Quaternion quantum mechanics: Second quantization and gauge fields

ANNALS

OF PHYSICS

157. 432-488

(1984)

Quaternion Quantum Mechanics: Second Quantization and Gauge Fields L. P. HORWITZ* Tel Aviv

Universiiv,

Ramat

Aviv.

Israel

AND

L.C. BIEDENHARN~ Center

for

Parlicle

Theory,

University

Received

January

of Texas,

Austin.

Texas

78712

9, 1984

Recent work on algebraic chromodynamics has indicated the importance of a systematic study of quaternion structures in quantum mechanics. A quaternionic Hilbert module, a closed linear vector space with many of the properties of a Hilbert space is studied. The propositional system formed by the subspaces of such a space satisfy the axioms of quantum theory. There is a hierarchy of scalar products and linear operators, defined in correspondence with the types of closed subspaces (with real, complex or quaternion linearity). Real, complex, and quaternion linear projection operators are constructed, and their application to the definition of quantum states is discussed. A quaternion linear momentum operator is defined as the generator of translations, and a complete description of the Euclidean symmetries is obtained. Tensor products of quaternion modules are constructed which preserve complex linearity. Annihilation-creation operators are constructed, corresponding to the second quan tization of the quaternion quantum theory with Bose-Einstein or Fermi-Dirac statistics. The tensor product spaces provide representations for algebras with dimensionality increasing with particle number. The algebraic structure of the gauge fields associated with these algebras is precisely that of the semi-classical fields introduced by Adler. c 1984 Academic Press. Inc.

I. INTRODUCTION

The existence of the phenomenon of asymptotic freedom in non-Abelian gauge field theories of the strong interaction suggeststhe possibility of the unification of such theories with the successful gauge theories of the weak and electromagnetic interactions. The minimum global gauge group, SU(3), x SU(2) X U(l), for such a unified theory, might be the residual symmetry of a larger group G which is “Supported in part by the Fund for Basic Research administered by the Israeli Academy of Sciences and Humanities Basic Research Foundation. ‘On sabbatical leave from the Physics Department, Duke University, Durham. North Carolina 27706. Work supported in part by the National Science Foundation.

432 0003.4916184 Copyright All rights

$7.50

fel 1984 by Academic Press, Inc. of reproduction m any form reserved.

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spontaneously broken. The search for such a larger group has involved the study of exceptional groups as well as groups such as SU(5) and SO(10). The exceptional groups were thought to be attractive because they admit only a restricted class of possibilities (there are only 5 exceptional groups), they are efficient in dealing with multiplets, and one might hope to argue about the nonobservability of the basic constituents of the hadrons because of the nonassociative behavior of the algebras which enter into their construction. Giinaydin and Giirsey [ 1 ], for example, have considered the possibility of constructing an octonionic Hilbert space. They identified a natural decomposition of the space. in terms of pieces of the wave function associated with different elements of the algebra, with a representation of leptons and quarks. Although Giinaydin, Piron, and Ruegg [2] showed that an octonion space is associated with a lattice of propositions with properties consistent with the axioms of quantum theory [3] (on a plane). the resulting structure cannot, however, be embedded in a Hilbert space. It is not possible to construct a scalar product which is nontrivially linear over octonions in an octonion vector space. It has been shown, however, that the associative closure of an octonion vector space is isomorphic to a vector space over the Clifford algebra C, [4 ] (a vector space which is also a module, and which may be completed to a Hilbert module [S]. and references therein), and that this type of vector space is also consistent with the axioms of quantum theory [6]. Following the ideas of Giinaydin and Giirsey in selecting a complex subalgebra. a decomposition of such a space was constructed [ 7 ] which may be identified with a representation of quarks and leptons in the same way as in the formal nonassociative case [ 1 ]. In a minimal ideal of the C’, algebra, one finds the same gauge and automorphism groups as well. Starting with a different point of view, Adler [S-l 1 1 has proposed an algebraic generalization of classical Yang-Mills fields with the goal of achieving a semiclassical understanding of the dynamics of non-Abelian gauge fields, and, in particular, confining potentials. In its application to prequark models of the type suggested by Harari [ 121 and Shupe [ 13 1, larger algebraic structures, encompassing higher Clifford algebras, such as C,, can be constructed starting with basic (prequark) objects carrying an intrinsic U(2) gauge symmetry. This work has indicated the importance of a systematic study of quaternionic structures in quantum mechanics. associated with intrinsically non-Abelian gauge fields. and their extension to higher algebraic structures through the tensor product. In this paper, we shall study quaternion quantum theory (the first extensive work on this subject was done by [ 141, see also [ 151) using techniques previously developed for the larger Clifford algebra C, [7 J, and develop a framework for its application to the domain of many-body states [ 161.’ There is a hierarchy of scalar products, real, complex (a proper subalgebra of the quaternions), and quaternion linear, which one can define on a space of this type. The invariance groups of these ‘J. Rembieliriski has defined a tensor product which. in common with the construction we shall use. is linear over a complex subalgebra of the quaternions. but is more restrictive. The scalar products used by Rembieliliski do not have the same linearity properties as ours (see footnote 4).

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scalar products and their intersections with the automorphism group of the quaternions will be discussed (these coincide with those found by Rembielinski [ 171). We obtain a definition of the momentum operator from the action of the translation group and study the corresponding uncertainty relations. These constructions are new. The seemingly technical and mathematical considerations are actually vital for developing a physical interpretation of the system. The complex linear scalar product plays a special role in the construction of tensor products. We show, by means of the covariant derivative, that the gauge fields associated with the many-body quantum theory belong to a sequence of algebras with dimensionality increasing with the number of particles in the Fock space, as also found in Adler’s constructions [ 181. The tensor product which we use, moreover, provides a Fock space which can be generated by means of annihilation-creation operators, which we shall construct, with Bose-Einstein or Fermi-Dirac statistics. This construction is also new. In Section II, we study the basic quantum theory associated with a vector space which is also a module over the skew-field of quaternions. The closure of this space is called a Hilbert IH-module. The hierarchy of scalar products is defined, and the invariance groups are discussed. An extension of the Riesz theorem to the quaternion case is demonstrated in order to discuss the adjoint of a linear operator. The general forms for operators linear over the reals, a complex subalgebra of the quaternions and the full quaternion algebra is worked out in terms of a useful decomposition (called symplectic by Finkelstein et al. [ 141) of the space into a representation in terms of “complex-valued” components. The structure of projection operators into subspaces closed over the three types of multipliers is given explicitly, and it is shown how to construct complete sets of states for insertion into scalar products. The Gleason theorem is discussed in order to interpret the quantum states. Horwitz and Soffer [ 19, 202] have shown that for a B * algebra containing a subalgebra isomorphic to quaternions, a GNS [21, 22, 231 type construction exists for its representation as a Hilbert IH-module. The state functional coincides with the quaternion linear Gleason form. (The proof of the Hahn-Banach theorem has also been carried out for the extension of positive linear functionals [ 241.) The structure of the pure states is found by applying Mackey’s procedure [25], and it is shown that these states have a residual structure due to the intrinsic degeneracy of a Hilbert IH-module. The pure states are not the simplest building blocks of the space; the special pure states associated with the simplest building blocks, subspaces generated by real or complex valued vectors, are defined and called primitive states. In Section III, properties of the translation group are used to obtain the explicit construction of the momentum operator for a Hilbert IH-module, and the basic commutation relation between momentum and coordinate is obtained. We then study the momentum-coordinate uncertainty relations. The minimum dispersion state is shown to be primitive. Fourier transforms are discussed in Appendix 2. In Section IV, the tensor product is constructed in terms of complex linear (the ‘Truini

et al. construct

the Hilbert

IH-module

using

the imprimitivity

theorem.

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435

maximum linearity available for tensor products) functionals, casting the structure of the tensor product of Hilbert IH-modules into the symplectic form [ 161’. Tensor product spaces are shown to have algebraic dimensionality increasing with the number of particles. Annihilation-creation operators are constructed for Bose-Einstein and Fermi-Dirac statistics. The construction of a covariant derivative is carried out in the second subsection of Section IV. It is shown that the introduction of a non-Abelian gauge compensation field is required, with dimensionality increasing with the number of particles. It is pointed out that these gauge fields have the algebraic structure of the semi-classical fields introduced by Adler [8-l 1 1.

II. 11.1. The Hilbert

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Iti-Module

We shall study a vector space V.. which quaternion algebra l-i, i.e., ifJ g E I’,H. then,

f41 +&22 E VI,.

is also a right module over the real

(f-t 8) 4 =.h + gq. f(4192) = (f41)42. f(s, + 42) =A, +A?.

(2.1)

Here. q, q,. q2 E H, the algebra of quaternions generated over the reals by the elementse, . ez. where et = e: = -1,

(e,,e2} =e,e, +e2e, =O.

(2.2)

This algebra has an involution f$

=-e

I’

*- --e2, e2

(e,e2)* = e*eT = -e,e2.

(2.3~

The elements of the algebra It4 are real linear combinations of e, = 1, e, , ez, and e, = e,e2. We suppose,moreover, that there exists a binary mapping (J g) of V,,, x V,,, into it~i with the following properties (.L s)* = (kf)~ CLg + h) = Cf. s> + (A h), (f. gq) = CLg) 49

(2.4i) (2.4ii) (2.4iii)

and

cm = Ilfll’ > 03

(2.4iv)

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and is zero if and only if f = 0. A right module V,, with the properties (2.4i)-(2.4iv), which is closed under the topology defined by the norm (1f 11will be called a Hilbert IH module, which we denote by qH. There exists a special set of mappings qH + qH representedby the left action of a set of linear operators, defined everywhere on &H, which obey an algebra *isomorphic3 to the quaternion algebra IH. There may, in general, be many such algebras; in the following we pick one which we call 9, generated over the reals by 1, E,, E, (and E, = E,, E2), which have the algebraic properties of Eqs. (2.2), (2.3). The elements of the algebra form a special class of bounded linear operators satisfying

Q(f4> = CQf) 43 Ql(Qzf) =
(2.5 1

and

Q
(2.6)

where, if Q=A,11 +/1,E,+~,E,+~,E3,11QI12=~~+n:+nf+n:. With this, we may define

I.e.. f,=d[f+E,feT

+E,fe$

+E,fe,*L

f,=f[fe,*-Elf-E,fe:+E,fe,*], fi=+[fe?+E,fe,*-E,f-E3fe,*15 f3 = d[fef - Elfe? + E2feF - E3f

I.

(2.7)

Direct calculation verifies that EiA. =f. ei

(2.8)

for all i, j = 0, 1, 2, 3. It then follows that f= 3 We shall

show in Subsection

II.3

5 Eif,=

ihei.

i=O

i=O

that this star operation

(2.9) is equivalent

to the Hermitian

conjugate.

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We shall call vectors& satisfying Eq. (2.8)f orrnal[~~ real, and, from Eq. (2.9), identify them as cornponerzts of the vector f. We emphasize that this decomposition into components depends on the choice of the left acting algebra. 11.2. Hierarch)? of Scalar Products We define a closed subspace A& of q;, as a set of elements Ag,..., for which fq, +gq2EM,, for allf,g in M,,,, and which is closed in the norm (2.4). Quaternion orthogonality between two closed subspaces M,. and Mz. _ is defined according to the property (2.10)

(f,.fz)=O for all& E M,.

. fz E M, _,,,: this statement is equivalent to tr((f,

.f,l9

(2.11)

1= 0

for all q E It. We have recognized that It-1 is isomorphic to a matrix algebra (4 x 4 real) in defining the trace. normalized so that tr 1 = 1. Since the e, anticommute, it is easy to see that tr ei = 0 (i = 1. 2. 3). Two subspaces M,. _. Al?, .. . closed under real linear combinations, are said to be reai orthogonal if (2.12)

tr(f, ..A) = 0 for all f, E Al,,

, f, E hf>,-.‘. The form (2.12) is linear over the reals. and we define (2.13)

C.6 gJc,=Wg) as the real scalar product. Note that (f,f):,= in the relative topology given by (2.4). Let us investigate the condition

Ilf’l’,

so that we may close M,,: . M?.:.

tr((f, .f2) ;) = 0

(2.14)

for all i E ‘i ( 1. e,). the complex subalgebra of lh generated over the reals by 1. e, . and f, E M,.:. .fi E M,., . subspaces linear and closed over the same complex subalgebra. It follows from (2.14) that (2.15) so that M,., that

and M, a are real orthogonal.

trI(f,tfL)e,I

Moreover,

=O

it is necessary

and sufficient

(2.16)

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to assure that what remains of (2.14) is satisfied. It then follows that

that the condition

(f,g>,=tr(f,g)-e,tr[(f,g)e,l vanishes is equivalent also true that

to (2.14) for f, g (=f, ,f,)

(2.17)

ranging over M,,,

(.Agz), = CLg>,z

and M,,c.

It is

(2.18)

for z E C(1, e,), and we shall therefore call (2.17) the complex scalar product.4 Note that, as for the real scalar product (f,f, = Ilf II*, so that we may close the complex subspaces M,.,, M,,, in the relative topology given by (2.4). Equation (2.17) may be compared with

(AS>= ,Ti + eitr[(f,g)eTl for the full scalar product. Using (2.17) and (2.19) (.A s> = (.L d, - U

(2.19)

one obtains

g> e2L e2.

(2.20)

The decomposition (2.20) has its analog in the abstract vector space. We may rewrite the formally real decomposition of the vector f given in Eq. (2.9) as (2.21) where yo =fo

are the components

+fi

‘i/]

=.f2

off in the so-called symplectic fz=

so that multiplication antilinear on v,.

e2 3

+f+,

representation

114, 151. Note that (2.23)

voz + v,z*e,

by elements of the complex subalgebra

II.3 Riesz Theorem and Symmetries

(2.22)

is linear on ‘i/O and

of the Scalar Products.

We have obtained, in subsection 11.2, a hierarchy of scalar products with real, complex, and quaternion values, each of which may be used to describe orthogonality of manifolds linear over the corresponding subalgebras, and closed under the 4Rembieliriski has also discussed detinition of linearity which is different is the complex part of a, and (J, g): is (f. geJC = -[(J g), + e,(J g), 1, which

the complex geometry from ours. Since in his his scalar product, one is. in general, not zero

of a Hilbert IH-module, but he uses a definition (A gu); = (s, g),(a),. where (a), would find (f, ge2)S = 0. In our definition. (see Eq. (2.17)).

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topology associated with the common norm (2.4). To prove the Riesz theorem and display the symmetries of these scalar products, and for our later use, we shall need two lemmas. LEMMA

11.1.

The Schwarz inequality is r!aIid for a Hilbert !ki-module.

The proof follows the lines of the well-known procedure for complex Hilbert spaces. We shall reproduce it here, since an intermediate result will be useful. Let J g E qi,, and A E IR. Then

i.e..

This inequality implies

ltCLg)l G l!fll We may now replace g by s(s~f)/l(gJ)~. quaternion norm /q 1= \/9*q. to obtain

IIgll.

(2.24)

w here the vertical bars signify the usual

IUg)l G llfll . II gll-

(2.25)

The condition for equality is that g =fq for some quaternion q. The second lemma that we shall need concerns the decomposition of a given vector into a part lying in a (quaternion) closed subspaceM,,,c cl, and a part (quaternion) orthogonal to M,,,. LEMMA

11.2. Let f be a given vector in CM. There is a unique decomposition f

=

g,,lH

+

h,

(2.26)

bchereg,,,+.and (h. g) = Ofor all g E M ,,,. The uniqueness of this decomposition follows from the fact that an alternative possibility,

f = GH + h’ implies that g;,,,, -g,+,,ti = h -h’ has zero norm, i.e., IIg:,,b -g,,jill’ = (h - h’. g.lt,p,- g,,,,J = 0. To construct the decomposition (2.26), we seek a greatest lower bound for iIf’- gll. g E M,,. Since M,, is closed, this minimum value can be realized by someg,,,,i,E M,,. and we call h = f - g ,,,,H. Then, for all g E M _,

llh fgll > llhll,

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from which it follows

AND

BIEDENHARN

that 2 trtk

(2.27)

g> + II g/l2 > 0.

Hence, tr(h, g) = 0 for all g E M,,, and since gq E M,, for any quaternion q, it follows that (h, g) = 0, and the decomposition (2.26) is proven. Remarks. If this construction were carried through for a subspace M,, closed under the subalgebra z E C(1, e,), the substitution in Eq. (2.27) would be restricted to g-+gz, and we would find the condition (2.14); one obtains in the way the unique decomposition f=&f,

(2.28)

+ k

where (h, g), = 0 for all g E M,, of a vector into complex orthogonal parts. If this construction were carried through for a subspace M,, closed under the reals, Eq. (2.27) results in the condition (2.12); one obtains in this way the unique decomposition f=

(2.29)

g,, + h,

where tr(h, g) = 0 for all g E M,, of a vector into real orthogonal

parts.

The decompositions (2.26). (2.28), and (2.29) define a class ofprojection operators on &, which are linear, respectively, over the quaternions, the complex subalgebra G(1, e,), and the reals. We shall discuss these systematically in Subsection 11.5. THEOREM

functional

II.1 (Riesz). of g E TH, i.e., ug,q,

Let

L(g)

be

a

bounded

+g,q,)=Lfg,)q,

quaternion-valued

linear

+L(g*)q2,

IL(g)1 GK II gll. Then, there exists an f E qH such that L(g)

= tf,g>*

(2.30)

Although the proof of this result is parallel to that for the complex Hilbert space (e.g., Kato [26], p. 252]), we shall give it in detail since some of the intermediate results will be useful. Let N E ;iv;H be the (closed) subspace of qti for which L(g) = 0, i.e.,

N= {glL(g)=OI.

(2.3 1)

If L(g) is not identically zero over all qH, then N’ (the subspace orthogonal to N) is not empty. SupposeSE N’, so that L(f)

= 1.

(2.32)

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Then, for any g E qH, g’=g-jZ(g)EN. SincefE

N-, (x g’) = 0 and hence

It follows from the definition (2.31) that N- consists of &}. Equation (2.30) follows by taking f=flli fil’. Let us define a quaternion linear operator as a mapping A:f-f’ such that A : fq -f/q. Formally, iff’ = Af.

The Riesz theorem enables us to define the adjoint of a quaternion linear operator A. For A a bounded quaternion linear operator (the definitions we shall give may be extended in the usual way for the unbounded case), the Schwarz inequality implies that

L(g) = (LAgI is a bounded linear functional.

and, by the Riesz theorem, there exists an f’ such that

L( 8) = CS’. 8). The adjoint of A is defined by f’=Atf: We now demonstrate a particularly important relation for the adjoint operators of the left quaternion algebra P defined in Eqs. (2.5). (2.6). THEOREM

11.2. Qt=

Q*.

(2.38)

Since Q is a bounded, quaternion

linear operator,

L( s> = Cf. Qg> is a bounded linear functional,

of the

(2.39)

and by the Riesz theorem, there is an f such that

L(g) = &-

(J g).

This f is determined by the condition that L(,f) = 1, and hence

f= Q*flllfll’

ilQIl’.

(2.41)

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From Eq. (2.40), we then obtain

-WI = (.L Qg) = “;f;;!” Settingf=f;:ei we obtain

(Q*A 8).

(2.42)

(no sum), and using (from Eqs. (2.8) and (2.4)) 11 Q*hll’ = /IQ\I* Ilfill’

(fiei, Qg) = (Q*hei, g>, for each i, and hence (s, Qg) =

+

(2.43)

We now discussthe explicit structure of the hierarchy of scalar products defined in Subsection 11.2. In particular, we first show that the quaternion scalar products of the formally real vectors 1, satisfying Eifo =.foei

(2.44)

defined in Eq. (2.7), are real valued. To prove this assertion, we note that

(fo,Qgo)=(Q’f,,go,=(Q*fo~go,. Using property (2.46), this implies (f,, go) 9 = uiq*~ go) or (fo~go)4=4uo~go) for all q E IH. i.e., (fO, g,) is real-valued. It then follows that %.A 8) = $ (L gi)

(2.45)

i=O

and tr((S, g> e,> = (hr g3>- (f3, g2) + (f, ygoI - (fo3 gl>. Equation (2.17) can therefore be expressed as (.A ET>,= (wo, x0) + 0117v/l>9

(2.46)

where wo, vu1and x0, x1 are, respectively, the symplectic components off and g, and the scalar products on the right-hand side are complex-valued. Although Eq. (2.46) has the form of a scalar product in direct sum complex

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MECHANICS

Hilbert spaces, it is evident from the definition disjoint. For example,

(2.21) that these spaces are not

ge2= xoe2-xl so that, by (2.46),

Use2L- = - (wo3xI) + ho. w,). This result differs from Rembielinski [ 151 (see footnote 4). With the help of Theorems I.1 and 11.2, we may explore a special set of symmetries of the scalar products. These are characterized by transformations generated by the right algebra lti and the left algebra 1. In particular, the real scalar product (2.45) is invariant under an SO(4) group which transforms the formally real components of the vectors in TV among themselves. The action of this group is generated by the multipliers in 1+-Iand the distinguished operator algebra I through

where !I Qliz = jq 1’ = 1. It then follows from Theorem II.2 that

W3q. Qm) = tr(fq. Q*Qgq) = W*(.C 8) 9) = W 8).

(2.48)

That this transformation generates SO(4) can be seenby comparing (Q = JJ LiEi, 4 = Xkillkek).

(2.49) = 2 ii fiph eiejek iih

with the action on a quaternion q' = x rjej of q and q" = c iiei: q"q'q

= 2 lie,2

vjeixpk

ek

(2.50) =

L

Ait>pk

eiejek

s

A subset of these transformations generates an algebraic automorphism of the vector space in the following sense. The decomposition (2.9) provides a decomposition of every vector uniquely (given 1’, IH) into four parts which have, formally, the algebraic value of the quaternion units. The (quaternion-valued) scalar product

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between two of these parts results in a quaternion number with value in the (starred) product of the two units, for example, (fie,,fie,)=e,*(fi,fi>ez

(2.5 I)

= (fiJi>e?e2

since (f, ,f,) is real. Now, let us put an element of 9 into correspondence with an element of IH according to

Q,=~LiEiES,

qO= \‘ liei E IH.

(2.52)

For this special choice, the parts of the transformed vector f’ =

Q&o’

(2.53)

obey binary relations of the type (2.51), but with units transformed to

e; = qOeiqO-1 .

(2.54)

In this sense,the transformations (2.53) induce an algebraic automorphism on the vector spaceqH. This automorphism is also induced on the algebras 9 and IH, since

Qh=QQ,'f'qoq and hence (2.55)

where

Q’ = QoQQ,‘?

(2.56)

9’ = 40440’.

We denote by S’ the group of algebraic automorphisms generated in this way. As is well known (from the action of (2.56)), ,d is isomorphic to SO(3). The invariance groups of the scalar products do not coincide with this group of automorphisms. We shall call their intersection the invariance automorphism subgroup.5 This subgroup corresponds to an intrinsic invariance of a theory associated with a particular type of scalar product (it is stronger than gauge invariance in the one-particle space since the phase is invariant as well). As we have seen,the invariance group of the real scalar product contains the full automorphism group s’, and hence & is the invariance automorphism group for this case. 5 This has been called for the group associated

the color group by Giinaydin with the “underlying algebra.”

and Giirsey

[ 11, but Adler

[ 111 reserves

this term

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MECHANICS

For the complex scalar product, Eq. (2.46) indicates the existence of a U(2) group of invariance which transforms v,,, wr and x0, xl* among themselves as the fundamental representation. The action of this group is generated by fwhere Q E A, zEC(l.e,) form (2.17) to find

and j/Ql/=lz/=l.

(erZ. Qsi), = WX

(2.57)

Q,fk To verify this assertion,

Qs) - e, tr((Qf

= tr(f. Q*Qg) - e, tr((f,

we use the

Qs) e, 1 Q*Qg) e, 1

= (A 81,. We remark that the symplectic

decomposition

(2.2 1) in the form

(2.58) where wf = f, -fie,

. satisfies E,.f = E, vu ~ ElE, WT.

EJ- = -VT + E,w<,.

(2.59)

fe,== E,y/, +EzEly/?.

(2.60)

and

The transformations (2.59), (2.60). and their product action of the matrices

E, Ez correspond

to the left

on the column vector

(2.62) The norm conserving algebra formed over the reals of the elements (2.61) generate U(2). The intersection of this U(2) with the automorphism group .-G corresponds to the subset of transformations (2.57) of the form Z-‘fz, where Z is in correspondence with z, i.e.. the invariance automorphism subgroup for the complex scalar product is U( 1). If we take t to be of the form ee16, Z-‘fz

= tyo + y, eCZr16 e,.

(2.63)

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The action (2.63) of this U(1) would admit one-half integral charges in the corresponding gauge theory. For the quaternion scalar product, the maximum invariance group generated by elements of _2 and IH is given by (2.64)

f-Qffor Q E 8, 11 Qll = 1, which follows from the relation

KZ Qg) = CLQ*Qg>; only the reals of IH commute with all (f, g). The symmetry generated by the transformation (2.64) is U(1; q) z SU(2) (containing the elements (2.61) without the fourth entry), and the intersection with the automorphism group is the trivial identity.6 Note that for the case of the complex scalar product, formally, U(2) n SO(3) z SO(3); however, the part of the automorphism group that leaves the scalar product invariant is just U(1). Similarly, for the quaternion scalar product, U( 1; q) CI SO(3) E SO(3), but the invariance automorphism group is trivial in this case. One should be careful not to use the abstract intersection of the associated groups without taking into account their action on the carrier space (they must be considered as transformation groups, which may act differently from the left and the right). 11.4. Operators

Linear operators on qM may be classified as real, complex, or quaternion linear according to the (right) subalgebra with which they commute. To discuss the structure of these operators, it is convenient to start with the representation (2.9). The most general operator which is linear over the reals, i.e., satisfying

for I E R, has the action

Af = s (aijfi) ei.

(2.66)

ii

where the elements CFijfj are formally real, and the gij are quaternion linear, To prove this result, we first remark that, for i fixed,

A(fieJ = Z: gijej, j ‘The invariance groups those given by Rembieliliski

and their [ 171.

intersections

with the automorphism

(2.67)

group

listed

here coincide

with

QUATERNION

QUANTUM

MECHANICS

441

where the g, are formally real, and are determined uniquely by the decomposition (2.9). We then define the operators fl’, by g, = CTjifi.

(2.68)

We specify the action of c?‘, on all of & by defining it to be quaternion linear: it then acts on the formally real components only, with the action (2.68). If A is bounded, the flii are also, since

Equation (2.66) then follows immediately. Since the g, are formally real, one obtains (using quaternion linearity last step) E,/‘(,,J;. = E, gij = giie, = (CT,,&) ek = Uji EJ,,

of ‘T/i in the (2.70)

and hence the operators /r, are in the commutant of 1. We may then express the result (2.66) in terms of the symplectic representation (2.21). It follows, from the relations (2.22), that hi=~wo+

WC?).

f,=-~(V/,-U/o*.)e,=--(“,~-WI;). (2.71)

Equation (2.66) can therefore be written

as

where the A,, ((1, p = 0, 1) are linear combinations of the cTij with coefficients in the subalgebra of -Y generated over the reals by the unit operator and E, . Equation (2.72) provides a convenient form for the action of a general operator on Vi, which is real linear. We are now in a position to consider complex linear operators, i.e., operators satisfying

A@) = @if) 22

(2.73)

where z E C( 1, e,). At first glance, it appears that (2.73) should be automatically valid from the associativity of the quaternion field. However, this is incorrect, since the right action of z on f can be represented as a (nontrivial) matrix action (see, e.g., Eq. (2.9)) from the left. and hence (2.73) actually involves a change in the order of two matrix actions. Note, for example, that the U(2) transformation (2.57) is complex but not quaternion linear.

448

HORWITZ

AND

BIEDENHARN

According to (2.23), w,, -+ wOz, v, + v, z* when f +fz, and hence all of the primed terms in (2.72) must vanish if (2.73) is valid. The most general complex linear operator therefore acts on elements ofTH in the form .4f=4,~~+4,1//~

+ @,,(ut

(2.74)

+A,,vJez.

Complex linearity therefore implies strong restrictions on the structure of an operator on ,&. We can summarize the action (2.74) using a column vector representation with the second element conjugated in the sense of Eq. (2.58), and an operator-valued matrix

(2.75 )

where A TO,A & differ from A iO, A ii in the sign of E, . We shall call this matrix form the symplectic representation [ 141 and an operator A, in this representation, will be denoted by (A), and vectors by (f). The adjoint of a complex linear operator in the symplectic representation can be obtained as follows. Consider the complex scalar product

(g,Af),=Olo+X1e2,AooV/o+AolWI*+(A,,Wo*+A,,~,)e,), =Olo~4ovo+4l ~/::)+(A,o~‘o*+A,,y/,,X1)~

(2.76)

where the last follows from Eq. (2.46). These complex-valued linear functionals can be expressed in terms of the adjoint operators (the A,, are quaternion linear, so the adjoint here coincides with that defined in Eq. (2.37)) as (glAf),=(At,,Xo+Af,*X::,Wo)+(W,,A:,XI+A,t~X~), and hence the adjoint of a complex linear operator (the complex adjoint) (A +&f),

is given by

= (g, 47,~

where

In the symplectic

representation, (2.78)

corresponding to an application of the usual formal conjugate of the matrix appearing in Eq. (2.75).

treatment

of the Hermitian

QUATERNION

449

QUANTUMMECHANICS

The existence of the complex adjoint. in the first of Eqs. (2.78) is, of course, also assuredby the complex analog of Theorem 11.4,the usual Riesz theorem. In caseA is also quaternion linear, Eq. (2.77) coincides with Eq. (2.37) (see Eqs. (2.92) and (2.104)). A Hermitian (symmetric) complex linear operator, satisfying H=H’ has the properties H:,,, = *ful -

H;, = H&,

Hi,, = H,,

(2.80)

and for a skew-Hermitian operator .‘$=-s+

(2.81)

= -As;,, s:, =-s,,. Cl,,= -so,,3 ST),

(2.82)

we have

LEMMA 11.3. A complex linear skewHermitian operator is related to a complex litlear Hertnitian operator bj? right tnultiplication bj? e, . equivalent to multiplication b?,E, in the s)wplectic representation.

It follows from Eq. (2.74) that right multiplication by e, is equivalent to left multiplication of every component of a vector in the symplectic representation by E, from 2 (E, v,, = v’oe,. E, r,~;~= VFe,); this is a complex linear, but not quaternion linear operation. Since

LW,,)+=E,&,,.

(E&J+=

E,S;;

(2.83)

the product (2.84) is Hermitian. As an operator on the original Hilbert l-l-module representation. Hf = Sfe,

(2.85)

THEOREM 11.3. (Stone’s theorem for complex linear unitary operators). A complex linear unitary* operator representing a one parameter group can be representedas an esponential of ati imaginar~~multiple of a complex linear Hertnitiatl operator.

595:157/2-10

450

HORWITZ

AND

BIEDENHARN

From (2.74) the product of two complex linear operators BAf = (BooA,,

+ Bo,AFo)

+ I(B,oA,*,

+B,,A,o)

v/o + (BooA,,

is given by

+ Bo,A :I) v/f

wX + (B,oA&

+B,d,,)

v,l

e2.

P-86)

Hence, the symplectic representation of BA is the product of the representations for B and A, i.e., @A) = (B)(A).

(2.87)

ll = (u+u) = (u+)(u),

(2.88)

For unitary operators,

Eq. (2.78) shows that (Ut) is the formal adjoint of the operator-valued matrix (U), so that the symplectic representation of a unitary operator is a unitary operator-valued matrix, and hence, by the usual Stone’s theorem for complex Hilbert spaces,for every (U(L)) which is the representation of a continuous one-parameter group, there is an (H) such that

(2.89)

(U(l)) = eElcHjA.

This E,(H) is of the form of the (S) of (2.84), and we seethat the action of U(k) on an analytic vector in the Hilbert IH-module is U(A)f

=f

+ AHfee, + ... = e’H’elS,

(2.90)

where we have introduced an abbreviated notation for the series. We now turn to a study of quaternion linear operators. Quaternion linear operators must have the form of Eq. (2.74), since they are also complex linear, but they must satisfy the additional restriction of linearity with respect to e2. Since fe, = -vl

+ voe2,

it follows from (2.74) that A(fe2)=--AoovI

+Aolvo*

+ (-A,,v,*

(Af)e2=--Al,vl

-AloG

+ (Aold

+4,vo)e2

and +Aoovo)e2.

(2.91)

Equating these two forms, we find the condition that A be quaternion linear,

AoO=Al,=Ao,

A,,=-A,,-A,.

(2.92)

451

QUATERNIONQUANTUMMECHANICS

Quaternion linear operators therefore act on elementsof;itq, in the form

This result may be expressedin terms of operators of the left algebra: THEOREM

11.4. Euery quaternion

linear operator has the unique decomposition (2.94)

A=A,+A,E?=(;ro+n,E,+~~Ez+n,E,,

vtshereA,,, A, are in the commutant of Cc(11,E ,), the subalgebra of 1 generated over the feals by E, and the unit operator. and CT,, ff ,, ff2, (r, are in the commutant of -1. EverJp complex linear operator acts as (2.95)

A,.f = Af + Bfe,

and ecery real linear operator as A,f=Af+Bfe,

(2.96)

+Cfez+Dfe,,

whereA, B, C. D are quaternion linear. The representation (2.94) follows directly from Eq. (2.93) and the quaternion linearity of A,, A,. To verify Eq. (2.96) we collect terms of Eq. (2.72) in the form A/J-= (A,, +A,,EJ + (A&, +A;&,)

v/o + (A,, +A,,EA ‘i/,7 + (A;, +AI,Ed

w? vv1

(2.97)

and compare to Eq. (2.96),

A,f=(A+BE,)y/,+(A--E,)E,y/: +(C+DE,)Ez~~+(DE,-cc)~l.

(2.98)

By taking sums and differences of the combinations occurring in (2.98), the relation between A, B, C, D and the operators defined in (2.72) can be established. If the last two terms identically vanish, one obtains the result (2.95) for complex linear operators. In the symplectic representation (2.7.5), the action of a quaternion linear operator is (2.99) Multiplication by an element of the left quaternion algebra 1 is a special case of a quaternion linear operator. Every Q E 2 can be written as Q=Q,+Qe,&,

(2.100)

452

HORWITZ

AND

BIEDENHARN

where Q,, Q, are elements of the algebra C(ll, E,). In the matrix form of (2.99), we have (2.101) which contains the first three elements in the set (2.6 1). The determinant of the matrix is unity if (1QII’ = 1; the set of all such transformations corresponds to SU(2). We further remark that the transformation (2.57) (complex linear) f- m is represented by (from (2.23)) (2.102) where Zyl, = wOz, Zty: = v?z. For 11Z112= IlQil’ = 1, Eq. (2.102) corresponds action of U(2) (containing all of the elements of (2.61)). Since

(2.103)

(f,(Ao+AlE,)g)=((A~-A:*E,)f,g),

the adjoint of a quaternion

linear operator Atf=

In the symplectic

representation,

to the

is represented

by

(A,t-A:*E,)f.

(2.104)

we have (2.105)

i.e., the formal Hermitian conjugate of the operator-valued matrix appearing in Eq. (2.99). A Hermitian quaternion linear operator H = Ht has the properties Hi,, = Ho,,

H,t,* = -H

10

(2.106)

and a skew-Hermitian quaternion linear operator S = --St has the properties s;, = -so07

s;o* = +s

10’

(2.107)

There is no simple general procedure for associating an arbitrary Hermitian quaternion linear operator with a skew-Hermitian quaternion linear operator. An arbitrary Hermitian quaternion linear operator can, however, be associated with a skew-Hermitian complex linear operator (a similar argument can be made for a quaternion linear skew-Hermitian and a complex linear Hermitian operator):

QUATERNIONQUANTUMMECHANICS THEOREM

11.5.

453

Let H be a Hermitian quaternion linear operator. Then 5” = Hfe,

(2.108)

defines a complex linear skew-Hermitian operator. The one-parameter complex linear unitary) group generated by S has the symplectic representation (2.89), with series e.upansio,lon (an analytic z’ecror of .<[, of the form (2.90). where H is quaternion linear. The proof is parallel to that operators are also complex clearly results in a operator The polar decomposition

of Lemma II.3 and Theorem 11.3, since quaternion linear linear. Multiplication of a quaternion linear (H) by E, which is complex. but not quaternion. linear. [27 1

where [J, HI = 0. J = -J’, J’ = -11, H = Ht= (-S’)“‘, provides a relation between Hermitian and skew-Hermitian quaternion linear operators, but there is an infinite number of possibilities unless the positive square root is chosen for H. This construction may be appropriate for a positive definite Hamiltonian. for example, but for many physical applications (e.g.. momentum and angular momentum). one must work with Hermitian operators with nondefinite spectrum. Furthermore, J will not, in general, be in the center of the algebra of operators in the commutant of 2. For the special class of Hermitian quaternion linear operators containing only one element of Y, e.g., only one of the last three terms in the second of (2.94) is nonzero. there is a simple relation to a skew-Hermitian quaternion linear operator. For example, for H = H, + H, E, , H, = Hi, H, = -H: in the commutant of 2. HE, = E, H = S = -St. This class includes some examples which are important for physical applications: we shall discussthese in later sections. The invariance groups of the scalar products and their associated invariance automorphism groups were discussedat the end of Subsection 11.3. The properties of operators invariant under the action of these groups are given in Appendix 1. 11.5. Projection Operators arld CompletenessSums We have discussedthe structure of real, complex and quaternion linear operators. Examples of such operators are the projection operators into manifolds that are linear over the corresponding subalgebras of IH, and closed in the norm common to all of the scalar products we have considered. We define projection operators into the quaternion complex and real closed linear manifolds M,,,, M,, and M, by means of Eqs. (2.26). (2.28), and (2.29), (2.109)

454 where a =Z,C,

HORWITZ

AND

BIEDENHARN

or R. Since in the unique decompositions

we have discussed,

fq, = g,noq,+ hq,

(2.1 IO)

for q, E IH, G, or R, respectively, the two terms on the right are orthogonal in the scalar product appropriate to the subalgebra a, it is clear that PM, is linear with respect to the corresponding subalgebra. In addition to the evident property that pif, = p,+f,, self-adjointness follows from (f, 5&J-do

= (?+,fi + h, 5f’,wJJ~ = (Gaf, 7f’Mof, + h,), = (&J-l

J2>, *

We shall say that such operators are, respectively, quaternion, complex, or real selfadjoint (PM,+is also compex and real self-adjoin& and P,,,c is also real self-adjoint). In the following, we shall give explicit expressions for the structure of these projection operators for one-dimensional manifolds. For the projection operator in a quaternion subspace spanned over the full quaternion algebra by a single vector g, i.e., M, = { gq}, q E IH we must find a q =x, + x,ei + x2e2 + x3e3 such that

llf-gqll* = llfll’ + II gll* Id* - 2 tr[df,g) 41

(2.111)

is a minimum. With this minimizing value, f - gq is the vector h of Eq. (2.26) orthogonal to M,H. Now, for i = 0, 1, 2,3, (2.112) Setting all these variations equal to zero (the second derivatives are diagonal and positive), we find, using Eq. (2.19), the minimizing solution

Hence, the quaternion linear projection operator into the one-dimensional manifold spanned by g over IH has the action

If the same minimization procedure is carried out with a = z E 6, only the indices 0, 1 appear in (2.112), and we obtain in this case, using Eq. (2.17), (2.115)

QUATERNION

QUANTUM

MECHANICS

455

Finally, if the minimization is carried out only over the reals, only the i = 0 term appears in Eq. (2.112), and we obtain, with the definition (2.13),

Corresponding to the three different types of projection operators, there are three types of completeness sums over intermediate states. Suppose that ,XId is separable, so that there is a denumerable sequence

f, Ji....,

(2.117)

which is dense in zti (in the sense of the common norm). Let us construct a complete orthonormal set according to the following procedure. The second element of our denumerable set has a unique decomposition into a part lying in the quaternion linear manifold {f,q}, and a part orthogonal to it,

f2 = P.++(f,,fi +

‘p23

(2.118)

where (pZ is quaternion orthogonal to f, = ql. Similarly, the next vector in the sequence./,, can be decomposed to a part in the manifold spanned over quaternions by q,. (pZ and a part quaternion orthogonal to this manifold (in particular, to q1 and 02).

f3 = P,H,ol.C1*,f3+ 93. Continuing in this way orthogonal elements and A similar procedure manifolds generated by (2.117) with projections orthogonal set, and for orthogonal set. LEMMA

(2.119)

indefinitely, we obtain a sequence (Do, ‘p*..., which contains whose span is the whole space (each can be normalized). can be carried out for the sequence of complex linear the decomposition of successive elements of the sequence on complex linear subspaces, resulting in a complex the sequence of real linear subspaces, resulting in a real

11.4. Completeness (2.120)

are, respectively, luli17 hiI, IVil sets. and 1 is the unit operator on A$.

quaternion,

The lemma follows from the well-known orthogonal linear subspaces.

property

where

complex, and real orthonormal P,w,, = P,,, + P,, where M, N are

HORWITZ AND BIEDENHARN

456

THEOREM 11.6. Let {vi), {xi), {vi} be, respectively, quaternion, complex, and real orthonormal sets. Then, the quaternion scalar product (f,g) can be expanded into intermediate state sums of the form

(f,

g)

=

X

(f,

uli>(rPi)

g>

=

y

t-6

XiN.Xi)

g>c

=

1

(S,

rli)(Vi,

g)iR.

(2.121)

The proof follows by inserting the resolutions of unity (2.120) into the scalar product (f,g) and using Eqs. (2.114)-(2.11(j). From the linearity properties of the scalar products, we obtain 11.1.

COROLLARY

(f,

g>c

=

T

(f,

Xi)cOri

1 g>c

5

(Jg)IR=~

(f,

rli)iR(rli~g)R*

(2.122)

i

Note that in the second and third lines of Eqs. (2.121), the first factor is a full quaternion scalar product (for which the orthogonality of the xi and the rli is not fully effective, e.g., &i 3g> = CjtJti 3Xj)kjY S>,>. 11.6. States and the Gleason Theorem The axioms of quantum theory [3] are concerned with the lattice of propositions which form a complete, weakly modular, orthocomplementary set, and are usually represented as the set of projection operators on a Hilbert space. These axioms are satisfied by projection operators on Hilbert Clifford-modules as well 161. The Gleason theorem [28] (see also [3]) for the representation of positive linear functionals (in terms of the trace with a density operator) on the set of projection operators in vector spaces closed over the real, complex, or quaternion algebras, corresponding to probability measureson the associated manifolds, may be applied to the Hilbert IH-module to provide a quantum mechanical interpretation. It has been shown [ 191 conversely, that the Hilbert IH-module arises in a natural way in the algebraic approach to quantum theory when the algebra generated by the observables contains a subalgebra *-isomorphic to quaternions. We state this result in THEOREM II. [ 191. Let ‘11be a B* algebra over the reals which contains a subalgebra ‘?I, *-isomorphic to the real quaternions IH, and w a two-sided quaternion linear state on ‘3. Then, there exists a representation lT, : ‘u + S(R;“), where AYE is a Hilbert IH-module,and ‘23(2Y$) is the set of bounded IH-linear operators on ZE.

QUATERNION

QUANTUM

451

MECHANICS

The two sided quaternion linear state of the theorem corresponds to the functional utilized by Gleason (in .F;), The Hahn-Banach theorem, assuring the existence of a bounded positive linear functional defined on a subset of ?I was proved in [ 19 ], and Soffer has more recently proven 1241 that a bounded positive linear functional defined on a subset has a bounded positive linear extension, a necessary property of the Gleason functional. Since the density matrix is a positive compact operator, the positive functionals on quaternion linear projections can be written as

m-(P,,, ,) = x Y,(S’* P,,,,S’),

(2.123)

where 1); > 0, Ciyi= 1, and ilf’]lZ = 1, and hence O
orthogonality. (cpj.

In Dirac’s

(2.124)

‘pJ-Uji*

A~;)

=

(2.125)

Uji.

notation, A can therefore be represented by (2.126)

Evidently.

the density matrix associated P =x

To demonstrate subset; then,

with the state (2.123) is, in this notation. If’)

l’i

(2.127)

VI.

this result, let us assume, for simplicity,

that if’}

is an orthogonal

(2.128)

458

HORWITZ

AND

BIEDENHARN

where we have taken the pi to coincide with as many of thef’ as occur in p (the rest are orthogonal to these). As we have seen in Eqs. (2.115) and (2.116), one-dimensional complex and real linear projection operators have the form (11gll = 1)

hf&) where we have generalized Dirac’s

(2.129)

= I g)L
notation with the definition

.klf)

(2.130)

= w-h

for u = C, R. In fact, for a complex complete set {Xi},

linear operator

A,,

acting on the complex

ACXi= x xjaji

(2.131)

and

afi= hj,AXi)c.

(2.132)

Hence, A, has the representation

AC=C

ij

IXj>aJicCXil

(2.133)

for aji E 6. Similarly, real linear operators have the representation, on the real complete set {Vi}
AR=)J

Illtjja:

d

Vi13

(2.134)

where aJy= (Vj,A,Vi)R*

(2.135)

Complex and real linear density matrices therefore have the representation (2.136) for a = C, R and complex and real linear states are then given by (the trace should be carried out with the appropriate orthogonal set; Tr A then corresponds to the diagonal sum of the aTi in a suitable representation) (2.137)

Q~ATERNI~N

Qu.4~wM

MECHANICS

459

Since quaternion linear projections are also complex and real linear, real and complex states are also positive functionals on quaternion linear projection operators; similarly. real linear states are positive functionals on complex linear projections. However, a quaternion state is not, for example, a positive functional on real or complex linear projections (these operators are not Hermitian in the quaternion scalar product). We now follow the procedure of Mackey [25] to study the structure ofpure states. Let us supposethat the state functional w”, achieves the value unity on P,,,p, defined as the projection operator on the one-dimensional (over IH, @, or RR)manifold generated by g (11g 11= 1). Then,

xI Yill(~ - PM&,) S’ll’=0. Hence

and, for I/fill2 = 1, j+=gcq, where ai has values in a and Iail = 1. We have therefore demonstrated THEOREM 11.8. An a-state (a = iH, C, or R) which achieves the value unity on a one-dimensional manifold generated by a vector g (11g/J= 1) over a has the unique form

%“(L,)

= (g, ~,w‘J), .

(2.138)

We shall call such a state a pure state; a mixed state is a positive weighted sum of pure states [29]. Every vector in T;, corresponds, up to a phase (in a), to a pure state, and we often call these “states” as well. As for the usual complex Hilbert space, it is the ray, elementsof the linear manifold generated over (I which are normalized to unity. which correspond, more precisely, to the state.

460

HORWITZ

AND

BIEDENHARN

The basic quaternionic structure of qH is evident in the structure evaluated on one-dimensional manifolds. Although

of the pure states

for a = C, R, the scalar product contains the coherent sum of more than one amplitude of complex or real type. In particular, for g = v/,, + w, e,, h =x0 + pie, (we rewrite Eqs. (2.45) and (2.46) for convenience here) (kg),=

olo, vo)+

‘i/l,~1)

and

(kg),=

;7 (hi,gi)R* i=o

(2.140)

Although the states 05 are formally pure, according to the standard definition 1291 and any general vector g generates a smallest linear manifold (an atom) of type a 131, it may actually span the full direct sum space, and is therefore not the smallest building block. In particular, if-g contains only one formally complex component, e.g., yo, the first of Eqs. (2.140) becomes (2.141)

(h, g>, = OlOl wok and if it contains (2.140) becomes

only one formally

real component,

e.g., go, the second of Eqs. (2.142)

(k gh? = (ho 3go>. We shall call minimal states of this type primitive. THEOREM

primitive

11.9. There exist complete orthogonal real states.

sets of primitive

complex and

The proof follows from the existence of a denumerable dense set (2.117). Since, for any g, there exists a subsequence f, such that

II g-“tx

= {iO II ‘!v-fnil12 = IlWo-Loll2 + llw, -Xn*l124

the (fki} are a dense set in the ith formally real component space, and (xkO}, {&+i}, where xkO and xk, are the complex components of fk, are dense in the formally complex component spaces of G%$. It is therefore adequate to take (2.143) or {XkO

9 XkO e2 1

(2.144)

QUANTUMMECHANICS

461

as dense sets. Since the components of (2.143) are complex orthogonal, the real orthogonalization of orthogonalization of (xkD} => (xj,} by the procedure sufficient to construct complete orthogonal sets (xjO. xjO e, } of primitive states.

real orthogonal, and of (2.144), (fkO} =P (11~~) and the complex described in Subsection II.5 is ( qjo, }f,,e, , ‘li,,ez, }T;,,e,} and

QUATERNION

THEOREM 11.10. A real pure state evaluated OH a complex or quaternion projection is equivalent to a mixed real state on real projections, and a complex pure state evaluated on a quaternion projection is equivalent to a mixed complex state on a complex projection.

This result is another manifestation of the essentially direct sum structure of P’+. The proof follows from application of Eqs. (2.17) and (2.20): Starting with a real state evaluated on a complex projection, we obtain (using (2.17)) ‘“;~(p,,eu,

) = (&f(f.

g),.h

= (s.fl(Lg)rr,= ‘fJK-@,,jt,,)

LLseI)Fe,l)r, + ~~‘(p,,~o,

1.

(2.145)

a state maximally mixed over g, ge,. Summing over all real manifolds generated by a real orthogonal set covers the set of complex manifolds generated by the same vector twice (since, e.g., as discussed in connection with Theorem 11.9, both f;, and f,,e, occur in the real orthogonal sequence.but both generate the same complex manifold). For a quaternion projection, using (2.20). u!.(P ~,,,,u,)=(g,J’l(f,g),-((f,gez)celI), = (kf(.f = ~W,f@J

g),b + (sezJT.Lgez)ch + dw!,,cm).

(2.146)

and (2.145 ) can be used to decomposethese further. Summing over all real manifolds M,..(f) covers the set of quaternion manifolds generated by the same vectors four times. Finally, for a complex pure state evaluated on a quaternion projection, we have (again using (2.20))

Now. applying (2.17). we obtain

(sJl.Lgez)Cez),. =tr(g~f(LgeZ)cez) -e, tr(g.f(Lge,),.e2e,) = -tr(ge,JXf, geJ,) - e, tr((sez3f(.f a),) e,) = 4 gezJ‘U gel ),)T = -(gezJM.Lged,3

462

HORWITZ

AND

BIEDENHARN

and hence 4(cq,cn>

= 4(bfccn)

+ QWM,,).

(2.147)

As before, summing over all complex subspaces generated by a complex orthogonal set covers the set of all quaternion subspaces generated by the same vectors twice. In the constructions discussed in Theorem 11.10, we understand the state as corresponding physically to the preparation of the system. The projection operator corresponds to the question posed by the laboratory equipment; if the system, for example, is prepared in a complex pure state, but the measurement can only inquire on a quaternion closed manifold (e.g., SU(2) gauge invariance), the predictions of the theory have the pattern of a mixed state on complex closed manifolds, where the states entering the mixture are in a definite relation (g and ge*). The effective definition of a “pure state,” as in the usual complex quantum theory, depends on the type of observables one actually has available to measure. A well-known example of this type of phenomenon occurs in the Stern-Gerlach experiment [30]. The split beam of silver atoms, actually a pure state when it reaches the detector plane, is effectively mixed because the expectation value of any ordinarily available observable in this state can be expressed as a sum of expectation values using the constituent pieces of the wave function separated by the inhomogeneous magnetic field (such observables do not connect the disjoint pieces). We close this section by discussing expectation values and dispersion. Operators self-adjoint with respect to the real, complex or quaternion scalar product have spectral resolution as in the case of the usual complex Hilbert space [4, 14, 161. For example, we shall assume that the self-adjoint position operator X is quaternion linear (to be able to specify the position of a particle independently of the quaternion phase of the wave function), and it therefore has a spectral representation of the form

X=~xdE,(x)=jxlx)(xl

dx,

(2.148)

where E,(x) is the (absolutely continuous) spectral family associated with X, and the formal Dirac bras and kets act in a quaternion linear way, w?)

= w->

4,

(XIX’)

= 6(x -x’),

qE IH,

(2.149) (2.150)

and (2.15 1) Complex spectrum,

linear self-adjoint operators, with, for example, absolutely continuous have a representation in terms of complex linear spectral families,

A,=jadE,c(a)=J&)

,(aldu,

(2.152)

QUATERNION

463

QUANTUM MECHANICS

where we have written the last expression in the modified Dirac form used in Eq. (2.129).

The (real linear) state functionals are defined on self-adjoint operators, since they are real linear combinations of projections, and we define the expectation value of the corresponding observables in pure states as (11gl/ = l), (2.153)

w~(A,)=(A)g,=(g,A.g),,

for a = R, C, IH (A, is self-adjoint in the corresponding scalar product). Clearly, AC and A,, are measureable (in the sensethat wi is real-valued) in 06, since they are real self-adjoint as well, and A ,cis measureablein 0:. THEOREM

II. 11. If an observable A, is dispersionfree in a state g, then g is an

eigenstate. The theorem follows in the usual way, 0 = @((A, - KXJ2)

= (g, (A, - W,XJ%,

= II(A, ~ (A,):) g/12.

III. TRANSLATIONS

AND LINEAR

MOMENTUM:

UNCERTAINTY

RELATIONS

In this section. we study the properties of the momentum operator in a Hilbert Ii-lmodule. We represent the operators which induce the action of the translation group by T(&); the momentum operator is defined as the corresponding Hermitian generator. The translation operator must preserve the norm

llfll’ = llm~)fll’

(3.1 )

for all J: Considering f + g in place off, we find that tr(f, g) = tr(T(&)f,

T(6.X)g).

(3.2

If T(6.x) is linear over a = R, C. IH, then (3.2) implies the corresponding unitarity,

(A g), = (T(hx) .L T(d-u)‘FYI,.

(3.3)

We assumethat on a dense set, for 6x small, 7-(6x) = 1 + 6x3 + 0(6x2),

(3.4)

where s is at least real anti-Hermitian, and it is to be associated with the observable Hermitian momentum operator P. We shall furthermore assume(to be able to specify the momentum of a free particle independently of the quaternionic phase of the wave

464

HORWITZ

AND

BIEDENHARN

function) that P is quaternion linear.’ As we have pointed out in Subsection 11.4, there is a simple association of a complex linear skew-Hermitian operator with a quaternion linear Hermitian operator (Theorem 11.5). We therefore define, on all f in the domain of P,

sf= where h is Planck’s THEOREM

constant

fPfeI,

(3.5)

divided by 27~.

III. 1.

Wf)=-e,+xlf)

(3.6)

or, in operator form, P=--h The theorem hence T(6x) a = C. Using X, defined in

-dxlx)e,g(x). J

(3.7)

follows from the interpretation of T(6x). From the definition (3.5), s and is complex linear; it is therefore unitary in the sense of Eq. (3.3) for the spectral representation of the quaternion linear self-adjoint operator Eq. (2.148), we may therefore only demand that’

(XIT(~X)fL = (x + WfL,

(3.8)

i.e., that proper translations are induced only in the complex scalar product (we recall here that according to Eq. (2.20), the complex scalar product is the complex part of the full quaternion scalar product). It follows from Eqs. (3.8) and (3.4) that (3.9) and hence (3.10)

‘The algebraic elements corresponding of the original B* algebra which, together Hilbert module as a representation [ 191 linear operators). ‘As we shall demonstrate later, there group. It is somewhat more instructive to

to the spectral families of both X and P may therefore be part with an algebra *-isomorphic to the quaternions, generates the (all elements of the B* algebra are represented as quaternion is also a quafernion start with the weaker

linear representation for assumptions made here.

the translation

QUATERNION

QUANTUM

MECHANICS

465

Since P is quaternion linear, and Eq. (3.10) is valid for all f in the domain of P, Eq. (3.6) follows immediately from this result and Eq. (2.20). Finally, the representation (3.7) is obtained by left multiplication of Eq. (3.6) by 1.x) and integration. It is easily seenthat translation of the form (3.8) is not induced by T(&) in the full quaternion wave function. Instead of (3.9), one would obtain

the right-hand side is quaternion linear, but the left side is not. The noncomplex part of the wave function transforms, in fact, according to the relation

The second term is

and hence (.ul(T(Gx)f)

e2)c = (x - 6s/fe,),.,

(3.11)

in contradistinction to Eq. (3.8). With the representation (3.7) for P. and (2.148) for X, we may immediately obtain the canonical commutation relation,

PXS=-hjdxlx)e&xl

=-h

(‘dx’~x’)x’(x’lf)

jd.rlx)e,(xlS)-h~d.xlx)e,x~(xlf),

XPf=-h.!.ds’Ix’).~‘(x’I

=-A

l~dxlx)e,~(x~f)

[dx~x)xe,&(xlf).

Hence, (XP-PX)f=

[X,P]f=h!‘rlxlx)e,(xlf.)

466

HORWITZ

AND

BIEDENHARN

or [K

PI

=

We,),

(3.12)

where (3.13) satisfies E(e,)*

= -1

(3.14)

but it is a nontrivial operator. As we have emphasized in Subsection 11.1, the decomposition of the elements of the Hilbert IH-module into formally real components, according to Eq. (2.7), depends on the choice of the left acting quaternion algebra 9. The result (3.12) suggests a particularly useful choice, in which the coordinate representation plays a decisive to quaternions, the role. Let us define, for the basis of 9, clearly *-isomorphic operators E(ei) = 1 dx Ix) e, (x I.

(3.15)

The consequencesof this choice for 9 are summarized in THEOREM 111.2. With the choice (3.15) for 9, formally a-valued elements of qH are represented, in coordinate representation, by a-valued wave functions, for a = R, 6, IH.

With this choice, there is therefore a direct algebraic correspondence between the abstract elements of the IH-module and their coordinate space representation in terms of L* functions. Multiplication of a-valued wave functions by any of the elements of IH is defined, but these spacesare naturally equipped with a-linear scalar products. We shall say that they are elements of Lz. To prove the theorem, consider a formally real component fi off obtained by means of Eq. (2.7). Then, by construction, E(ej) fi =fi ej and hence, with the help of Eq. (3.15), (3.16) It therefore follows that (xlfi) is real-valued in IH, and that a formally ej-valued element of Zx is ej-valued in IH in the coordinate representation.

QUATERNION

This precise algebraic tation, since

QUANTUM

correspondence

467

MECHANICS

is not maintained

in momentum

(Pm = j_(PI-X)(-~I./-) d-x,

represen-

(3.17)

and the unitary coefficients (pls) contain the element e,. The properties of the Fourier transform (3.17) are discussed in Appendix 2. Returning to the representation (3.7) for the momentum operator, we see that with the choice (3.15). it can be written as P = -AE(e,)

1’ Ix) & (.KI ds.

Since (along with [E(e,), X] = 0), [E(e, 1.P] = 0,

(3.19)

the operator P falls into the special class of quaternion linear operators which contains only one antisymmetric element of I. and hence, as pointed out at the end of Subsection 11.4, a quaternion linear skew-Hermitian operator (3.20) can immediately be associated with it, The reversal in the sense of translation induced by the complex linear representation T(6.x) can be easily seen using Eqs. (3.18) and (3.20).

sf= +J%,= --SE(e,)(v, = -S(y/,e,

+ v/, e2) e,

+ ~/~e,ede~

= +S(vo - ‘i/,e,) so that the term sf in the expansion of T(6x) adds a derivative with a relative negative sign to the quaternionic component. Both symplectic components are treated in the same way by the quaternion linear S. THEOREM

III.3 T,,(dx) = e (l/A)E(e,)Pbx

is a quaternion linear unitary representation

of the translation

The theorem is a direct consequence of Eq. (3.20).

(3.22) group on qH.

468

HORWITZ

AND

BIEDENHARN

Remark. The existence of position and translation operators allows us to construct the generators of infinitesimal rotations in 3-space, and, following the procedure leading to Eq. (3.22), quaternion linear unitary representations of the rotation group. We have therefore achieved a complete description of the basic Euclidean symmetries. In a space-time (Minkowski) manifold R4, an extension of these results permits the construction of the operators representing the Poincare group. We now turn to study the uncertainty variables X, P. Let 2=X-(X),

relation for the quaternion

linear canonical

&P-(P)

(3.23)

P> = tf, w>*

(3.24)

where, for llfll = 1, w Then, according

to the Schwarz AXzAP

The usual procedure

= (.A 4-f-h inequality = ilJ?ll’

(2.25), lI&ll’>

1(xJ&-)12.

(3.25)

to obtain a lower bound for AXAP is to write

lGo23-)12=blU KW)12 + a1c.LP3v-)I’~

(3.26)

where the second term contains the symmetric anticommutator, and to retain the first (commutator) term alone. It is easy to show that in the Hilbert IH-module, this procedure leads to a very poor lower bound. In particular, there are states for which the expectation value of E(e,) is zero. Let (we restrict ourselves to one dimension for this example) (x/f) = e, (xl g) for -co
+jm(glx)e,(-xlg)dx=O.

(3.27)

0

It is a further remarkable

property

of this example that

(L{JLa f) =0.

(3.28)

Since

jmtglx)x(4g)~=O~ (X> =j;* -(glx)e,xe,(xlg)~x+ 0

QUATERNION

QUANTUM

469

MECHANICS

it suffices to calculate (f, (X, P) f). From the equations preceding (3.12),

r -2h

/‘dx f\x)e,x&f -’ r

1

.

(3.29)

The first term vanishes by the argument given above for Eq. (3.27), and the second is dx(glx)e2e,

x-&e2(xlg)+]~dx(glx)e,x$xlg)}=0.

since s( a/?.Y) (X 1g) 1s even. Hence, both positive terms of the lower bound for AX’AP’ vanish. This does not imply, however, that the product of dispersions can attain that lower bound. In fact, in the example treated above,

(P) = -2A -11 dx ( glx) e, $ (sl g). In the special case for which (.ujg) is real-valued.

(P) vanishes and

and hence AXAP # 0. We now state and prove the uncertainty principle for a Hilbert IH-module. THEOREM 111.4. The greatest lower bound for AXAP is h/2; this bawd can ortl~ be achieved when the state is complex primitive.

For A=Xor

P,f=y/,+

v,ez, ~lf~~2=Il~o~12+ ll~,ll~=

1,

AA’ = llaflj’ = (A Ay) - (.L Afj2 = (A’),1

+ (A’), (1 -A)-

(L(A), + (1 -WA),)‘.

(3.3 1)

where. for a = 0. 1. (A), = kA~)/llyl~ll~

(3.32)

1 = IIwol12.

(3.33)

and

We may scale w. and y/, so that 0 < J. < 1 without changing the first two moments of X. P in these states (starting with Ao, call the scale factors yo, y1; then, the

470

HORWITZAND

BIEDENHARN

normalization condition y:L,, + $( 1 - 1,) = 1 establishes a linear relation between yz, yf). In fact, AA2 - w*hr

ww

+ ((-J12)t-@X)U

-A>1

= A(1 - L)((A), - (A),)2 > 0.

(3.34)

Since the product of two positive convex functions of one parameter is minimized only at one or the other of the end points,

AX2AP2 > min ((X2)0 - m>v’h ! W’h

- @X>((P’>l

- (PX)? - (PXh

(3.35)

each of which is greater or equal to h2/4 by the usual results for complex Hilbert spaces. Since the end points correspond to f = w,, or w1e,, the minimizing wave function is primitive. If AXAP achieves this lower bound, then the usual conditions on the complexvalued primitive states must be valid, Ply0 = 2ty,z

(WI = 0)

Ply, = 2ly,z

(v/o = 0).

or (3.36)

Using the representation (3.6) for P, these equations have the well-known minimal uncertainty solutions for Re z = 0 and Im z > 0. IV. TENSOR PRODUCT,SECOND

QUANTIZATION

AND GAUGE

FIELDS

[ 311

IV.1. Tensor Product and Second Quantization The construction of tensor product spaces is essential for the treatment of the many-body problem and for a constructive approach to quantum field theory. A tensor product must be (a) well-balanced, i.e., linear in each factor, (b) for sufficiently well-behaved linear operators acting on each factor, there must exist a linear operator on the tensor product with equivalent action, and (c) the norm of a tensor product state should be factorizable in terms of the norms (and scalar products for identical particles) of the constituents so that second quantization can be achieved. The first of these requirements for a tensor product, the well-balanced condition, is extremely restrictive. It is so restrictive, in fact, that for Hilbert IH-modules there exists no quaternion linear tensor product [ 171.9 ‘If one has two vector spaces, one linear over IH on the juxtaposed product is then linear over IH on both acceptable tensor product since half of the quaternion For more than two vector spaces, even a tensor product Professor Jacques Tits for discussing this subject with

the left and the other linear over IH on the right, the left and the right. However, this is not an action on each of the two vector spaces is lost. of this type connot be defined. We wish to thank us.

QUATERNION

QUANTUM

471

MECHANICS

We conclude from the impossibility of quaternion linearity for a tensor product that one is forced to accept complex linearity as the largest possible linearity. Consequently, one must take also the complex inner product. In this section, we shall explicitly construct a complex linear tensor product for Hilbert IH-modules [32, 331.” Before carrying out this construction, let us discuss the significance, and structural implications, of the results to be obtained. Let us imagine, for the moment, that a quaternion linear tensor product could have been defined. This would have meant that the right quaternion algebra would have been a universal structure, in the sense that a multiplier from IH on a many-body state could be applied equally well to any of the one-particle factors in the tensor product, just as, for example, the complex unit in ordinary many-body quantum mechanics is universal. The lack of a quaternion linear tensor product in effect means that the universality of the quaternion units is lost, with only one surviving complex unit (e, ~say) retaining universality. Each factor in the tensor product provides, in principle, a representation of SO(4) by means of the transformationfj QJq. as pointed out in Section II. an invariance of the real scalar product for one-particle states. and for the one-particle probability density. As we shall show, however, the probability density for identical particle symmetrized (BE) or antisymmetrized (FD) many-body states in the complex tensor product is invariant only under the U(2) generated by f + @, which is represented linearly, and a discrete part of the SO(4), right multiplication by ez, which is represented antilinearly (charge conjugation). Accordingly, each subsystem in the tensor product carries with it an individual (complex) two-space linearly representing the left action of .9 along with the (universal) unit e, from IH. The algebraic dimensionality of the tensor product, in consequence, depends on the number of subsystems,increasing with particle number as 2”. It is precisely this phenomenon of increasing dimensionality that Adler has found to be characteristic of operator-valued semiclassicalYang-Mills fields 18-I 11. where the gauge fields themselves have algebraic properties which depend on the number of source particles. We find it particularly interesting that one of the most puzzling aspects of Adler’s work on algebraic chromodynamics is naturally explained by the tensor product structure of Hilbert IH-modules. We shall explicitly discuss gauge fields in Subsection IV.2. We now proceed to construct a general form for the complex linear tensor product of Hilbert IH-modules, and define annihilation-creation operators on the Fock space obtained in this way (we do not maintain the theorem-proof format in the sequel). We recall that for f = vO + vle2, if f-fz, v. + v/~z, but v/, + wlz* for z E C( 1, ei). Hence, v. and vr are linear under this action, and if we wish to form a complex linear direct product of the elementsf and g = x0 + x, e, of -;it”;,, we must

‘OK. Morita has applied quaternionic Weinberg-Salem

these results with a somewhat theory.

different

interpretation.

in his construction

of a

472

HORWITZ

AND

BIEDENHARN

therefore utilize only the combinations ~&,, ~,,x~, I&,, IJ$X~. We define the complex linear tensor product of two Hilbert IH-modulesby means of the functional wig)=

K,,,um

= w::x::~

E (4xjFqH)o

(4.1)

where ai = 0, 1 and ci corresponds to (formal) complex conjugation when ai = 1, and is of no effect when cli = 0. The functional (4.1) has the property Y(.L gz) = q/k

s> = wt

g> z.

(4.2)

We define its norm squared as (4.3)

where )I ylala2(12is the usual norm squared for the direct product, i.e.,

II ~a,,,uh/12 = IIw::ll’ Ilx~:II’.

(4.4)

Hence,

IIwig)ll’=,y

IIw::lI’~ Ilx~:II’= Ilfll’ . II gl12. a1 a2

(4.5)

Moreover, we have for the scalar product vu

g),

w-‘3

6))

=

lala2

K,&

g>,

~cw2u-‘d))

since, by Eq. (2.46)

The invariance of the scalar product (4.6) is U(2) X U(2) under the action of 9 X -9 and the residual C(1, e,) of IH. The functional YaIa,(f; g) is a second rank tensor under the invariance automorphism group for complex scalar products, U(1) (under the transformation (2.63), w; and vi* acquire a factor ezels). Applying the automorphism independently to the two spaces, with different parameters, will not preserve the invariance of the scalar product (4.6). Note that applying the definition (4.1) to a single Hilbert IH-module, one has

QUATERNION

which is, in fact, the symplectic

QUANTUM

MECHANICS

representation,

413

and

For the case of identical particles, one must deal. as usual, with totally symmetric or antisymmetric linear functionals for the Bose-Einstein or Fermi-Dirac cases. We define (4.10) where it will be noted that the interchange of particle states does index labels. They are not properties of the physical states, such labels on the component Hilbert spaces contained in the Xl,., 0 flu,, 3 and may be thought of as part of the manifold functions are defined. It follows from Eq. (4.10) that

not interchange the as spin indices, but direct sum & = on which the wave

(4.11) The scalar product in ( qlH

x

9’*)c

is defined as

These results are parallel to the corresponding relations for the tensor products for complex Hilbert spaces, except that all scalar products have the form of the complex scalar product defined in (2.46). We now turn to criterion (b) (we shall work here, for simplicity, with nonsymmetrized functionals). According to (2.74). a complex linear operator has the action (we have redefined A 1,,, A ii as A &, A ,” for convenience) Af=A,o~o+A,,v:‘+

(AF,,vo* +AT,y/,)ez.

(4.13)

474 According

HORWITZ

to the definition

AND

BIEDENHARN

(4.8)

‘Y,w)=Aoo~o

+Ao,W:~

~Y,w-)=~lovo+~l,v:~

i.e.,

Since (4.16) it follows

that

where, in the sense of the complex direct product, (4.18) We shall now construct annihilation-creation operators for the complex tensor product Fock space. To do this, we consider the extension of the definition (4.10) to the description of a many body state (BE or P’D) by means of the complex linear functional

where the permutations

P act

onfy on the indices i,,

i,

,..., i,,,. The scalar product is

(Yu(w’, w*,.... w”>,WA’,x2Y..,xN>>

= T (*>‘(w’,xi9&Y2, xi%**.(w”,xi”),,

(4.20)

QUATERNION

QUANTUM

415

MECHANICS

the usual (symmetric or antisymmetric) Slater determinant, except that all scalar products among constituents are of complex quaternion form. We now define (af(‘YNf’)

yY(‘YN-

‘Y’Na,v*l,a* ,..., (II = Kr+l.a,v .,.., a,(‘YN+l, ‘!JNI...3‘$1

(4.2 1) The adjoint of this creation operator, obtained by means of the relation

the annihilation

operator, a($+

‘), may be

(Yucx”+l,x’v )..., xl), u+(ly+ ‘) Y(lyV ).... ly’)) = (a(yP+ ‘) yloy+ ‘,XN ,..., x’), Y(ly ,..., I$/‘)).

(4.22)

Now, by (4.21). (a(w”“)

Y(y+‘.y

=

\“-

,..., x’), Y(ly‘V ,..., I/‘))

(*y’(x.V+l-j,

\’

vyN+l)c

JZO

(*)‘fJ&

itN+

‘y’v))c

...

fy,

If’),

I -j P

,v

= ” (f)+V+‘-j, ,TO Comparing

I//~~+~)~(Y(X~~+’ ,..., f+‘-j,~.~-j

,..., x1), Y(li/‘,..., I,!/‘)).

(4.23)

the first and last of the relations (4.23) we find that

j=O

(4.24) which defines the annihilation It now follows from

af(‘Y2)a(‘y’)

operator a(@‘+‘)

YtjN+‘,xN,...,

= 2 (*)j(wl,x~~‘+l-j), i=O

on the entire Fock space.

x’) Y/(v2,XN+1 ,..., x.V+2-j,xlV-j

,..., xl)

(4.25)

476

HORWITZ

AND

BIEDENHARN

and a($> a+@*) yl(XNf’, XN,...,x1> = ($3 w’>, yOIN+ l, XN,..., x1) f 5 (*y’(ly’,~“+‘-‘),

!P(y12JNfl )..., XN+*--j,XN-j )...) x’)

(4.26)

i=O

that

[fz(J$> a+@*) f a+(ul’> a($)] Ytj”’ ‘,...,x’)= (w’,wycY(XN+ ‘,...,x’), i.e., the annihilation-creation operators satisfy the BE or FD relations

[4w’), a+(W*>lF = (w’, w’>c*

(4.27)

IV.2. Gauge Fields With the special choice of left quaternion algebra given by Eq. (3.15), Theorem III.2 is applicable, and the wave functions f(x) = (xlf) have algebraic properties which coincide with the formal algebraic properties of fE &. The results of the previous section can be taken over with this choice, if we replace the elementsff qH by f(x) E LL (and hence w, by V,(X) E Lk) everywhere. According to Eq. (4.20) the amplitude for the norm of a many body wave function, with identical particles, is

x ifgyx*) ... lygyx,)* l&y (XN)

(4.28)

and is invariant under the local gauge transformations

v’(x)’ = q(x) v’(x) z(x) = q(x) Y+>,

(4.29)

where q(x) E IH, z(x) E C(1, e,) c IH, and Iq(x)l’ = Iz(x)l’ = 1. The transformations (4.29) are a local generalization of the U(2) transformations defined by Eq. (2.57), which leave the complex scalar product invariant. The larger SO(4) invariance of the one-particle norm (or density) is not an invariance of the norm of a many body state due to the presence of complex scalar products between the wave functions of the constituents (see, e.g., Eq. (4.11)); this statement applies for the densities as well.

QUATERNION

QUANTUM

MECHANICS

477

Consider the action of the full SO(4) invariance group of the one-particle norm on the complex scalar product, i.e., for Q E 1’, q’ E IH,

KWt Qm'), = (fq'- ix'), = tr(fs'- ix') - e, tr[t.W3 gq') e, I =tr(.Lg) -e, tr((.Lg)q'e,q'*).

(4.30)

The imaginary part therefore obtains a contribution in a direction of the algebra rotated away from e, . The two-body norm (4.11) is therefore, in general, altered under the SO(4) transformations due to the change induced in I(f,g),.l’. In the special case that q’ = e,, however, the imaginary part of Eq. (4.30) only changes sign, and I(f,g),(’ is left invariant (as is the density before integration). This is a general result. The density (5.1) is a determinant (FD) or symmetrant (BE) of an N x N matrix whose elements are densities of complex scalar products. Since right multiplication by ez has the action (U7.u) e2)*g@)

e,), = (f*(-u)

&I),?:

= (s(x)*.&y)),,

(4.3 1)

the matrix is altered to its transpose, leaving the determinant or symmetrant invariant. This remaining symmetry of SO(4) is discrete, and therefore it cannot form a part of the gauge group. It is, furthermore, not realized linearly in the symplectic representation. In particular, for the functional (4.8) (using the relation (2.61) and the definitions (4.3X)),

= - e, r2 Y(f)*

(4.32)

corresponding to the charge conjugate fundamental representation of U(2). The transformation (4.29) can be realized in operator form. Let us define the quaternion linear operator QL=)

/x)q(x)(.u/&E

2

(4.33)

with q(s) E IH and Iq(,u)l* = 1, and the complex linear operator Z,. such that (see Eqs. (2.74) and (2.93)) Z,.f = zL v. + z.? vl e2 3

(4.34)

where zL = / Ix)z(x)(.K~

d.xE C((l,E(e,))c.~

(4.35)

478

HORWITZ

AND

BIEDENHARN

with z(x) E C(1, e,) c IH and lz(x)~’ = 1. By Theorem II.2 quaternion linear), zL* = z&

I

(Eq. (2.38); zL is

Ix) z(x)” (xldx.

(4.36)

The action of the complex linear operator

QL=QLZL=ZLQL.

(4.37)

in the x representation is precisely that of Eq. (4.29). In the symplectic representation, the transformation (4.37) takes the form of Eq. (2.102); with the special choice of _2, the action of Z, is just that of multiplication by z(x). From the results (2.61), corresponding to the action induced on the symplectic representation (4.8) by the E(eJ (and the right action of e,) on qH, it follows that

where ri, r2, r3 are the conventional Pauli matrices (with e, playing the role of i) acting on the indices a = 0, 1 of Y. The local gauge transformations (4.29) can therefore be expressed in terms of unitary complex linear combinations of Pauli matrices (the fundamental representation of U(2)), which we shall denote as u(q). Each constituent in the tensor product must have such a factor acting on it at the corresponding points of its manifold, independently of the form of the wave function. Hence, it is also true for identical particles that we have the relation ~Y(d,

9W2Y.~wN)(xl = %(W,))

uz(q(x2))

. . . XN) *-* ~NWN)) YY(yl’, .** WNMl ..* XN),

(4.39)

where uj(q(xj)) acts on the indices aj = 0, 1 of Y (corresponding to the jth factor of L:,XL:,X-*XL:,). Under a motion of the manifold xi .a. x, of the N-body system, the factors uj(q(xj)) must be altered in Eq. (4.39), along with the many-body wave function. We must therefore construct, as in any gauge theory, a covariant derivative D” which satisfies D”‘Yy’ = (D” Y)‘,

(4.40)

where the prime denotes the gauge transformation defined by Eq. (4.39). TO satisfy the requirement (4.40), D” must contain, in addition to derivatives acting on each

QUATERNION

QUANTUM

479

MECHANICS

constituent wave function, a gauge compensation field Bfi(x), whose transformation properties compensate the changes induced by the derivatives acting on the Uj. Since derivatives act on each term of the product in Eq. (4.39), gauge transformations of the compensation field must include a direct sum of functions in the algebras acting on each index. It may, however, have a richer structure. As an algebra valued field acting on the N-body states, it may locally (in the manifold X i ,..., x,~) alter the structure of the tensor product, i.e., the action of the field on one of the constituents can couple algebraically to the charges on the others. This mechanism is the new feature of the construction, and coincides with the nonlocality found by Adler. It offers the possibility of maintaining the essential nonlinearity in the Yang-Mills equations on a semiclassical leveL8-‘I’. We seek a solution of the form

- e,gB UUI...dJ...a’(Xj)sa,(x,)... ra,(Xi) ... r.,&.) x Y(y/‘....) l+!P)(X,... X,#)},

(4.4 1)

where the carat denotes deletion, and the (sai(xi)} are x,-dependent automorphisms of the Pauli algebras associated with each index. In this form, we have separated the nonlocal structure of the compensation field as (gauge-dependent) orientations of the algebraic basis of the field on the full direct product space. The coefficients h ave values in the Pauli algebra of the jth constituent. Defining, for B ““‘...+.“‘(,uj) brevity, (4.42)

A i = (a, , a2 ,..., Gj... u,~),

one finds that the covariance property (4.40) is satisfied if BL(.‘j(s)’ = uj(q(x)) Bw”qx) uj(q(x))-’ -el

g-'a~uj(q(x))

uj(q(s))p'

6di,o,

(4.43)

where 6Ai,O= do,.08nz,0‘.. Sai.+... da,,,, and for all i = 1, 2,..., N, rAi(xi) =

Uj(Xi)

r,i(Xi)

Ui(Xi)-

‘.

(4.44)

Since q(xj) is an algebra-valued field evaluated at xj, we have extended the transformation law of the compensation field to all x. It was for this reason that we included the raj(xj) into the definition of the coefficients in Eq. (4.41) explicitly. The homogeneouspart of the transformation in Eq. (4.43) is effective only for the part of the gauge field for which all indices referring to the charges i #j vanish. Since rO(xi) = 1, this part of the gauge field is local, and acts as a direct sum. We remark that according to the transformation property (4.44). there always exists a gauge (given the (xi}) such that soi = rai and hence the dependence of soi on xi is through a gauge induced unitary transformation.

480

HORWITZ

The separation strength tensor,

of kinematical

AND

BIEDENHARN

nonlocality

persists

in the gauge covariant

field

-. - faj(xj) aa. raN(xN) = {PB uAj(x) - PBuAj(x)

FpuAj(x) t,,(xl)

- e, g(BpBj(x) BuCj(x) _ BuBj(x) BwCi(x)) . * * r$xj)

x qxl)

qAjBjCj} (4.45)

* * ’ r~,(xN>Y

where we have used the Pauli multiplication sb5,=q

rules abc

(4.46)

z,

with 4

AQFj

s

qalblC1

qazbsC2

. . . q&&j

, . . qaNb,+,v

(4.47)

to extract the product of Pauli matrices from the commutator. after gauge transformation, F”“Aj(x)‘r;,(xl)

that,

. . . r;i
= uj(q(x))Ffl”Aj(x) Although

It then follows

uj(q(x))-’

r&(x,) .*. r$xj)

. . ’ t&(xJ.

(4.48)

the gauge fields Bjyx;x,

. ..zj .*. x,) = BWAj(X) t,JxJ

*. . QXj)

. *. ?7,kv>~

(4.49)

*. * ~qJTv)

(4.50)

and the field strengths Fju”(x; x1 ... Zj . . . xN) E FwuAj(x)

...

r,,(xl)

raTxj)

which act on the tensor product wave functions as elements of the full tensor product algebra, are nonlocal, we may treat the coefficients WA)(x), FWrAj(x) as local fields which can be assumed to satisfy Yang-Mills equations. The structure of the covariant derivative acting on gauge fields may be identified from the relation 6F;” = D”&y

- D”&#;

(4.5 1)

yielding the usual form Du WAj = a@WAi _ e,g(BwBj wj It is also straightforward [D@“, D”] Assuming

_ WBiBKj)

qAPFj.

(4.52)

- WBiFfl”ci)

qAiBjC’.

(4.53)

to show that WAi = -q

g(F”u”j vj

the equations of motion D,PuAj(x)

= pAi(

QUATERNION

QUANTUMMECHANICS

481

it follows from (4.53) that the current must satisfy

D,.Pj(x)

= 0.

It is also an identity (from the Jacobi identity) that D, F$ + D,, F$, + D.&i.

V. SUMMARY

= 0.

AND CONCLUSIONS

We have constructed and studied some of the properties of a quaternionic quantum mechanics, realized in terms of a Hilbert IH-module c,, a closed vector space, linear over quaternions, with many of the properties of a Hilbert space. Along with the right multiplication of elementsby quaternions from lH, we defined left multiplication from an algebra -/ of quaternion linear operators which are *-isomorphic to quaternions. Linear combinations of left and right multiplications from ._i/ and 11-1 permit us to decompose K into the direct sum of four real linear spaces,or, in a form which is somewhat more useful. into the direct sum of two complex linear spaces, where the imaginary unit is taken to be that of the complex subalgebra i(1. e,) of Ill. Corresponding to real, complex, and quaternion closed linear manifolds (subspaces) and their associated projection operators, there are real, complex and quaternion linear scalar products. The invariance groups of these scalar products are, respectively. SO(4). U(2). and SU(2). The minimal subspacescorresponding to pure states in the senseof Gleason I28 1 and Mackey [25 1 were shown to have more structure than the minimal elements. which we call primitive, for the Hilbert lh-module. It has been shown elsewhere [ 19) that a B* algebra containing a subalgebra *-isomorphic to quaternions. with twosided quaternion linear positive states, has a Gel’fand-Naimark-Segal-type represen tation as a Hilbert @module: these states correspond to the (quaternion linear) Gleason-Mackey-von Neumann states used here. Operators from the B* algebra are represented as quaternion linear operators on the Hilbert IH-module, but in the module, there are also real and complex linear operators, useful in relating unitary groups and quaternion linear observables (Stone’s theorem). Defining a quaternion linear momentum operator as the generator of translations. we obtained its representation in coordinate space (and all of the Euclidean symmetries). and found that the canonical momentum-coordinate commutator is nontrivially operatorvalued. The minimum uncertainty state, for which APAX = h/2. however, exists and is complex primitive. Adjoining to this anti-Hermitian operator two others that anticommute with it, we find a special algebra *-isomorphic to quaternions. which we may take to be 2. The x-representation of the Hilbert IHmodule as quaternion-valued functions in Li is then algebraically isomorphic to the abstract space. in that abstract elements with “formal” algebraic properties go over into Li, functions with precisely those properties.

482

HORWITZ

AND

BIEDENHARN

There is no possibility of constructing a quaternion linear tensor product; we have therefore constructed a complex linear tensor product of Hilbert IH-modules for the representation of many-body states, and obtained annihilation-creation operators for the second quantization of the quaternion quantum theory with Bose-Einstein or Fermi-Dirac statistics. The complex scalar product of constituents occurring in the Fock space scalar product is of the form of a direct sum of contributions from the complex decomposition of qH to two complex Hilbert spaces. The left quaternion actions from 9 are represented on the one-particle complex linear functional Y(f) (the symplectic representation (4.8)) linearly in terms of 2 x 2 complex valued matrices, and the right multiplication by a complex number from IH is represented by a central, universal, complex number. The complex scalar product is invariant under the combined action of these multiplications, specialized to U(2). Since the complex linear tensor product is a tensor product of linear functionals of this type, and the norm of BE or FD symmetrized products is the symmetrant or determinant of complex scalar products, one obtains a natural framework for the introduction of non-Abelian local gauge symmetries which increase in dimensionality with particle number. To make this new gauge field structure explicit, we have defined covariant derivatives and shown that Yang-Mills-type equations can be imposed even though the gauge fields carry intrinsic nonlocality. This realizes a structure first discussed by Adler [8-l 11. For two particle states, the gauge fields are in the space of the fundamental representation of U(4) (the complex extension of the representation of the Clifford algebra C,), and for three particle states, of U(8) (the complex extension of the Clifford algebra C, ; see [ 71 for a discussion of quantum theory on a Hilbert module over this algebra). Subspaces corresponding to certain subalgebras of these groups appear to be useful in phenomenological models presently under study. We have discussed the mechanism, in Ref. 3, by which Adler’s construction, adjoining an additional set of generators to the algebra-valued gauge field, ‘i imposes a constraint which reduces the size of the U(29 gauge group.

APPENDIX

1:

OPERATOR

INVARIANCE

In this Appendix, we consider the properties of operators which are invariant under the action of invariance groups of the scalar product and the associated invariance automorphism subgroups discussed in Subsection 11.3. The quaternion scalar product is invariant under the SU(2) generated by f -+ @ An operator invariant under this SU(2) satisfies

Am-= c2.G ‘I See [lo]

for some physical

insight

into this construction.

(All)

QUATERNION

or, with the representation

QUANTUM

483

MECHANICS

(2.72), using

Qf= (Q,,+ Q,&)(Ic/~ + ‘i/,ed = Qo’i/,,- Q,w,* + (QIvo* + Qow,)ez

(A1.2)

we have

Comparing

coefficients

of Q,. Q,, v~,, ‘i/, and their conjugates, one finds A,,=A~,,

(A1.4)

Ai, =-A;,.

and all other elements must vanish. If we require that A be linear over complex scalar multiplication according to Eq. (2.74), Ah, = Ai = 0, leaving the condition

as well, then.

(A1.5)

Aoo=A?, and all other components vanish. If, furthermore, A is quaternion

linear, it follows

from Eq. (2.92) that (A1.6)

‘4 on = A 1 1 = A 7, .

The invariance automorphism group associated with the quaternion scalar product is trivial. For the U(2) invariance of the complex scalar product, we consider operators linear over the reals satisfying

A CQb, = QM z. Such operators are complex linear and satisfied by operators with the property The invariance automorphism group complex scalar product is U( 1). and we the reals for which A(Z*fz)

(A.1.7)

satisfy (Al.l) as well; these conditions are (A1.5). associated with the U(2) invariance of the ask for the structure of operators linear over = Z”(Af)

z,

(A1.8)

484

HORWITZ

AND

BIEDENHARN

where Z = x + E,y if z =x + e,y, x,y real and (~1’ = 1. It follows that this operator must act as 4f=&vo

+Ahv’o*

from Eq. (2.72)

(A1.9)

+A,,y/,e,.

If such an operator is complex linear, then A&, = 0, and the U(1) invariant acts as

operator (A1.lO)

If it is. furthermore,

quaternion linear, (Al.1 1)

A00=A,,* For the SO(4) require

invariance

of the real scalar product

(or one-particle

norm),

we

(A1.12)

A(Ql*fsJ = QWf> qz

and is therefore quaternion linear and satisfies (Al.l) as well. The result is given by Eq. (1.6). The invariance automorphism group SO(3) associated with the real scalar product requires

A(Q*f4) = Q*W-1 q

TABLE Summary

of Properties Products

Scalar Product

of Operators Invariant and Their Associated

(A1.13)

I

under the Action of the Invariance Invariance Automorphism Groups

Groups (IAG)”

(s, Sk

(.f .sh

of the Scalar

Operator Linearity

cf. g) Invariance Group: SU(2) IAG: trivial

Real linearity

Ao,=Af, A;, = -A ,‘d”

&,=A?,

Aoo.Ah. A,,#0

A,,=A,,=A$

A,, =Ao,=Ah all real.

Complex linearity

A,,=A?,

&,=A:,

Aoo.A,, #O

A,,=A,,=A,*,

A,,=A,,

Quaternion linearity

A,,=A,,

A,,=A,,=A,*

Aoo=A,,

A,,=A,,

=A$

Invariance

U(2)

IAG: Y’)

Invariance Group:S0(4)

“All elements A$ (a,/3 = 0, I), on which conditions are not given should Eq. (2.72) for the definition of the components A$ of the operator A).

IAG: W3)

=A$

be taken

=A:,

A,,=A,,=A&

to be zero

(see

QUATERNIONQUANTUMMECHANICS

485

for Q in correspondence with q (coefficients of the quaternion units are equal), and 1q12= 1. One finds that A,,--A;,-A,,=0

(A1.14)

and that these components are formally real, and all others vanish. If an SO(3) invariant operator is complex linear, A&, = 0, and Aoo=A,,.

(A1.15)

The resulting operator is also quaternion linear. Table I summarizes the above results. APPENDIX

2: FOURIER TRANSFORM

From the representation (3.7) for the momentum operator, it follows that (A2.1 ) so that (sip) =erlp_rlhy(p).

(A2.2)

where y(p) is some function of p in IH. To satisfy the completenessrelation (A2.3) it is necessary that (A2.4) where ?I is the dimensionality of the manifold (-u or p). Let us now define

~p).=,p)$g

(A2.5)

so that (A2.6) Then,

(A2.7)

486

HORWITZ

AND

BIEDENHARN

and we may use (p)’ for the spectral representation on the momentum in place of 1~). The operator P remains multiplication in the 1~)’ representation (determined only up to a phase, such as y(p)/&),

=p 1 y(p)*

Equation

(A2.8) remains valid if one multiplies

‘(Plfn The momentum representation,

(A2.8)

e~‘~X~h{x~f)dx.

=P

on both sides by y(p)/&,

so that (A2.9)

‘(Plf).

in terms of ((p)‘}

of an elementfE

qH is given by (A2. IO)

As we have shown (Theorem f = I,Y~+ ly,e, is represented by

111.2), in the special

f(x) where V,(X), v,(x) the decomposition

= v,(x)

are complex-valued.

‘Mf)

+ v,(x)

choice of 2,

the element

(A2.11)

e,,

It is evident from the relation

=f(p) = v~(P) + V,(P) ez9

(A2.10)

that

(A2.12)

where W,(P) = ‘
Furthermore,

= f b)‘e,

‘Wh

W,(P) = ‘(PI WA

from the definition

(A2.13)

(3.15), we have

(A2.14)

QUATERNION

QUANTUM

481

MECHANICS

and hence ‘(PIE@,)

(A2.15)

cd = el w~P).

The action of the Fourier transform is, however, representation (3.15) for E(e,) becomes

not trivial

algebraically.

The

E(e2) = [ / 1p)’ e-e@.“he, ee@‘X’R6(p’l dxdpdp’

= f IP)’ e, ‘(6pldp,

(A2.16)

and hence, while

it follows

from (A2.16)

WW2> vo) = e, v,b)

(A2.1’7)

‘(PIE@,) vd = e, v&p).

(A2.18)

that

ACKNOWLEDGMENTS

We are grateful to Duke University and Tel Aviv University for their hospitality during exchanges of visits both at early and later stages of this work. We engaged in further collaboration at the Institut des Hautes Etudes Scientifiques and while one of us (MPH) was on sabbatical leave at Syracuse University. and at the University of Texas at Austin. We thank these institutions for their hospitality and support. The authors gratefully acknowledge useful discussions with Professor L. Michel. We thank Professor S. L. Adler for reading an earlier version of the manuscript and for his helpful comments. We also wish to thank D. Sepunaru and A. Soffer for many valuable conversations during the course of the research, and U. Wolff for a helpful discussion.

REFERENCES I. M. G~~NAYDIN

AND

Lett. Nuooo Cimento 6 (1973). 401; Phys. Rev. D9 (1974), 3387; J. 1651: F. G~IJRSEY, “Johns Hopkins Workshop on Current Problems in High p. 15. Johns Hopkins University, 1974; M. G~~NAYDIN, J. Math. Phys. 17

F. G~~RSEY.

Math. Phys. 14 (1973). Energy Particle Theory.” (1976). 1875.

2. M. G~~NAYDIN, C. PIRON, AND H. RUEGG. Comm. Math. Phys. 61 (1978). 69. 3. C. PIRON. “Foundations of Quantum Physics.” p. 75. Benjamin, New York, 1976. 3. H. H. GOLDSTINE AND L. P. HORWITZ, Math. Ann. 154 (1964). 1: 164 (1966), 291. 5. M. J. DUPR~ AND P. A. FILLMORE, in “Topics in Modern Operator Theory,” (C. Eds.). Operator Theory: Advances and Applications. Vol. 2. Birkhliuser-Verlag, Stuttgart. 198 1. 6. L. P. HORWITZ AND L. C. BIEDENHARN, Helv. Phvs. Acta 38 (1965), 385. 7. L. P. HORWITZ AND L. C. BIEDENHARN, J. Math. Phvs. 20 (1979). 269.

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8. S. L. ADLER, Phys. Rev. D 17 (1978). 3212; R. GILES AND L. MCLERRAN, Phys. Left. 79B (1978), 447. 9. S. L. ADLER, Phys. Rev. D18 (1978), 411. 10. S. L. ADLER, Phys. Left. 86B (1979), 203. 11. S. L. ADLER, PhJx Rev. D21 (1980). 550. 12. H. HARARI, Ph.vs. Lett. 86B (1979). 83. 13. M. A. SHUPE, Ph.vs. Lett. 86B (1979), 87. 14. D. FINKELSTEIN, J. M. JAUCH, S. SCHIMINOVICH, AND D. SPEISER, J. Math. Phys. 3 (1962). 207; 4 (1963) 788. 15. G. BIRKHOFF AND J. VON NEUMANN, Ann. Math. 37 (1936), 823. 16. J. REMBIEL&KI, “Notes on the structure of the octonion and quaternion Hilbert spaces,” Institute of Physics, University of Lodz. unpublished, 1979; J. Phys. A: Math. Gen. 13 (1980). 15 and 23; 14 (1981), 2609. 17. J. REMBIELI~~SKI, Phys. Lett. 88B (1979), 279. 18. L. P. HORWITZ AND L. C. BIEDENHARN, in “Weak Interactions as Probes of Unification,” Virginia Polytechnic Institute 1980, (G. B. Collins, L. N. Chang, and J. R. Ficence, eds.). AIP Conf. Proc.. Number 72, Particles and Fields Subseries No. 23, p. 553. Amer. Inst. of Physics, New York, 1981. 19. L. P. HORWITZ AND A. SOFFER, J. Math. Phys. 24 (1983). 2780. 20. P. TRUINI, L. C. BIEDENHARN, AND G. CASSINELLI, Hadronic J. 4 (1981), 981 in “Proc. of the Third Workshop on Lie Admissible Formulations,” Boston, Mass., 4-9 August 1980). 21. I. M. GELFAND AND M. A. NAIMARK, Izo. Akad. Nauk. S.S.S.R. 12 (1943). 445; Mat. Sb. 12 (1943). 197, English translation. 22. I. E. SEGAL, Bull. Amer. Math. Sot. 53 (1947). 73. 23. J. DIXMEIER, “Les C*-Algebres et Leurs Representations,” Gauthier-Villars, Paris, 1964. 24. A. SOFFER, Doctoral dissertation, Tel Aviv University, 1984. 25. G. W. MACKEY, “Mathematical Foundations of Quantum Mechanics,” p. 75. Benjamin. New York, 1963. Springer-Verlag, Berlin/Heidelberg/New 26. T. KATO, “Perturbation Theory for Linear Operators,” York, 1976. 27. G. EMCH. Helv. Phys. Acta 36 (1963), 739, 770. 28. A. M. GLEASON, J. Math. Mech. 6 (1957), 885. 29. J. M. JAUCH, “Foundations of Quantum Mechanics,” Addison-Wesley, Reading, Mass., 1968. 30. K. GOTTFRIED, “Quantum Mechanics,” Benjamin, New York, 1966. on Group Theoretical 31. L. C. BIEDENHARN, D. SEPUNARU, AND L. P. HORWITZ. “IX Int. Colloq. Methods in Physics, Cocoyoc, June 23-27, 1980” (K. B. Wolf, ed.), Lecture Notes in Physics Vol. 135, p. 5 1. Springer-Verlag, Berlin/New York, 1981. 32. K. MORITA, Prog. Theor. Phys. 67 (1982), 1860. 33. K. MORITA. Prog. Theor. Phys. 68 (1982), 2159; Nagoya preprint DPNU83-13. August 1983.