. L.: Generalised translation-inva-
riant mechanics. D. Phil. Thesis, Bodleian Li-
brary, Oxford, 1966.

19



Acta Polytechnica Vol. 51 No. 4/2011

[28] Kocik, J.: Duplex numbers, diffusion systems,
and generalized quantum mechanics, Int. J.
Theo. Phys. 38, 1999, 2221.

Dorje C. Brody
Mathematical Sciences
Brunel University
Uxbridge UB8 3PH, UK

Eva-Maria Graefe
Department of Mathematics
Imperial College London
London SW7 2AZ, UK

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