Quaternionic quantum mechanics and quantum fields

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 mechan

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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 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 2.7 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 litcrature2 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 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 inrtion is called a Hrlbcrt module but we will not follmv this terminology

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INTRODUCTION 11

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 quater-nionic Hilbert space The second type consists of one case, the so-called excep-tional Jordan algebra, consisting of the 27-dimensional11 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 infi-nite-dimensional case one finds no new simple12 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

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

£123 = I To verify associativity of the quaternion algebra, we find by direct calculation from Eq ( 1.18) that

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 = )2 = k2 = -I

¢ = i¢1 + j¢2 + k¢ 3, with ¢0 = 0 The operation of extracting the real part of

¢ is denoted by tr,

( 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([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 (1.17), 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

Ll

representatwn ·

(1.23a) with the symplectic components ¢,_11 E <C( I, i) defined by

(1.23b)

Note that the use of -i in ¢r 1 in Eq (1.23b) is a direct consequence of the fact that j in Eq ( 1.23a) 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

which acts as an antiautomorphism within the complex <C( I, i) subalgebra; since

i and j anticommute, we have

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) see Fomenko ( 1988)

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14 l!':TRODUCT!Ol'.' Al':D GE:\TERAL FORMALISM

which by Eq (1.26) satisfies

Again using 1¢1, we can write the quaternion ¢in polar form14

From the algebra of Eqs ( 1.18) or ( 1.22a), 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,

which in general is unequal to {!cp

Introducing an n x n quaternion matrix M,.s,r,s= l, ,n, the matrix elements of which are quaternions, we define the adjoint matrix Mt by

14

The polar form can be used for example to find the nth root; of the quaternion ¢ If pis an nth root of

¢.so that q1 = p" then 1''/' = p"' 1 = '/'f!· and sop commutes with¢: hence if,in04, cJ 0 (so that¢ is not real) p must lie in the C( I e4,) subalgebra In this case there are exactly n nth roots of q1 given by

f=O.i n-1

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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

finite-In certain applications involving fermions (see, e.g., Adler, 1985a) one duces Grassmann quaternions x, defined by

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16 INTRODlJCTJON AND GENERAL FORMALISM

(t.At.B '~ '~ )T _ XTXT - B ·A (1.3le)

Equation (L3le) implies that for two Grassmann quaternions x and ~ the analog of the product conjugation rule of Eq (L28b) is

because T reverses the order of the real Grassmann elements 15

In addition to the quaternion conjugation operation of Eq ( 1.24), we will make frequent use of the quaternion automorphism transformation16

15

This differs from the adjoint rule for Grassmann elements proposed in Adler ( 1986c) where it was incorrectly assumed that transposition T acts trivially on real Grassmann elements Note also that in using Grassmann variables in functional integrals in complex quantum mechanics, the operator adjoint x 1 is

replaced by a new Grassmann variable (often denoted by x, but not the same as x of the text) that is completely independent of X and that anticommutes with z

16

It can be shown that the transformations of Eq (1.32) constitute the only automorphisms of the

quaternion algebra In particular, because quaternion conjugation reverses the order of products [see

Eq ( 1.28)[, it is not un automor-phism, but rather what is called an antiautomorphism This contrasts with the complex case, where complex conjugation is an automorphism of the algebra of complex numbers

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for an appropriate unit vector£' In fact, by an appropriate choice of w we can

make i' be any arbitrarily assigned unit vector To see this, let

with ii the unit vector orthogonal to the plane containing i and the desired f'

Then

¢w =(cos f)- sin Oe· ii)e· i(cos 0 +sin Oe· ii)

which using Eq (1.18) and fl· ii = 0 reduces to

¢oJ = e · f( cos 0 + sin Oe' · ii)2

= e i( cos 2 0 - sin2 8) +e ie ii 2 sine cos e

then we necessarily have

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18 INTRODLCTJOl\ Al\D GENERAL FORMALISM

To conclude, we note that in analogy with complex analyticity, a much more restricted concept of quaternion analyticity has been developed in the mathe-matical literature Although we use complex analytic methods in our quater-nionic calculations involving symplectic components, we have not found any context in our development of quaternionic quantum mechanics in which the use of quaternion analyticity seems natural (but there could be one).17

17 The reader interested in pedagogical t·eviews of the methods of quaternion analysis should consult Giirsey and Tze (1979) Deavors (1973) Sudbery (1979), and Brackx Delanghe and Sommen (1982)

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2

General Framework of Quaternionic Quantum Mechanics

We proceed now to give the basic kinematic and dynamical framework of quaternionic quantum mechanics In much of what follows there is a close analogy with the familiar framework of complex quantum mechanics, but there are a number of characteristic features of the quatcrnionic case that play a significant role in the sequel, and to which we alert the reader First of all, since quaternionic multiplication is noncommutative, we must specify whether the quaternionic Hilbert space is to be formed by right or by left multiplication of vectors by quaternionic scalars; the two different conventions give isomorphic1 versions of the theory Fallowing Finkelstein, Jauch, and Speiser (19 59) and Kaneno (1960), we adopt in Sec 2.1 the convention of right multiplication by scalars, since this is the one appropriate to the usual conventions of matrix operations and to the Dirac hra and ket notation for state vectors Second, although the spectral theory for quaternion self-adjoint operators (see Sec 2.2)

is a straightforward extension of the complex case, significant differences from the complex case arise in the spectral theory for quaternion anti-self-adjoint operators given in Sec 2.3 Because anti-self-adjoint operators make a natural appearance in quantum mechanics in the role of symmetry generators, and in particular as the time translation generator or Hamiltonian, as discussed in Sec 2.4, the characteristic features of their spectral theory have important consequences for the overall structure of quaternionic quantum mechanics Third, we saw in Sec 1.4 that a quaternion can always be represented, through the symplectic component formalism, as a pair of complex numbers Despite this fact, however, quaternionic quantum mechanics is inequivalent to complex quantum mechanics with two internal wave function components, as is discus-sed in detail in Sees 2.5 and 2.6 Finally, the formulation of quaternionic quantum mechanics given later makes essential use of the fact that quaternion

1

Specifically, the theory with right multiplication by scalars and left multiplication by operators can be mapped into the theory with left multiplication by scalars and right multiplication by operators as discussed in Sharma and Coulson (1987) Sec VI The isomorphism requires the reversal of the order of multiplication in the definition of the quaternion algebra or equivalently, mapping the quaternion units

i.j k into their conjugates i.j.k When multiplication by scalars and by operators are both taken to act

from the left the structure of linear matrix operators is restricted as discussed in Horwitz and Biedenharn (1965) Appendix 2; this corresponds to the restricted structure of linear matrix operators which in the conventions used in this book act from the right, as can be inferred from Horwitz and Biedenharn (1984), Sec 11.4

19

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20 INTRODCCTION AND GENERAL FOR~ALISM

multiplication is associative; to emphasize this point, we show in Sec 2.7 that key features of the formalism of Sees 2.1-2.4 fail in an attempted octonionic extension, as a consequence of the fact that octonionic multiplication is nonas-sociative

2.1 STATES, OPERATORS, WAVE FUNCTIONS, AND INNER

PRODUCTS

The states of quaternionic quantum mechanics will be described by vectors of a quaternionic Hilbert space VJJ-1, defined2 by the following axioms

(i) V1H is a linear vector space under right multiplication by quaternionic

scalars Thus for vectors .1; g E V1H and scalars ¢, ¢ 1 ¢ 2 E IH, one has f¢, + g¢2 E v,H and

U+ g)¢=.!¢+ g¢

f(¢,¢2) = (f¢,)¢2

which can be used to define a real-valued norm II f II, with the properties3

(fg) = (gJ)

llff = (ff) > 0

(!~ g + h) = (f g) + u h) (f;gqJ) = (fg)¢

unlcssf = 0

which on combining Eqs.(2.2a) and (2.2d) also gives

(/¢,g)= (/J(fg)

(2.2a) (2.2b) (2.2c) (2.2d)

(2.2e)

arbi-trarily closely approximates any f E V111 ) and complete (every Cauchy sequence {/;1 } E VIH has a limit fE VIH) under the topology defined by

II f II- (These two assumptions, which are traditional properties of quantum mechanical Hilbert spaces, permit the use of standard limiting operations in quaternionic Hilbert space.)

In the quaternionic case an inner product obeying (/.J<) =-(g./) (as will be encountered

in our dtscussion of the quaternionic Klein -Gordon equation in Sec 11.1) cannot be analogously redefined

to satisfy Eq.(2.2a) as a result of"noncommutativity of the quaternionic multiplication For example, if we try ( f.R) 1 = i( f,J<) we get (t:g)1 = i( f.R)- ( t:R)) = 1-(R-f)](- i) cc (J<./)icj i(J<./) ~ (R./)1 and we clearly also violate Eq (2.2e) Thus the conditions or Eq.(2.2) arc more restrictive than they might at first seem If one wishes to discuss indefinite metric quaterntonic Hilbert spaces, in which the condition of Eq.(2.2b) is relaxed, one should also drop the condition of Eq.(2.2a) and consider the two separate cases

fTiJ = ±(g.IJ

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GENERAL FRA'\1EWORK OF QUATER:'iiO:\TIC QUANTlJ'\1 MECHANICS 21

From the scalar product and norm properties of Eq (2.2) we can immediately prove a quaternionic Schwarz inequality (Finkelstein, Jauch, Schiminovich, and Speiser, 1962, Appendix A) We start from

and substitute¢= (g,g),l/l = (gJ) to get

0 ~ (g.g)[(JJ)(g,g) ~ (f,g)(g f)] (2 3b) Since (g,g) ~ 0, Eq (2.3b) implies the desired inequality,

(fg)(gJ) = l(f.g)l2 ~ Uf)(g.g) =IIIII2II g 112 (2.3c)

We note that the equality can hold in Eqs (2.3a-c) only if/¢ ~ gljl = 0, that is, only if the vectors f and g arc proportional (in the sense of quaterni onic scalar multiplication)

It will be convenient to use the Dirac bra-ket notation for the states and inner product in VlJ-1 Hence we define kel states If) that obey

and bra states (II as their adjoints in a matrix sense,

22 1:\TTRODUCTJ0:\1 A:\TD GENERAL FOR:\1ALIS:\1

as extended to nonsquare matrices; that is, the bra state (fl is the row vector obtained by taking the transpose of the quaternion conjugate of Eq (2.6a):

and is readily seen to obey all the postulates of Eqs (2.2a-e)

According to the discussion of Sec 1.1, with the inner product or probability amplitude ( gl f) one associates a probability

(2.7) From Eqs.(2.4a,c) and (2.7), we sec that the association between physical states and Hilbert space vectors is not one to one, for if we replace the unit-normalized vector If) by the inequivalent vector lfw), with lwl = 1, the probabilities P,e;!

arc unchanged for all g Physical states are thus in one-to-one correspondence with unit rays of the quaternionic Hilbert space of the form

When the term operator is used without further qualification, it will be assumed

to be a quaternion linear operator obeying

GENERAL FRAMEWORK OF QUATERNIONIC QCANTUM MECHANICS :i!3

for arbitrary state vectors J,g in suitable domains Following Teichmuller (1935) and Horwitz and Biedenharn (1984), we introduce a set of left-acting operators

01 = - ~ (IO +OJ JOK + KOJ)

02 = -~(JO +OJ- KOI +!OK)

03 = - ~ (KO +OK- !OJ+ JOI)

so termed because they obey

A= 0, 1,2,3 and in terms of which 0 has the decomposition

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24 INTRODUCTION AND GE:\ERAL FORMALISM

Hence the relationship between operators, their adjoints, and their formally real components in quaternionic quantum mechanics is analogous to that familiar for real and imaginary parts in complex quantum mechanics

The set of left-acting operators I, J, K and right-acting scalars i,j, k can also be used (Teichmiiller, 1935, and Horwitz and Biedcnharn, 1984) to define "formally real'" components I fil,l 2.3) for an arbitrary state I f), as follows:

If)= 116) + Il/1) + Jl/2) + Klh) =I /6) + lf1)i + 1/2)/ + lh)k (2.12c)

Clearly, Eqs.(2.11 b-d) are just specializations of Eqs (2.12a-c) to the case when the left-acting algebra I, J, K and the right-acting algebra i,j, k are the same Equations (2.12b) and (2.12c) can again be verified by direct calculation This is most succinctly done (following a method of L P Horwitz) by reverting to the notation EA, eA, A = 0, 1 2, 3 for the elements of the left and right algebras, in terms of which Eqs (2.12a) become

(2.12d)

for example,

I /1 ) = ~ [I f) ( ~ i) + II f) ( ~ i) ( ~ i) + J I f) ( ~ i) ( ~ j) + Kl f) ( ~ i) ( ~ k)] ( 2 12e) Multiplying Eq (2.12d) from the left by Ec gives

which defines a unique D = D( C, B) as a function of C and B Moreover, as

B ranges from 0 to 3, the unsigned part of D assumes exactly once each of

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GE!';ERAL FRAMEWORK OF QUATERNIONIC QUANTUM MECHA"'ICS 25

the values 0, 1,2,3 From Eq (2.13b) we have Es = EcED, the conjugate of which is Es = EDEc, which in turn has the isomorphic image in the right algebra

(2.13c) Substituting Eqs (2.13b,c) into Eq (2.13a), we get

giving Eq (2.12b) Multiplying Eq (2.12d) by EA from the left and summing,

we get

L EAI !A)= 4 L L EAEslf)eAeB

= 4 L(EA) 2 1 f) (eA)2 + 4 L LEAEsl f)eAes

E3Eolf)e3eo + E2E1I f)e2e1 = O (2.13f)

and similarly for the other index values obtained by cyclically permuting 1, 2, and 3 So Eq (2.13c) reduces to the first half of Eq (2.12c), and use of Eq (2.12b) then gives the second half of Eq (2.12c)

We now can show that the inner product (.fAigs) of any two formally real components of the state vectors I f) and I g) is a real number, a result proved as follows From the adjoint of Eq (2.12b), together with Eq (2.12b) rewritten with I fA) replaced by lgs), we have

26 l!'.TRODl!CTION A:-iD Glc:NERAL FOR:VIALISM

An operator of particular importance in what follows is the coordinate operator x, which has a complete set of cigenstates ix') obeying5

(Even when eigenvalues arc real, we will write them to the right of the sponding eigenstate, in accordance with our convention of right multiplication

corre-by quaternionic scalars.) Introducing the completeness relation

inner product (f ig)c defined by

and a real inner product (fig) R defined by

(2.20)

where tr is the operation of taking the quaternion real part defined in

Eq (1.22b) To interpret Eqs (2.19) and (2.20) in more concrete form, let us express the quaternionic inner product (fig) in terms of wave functions, as in

Eq (2.18), and then write the wave functions in terms of their real or symplectic components,

f(x) f=h + i.f1 +J.h + kh =f~ +Jfi1 g(x) g =go+ ig1 +Jg2 + kg3 = g~ + jgfJ (2.21)

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GENERAL FRAMEWORK OF QUATEIRNIONJC QUA~TCM MECHANICS 27

(fig)= j d 3 x[f0go +figl +hg2 +f3g3

+ i(f'ogl - f1go +hg2- hg3)

+J(f'og2 -f2go +fig3 -/3gJ)

+ k(f'og3- f3go + /2g1 -/ig2)]

projec-2.2 OBSERVABLES AND SELF-AD.JICIINT OPERATORS

In analogy with complex quantum mechanics, observables in quaternionic quantum mechanics will be represented by quaternion self-adjoint operators, that is, by operators H that arc both quaternion linear and self-adjoint or Hermitian, so that

If ih) is an eigenstate of H with eigenvalue h,

then we have

Hih) = ih)h

h = (hiHih) (hih)