com 10 1'\'TIWDUC:TION AND GE'\'ERAL FORMALISM For physical purposes, we are interested in number fields over the reals; since by Eqs.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.16) and so constitute the complete class of number fields over the reals.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.2 only required that the probability amplitudes (i., the superposition coefficients) belong to one of the four classical division algebras, in fact the Hilbert space formulation of quantum mechanics further requires the associa- tive 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.
The presentation of quatcrnionic quantum mechanics given in this book is based in its entirety on the Dirac, or quaterni- onic 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 purpo- ses 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. 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. 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. a llilbert space is by definition a complex vector space. and its quaterniomc gcncral- inrtion is called a Hrlbcrt module. but we will not follmv this terminology.com INTRODUCTION 11 mental entities, and the probability amplitudes introduced in Sec.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 quater- nionic Hilbert space. The second type consists of one case, the so-called excep- 11 tional Jordan algebra, consisting of the 27-dimensional nonassociativc algebra of 3 x 3 octonionic Hermitian matrices.
The independence of the exceptional algebra (i., 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 infi- nite-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.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.2, a quaternion ¢ has the form ( 1. 3 real and with the quaternion units eA obeying the associative but noncommutative algebra 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.com 12 INTRODUCTION AND GENERAL FORMALISM where £ABC is the usual completely antisymmetric three-index tensor with £123= I.
To verify associativity of the quaternion algebra, we find by direct calculation from Eq.18) that 3 (eAes)eD- eA(eseD) =-bAseD+ L t:Asc£cD£C£ C.E=i 3 + 6sD eA - L ssDc £ACE e£ ( 1.£=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.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.17) and the quaternion algebra ofEq.18) take the form ¢ = ¢o + i¢, + Jcf>2 + k¢3 i2 = )2 = k2 = -I ij = -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 be termed real if ¢ = ¢ 0 , with ¢ 1 = ¢ 2 = ¢ 3 = 0, and imaginary if ¢ = i¢ 1 + j¢ 2 + k¢ 3, with ¢ 0 = 0. The operation of extracting the real part of ¢ is denoted by tr, ( 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.com INTRODUCTION 13 Equations (1.22e) have a number of useful applications. For example, letting [¢, p] denote, as usual, the commutator [¢,p] == ¢p- p¢ ( 1.22f) we have tr([¢, p]) = 0 tr([¢,p]rJ) = tr(¢pT)- p¢1J) = tr(prJ¢- P¢T7) = tr([r), ¢]p) =~ lr(pT)¢- TJP¢) = tr([p, rJ]¢) ( 1.22g) Instead of writing a quaternion in terms of its four real components, as in Eq.17), it will often be convenient to write it in terms of two components lying in a complex subspace of the quaternion algebra. Taking this subspace to be the one spanned by I and i, denoted by <C( I, i), we get the so-called symplectic .23a) with the symplectic components ¢,_ 11 E <C( I, i) defined by (1.23b) Note that the use of -i in ¢r1 in Eq.23b) is a direct consequence of the fact that j in Eq.23a) is ordered to the left; that is, j( -i) = U= k.
When dealing with symplectic components, we will use the notation* to denote the complex conjugation operation I* = I ' z'* = -z.24a) which acts as an antiautomorphism within the complex <C( I, i) subalgebra; since i and j anticommute, we have ,;.24b) Following the discussion of Sec.2, we introduce the quaternion conjuga- tion operation denoted by ·· and defined by T= -·i, J = -J, !( = -k (1.25b) and the conjugate of (/J is ¢, 11 For a discussion of the relationship between the symplectic representation of quaternions and the symplectic group Sp(n).com 14 l!':TRODUCT!Ol'.' Al':D GE:\TERAL FORMALISM ((/;) = ¢ ( 1.25c) The quaternion norm 1¢1 = 1¢1 is then defined by l¢l- N(¢) = (¢¢)'/2 = ((fJ¢)'/2 = (l¢~l2 + l¢r;I2)I/2 = (¢~ + ¢t + ¢~ + ¢~)'/2 ( 1.26) and vanishes only when (p is zero. Using 1¢1, we can explicitly construct the unique inverse ¢ ~ 1 of any nonzero quaternion ¢ as (1.27a) which by Eq.27b) Again using 1¢1, we can write the quaternion ¢in polar form 14 ( 1.27e) From the algebra of Eqs.22a), we find that the conjugate of the product of two quaternion units (say i and;) is Tj = k = -k = ( -j)( -i) = ji ( 1.28a) and similarly for cyclic permutations of i,j, k, as a consequence of which the conjugate of a product of two quaternions p and¢ is the product of the conju- gate quaternions in reverse order, p¢ = (/Jp (1.28b) which in general is unequal to {!cp. Introducing an n x n quaternion matrix M,. ,n, the matrix elements of which are quaternions, we define the adjoint matrix Mt by t - M 15 = M 11 (1.29a) 14 The polar form can be used.
to find the nth root; of the quaternion ¢. If pis an nth root of 1 ¢. then 1''/' = p"' = '/'f!· and sop commutes with¢: hence if,in0 4, cJ 0 (so that¢ is not real). p must lie in the C( I.
In this case there are exactly n nth roots of q1.com INTRODIUCTION 15 Then using Eq. = L M,p Npr = ~= Npr Msp = L N1.29b) and so the adjoint of the product of two quaternion matrices M and N obeys the usual rule ( 1.29c) We will later use the customary convention of defining the transpose MT of the matrix M by ( 1.29d) so that Eqs.29c) become Mt = MT, (MN)t = (MN( = JVTMT = NtMt ( 1.29e) In general, however, for quaternionic matrices MN one has (1.29f) whereas these statements hold as equalities for complex matrices M, N. Defining a quaternionic column vector v" s = l,. ,nand its adjoint v! = i'_; = v_,, we also have (1.29g) s s s giving (Mv)t = vtMt as expected.
We define the trace operation Tr acting on a quaternion matrix M by (Finkelstein, Jauch, and Speiser, 1959) Tr M = tr L,.