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10

1'\'TIWDUC:TION AND GE'\'ERAL FORMALISM

For physical purposes, we are interested in number fields over the reals; since by
Eqs. (1.13b) and (1.16b) these must be associative division algebras over the
rcals, they can only be the reaL complex. and quatcrnion numbers IR, C, and !H.
It is easily verified that IR, C, and II-I do in fact satisfy all the postulates of Eqs.
(1.13) ( 1.16) and so constitute the complete class of number fields over the
reals. 9

1.3 ALTERNATIVE FORMULATIONS OF QUANTUM

MECHANICS
In this section we will very briefly describe three alternative formulations of
quantum mechanics that appear in the literature. The first is the Dirac ( 1930)
formulation of quantum mechanics in terms of state (ket) vectors that obey a
superposition principle with complex coefficients: This is standard quantum
mechanics in a complex Hilbert space. When the allowed superpositions are
restricted to real coefficients or extended to quatcrnionic coefficients one gets.
respectively, ~uantum mechanics as formulated in a real or in a quatcrnionic
Hilbert space. 0 Although the analysis of the probability interpretation given in
Sec. 1.2 only required that the probability amplitudes (i.e., the superposition
coefficients) belong to one of the four classical division algebras, in fact the
Hilbert space formulation of quantum mechanics further requires the associative law of multiplication, and so admits no extension to quantum mechanics in
an octonionic Hilbert space. Specific features of the Hilbert space formulation
of quantum mechanics which fail in an attempted octonionic extension arc

described in detail in Sec. 2.7. The presentation of quatcrnionic quantum
mechanics given in this book is based in its entirety on the Dirac, or quaternionic Hilbert space, formulation.
To establish an axiomatic foundation for complex quantum mechanics,
Birkhoff and von Neumann (1936) abstracted a set of axioms obeyed by the
true-false propositions of quantum theory. This "propositional calculus" leads
to a ''lattice of propositions" obeying the laws of projective geometry, which
can be analyzed as a mathematical system in its own right. and is the basis for
much of the litcrature 2 on the foundations of quantum mechanics. Concrete
realizations of the lattice of propositions are provided by quantum mechanics
over a real, complex, or quaternionic Hilbert space. and so for practical purposes the propositional lattice is equivalent to the Hilbert space approach.
Historically, the possibility of a quaternionic quantum mechanics was first
pointed out in the paper of Birkhoff and von Neumann ( 1936), and the subject
was further explored in an important article by Finkelstein, Jauch. and Speiser
(1959).
Yet a third formulation of quantum mechanics was given by Jordan (1932,
1933a, b), based on an algebra abstracted from the properties of the projection
operators on pure states. Pa = la)(al, of the Dirac formulation. In the Jordan
formulation of quantum mechanics these projection operators are the funda'' J·or a topo.Jogrcal characterization of the number fields IR, G:. lH sec Pontryagin (1946). Yet another
characteri?ation of JR. Q' and (less trivially) IH rs that they form Clifford algebras; for a clctailcd discussion
see Brackx. Dclanghc. and Sommcn (19H2). As an example of the application of the Clifford algebra
tepn:scntation. iC one wishc:) to classify the finite dimcn:.,ional real matrix representations of the quaternion
algebr·a. one can usc the fact that the real representations or finite Clilford algebras have been classified
and explicitly constructed; sec Okubo (199Ja,b). and references cited therein.
10

Strictly speaking. a llilbert space is by definition a complex vector space. and its quaterniomc gcncralinrtion is called a Hrlbcrt module. but we will not follmv this terminology.

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11

INTRODUCTION

mental entities, and the probability amplitudes introduced in Sec. 1.1 play no
role. The representation theory of the finite dimensional Jordan algebras was
studied by Jordan, von Neumann, and Wigner (1934), who concluded that the
representations are of two basic types. The first type, known as special Jordan
algebras, can be constructed with the product operation in the Jordan algebra
defined as symmetrized multiplication, ~ (ab + ba), in an associative algebra of
real, complex, or quaternion Hermitian matrices. The special Jordan algebras are
equivalent (sec Gursey, 1977, and Niederle, 1980, for an exposition) to the Dirac
formulation of quantum mechanics in, respectively, a real, complex, or quaternionic Hilbert space. The second type consists of one case, the so-called excep11
tional Jordan algebra, consisting of the 27-dimensional nonassociativc algebra
of 3 x 3 octonionic Hermitian matrices. The independence of the exceptional
algebra (i.e., the fact that it cannot be obtained by symmetrized multiplication of
the elements of any associative algebra) has been proved by Albert (1933), while
Gunaydin, Pi ron, and Ruegg ( 1978) have shown that the Birkhoff-von Neumann
axioms arc satisfied over the exceptional algebra, corresponding to a quantum
mechanical system over a two- (and no higher) dimensional projective geometry
that cannot be given a Hilbert space formulation. and constitutes the only known
example of an octonionic quantum mechanics.
In any quantum mechanical system with continuum variables, the algebra of
observables is in fact infinite dimensional, and so the classification theorem of
Jordan, Wigner, and von Neumann is not directly relevant. An investigation of
infinite-dimensional Jordan algebras was initiated by von Neumann (1936), but
it was not until recently that decisive results were obtained by Zel'manov (1983)
(for a pedagogical review, see McCrimmon, 1984), who proved that in the infinite-dimensional case one finds no new simple 12 exceptional Jordan algebras!
Hence an infinite simple Jordan algebra of observables must be of the first or
special type and is realizable as a Hilbert space quantum mechanics. We

conclude that the Jordan formulation of quantum mechanics does not suggest
any physically relevant extension of standard quantum mechanics, other than
the replacement of complex Hilbert space by quaternionic Hilbert space in the
Dirac formulation.
1.4 NOTATION AND INTRODUCTIOIN TO QUATERNIONIC
ARITHMETIC
To conclude the Introduction, we summarize our notation for the quaternion
algebra and introduce some elementary properties of quaternion arithmetic. As
stated in Sec. 1.2, a quaternion ¢ has the form
( 1.17)
with ¢o.u. 3 real and with the quaternion units
noncommutative algebra

eA

obeying the associative but

3

eAeB

=

-6AB

+L

£ABC ec,

A,B= 1,2,3


( 1.18)

C=l
11

The exceptional algebra is 27-dimensional because a 3 x 3 octonionic Hermitian matrix has 3 real
numbers along the principal diagonal, and three independent octonions as upper-right off-diagonal matrix
clements, giving 3 + 3 x 8 = 27 real parameters in all.
12

A simple algebra is not decomposable into independent subalgebras.

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12

INTRODUCTION AND GENERAL FORMALISM

where £ABC is the usual completely antisymmetric three-index tensor with
= I. To verify associativity of the quaternion algebra, we find by direct
calculation from Eq. ( 1.18) that

£123

3

(eAes)eD- eA(eseD) =-bAseD+


L

t:Asc£cD£C£

C.E=i
3

+ 6sD eA

-

L

ssDc £ACE e£

( 1.19)

C.£=1

which vanishes when use is made of the identity satisfied by CABC (but not by
any more general three-index antisymmetric tensor)
3

L t:Asc £CDE = 6 AD[JBE- 6AE r5sD

( 1.20)

c~I

Since, as emphasized in Sec. 1.2, we will never employ complexified quaternions,

no confusion arises from use of the notation
(1.21)
for the three quaternion units, in terms of which the general quaternion of
Eq. (1.17) and the quaternion algebra ofEq. (1.18) take the form
¢

= ¢o + i¢, + Jcf>2 + k¢3

i2

=

ij

)2

=

k2

= -I

= -ji = k

jk = -kj = i

ki = -ik =j

( 1.22a)


The sum i¢ 1 + j¢ 2 + k¢ 1 is called the imaginary part of the quaternion
while ¢ 0 is called the real part, and correspondingly, the quaternion ¢ will
termed real if ¢ = ¢ 0 , with ¢ 1 = ¢ 2 = ¢ 3 = 0, and imaginary
¢ = i¢ 1 + j¢ 2 + k¢ 3, with ¢ 0 = 0. The operation of extracting the real part
¢ is denoted by tr,

¢,
be
if
of

( 1.22b)
From Eq. (1.18) we see that
tr(eAes)

=

-6AB = tr(eseA)

(1.22c)

which implies that for any two quaternions p and ¢we have
tr(p¢) = tr(¢p)

(1.22d)

which immediately generalizes to cyclic invariance of the trace of a product of
any number of quaternionic factors,
( 1.22e)


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