a quantum mechanical theory of local observables and local operations

R c p r i n t e d f r o m F o L \ D A T T O N S o F P H y s r c s

A Quantum Mechanical Theory of LocalObservables and Local Operations

Willem M. de Muynck'

Receiued May 2, 1983

Local operators are characterized mathematically b1' means oJ projection

operators on the Banach space of bounded operators. The idea of microlocalitl '.

as opposed to macrolocality, is implemented into the theory 56 as to enable us to

deJ'ine operations that are strictly local. Necessary and sufJicient conditions are

intestigated in order that the interaction of a local measurement instrument tt'i lh

o local quantum Jield is such a strictly local (or microlocal) operation.

.lpplication of the theory to quailtum eleclrodynamics rereals that this theorl'

Liolates microlocalitl: as dejined here. Implications which our theorl'may hate

on the issue of quantum nonlocalitl ' as studied in relatiorr to the Bell inequalities

ure discussed.

I . INTRODUCTION

In recent years the problem of the locality or nonlocality of themicrophy'sical world has received much attention, mainly in the context ofexperiments such as those considered for the first t ime by Einstein, Podolsky,

and Rosen.(') The study of this problem has picked up considerable impetusby the discovery of the Bell inequalit ies.(2) These inequalit ies, which arederived on the basis of a theory of so-called locql hidden variables, expressthe fact that such theories do not allow correlations between distant particles

to exceed a certain value. From the experimental violation of the Bellinequalit ies,(r '1)it is often inferred that the microphysical world is nonlocal,and that there can exist influences(') or even signals(o)that propagate faster

than light. Since quantum mechanics gives a good description of the excesscorrelation revealed by the experiments, and hence seems to provide an

t D.purtrn.nt of Theoretical Physics, Eindhoven University of Technology. Eindhoven. The

Nether lands.

Vol . 1 ,1 , No.3 , March 1984Printed in Belgium

r99

0Ol5'9018/84/0300 0199$03.50/0 c 1984 Plenum Publishing Corporation

200 de Muynck

adequate representation of a nonlocal world. it is often considered to be itself

a nonlocal, or even noncausal. theory (see. e.g., Ref. 7 and references cited

there ).This alleged nonlocality of quantum mechanics has not remained

unchallenged. Both on the basis of the usual postulates of quantum

mechanics.(8) as well as by means ol a quantum mechanical analysis of the

measuring pocess,(n' it was concluded that the measurement results of a

measurement performed on one subsystem are completely independent of

which measurement is performed simultaneously on another subsystem thatis located at a big distance. In these anaiyses quantum nonlocality, if i t

exists. does not manifest itself on the level of the measurement results.l 'he idea that quantum mechanics is basically local is also transparent

in the theory of quantum f ie lds. ( r0 ' ' r ) where local observables are postu lated

to commute if they pertain to regions that are causally disconnected lthepostulate of local commutativity), This postulate expresses the expectationthat, because of relativistic causality, mutually distant measurements wil l not

disturb each other. In accordance with this expectation a reiation betweenlocal commutativity and mutual nondisturbance of distant measurements was

demonstrated bv means of a quantum mechanical analysis of the joint

measurement of two observables.( r2) Here, local commutat iv i ty was der ivedas a consequence of mutual nondisturbance of trvo distant measurements.which nondisturbance was required to hold for the measuring process.

Since quantum nonlocality can be demonstrated in a direct way only ifi t manifests itself as a disturbance of a distant measurement. by the presup-positiott of nondisturbance we restricted ourselves in Ref. 12 to thosemeasurement procedures in which the quantum nonlocality, if i t exists, hasno effect. By this approach the problem of quantum nonlocality is placed

more or less out of sight. In order to attack this problem from a differentpoint of view in the present article, the converse question is considered, viz.whether local commutativity is sfficient for nondisturbance. It is

investigated whether a quantum mechanical description can be given of ajoint measurement procedure in which two distant measuring instrumentsinteract locally with one and the same object system. in such a way that theinteraction processes take place in a mutuall,v nondisturbing way.

ln this investigation both object and measuring instruments aredescribed by quantum fields, thus avoiding the serious l imitation inherent in

the treatments of Refs. 8 and 9, that each of the two measuring instrumentsis taken to be sensitive to one of the subsystems only. It seems that by thislimitation the possibil i ty of obtaining experimental evidence of nonlocality is

excluded beforehand. As a matter of fact. since there is no reference to thedistance between the measuring instruments, the reasonings of Refs. 8 and 9give the same results if the measuring instruments are not far apart. This

Quantum Mechanical Theory of Local Observables and Local Operations

indicates that these analyses have no bearing on the problem of quantum

nonlocality. A genuine proof of nondisturbance of mutually distant

measurements should demonstrate in which way the disjunction between

measuring instrument and distant subsystem makes the interaction between

these two ineffective. It is the purpose of the present article to study quantum

(non)locality from this point of view.ln order to be able to perform such an investigation. we wil l f irst have

to clear up the nature of the locality involved in the interaction between the

quantum f ie lds. In the theory of local quantum f ie lds, ( r0 ' r r ) local observables.

or. more generally. local operators, are defined to belong to some reglon

either of space,time or of t i lr, and are interpreted(r0) as representing physical

operations perlormed in this region. Thus. the field operators,y*i-r) and ry(x)

create and annihilate, respectively, a particle in x, and hence are operators

pertaining to this point x.Although there exists a relation between local commutativity and

localit.v or causality, from our discussion in Ref. 12 it lollows that this

relation is far from clear. From the anticommutativity of fermion fleld

operators. we mav deduce that local commutati[ i/.f is not necessary for an

interpretation of f ield operators as representing local operations. So, it seems

necessary to devise a locality criterion that is dilferent from local

commutativity. by which the idea of a local operation is represented more

faithfully'. Such a criterion is developed in Sections 2-4. Operators obeying

this criterion wil l be called microlocal operators. because it derives from the

tnicroscopic properties of the quantum mechanical states. By way of contrast.

local commutativity, being derivable from the nondisturbance of

measurements. essentially is a mqcroscopic property. which probably has

reference onl\ to operators that can play the roles of observables. If mutual

nondisturhance of d is tant measurements is in terpreted as a consequence of

macrolocalitv or macrocausality. then local commutativity of observables is

the reflection of this macrocausality in the quantum mechanical formalism.It is the purpose of this article to investigate whether it is possible to

devise. u'ith the help of microlocal f ields, joint measuring processes that aremutually' nondisturbing. In doing so we are able to study the interrelation

between the concepts of microlocality and macrolocality, and to implement

the suggest ion(r ' r r6) that quantum nonlocal i ty merely is a v io lat ion of

microlocalit l ' or microcausality. without macrocausality being violated. To

this end in Sections 5 and 6 definit ions of the four notions micro- andmacrolocality, and micro- and macrocausality are proposed, in order to

make these notions more precise. After defining in Section 7 what is to be

understood by a microlocal interaction between quantum fields, in Section 8necessary and sufficient conditions are investigated for this interaction to bemicrocausal. ln Section 9 the theory that is developed in the preceding

201

202 de Muynck

sections is applied, in a rather provisional way! to the interaction of quantumelectrodynamics. Finally, in Section l0 implications which our theory mayhave on the issue of quantum nonlocality as studied in relation to the Bellinequalit ies are discussed.

We close this Introduction by noting that, in developing the notionsmentioned above and in proving the theorems, we did not pursue maximalmathematical rigor, in order not to obscure the conceptual issues. Thus,theorems are proven only for bounded operators with a discrete spectrum,whereas in Section 2 use is made of a basis for Fock space which only existsin the improper Dirac sense. (In Appendix A it is shown that the analysis ofSection 2 can also be carried throush without reference to such a basis.)

2. LOCAL ALGEBRAS: EXPLICIT REPRESENTATION FORA LOCAL SCALAR FIELD

The present section wil l have mainly a heuristic character. We considerthe simplest model of a quantum field, viz. the scalar f ield obeying canonicalcommutation relations, in order to develop a mathemaliccl criterion for anoperator to be a local operator pertaining to some region C of conhgurationspace ]f i3. In this way we can obtain a better insight into the nature of thequasilocal algebra generated by these operators. By exhibit ing a concreterepresentation it is, moreover. shown that the more general axiomatic theoryto be developed in Section 3, is not an empty structure.

2.1. Localized Quanta

The Hilbert space .7 of the system is taken to be the usual Fock space.A general state of the field is given as(")

l y l ) : \ ' ( N ! ) ' ' ' | t u , . . . 1 d r . n p " ( r , , . . . , r , r ) v r * ( r , ) . . . y t ( r n ) l 0 ) ( l )N : 0 ' l l l l ' l i r l

In ( l ) P.( r , , . . . , r . ) isthe coordinates. Thecommutation relations

a function which is symmetricfield operators 14*1r) and r4(r)

Iv$) , vr ( r ' ) ) : d( r - r ' )

IvG), v$') l : [ , r*(.) , y '( ' ' ) l : o

under permutation ofsatisfy the canonical

(2)


The N:0 term in ( l ) should be in terpreted as Yol0) , Yo a constant , l0)being the vacuum state of the field, defined by y(r)10):0. The set ofvectors

] r , , . . . , r " ) : ( N ! ) t ' ' r l t ' ( : r , ) . . . r t i * ( . , u ) 1 0 ) ( 3 )

can be considered as a complete, orthonormal set of improper vectors inFock space.( r8)

Our heuristics starts from the observation that the vectors (3) might be

interpreted as describing localized states of the field. More generally, a statein which all quanta are localized in some region C of lF, r, might be obtained

analogously to (1) as

f -

y ) c : \ ' ( N l ) " ' I a r r . . . l a . r P , ( r ' , . . . , r r ) y * ( r , ) . ' . , i r t ( t r ) 1 0 ) ( 4 )\ : 0 ' ( '

Since the creation ofsuch a localized state in C from the vacuum ]0) is aphysical operation performed locally in region C, it seems natural to requirethat the operators U'(r), r € C be local operators, pertaining to region C (seealso Ref . 19) .

Although the interpretation of a local operator as representing a localph-vsical operation is analogous to the one adopted by Haag and Kastler,(r0)it should be mentioned here that the interpretations are not identical.Whereas in Ref. l0 localization is defined with respect to regions ofMinkowski space, we take here IRr as the conhguration space (in Ref. l lthese two interpretations are considered side by side). lt is clear that thischoice restricts our treatment to a nonrelativistic one. For this reason weshall not require our local operators to obey Lorentz covariance as is done inRef. 11. Also local commutativity, being defined here as commutativity ofoperators pertaining to disjoint regions of Rr. is not required from the outset

Ialthough eventually local commutativity wil l turn out to be necessary lorthe existence of certain microlocal interactions (cf. Section 8)1. Instead, inthe following we shall try to give a mathematical characterization of a localoperator pertaining to some region C of lRr. which is in l ine with the idea ofa physical operation performed in C.

To this end we denote the states (3) in the following according to

l i t i ' ln l ) . E(n+n): Iy ' . In th is notat ion n can be in terpreted as theoccupation number of the single-particle state d(r - ro), ro € C, and {n} asthe set of these numbers if ro ranges over C. For {l i} an analogous inter-pretat ion obta ins wi th respect to C. the complement of C. l t is in terest ing tonote here that, although in the following the heuristics is based on the deltafunction representation, the notions to be developed are independent of thisspecial representation. For this reason our theory encompasses both the

203

204 de Muynck

notions of strict locality and of essential locality (Ref. 20; see alsoSection 4), the latter corresponding to a representation in which thedeltafunctions are replaced by a complete orthogonal set of nearlt, localizedsingle-particle states.

2.2, Local Operators

We shall now give a characterization of which operators on the Hilbertspace,,7', having l1nl, 1ri)) as a complete orthonormal set of states. are to beconsidered as local operators pertinent to region C of Pr. Such operatorsshould obey the following two requirements:

l. They should not affect the number of quanta outside C.

2. The effect of the operator inside C should be independent of thepresence of quanta outside C.

In order to meet these requirements a local operator A1, pertinent to C.should have the property

( 1n1 , l i a i A , . \m f , 1n f ) : 6 ,7 , , . 1 ^ / \ n l , { 0 l i l , l { n l , 1o } ) ( s )

Operators l. obeying (5)can be characterized as follows. Consider the set.d(f) of bounded operators A on .7. Define a mapping PI ol ,19(7) intoitself, according to

PtA : \ - ( { n l . 10 } A I \m l , 10 l ) 1 r f , { i 712 ( l n r f , { n - } ( 6 )( n ) \ i l l m l

Since the operator P|, defined bV (6).is easily seen to be idempotent,

(PI) ' : PI (7)

and is continuous in the uniform topology of ,4(f), it is a projectionoperator on.7(7). The range of Pf follows from (5) and (6) to be preciselythe set of local operators,4. pertinent to C. So, we can characterize this setby means of the equality

PIAr ' : t r ,

It is easily seen that

PII : I, for all C (9)

that is, the unit operator 1 is a local operator, perinent to any region C.Let 7(7) denote the Banach space of traceclass operators B on,,{.

Then Tr AB, A e 3(f) is a l inear functional on 7(.7).In fact, taking the

( 8 )


usual operator norm as a norm in ,*(7), this Banach space is the dual of7(f) (see Refs. 21,22).It is now straightforward to show that P/ is the

adjoint with respect to the l inear functional TrAB, of an operator P(- actingon / ( 7) and defined by

PcB : \ ' ( 1n f , { , r i l B \m \ ,1 ,41 ) l 1n f , l 0 f ) ( l r r f . { 0 il n l l n l l m l

Thus.

B e v ( r )

T r P I A . B : T r A P ( . 8

From (7) and ( l l ) i t is c lear that a lsoP, , is a pro ject ion operator .

Pi _ P,

( 1 0 )

( l l )

( 1 2 )

Moreover. tak ing I :1 in ( l l ) . i t fo l lows f rom (9) that

T r P r . B : T r B . B e ; ( t f ) ( 1 3 )

Restr ic t ing B to the posi t ive cone { ( ; t ' ) , o f 7 ' (7)^ (10) impl ies that a lsoP, B belongs to 7(-Z)* . Thus.

B € 7(7) t - . : , Pr .B e

'd(7) t ( l 4 )

The mapping (14) , i f rest r ic ted to the operators.B e ' t - ( f ) * wi th TrA: l .

is an example of a mapping, somet imes cal led a mimorphism,(2r) which is alinear mapping of the base of a base norm space into the base of anotherbase norm space. Indeed. l rom (13) and (14) we see that . i l B is a densi tyoperator. P, B is also a density operator. Since

T r A . P : T r A r . P t P ( 1 5 )

it is clear that Prp embodies the same information as p concerning theexpectat ions of operators 1, . per t inent to C. From (6) and (10) i t is a lsodi rect lv shown that

Tr A7,Prp : (0 ,4 . 0 ) ( l 6 )

0) being the vacuum state ( j0) : ] {01, {0f ) ) . So the in format ion content ofP,.p. relating to operators 16" pertinent to the complement C of C, isequivalent to that of the Fock space vacuum. This suggests that the densityoperator P..p describes a state in which all particles are localized in C. Weshall now show that this is indeed the case.

de Muynck

2.3. Localized States

In order to give a definition of a localized state we introduce theprojection operator P.' operating on 6 (7) according to

P i p : P r p P s , p c : \ - l { n } , { 0 } ) ( { n } , { 0 } , p e r Q n ( 1 7 )

We shall say that p represents u'1,*. localized in C, if and only if

P i p : p ( 1 8 )

that is, p belongs to the range ofP/, From (10) and (17) it can directly beproven that

P I P ] : n r - ( 1 9 )

Since (19) implies that Pr.p is in the range of P,l, we conclude that for anarbitrary density operator p the projected state described by Pcp is localizedin C.

From the definit ions (10) and (17) we can, equivalently, prove that

PrP; : P/ (20 )

The two equalit ies (19) and (20) imply that the projection operators P,. andP| have equal ranges. As a matter of fact, in Hilbert-Schmidt space (withinner product Tr A+B) the Hermitian projection Pt- would be related to thenon-Hermitian projection P. as orthogonal and nonorthogonal projections,respectively, onto the same subspace. For the characterization of a state p..as a state localized in C, i.e., having only quanta inside C. the projection P,.is as suitable as P. is. For this reason, the relation

P , ' 9 : P Q l )

can be used as an alternative to (18) as a definit ion of a state which islocalized in C. By choosing the definit ion (21)we can take advantage of thefact that not Ptp but Prp represents the information contained in region C:contrary to (15) we have in general TrArp*TrArP;p. From (10) i tfollows directly that

Pc 10 ) (01 : 10 ) (01 , f o r a l l C (22 )

that is, the Fock space vacuum is localized in any region of lFi. Thissomewhat metaphorical though not inappropriate result stems from ourdefinit ion (6) of a local observable. It is shown in Appendix B that an alter-native dehnition is possible on which the vacuum is localized nowhere. Since


by this alternative definit ion certain observables are excluded that arephysically relevant, we stick to the definit ion presented in this section.

2.4. Local Observables and Local Algebras

From the definit ion (6) of Pf it is easily seen that the local operators1, . sat is fy ing (8) const i tu te a * a lgebra.( r0 ' r r ) Hencefor th, we shal l ind icatethis algebra as the local *-algebra (/, of local operators pertinent to regionC. Note that a lso (21)associates an a lgebra wi th region C.

The selfadjoint operators of (1c. are the local obseruables pertinent to C.I f

Q r P ^

is the spectral representation, it is easily shown that

P ̂ : \ P ̂ . r a tI n l

(24)

in which P,,,u, is the projection operator of a subspace of Fock spacespanned by vectors having definite numbers {l i} of quanta in C. From (24) itfo l lows that

PIP^- P^

It is seen lrom (24) that each eigenvalue a,, of the local observable is highlydegenerate s ince e igenvectors have the form L, ,C, , , l 1n i . ln l ) , which belongto the same eigenvalue a* for a l l { t r f . Because of (15) and (16) i t seemsreasonable to associate local observables 1,. with local measurementsperformed in region C.

Ii D is another region of l lrr, the projections Pf and P|.,, are definedanalogously to (6). By specifying the states (3) according to whether quantaare located either in C\C a D, D\C ^ D, C o D, or C l) D, it can be shownthat

Ptno: PIPf i : P|PT ( .26)

From (26) it follows that the local operators satisfy the properties

t . l P t A : A & P ; A : A l = > P [ n , A : A(set-theoretical inclusion) (21)

2. C c- D = \PtA: A => PtA: Al ( isotony)

207

(23)

(2s )

(28)

de Muynck

These properties make it possible('0'tr)to define the quasilocal algebra r1 as

the C*-inductive l imit of the union of the local algebras 17. pertinent to ail

bounded regions C of rl 3. Then, t1 is a C*-algebra. From the definit ion (6)

it is straightforwardly demonstrated that all operators pertinent to disjoint

regions are mutually commutative. This verif ies the property of local

commutativity of the local observables of f 1, which, here, is a direct conse-

quence of the model.It is possible to construct in an analogous way a quasilocal algebra

based on the local x-algebras generated by the projections P,. Although

from a mathematical point of view both algebras are equally interesting'

because of the physical interpretation we shail restict our attention to the

quasilocal algebra generated by PI.The first reason for doing so is that

generally the product of two density operators is not a density operator. So.

there cannot exist a local algebra of densitl 'operalors.

2.5. The Schlieder Condition

A second reason is der ived f rom the Schl ieder condi t ion. l r t )which. for

our definit ion of a local algebra, reads

A ( . . A r ) : 0 , C ^ D : A . > A c : 0 o r A r , : 0 ( 2 9 )

This condition which, in its relativistically generalized form. can bedemonstrated to hold for f ield theories of the Wightman type.(tt '25) plays aprominent role in recent research on the causality properties ol local f ieldtheories (for a review, see Ref. l2). It is easily seen that (29) is satisfied bythe local operators defined bV (8).

That the Schlieder condition (29) is not automatically fulf i l led for

arbitrary quasilocal algebras can be seen from the fact that. contrary to theone based on P$. the quasilocal algebra based on P,. does nol obey theSchlieder condition. This follows most easily from a consideration of theope ra to r s { n f . { 0 f ) ( { n i . { 0 i 1 , { n l+ 101 and l { 01 . 1 t l ) ( 10 f . 1 t l , 1 t l + { 01 ,which belong to the ranges of P.: snd P.=, respectively. Multiplication of theseoperators gives zero. without vanishing of either of the two operators.

This example answers an observation made in Ref. 26 regarding thequestion of the universal validity of the Schlieder condition for quasilocal

algebras. Evidently, it is possible to conceive of such a definit ion of localoperators that the Schlieder condition is not satisfied. ln order to obey this

condition. it is not sufficient that the operators are merely pertinent todisjoint regions.

It is suggested by these considerations that the Schlieder conditioncould be used as a requirement to be fulf l l led by a quasilocal algebra inorder that the operators of this algebra represent strictly local operations.

Quantum Mechanical Theory of Local Observables and Local Operations 2Og

Such operators should be equivalent to the unit operator , l outside their

domain of operat ion. So, .4 . ' ,4 , should be in D equivalent to A, , i l

CaD:g , show ing tha t A r . 'An canno t van i sh un less l 1 ) van i shes . So ,

from a physical point of view, the Schlieder condition is compulsory for

local operators representing strictly local operations.Notwithstanding the importance of the Schlieder condition as exem

plif ied by the above considerations, in the general treatment of quasilocal

algebras to be taken up in the next sections, we do not resort to this

condition. The reason for this is twofold. In the first place. not all local

operators correspond to local operations. As a matter of f lact, with.4.'. also

uAr. . a a constant . belongs to the local a lgebra ( / t . l t is then c lear that in

general a local operator ,4(. also changes the state outside C, thus

invalidating the heuristic argument which was based on the interpretation ol

a local operator as representing a local operation. The second reason is that

local commutativity'. which is often taken as one of the defining charac-

rer is t ics of a quasi local a lgebra (and which wi l l be shown to be necessary

also in our theory) , is qui te independent of the Schl ieder condi t ion. This can

be seen as fo l lows.The quasi local a lgebra based on P( (10) . not obeying the Schl ieder

condi t ion. turns out to be a lso not local ly commutat ive. That is . in general

l P ( . A . P D B l + 0 . C . D : A ( 3 0 )

A deta i led inspect ion. however, of the commutator (30) shou's that the

ta i lurc to obe-v both the Schl ieder condi t ion and local commutat iv i ty does

not hale a common origin. The relative independence of these two propertles

can be demonstrated by considering the slightly modified quasilocal algebra

del lned b l lormula (10 ' ) o f Appendix B. I t is easi ly seen that th is a lgebra is

iocal lv commutat ive but does not obey the Schl ieder condi t ion.The Schlieder condition seeming to be too strong in one respect. and too

u.eak in another. in order to provide a characterization of local operators. we

shal l look in the next sect ions for a bet ter cr i ter ion.

2.6. Theorems

For l a te r compar i son we c lose th i s sec t i on by p rov ing the fo l l ow ing twotheorems.

Theorem 2.1. The projection P,. can be defined by

p (B : \ - A ,u ,BA l r , , B € r ( r )l x l

( 3 1 )

210 de Muynck

in which the operators A1n1 and A,f,-, are partial isometries which arepertinent to C, i.e.,

P f A ,7 , : A 171

P t A [ , t : A [ , t ( 3 2 )

and satisfy the equality

Y A { r , A , r , : t ( 3 3 )t n-]

Proof. Taking

Ara t : ) ' l { ' z f , { o f ) ( { n f , { a } l ( 34 )I n'J

(31) and (32) follow by direct inspection.S i n c e

A [ r , A , u , : ) ' i { n } , { i a } ) ( { n f , ] i z } l ( 3 5 )

is a projection operator, it i , ,.. 'n that Alny and Af^are partial isometries.Equation (33) is directly entailed by (35). I

Theorem 2.2. If I is an operator on Fock space. obeying I 0): a,then

P tgPFA) :o f o r a rb i t r a r y B (36 )

Proof. Since

PtA: I ( {0 i , I r i p * r i {0 t . \41 ) \p l , i f l ) ( l p l , \q } lI p i l F l t s )

we get

BPtA - : ( \m l , \m l lB \ n t , { t i ) ( 10 i . \ p l p {A i 0 } , i r i )I m l I m l I p l

tn l ln l

. l \m | , \m f ) ( {n } , 1 , ' t - f l

and

PI(BPFA) : : (1 "21 , i0 l l B l \n l , i t i ) ( {0 i , \F l P tA 110} , 101)t m l l m l l p l l n )

. l \m l , \n l ) ( \n l , \n f l (37 )

Since ,4 0):0 impl ies PtAIO>:0, the theorem fol lows. I


Corollary. If ,4 l0) : 0, then also

P t A ' P c P : O

Proof. Equation (38) is the adjoint of (36) with respect tofunctional Tr AB.

3. LOCAL OBSERVABLES AND LOCAL OPERATORS:GENERAL THEORY

3.1. Definit ions

In the foregoing section we considered properties of the local algebrasdefined bV (6) or (10). In the sequel of this article we shall relinquish thespecial representation presented there. The theory of Section 2 wil l begeneralized so as to be valid for more general f ields. although it wil lmaintain its nonrelativistic character.

Taking Fock space as the Hilbert space .7 of the field states, thevacuum state 0) is defined as the state without freld quanta. If 1/ is theobservable measuring the number of quanta, then

1 i l 0 ) : 0 ( 3e )

As to the operators working on ,j/ ' we restrict ourselves. as before, tothe set l(7) of bounded operators oD.7, which can be considered as aBanach space with respect to the operator norm. As an algebra,.V(7) is aC'f -algebra.(rr) Also F(.f) is defined, as in Section 2, as the Banach spaceof traceclass operators containing the density operators in its positive conez-( ; f ) * . Then the expectat ion values

(A ' ) :Tr Ap, A e.4(z) , p e 7(7) (40)

are bil inear functionals. The operator A is completely determined by therestriction of the functional to g(f)*. Also, p is determined completely if(l ) is given lor all A € .'tGf).

In characterizing how the locality of an operator has to be specified inthe general theory, we first turn our attention to self-adjoint operators. If C issome (bounded) region of []r. then a (bounded) self-adjoint operator,4,- wil lbe taken to be pertinent to C, if the measurement of the corresponding obser-vable is a local measurement in C. This means that the measuring instrumentfor A, draws its information entirely from C. In Section 2 it was seen thatthe information content of C can be represented by the density operator P(.p,

(38 )

theI

212 de MuYnck

in which P. obeys (12) through (15) . These re lat ions are independent of the

special representation used in Section 2. and hence can be generalized.

Taking, as before, Pf as the adjoint operator to P. with respect to the linear

functional (40), we give the following definit ion.

Definit ion 3. l. (Local Operators). If P, is a mimorphism which

projects rQf ) into itself. that is.

B € r ( 7 ) * . . > P C B e { ( 7 ) r

T r P . B : T r B . B e 7 ( f )

P'r: P.

then .4. is a local operator pertinent to C il

PIA , :A ,

( 4 i )

(42)

(43 )

(44)

From (44) the equality (15) for general f ields directly follows, showing that,

if ,4. is a local observable pertinent to C. the density operators p and Prp

give the same expectation values.Generalizing analogously relation (2 1), we arrive at the following.

Definit ion 3.2 (Localized States). If the mimorphism P, is defined asin (a1)-(a3), then p.. is the density operator of a state localized in C if

P t p c : p r ( 4 5 )

For states localized in C we have. for arbitrary A € . 't(,f),

T r A P r : l r P t A ' P r (46 )

So. as far as region C is concerned, I and P.*,4 represent the same infor-

mation about the system. If I is an observable. then PI A can be interpretedas the restriction of this observable to C. Thus. if l{ is the total number ofquanta, PIN is the number of quanta contained in C (in Theorem 3.1 it isshown that Pf ,.{ is self-adjoint i l l isr see also Theorems 3.8 and 3.9).

3.2. Postulates with Respect to Locality

We now formulate the basic assumption of the present article.

Postulate 3.1. With every (bounded) region C of Fl 3 is associated amimorphism P. obeying


( i ) ( 4 1 f ( 4 3 ) r( i i ) P7r, : I , / being the unit operator on l (7);

(ii i) if C and D are two regions, then

(41)

Pt -a t r : P rP , r : PnPr . ( 48 )

Equation (47) expresses [cf. (45)] that on1' state is localized in F r. The

equality (48), which is equivalent to the adjoint relation Icf. (26)l

Pr . , : P t P t : P IP [ (4e)

expresses that the information rvith respect to an observable I € l(7) which

is contained in region C a D is independent of the way the region C O D is

singled out. thus making this information a unique propertlr of this region'

From (48) the propert ies of set- theoret ica l inc lus ion (27) and isotony (28)

f ollou'. also in the general case. for local measurements performed in C o D.

From (28) we d i rect ly f ind the fo l lowing general izat ion of (15) :

T r A r . P : T r A c . P r P . C c D ( s0)

From the phy 's ica l in terpretat ion i t seems reasonable that (50) remains t rue i f

.1 , is replaced by a product Ar Br . o l operators per ta in ing to C. This asks for

a second postu late, which is commonly requi red lor local operators. ( r0 ' r r )

Postulate 3.2. For any region C the operators pertaining to C

const i tu te an a lgebra, that is ,

P t ( A r * 8 , . ) : A ( + B (( s l )

P[ (A , . B r ) : A , . B ,

Actually. it wil l be shown in Theorem 3.2 that this algebra is a x algebra.We shal l refer to th is a lgebra as the local x-a lgebra / /c . .S ince Pf is aproJection operator on 4(7), it should be continuous in the uniformtopolog-v. From this it follows that t lc includes all of its l imit points with

respect to this topology. Then we also have

Pt- f @r.) : " f (Ar) ' / an arb i t rary bounded funct ion (52)

B.v means of (52)the equality (25) for the spectral representation of the local

observable (23) can be shown to hold also in the general case. Then, theequal i ty (9 ) .

P I I : I ( 5 3 )

214 de Muvnck

which follows directly from (42), can also be derived from the representationI : l ,^ P. '

As in Section 2, the quasilocal Cx-algebra (1 can be defined as the C*-inductive l imit of the union of the local algebras t '7c. As wil l be discussed inmore detail in Section 4, the present definition of local operators does notimply local commutativity for operators pertaining to disjoint regions of lRr.So, according to this definition, local commutativity may be fulfilled, or itmay not be so. As a matter of fact, up until now the meaning of locality isnot uniquely fixed. Different notions of locality may be accommodated byour definit ion (cf. Section2.l). By ascribing additional properties to theprojections P. and PF, the class of localization definitions may be furtherrestricted.

One such restriction, which is encountered also in Ref. 20, is based onthe premise that a state Prp,being localized in C, should be equivalent to thevacuum state l0) outside C. We take this premise as the following.

Postulate 3.3. If C is the complement of C in tr r. then

Tr P{ A . Prp : (0 | PF,4 l0) (54)

As wil l be seen in Section 4, also with this additional postulate locality is notspecified so as to warrant local commutativity.

3.3. Theorems

We shall now derive some useful theorems.

Theorem 3.1. I f A : l + . then Pt A : (P[ A) ' . (55)

Proof. The operator ,4 is Hermitian if and only if the functional (40)is real for all p€6(,7)*. Then, because of (4 l), also Tr A prp is real, andhence Tr PI A . p is real for all p e 6 (-T) -. I

Theorem 3.2. l f PI A: l , then PI Ar : Ar. (56)

Proof. Since Tr 13 : (Tr B'A'1*, we have

Tr P IA ' ' p :T , A 'P rp : (T r (p .p ) t .A ) * : (T rAp ) * :T r A tp I

Theorem 3.3. If p is a density operator, then

P "p

: 10)(01 (57)

Quantum Mechanical Theory of Local Observables and Local Operations 215

Proof . Taking in (54) C:A g ives C: lRr . Because of (47) we have

P#A: A , A e . " ( 7 )

Then (54) may be written according to

Tr AP.p :rr A l0)(0 , A e 1J(r) (s8)

This directly entails (57). I

Theorem 3.4.

P;A: (0 l r l 0 )1 (se )

Proof. This follows directly from (58) srnce

T r A P . p : T r P I A . p : ( 0 A O ) T r p a

Using (59) we can general ize (57) to arb i t rary B € { (7) :

Theorem 3.5.

P t B : ( r r B ) 1 0 ) ( 0 1 , B € r ( 7 ) ( 6 0 )

Proof.

Tr AP,B -Tr P$A. B : (0 A O' )Tr B : (Trz l0) (01)( r r r ) I

Theorem 3.6.

P c 0 ) ( 0 : 1 0 ) ( 0 1 , C c F r ( 6 1 )

Proof. From (48) and (57) it follows that for arbitrary A:

T r APr . 0 ) (0 : T r APr .Pnp :T r APap : T r , . 1 l 0 ) (01 I

Corollary.

\o Pt A 0) : (01.4 0), A € ?Qn 62)

Theorem 3.7.

T r A r P r . p : ( 0 l D l 0 ) , C o D : A ( 6 3 )

Proof. This generalization of (54) directly follows from (48)and (57) . I

de Muynck

Theorem 3.8. If the operator ,4 is given as

dr A(r) , PI A(r) : A(r) , r € C

PT A : dr(01 , l(r) l0)

Proof. Equation (65) follows immediately from Pf Pf :Pj and(se). I

Theorem 3.9. I f N is a sel f -adjo int operator N > 0. N 0) :0. then

PJN>0. PrN l0 ) :0 (66 )

Proof. By (55) Pf N is self-adjoint. Tr l,{p) 0V, > Tr NPr.p:TrPIN .p>OV

" . .> P I N)0 . Equat ion (61) imp l ies (0 lPd l i ]0 ) :

( 0 1 r / 1 0 ) . F i n a l l y , ( 0 l P F r / 1 0 ) : 0 , P t N > 0 > P l N 0 ) : 0 . I

4. MICROLOCAL OPERATIONS AND (MICRO)LOCAL

OPERATORS

4.1. Microlocal Operations

In Section 3 the problem of local operators was tackled on the basis ofthe notion of local measurements. A complementary l ine of approaching thisproblem presents itself if we start from the idea of local state preparation orlocai change of state. A local operation in region C, then, should change thedensity operator of the system only in C. It wil l turn out that this latterrequirement adds a new element to the notion of locality, which is notembodied in the idea of local measuremenr. Since this new element is essen-tially of a microscopic character, we shall refer to that notion of localitywhich includes this new element as microlocality.

If by some physical operation the state of a physical system is changed.in general, both init ial and final states of the system should be describable bydensity operators. For this reason an operation should be a trace-preservingmapping of (a subset of) 6(.7)* into 6(.7)*: p-+T(p). If the operationcorresponds with a linear mapping, it is mimorphism. However, in the sequelwe shall encounter also onerations which are nonlinear.

t _ l'J lltl

(64)

(6s )

then

lara6+iJ c J a


In the present section we shall ignore the fact that physical operationsgenerally are not instantaneous but need some time in order to be completed.Time-dependent operations wil l be dealt with in later sections, in whichoperations are studied which are brought about by the interaction withanother system.

We shall give now a definit ion of a microlocal operation in region C.

Definit ion 4.1 (Microlocal Operation). A microlocal operation 7,.:p - Tc@), in region C of F 3 is a trace-preserving mapping of (a subset of)'a(7'). into 7(,7)*, obeying, on its domain, the two requirements

( i) | r. ,P.. l : 0. (61)

( i i ) P r T r : P o . C . D : O ( 6 8 )

We shall f irst discuss the two requirements (67) and (68). Equality (6i)seems a reasonable requirement because a microlocal operation T, shouldhave to do only with that part of the state which is localized in C. Theequality is equivalent to the two relations

T(.P( . : PcTcP(

PcTc: Pc T( .P(

(6ea)

(6eb)

Then. (69a) signifies that a microlocal operation I,.. operating on alocalized state Pcp, does not spoil the localization of this state. Also (69b) isplausible if we assume that a microlocal operation 2. is constituted out oflocal operators pertinent to C. Then, for any B € 4(f ).

Tt B PrTr( .PrP): f r PI B ' Tr(PrP)

: rr Pf (r](Pf 8)) . p :rr r{QI B) . p- Tr B PrTr(p)

since the local operators P$B are in the same local * -a lgebra /1r .as those ofrf lci. (s l ) l.

The requirement (68) is equivalent to

Tr P IB . r r (p ) :T r P tB . p , C . D : O

or

expressing that thement outcomes ofdisjoint region D.

TtetrBt: PtB. C .\ D -- a

microlocal operation in C does not influence measure-measurements which are performed simultaneouslv in a

(70)

2 1 8 de Muynck

The two properties (67) and (68) are not completely independent. From(69a) we get, because of (57),

C a D : a => PoTr(Prp) : PoPcTc(P.p) : l0) (01 : PoPcp

which is obtained also if (68) is applied to states localized in C. However,this does not imply that either of the two requirements would be superfluous.On the one hand, (68) cannot be derived from (67) for states that are notlocalized in C. So (68) should be postulated to hold at least in such states.On the other hand, also (68) does not imply (67). This can be seen from asimple counterexample, constructed for the local scaiar f ield discussed inSection 2. If we define an operation Z by

r ( l {p } , 101) ( {p } , {0 } l ) : l {0 } , 10 } ) ( {0 } , {01r ( l \p l , {1 } ) ( {p } , ( t } l ) : l l p } , { / f ) ( { p \ , \p l l , { t }+ {0 }

then it is easily shown that (70), and hence (68), holds. However, (67) is notfulf i l led, since, for \ pl + O,

r@,1 \p l , {p - }X{p } , { t } l ) : l {01 , {01 ) (101 , {0 } l

but

P, r ( l \p l , {1 } ) ( {p } , { t } l ) : l l p l , {o } ) ( {p l , {0 l l

We close this section by remarking that, by virtue of (a8), the equalit ies (67)and (68) remain valid if T. is replaced by P6, thus showing that theprojection P7 is a microlocal operation performed in C.

4.2. The Kraus Representation; Microlocal Operators

A very common kind of operations is given by the mimorphisms p -r@):ApAr, A uni tary. More general ly , i t was shown by Kraus(27) thatoperations as defined by us (which correspond to the nonselective operationsof Ref. 27) can generally be represented according to

r@:\ - AupAl , , Y ,q I ,4o: tk k

Note that the adjoint T* of T [cf. (70)],

r * (B) : l t la t o

( 7 1 )

(72)


is a transition map as defined by Mercer(28) (see also Ref. 12, Section 3.1).The projection operator P. of the scalar field representation was shown inTheorem 2. I to have precisely this form. Moreover, from (3 1) and (32) wesee that in this_special case the operators Ao of (7 1) are local operatorspertaining to C, This seems to reflect the fact that P. is a microlocaloperation in C-.

It is tempting to extend this to microlocal operations indefining a microlocal operation by the expression (71) in whichand A! are required to pertain to the same region, that is,

T t n \ : \ ' p t , a . n P \ a +- w , _ - ( , - k r - ( , . 4

general, bynow all I o

( 7 3 )

With this possibil i ty in mind we prove the following two theorems.

Theorem 4.1. Local commutativity of the operators pertaining toregions C and D (with C )D: O) is necessary and sufficient in order thatall operations of the form (73) obey (68).

Proof.

( i ) Inser t ing in (70) the specia l operat ions r@): ArpA[ . , A[ .Ar . : 1 ,we immediately obtain

At rP tB .Ac :P iB , C^D:a

Hence 11.., PtB):0, C n D : A for arbitrary unitary operators 1. .(ii) If local commutativity is assumed, we get for any B (with

A * : P t A * ) :

Tr B Prr@l:Tr B Po 5- O r re l \\ T t

: ' ' }

ATPTB 'A1 , . P

: T rk

A i .Ao .P tB .p

:T r B png

which directly entails (68). In view of (70) and (72) this rheorem can be seento coincide with Proposition 3.1 of Ref.28, as far as applied to transitionmaps. I

Theorem 4.2. The operations (73) satisfy equality (69b).

220 de MuYnck

Proof. Since the operators pertinent to C constitute a x-algebra, it

follows that for any B and any operator .4. pertinent to C

Pt@trPtB . A,): AtrPtB ' AcSo.

Tr BP, . (ArP,p . A l ) :Tr r [@[r tB 'Ar) ' P

:Tr ALptB . Arp:Tr BPr. (ArpA[)

Hence

P r (A rP , p . A [ ) : r r@, pA [ )

which implies (69b) also for the more general operations (73). I

Because in the general theory, up unti l now, we did not specify anyproperty of local operators characterizing these as entit ies which are related

to microlocal operations, it is not possible to prove that the operations (73)

also satisfy equality (69a) (although it is easily proven that they do so in the

scalar f ield theory of Section 2). In order that the operations (7-l) wil l fulf i l l

al l requirements of a microlocal operation, we shall now restrict somewhatfurther the notion of locality as embodied by the projections P. and Pf . This

wil l be done by generalizing the requirement (69a), viz. that microlocal

operations do not spoil the localization of a state, to operators. To this end

we first give the following definition.

Definition 4.2 (Microlocal Operators). The local operators defined inSection 3 will be called microlocal operators if the projections P. are suchthat

A c P c B : P c ( A c P c B ) Qaa)

ano

P c B . A c : P r ( P r B . A ) ( 7 4 b )

The property (14a), (74b) will be called the miuolocality property. We now

can prove the following.

Theorem 4.3. If the operators Pf,4* of the operation (73) have themicrolocality property, then Z(p) satisfies (69a).

Proof. Since by Qaa), Qab)

ArPrp . AI: Pr(ArPrp) . AI: Pc(AcPcp . Atr)

the theorem follows by inspection.


In the theory to be developed in the sequel of this article we shall only

consider fields that satisfy the microlocality property. Since this property

obtains for the fields introduced in Section 2, such microlocal f ields evidently

exist. In considering microlocal operations we shall, however, not rely on the

Kraus representation (73), because it is not clear whether this is the most

general representation of a microlocal operation. As a matter of fact, the

results obtained in the following set of theorems suggest that the properties

(54) and Oaa), QaV, which specify the more characteristic properties of our

microlocal operators, are not sufficient to derive local commutativity. This

suggests that it might be possible to devise microlocal operations, which are

constituted out of microlocal operators that need not obey local

commutativity, and which, because of Theorem4'1, should have a more

general form than (73). So, we shall stick in the following to the definit ion of

microlocal operations as given in Definit ion4.l. This also implies that we do

not take local commutativity as an 4 priori property of (micro)local

operators. Instead, the properties (54) and Qaa), Qab) wil l be taken as the

starting point of the present investigation.

4.3. Properties of Microlocal Operators

We shall now derive some properties of microlocal operators.

Theorem 4.4.

Pt@Ac) : P rB .A (

P t@cB) : Ac . ' P tB

.B arbitrary.

Proof. From (74a) it follows that

(1s )

Tr BA rP r. P : Tr BP c(A <.P c P)

This impl ies

Pt@A) : Pr (PrB . A , ) : P tB . A(

because of (51). The second equality is proven analogously. I

The assumption that local operators constitute a local algebra allows us

also to derive the following generalization of (74a), (74b).

Theorem 4.5.

Pr(A rpB r) : AcPcp . B c (16)

221

de Muynck

Proof.

Tr DP,(ArpB.): Tr BrP[D .Arp

:Tr P t (BcPtD . Ar ) . p :Tr DPr (ArPrp . Br )

:Tr DAcpcp . Bc I

Theorem 4.6.

Pr(APrP): PtA ' PcP

Pr(PrP . A): Pcp . PtA

Proof. From (75) we get

PtetB . A) : Pt(BPtA) : PtB . PtA

The first l ine of (77) is an adjoint relation of this expression.

The equality (77) states that only the local paft PtA of Aeffective in C. Note, however. that

Pt(APtB .D)+ PtA .P tB .P tD

Theorem 4.7.

[email protected] ) : P t@cAd: (0 l . l 0 )8 .

Proof. Equation (78) follows directly from (75) and (59).

Remark. Since (78) entails

( 7 1 )

Iis locally

Pr(APrp . B)+ PtA . Prp . PIB

because APrp is not localized in general. This inequality is equivalent to thegeneral inequality

(78 )

T

Tr ATBrPrp :T r B .ATPrp

it is seen that a restricted kind of local commutativity follows from ourassumptions, namely a local commutativity restricted to appropriatelylocalized states. Since, however, only such localized states are involved in theexpressions (54) and (74a), (74b), it seems impossible to derive general localcommutativity from them in a direct way.

We are now able to prove a number of properties which are, in a certainsense, complementary to the locality property (74a), (74b).


Theorem 4.8.

Pr(Pcp . A): Pr(A7Prp): (0 Ae l0) Prp

P lAr .Prp .Br ) : (T r l .pB. ) l0 ) (01

P7{A7PrP) : l . l0X0

P d P , p ' A ) : l 0 ) ( 0 l .

Proof. Equation (79) is obtained as an adjoint relarion of (78) if in thelatter equality.B. is equated to PIB for arbitrary B. The equality (g0)follows from an application of the projection P1 to (76), transforming thelef t -hand s ide of (76) in to PTPr(ArpBr) :Pu(A.pB.) which, by (60) ,reduces to (80). Finally, irom (59) we ger for arbitrary B

p [ (p tB .Ad :p tp teg .Ad :p t@tB .A ) : ( 0 pFB . l e i 0 ) ( 82 )

which has (8 la) as an adjo int , s ince by (76) and (61)

Pe(r .10) (01) : AePe 0) (0 : l . 0 ) (0 I 1s:1By (78) and (82) the absence is implied of correlations between measuremenrresults of local observables in C and c ir tne state is localized in c. As amatter of fact, because of (54) the equality (78) is equivalent to

(7e)

(80)

(8 1a)

( 8 l b )

(8s)

(86)

Tr ATBrP .p :T r AeP .p .T r B r .P r .p

indicating a certain independence of A7 and 8.. The equalitygeneralized somewhat. By successive application of (75) it isshow that

Pt(A(B.D( ) : (0 B. l0) A( .D( .

which is equivalent to

T r A . .BTDrPc .p :T r BePcp .T r A r .D r | r . p

However, it is presumably not possible to derive analogous relations formore general alternating products of local operators, as would be necessaryif operations c.q. measurements in disjoint regions are to be completelyuncorrelated.

we close this section by proving a theorem which can be viewed as aweakened version of the schlieder condition discussed in Section 2.5.

Theorem 4.9.

A ( . . B . : 0 = > A r : g o r B e : 0 o r ( 0 , 4 c 1 0 ) : ( 0 l B e l 0 ) : 0

(84 )

(84) can bepossib le to

224 de Muynck

Proof. Since, by assumption, Tr ArBTp:0 for all p'

T t ArPr (B7P) :Tr B7PfuAr ) :0 Ve

Taking in either of these expressions p: Pcp or p: P6p, this gives' by (79)'

( 0 1 , 4 c 1 0 ) . T r B 7 p : Q v e

and

( 0 l 8 6 l 0 ) . T r A r p : Q

This is equivalent to the desired result.

5. MICRO. AND MACROLOCALITY

ln this section we draw a distinction between the idea of microlocality

as introduced in Section 4, and a notion of macrolocality, which is defined in

the following as the implementation of microlocality on the macroscopic

scale of measurement results. This distinction is inspired by the diff iculties

which abound in relativistic quantum theory as regards a covariant definit ion

of a position observable, which seem to put the idea of locality into sertous

doubt. Since there is no direct evidence whatsoever of a violation of locaiity

on the scale of (macroscopic) experimentation, it is sometimes

conjectured(r3-r6) that the above-mentioned diff iculties are characteristic for

the microscopic level only, and can be solved by replacing the requirement of

microlocality (or -causality) by the weaker requirement of macrolocality (or-causality ).

In order to implement the distinction between micro- and macrolocality,

we resort to the theory of quantum measurement as developed in Ref. 12. In

this theory properties of the object system (which are represented by self-

adjoint operators A:1,^a^P^) are mapped onto properties of the

measuring instrument according to (cf. Eq' (7), (8) and (10) of Ref. 12)

T.rU^Sr OpoS*:Tur pP^ (87)

in which p and po are the init ial states of object o and measuring instrument

c, respectively, Sp@poS'is the final state of the combined system, and Em

is the macroscopic property of the measuring instrument corresponding to

P^.By means of (87) i t is possib le to t ie the Def in i t ion 4.1 of a micro local

operation to the macroscopic level:

and


Definition 5.1 (Macrolocal Operation). An operation T: p-+ 7(p) is a

macrolocal operation in region C of R' if, up to experimental error,

( i ) T ro E ̂ ST (P rp ) A p "S+ : T . r E ̂ SPrT (p ) @ p .S ' ( 88 )

lor all .8..

( i i ) T . ru E .ST(p )O pos t : T . ,

Z ,S , @ PoSo (89 )

for a l l E- correspondingto a P^ which is per t inent to D, wi th D'C:A'

Since from (67) and (68) we get, for arbitrary P^'

Tr P ̂ Tr(P r r l : Tr P ̂ P cT c(P) (e0)

T,,r PIP^. Tr(P):Tor Pf iP^' P' c . D : Q ( 9 1 )

i t follows irom (87) that, according to our definit ion, microlocal operatlons

are also macrolocal. Conversely, since (67) and (68) follow from (90) and

(91) i f in the la t ter expessions P^can be taken to be arb i t rary, macrolocal i ty

would also seem to entail microlocality. This, however, need not be the case,

because the arbitrariness of E. in (88) and (89) does not necessarily imply

that the corresponding set of P.'s separates the states ol the object system.

Yet. it is clear that a distinction between microlocality and macrolocality, as

is made here. makes sense only by virtue of the assumption that it is

impossible to determine the quantum state completely by means of

macroscopic measurements. Since all measurement is subject to a certain

inaccuracy. this assumption, however, does not seem to be unreasonable. So,

macrolocality is reconciled with a possible violation of microlocality because

there exist such limitations on the possibil i ty of measuring local observables

that the restriction of (88) and (89) to actually possible measurements does

not entail (67) and (68). Yet, these limitations seems to have rather a prac-

tical than a fundamental origin.The distinction of micro- and macrolocality as given above could be

implemented by an idea encountered in the relativistic theory of quantum

fields.(2e'r0) t: iz. that the notion of strict localization, as defined bv (45)'

should be replaced by the weaker notion of essential localization of the state

of the quantum fie1d. That is to say, there do not exist local operationspreparing the init ial states p and po of object and measuring instrument,

before the measurement, to be contained exactly in some bounded region ofFl3. Then the equalit ies (88) and (89) obtain, because the essentially

226 de Muynck

localized state is supposed to yield for all local measurements, up toexperimental error, the same results as can be calculated for the strictlylocalized state of which it is an approximation.

Although, for reasons not to be discussed here, it seems very reasonableto assume that the init ial states of the quantum measuring process cannot bestrictly localized but are at best essentially localized, we shall, for the sake ofclarity, in this article stick to the notion of strict localization as defined by(45). As a matter of fact, the mathematical possibil i ty of strictly localizedstates can be demonstr&ted,(zo':01 though the physical interpretation of suchstates remains somewhat obscure. So, although it seems possible to have atheory of macrolocal operations which are not microlocal, in the followingwe shall aim at a quantum measurement theory in which the (instantaneous)influence of the localized state of the measuring instrument on the objectsystem is a microlocal operation. If i t is possible to devise such a theory, themeasurement interaction wil l then automatically obey also macrolocality,and simultaneous measurements in disjoint regions wil l be nondisturbing (ifthese measurements are instantaneous). In Sections 7 and 8 it wil l be shownthat such a theory can be constructed on the basis of the assumption that thefield operators constituting the interaction hamiltonian between object andmeasuring instrument are microlocal operators that commute with operarorspertaining to disjoint regions of R3; that is, both local commutativitv andmicrolocality are assumed.

As we saw in Section 4, microlocality and local commutativity seem robe more or less independent properties of operators. Hence. we evidently donot completely succeed in what we set out to do. viz. derive nondisturbancefrom local commutativity. As a matter of fact, local commutativity is onlysufficient for nondisturbance, in our derivation, if i t is supplemented withother, possibly independent requirements. This state of aflairs may be-andhopefully wil l be-improved on, to the effect that nondisturbance is derivedfrom local commutativity without supplementary requirements. yet, ourpresent derivation appears to be i l luminating. By constructing a theory whichis not only macrolocal but which is also microlocal, we get at our disposal amodel theory to which we can compare our physical theories in order to seehow far in these theories microlocality is satisfied or violated. Thus, quanrumelectrodynamics wil l be seen, in Section 9, to be a theory which, ln ourdefinit ion, is nol microlocal. From our analysis it is also possible to infer inwhich way the measuring process has to be restricted in order that quantumelectrodynamics can be a macrolocal theory, although violatingmicrolocality. The possibil i ty of obtaining these results seems to be suffrcientjustif ication to consider theories in which the idea of strict localization playsa role. The replacement of strict localization by essential localization wouldintroduce an additional source of nonlocality by which the violation of


microlocality is enhanced. In the present analysis the possibil i ty of this kindof contribution to nonlocalitv is left out of consideration.

6. MICRO- AND MACROCAUSALITY

Up until now we did not take into account possible time dependence ofthe operations. Thus, the transformation p . T(p) could be interpreted as aninstantaneous change of the density operator. However, if such a transfor,mation is the result of a causal physical process, the corresponding change ofthe density operator wil l take some time. We shall discuss in this sectionsuch time-dependent operations, with special attention directed towardcausality properties.

LetT ( p ( t ) , t 2 t , ) - p ( t r ) . t z ) t t (e2)

describe a general t ime-dependent operation transforming the densityoperator at /, into p(tr). lt is noted that the representation (7 1 ) of Kraus alsoaccommodates this kind of operations, but we shall not make any use of it.A very common operation of the type (92) is given by

T ' ( p , t ) : T r s ( r ) p , 6 ; p r s ( r ) * , s ( l ) : e t t ( I I t + I I t + I I t l )

describing the influence on system l, caused by the interaction with someother system 2. This example provides an excellent opportunity to discuss therelation between the notions of microlocality and microcausality. InSection 4 microlocality was introduced as a property of operations on thedensity operator of the object system. such operations can be brought abouteither through the free evolution of the state ol the object system, or bymeans of an interaction with a second system. The properties of these aregoverned by the free field hamiltonian H, and the total hamiltonianHt+ H21- Hrz, respect ive ly . In both cases the propagat iv i ty of the solut ionsof the field equations wil l induce a violation of the microlocality conditions(67) and (68) , s ince a system which, at t :0 , is local ized in a region C wi l loccupy a different region at a later t ime. Relativistic (Einstein) causalitydemands that this region should be contained in the domain of influence c,of C, defined by

C , : Q(r, ct) (e4)

radius c/. Implementing thismicrolocal operation we get

) ) 1

(e3 )

r € C

9(r, ct) being a spherical neighborhood of r withidea of relativistic causality into the notion of athe followins.

228 de Muynck

(esa)

(esb)

(e6)

( 9 7 a )

(e7b)

Definition 6.1 (Propagative Microlocal Operation). A propagatiue

microlocal operation Tr(',t) is an operation satisfying

(i) Tr(Pr.p, t): Pr,Tr(PrP, t)

( i i ) PrTr(Pr ,p. t l : PcTr(p. t l

( i i i ) P D T c ( P , t ) : P o U ( t ) P ' D A C , : g

U(r) representing the free evolution of the system.

The equalit ies (95a), (95b) and (96) reduce to (67) and (68)' respec-

tively, in case there is no propagativity. More specifically, (95a) and (95b)

are the generalizations of (69a) and (69b), respectively, to the propagative

situation. If all operations that can be applied to a system are propagative

microlocal operations, we shall say that the system is microcausal.

If T. is a nonpropagative microlocal operation' then U(l)T. and T('Ult)

are examples of propagative ones if U(/) satisfies certain conditions.

Inserting U(t)T, into (95a) we get, using (67),

IJ( t ) PrTc: Pc,U(t ) Pc.Tc

which, for 7. :1, implies the equalitY

lU(t). Pr,lPc : 0

An analogous expression can be derived from (95b), viz.

Pc lu ( t ) ,p . , ] :0

In Theorem 6.2 it wil l be shown that, for a convex set C' (97b) is also a

consequence of (96). Equalit ies (97a), (97b) express the miuocausal

evolution of localized states of the free field.Although (97a), (97b) is a necessary condition for microcausality of a

system, it is not sufficient. If the system interacts with another system, the

locality properties of the operation (93) describing this interaction are a

consequence of both the free evolution of the two systems and the propertles

of the interaction operatot Htz.If I/r, does not have such properties that it

can be characterized as a local interaction between systems I and 2, the

whole idea of the possibil i ty of a microlocal operation evaporates. In order

that the operation (93) be a propagative microlocal operation, it is sufficient,

as wil l be demonstrated in the next sections, that (i) Hrrbe a microlocal

interaction hamiltonian (cf. Section 7 for the definition), and (ii) the free field

propagation be microcausal. Then, the system is microcausal if i t has only

microlocal interactions with other systems.


As is well known, microcausality is one of the outstanding problematic

features of relativistic quantum theory. Especially the condition (97a) is

often judged to be impossible, because of the problems encountered in

constructing physically relevant solutions of the wave equations which do

not spread with a velocity exceeding the velocity of l ight.(3t) In the context

of local observables this problem is demonstrated especially lucidly by

Schlieder(32) on the basis of a theorem proved by Borchers,G3) viz. that for

any local projection operator [cf. (25)] Tr P^U(t)p cannot be zero in any

open subset of the t-axis, even if the init ial state p is localized in a region

which is located arbitrari ly far away from the region C of the measurement.

Because of the above-mentioned data the requirement of microcausaiity

is sometimes dropped, and replaced by a requirement of macrocausality, on

the basis of argumentations similar to those given by us in Section 5 on

account of the locality problem.(r3-r5) At this moment, however, it is not

completely clear whether this weakening of the general demands to be met

by the theory is really necessary. As a matter of fact, both the infinite-

velocity wave packet spreading and the Schlieder-Borchers theorem are

consequences of a presupposed one-sided boundedness of the spectrum of the

free-field hamiltonian. If the spectrum is not bounded from below, it is often

possible to construct wave packets having sharp wave fronts and behaving

microcausal ly . ( r3 '3r) Indeed, i t is somet imes thought t rs ' : ' t ) that cer ta in

paradoxes with respect to causality can be solved by allowing the spectrum

of the hamiltonian to be unbounded also from below.

The above-mentioned considerations show that it is not possible, as yet!

to disqualify microcausality as a property of relativistic quantum fields, on

the basis of its f lree-field behavior. For this reason we feel free to postulate

(97a), (97b), thus assuming microcausality of the free field. By doing this we

are able to disentangle possible violations of microcausality originating from

its two possible sources, viz. free-field evolution and interactions, the latter

being the proper subject of this article. Of course, this procedure is only of

pragmatic value, since both sources possibly contribute to the violation of

microcausality. In a more "realistic" treatment we have to take into account

also the violation of microcausality by the free-field evolution. The analysis

of the measuring process wil l have to explain, then, how it is possible that

this latter violation of microcausality does not show up in actual experience

as a violation of macrocausality. Ruijsenaars(tu) has shown that this,

presumably, is the case, since the deviation from microcausality can be

calculated to be too small to be detected under present laboratory conditions.

So, it seems that a possible violation of microcausality by the free-field

evolution will not spoil our derivation of nondisturbance of joint local

measurements in causally disjoint regions'We conclude this section by deriving the following theorems,

2Jo de Muvnck

Theorem 6.1. If the free-held evolution satisfies (97b), then theHeisenberg operator Br(t): U(t)* B, is a local operator in C,, i,e.,

Pt,B cQ) : B r(t), B r(t) : U(t)* B, (e8)

Proof. Equation (98) follows directly by taking the adjoint of (97b).1

Theorem 6.2. If all operations of the form U(t)Tc,I. a microlocaloperation, C convex, are propagative microlocal operations, then (97b)follows from (96).

Proof. The projection P., is a microlocal operation in 1-,,). Since, forconvex C, (e ,),: C-, it follows irom (96) that

P D U ( t ) P c , : P o U ( t ) , D - C

Then (97b) follows by taking D: C.

7. LOCAL INTERACTIONS

In order to be able to handle compound systems, we have to extend theformalism of local operators in the usual way to encompass direct productsof operators of the different systems. Thus, if At, and Bi, are local operatorsof systems I and 2, respectively, we have

(Ptc @ P'z;*@i @ Bi l : P l* g P, ,* (AL @ Br, )

: P[* A ' , @ Pl , * B ' r : A i @ B' , ( loo)

This allows a straightforward generalization toward the tensor product of theBanach spaces of the operators of systems I and, 2, which contains theinteraction operators of these systems. we shall take such interactionoperators to have the form

g r r : . l * , d r A t $ ) 8 l B ' ( r ) ( l 0 l )

Moreover, ̂ I1,, wil l be called a local interaction if l '(r) and B2(r) are localoperators, satisfying

(ee)

t

P[ * A r ( r ) : A t ( r ) , r € C

P?* B'1G): B ' ( r ) , r € C( 1 0 2 )

Quantum Mechanical Theory of Local Observables and Local Operations 2Jl

In the following it is also assume that At(r) and B2(r) satisfy themicrolocality property (7 4a), (7 4b).

If we restrict our attention, as we will do in the following sections, to aspecial kind of microlocal interactions, namely interactions describing onlythe scattering of two field quanta, we may assume that the operators,4'(r)and B2(r) of (l0l) both consist of an ordered product ofa creation and anannihilation oDerator. In this case we have

l ' ( r ) l0) : n 'zG) l 0) : 0

(0 l l ' ( r ) : (0 B '?( r ) : 0

( 103a)

( l03b)

( 104)

For the scalar f ield theory of Section 2, Theorem 2.2 and its corollary can begeneralized to compound systems such that from (103a), (103b) thefollowing equalit ies can be derived:

A t ( r ) P t r p : f ' ? ( r ) P ' r p : 0 , d C

P ' rp . , q ' t ) : P ' rp . - 82 ( r ; : I

In the general case it is possible to derive from (103a), (103b), (79), and(8la) that Pi ( l 'G) p ' rp) :PNAft )p ' rp) :0, r€ C etc. Analogously tothe remark made after Theorem 4.7, it does not seem possible to derive themore general relations (104) also here. In the following it wil l be convenientto restrict the class of local interactions which wil l be considered to thoseHrr in which the operators l t ( r ) and B'z(r ) sat is fy (104) .

It is noted here that the restriction of local interactions to interacuons(101) satisfying (104) is, strictly speaking, too severe, because this restrictionmakes it difficult to apply the theory to, for instance, the interaction ofquantum electrodynamics which does not obey (104). Since, however, in thepresent investigation it is our main intention to find sulficient conditions fora field theory to be microcausal, this is not a serious drawback.

For the interaction hamiltonian (101) obeying (104), the followingtheorems are now derived.

Theorem 7.1. If the systems I and 2 arelocalized in regions C and D,respectively, which do not intersect, then

T r H r r P [ @ P i p : 0 , C ^ D + A

or, equivalently,

P [ * e ; P ] o * H , r : 9 . C a D : a ( 1 0 5 )

232 de Muynck

Proof.

Pl* @ P'o* H,r: I a, Ptr*A'tr l O P3*PZ-x82(r)

* | a, Pl" Pl,* A'1r; 5l r l* 4':111J D

+ l ' dr P!* A' 1r1 6 ri * r '1r;J c o D

: l , ar,4'G)(o I B'z(r) | o)

+ { a '1o1,ar(r) lo) B'z(r)

+ [ -

dr(olA'( . ) lo)(o]8 'z(r) lo) : o

because of (49) , (59) , and (103a), (103b).

Theorem 7.2.

P l * @ P 2 D * H i 2 : 0 , C . D * A , n ) 2 ( 1 0 6 )

Proof.

f ,Hi r : l l t u r [ * , t ' ( . )OP i *B ' ( r ) + | d r P lx A ' ( . )Op l * r ' t r t l

L ' c ' ' ( \ ' v

I

So, Pf.* @ P'o* Hi, is a multiple integral of a sum of terms of the form

Pi .*etc f A ' ( t , ) . ' . P[ : A ' ( . , ) ) @ Pl , * (Pl l B ' ( r ' ) ' " r l - * n16,11

wi th C , : C o r C , i : 1 , . . . , n .From (104) and the microlocality property (14a), (74b), it follows that'

for this choice of the C;, we have

Y ,T r P ! . ! A ' ( . , ) . . . P ' r : A ' ( r , ) P f ' p : g

i f not a l l Cr : C. So,

PL*(PLf A ' ( t , ) " . PL: A ' ( r " ) ) : 0 , unless V'C' : C

Since Pj*(Pt * B' (r r) .. ' P2r* 82 (1)) : 0 if C a D : g, the theorem follows.T


From (105) and (106) we get the equality

p l * @ p 2 D * e i H t 2 t - 1 , C ^ D : A ( 1 0 7 )

These results can easily be generalized to interaction hamiltonians that are

sums of terms of the form (101) with microlocal operators obeying (104).

Theorem 7.3. The equality (107) is equivaient to

v "P l @ [email protected]" t PLa p "p ) :p i e PL (P ' , @ PLp ' e i ' ' 1 t )

: P L @ P ' o p , C a D : Q ( l o 8 )

Proof. Equation (108) follows directly from a generalization of (77) to

tensor product space, glvlng

PL @ Pl)@tH "r Pl- E pii l : P:* I P2o* einrtt ' pi @ pLp

: P L @ P L p , C ^ D : a

Conversely. from (108) we get the adjoint relation

v r p l * @ p : * ( p l * @ p ' o * B . e i I I ' , t ) : p l * @ P ? , * 8 . C a D : a

Taking -B:1, th is enta i ls (107) .

Theorem 7.4.

p tc T2r (eiH,,t p[ @ pL p) : rf Tr (pf @ p'" p . eiH'. l )

: P t r T r r p , C a D : @

Proof. Since

Atr: P[* @ P?,* A',

i t follows, by a generalization of (75) to tensor products. that

p l* g prr* (Atre iHnt) - At t t r * @p2* e iHt l t

So, i f C ) D: e, we have by (107)

( 1 1 0 )

P l . * @ P " * ( A L u i n t t t ) : A t 6 ; c ^ D : A ( l 1 l )

Since ( I I I ) obtains for AI: Ptr* At with arbitrary A', (l10) followsstraightforwardly as the adjoint of (11l). I

( r0e)

I

de Muynck

The equality (110) clearly admits the interpretation that, if systems 1and 2 are both localized in nonintersecting regions, the interaction is noteffective, at least not as far as local measurements in C on system I areconcerned. In the next section we shall relax the restriction of the states ofsystem I to localized states, and keep only system 2localized. We shall thendemonstrate the more general property of a local interaction, viz. that theinfluence of system 2 on system I obeys all the properties of a microlocaloperation as defined by Definit ion 4.l. l f a local interaction operator I l,, hasthis property, we shall term it a microlocal interaction operator.

8. MICROLOCAL INTERACTIONS AND MICROLOCALOPERATIONS

Theorem 8.1. I f 11, , is a local in teracr ion def ined by ( l0 l ) , (102) , and(104), then

g i v r l t p i O , i l r 12 t2 : p t . ( g iH r t t t p i r p . e i ? r t t r l ,

Proof. Writing

H r r : H , r ^ l H , r -

j : r , 2 ( l 1 2 )

I: I d, P'r*A'(r) Q Pf xB'z(r) + l_dr p[*AtF) @ pa*B'z(r)

' ( . c

it follows from (104) that

H , r .P [ p : P Ip . H , , - : 0 ( r r3)

H , r P L p : H r r , P L p : f [ ( n , r , r ' r p ) ( l t 4 )

the latter equality following from the microlocality property (74a), (14b).which can be shown to be valid also in the tensor product space. A similargeneralization of (78) yields for arbitrary operators Dt2 on the tensorproduct space

PL![*A' . PLD"): <01P[*A' lo> pLDt '? ( l 1s )

From (115) we der ive

pL(H,, !Lpt : i a.qo1 t t ( r t l0) p lxa,(r) . pLp :0 ( l 16)

because of (103a), (103b).

So,


Adding ( l14) and (116) we f ind

H , r P L p : P L ( H n P t c p ) ( 1 1 7 )

From (l l7) it is proved by induction that

HirP ' rP : P i ' (Hi rPLP), n :0, 1,2," '

Hence.

e i r , t , p l .p : pt (e, r t r , PLi l ( l l8a)

In a completely analogous way we also find

P t rp . s i n ' t t : P [ (P ' rp . e ' ' , r ' ) ( 118b )

and analogous expressions i f Pf is replaced bV p2. .The resul t ( l l2) f ina l lyfo l lowing f rom a successive appl icat ion of ( l l8a) and (118b). I

Tak ing the ad jo in t s o f ( l l 8a ) , ( l l 8b ) , we ge t , f o r j : 1 ,2 ,

pt*(g igrzt Btr ) : prr* (s iHr t t p t . * Brr )

p i r *@tz e iH t2 t ) : p r ( i ( p t j B t2 . e i | t l t ) ( 1 19 )

which equalit ies are valid for any region C and operator,8r2.

Theorem 8.2. If H,, is a local interaction, obeying the microlocalityproperty and (104), then

e i H t 2 t t P t g P L p . r i H 1 2 t 2 : p l @ P \ r p , C ^ D : A ( 1 2 0 )

Proof. From (l l2) we obtain, for arbitrary regions C and D,

, i H r t t t p t @ p L p . , t H 1 2 t 2

: p i e p2o(e iH, r t , p [ q p t "p . s iH t t t t l ( 121)

: Pf @ P|(PL @ P2o(einnt ' pL @ p'rp) ' ei I I t2tz)

The theorem then fol lows from an application of (108). I

The adjoint relation

Pl . * @ Pzo*(s i t r t t t t g t2 t in t l r l : p l * @ p?, * B t , ( l2Z)

Br2 arb i t ra ry , C o D : g , i s equ iva len t to (120) .

236 de Muynck

Theorem 8.3. The mimorphism Z: g(4)-{(4), defined by

T(p) , : T; e IH nt P2rp . s in, , t . p e 4 (7, r ) ( 123)

satisfies

l r ,P l . l :0 ( r24)

Proof . By ( l l2) ,

r (P ' rp) : ^ l -1

s- iH rz t P ' - @ Pl p . , iH t l tI

- P L T : e - i H t 2 1 P t ( . @ P Z p e i l t t ) t

Because o f (113) and (76) th is can be wr i t ten accord ing to

rQL p): Pi T. , i I I rz, t Pl @ P? p ' , i I I 11, I

- P t r T : , i H v 2 r ' t P 2 r . p . e i l t , z , t

By once more app ly ing (113) we f ina l l y ge t

f@Li l - P : 1 .

, iH12t P l p . s i t t " t : P l f@) I

F o r a n y p r e { ( 7 . ' r ) * a n d a n y p e { ( 4 ) * s u c h t h a t p r : T r z p . t h em i m o r p h i s m T ( 1 2 3 ) d e f i n e s a n o p e r a r i o n T ' ( p . . ) w h i c h m a p s p , i n r oa(4)- according to

P r - - T t ( p . p , ) : T \ p ) , p r : T r , p ( 1 2 5 )

(note that T ' (p, . ) is def ined on one densi ty operator p, only) .The result (l2a) is actually a property of this operation. which can be

seen by rewr i t ing (124) as

T ' ( P l . p , P ' , . p r ) : P ' r T ' ( p , p r ) ( 1 2 6 )

This expresses the commutativity of the operators Z' and Pl . Consequently.the operat ion f r , def ined by (125) , sat is f ies one of the necessary condi t ions.viz. (67), in order that it be a microlocal operation in C. We shall now alsoderive a theorem that is pertinent to the second condition, viz. (68).

Theorem 8.4. If H,, is a local interaction, satisfying the microlocalityproperty and (104) , then commutat iv i ty of .4 ' ( r ) , r€ C, wi th a l l localoperators P],* B', C a D : A, is necessary and sufficient in order that

P ' r * ( r * tH t ) t P ] ) *B t . e iF t z t ) : p ] , , *B t , C aD :A ( . 12 i )


Proof.

(a) From (119) and (75) i t fo l lows that

"n =',';'.:,I:,' r''"'-"'''''|;* '' ' P?'* e i,r'\)'|)

Since our local interaction operator 11,, satisfies

P j r * H r r : P l * @ P ' r * H r r : H r 2 t . , j : 1 , 2 ( 1 2 8 )

we can, by the method employed in Theorem 8.1, derive the equality

p t * g i n t 2 t : P l * @ P 2 r * , i n r r t : g i H r t r . t , j : 1 , 2 ( 1 2 9 )

Inserting this in the left-hand side of (127), we get

P2cx (e+ iH t i P)r* nt . e- i I I t1t )

: P 2 c * ( e i u t ) c , , t o * B t . e - i t i t ) ( t )

( 1 3 0 )

Since in 11,r, the spatial integration only extends over region C, theexponents only contain operators l,(r) and B2(r) pertinent to region C. So,f rom (130) i t is d i rect ly seen that commutat iv i ty of Pj*Brand f f . *Z '1r ; issufficient ro obtain (127).

(b) The necessity is proved as follows. The equality (121) isequivalent to the adjoint relation

P ) r T r r e i u , r r t P L p e + i H t 4 t : P t o T r r p , C a D : A ( 1 3 1 )

Since this is an identity in r. it follows that

P L T 2 r l H n , . p ' r p ) : P L T 2 r l H n , , p J : 0 , C ^ D : A ( 1 3 2 )

Taking again the adjo int of (132) , we obta in for arb i t rary.Bl :

T r [ 3 f , H r r , ] p : 0 , C a D : Al . -

which, since p is arbitrary, entails

l B t > H r r , . l : 0 , C ^ D : a ( i 3 3 )

Final ly , (133) is only possib le i f

IBL ,A ' ( . ) ] : 0 , r €C I ( 134 )

238 de Muynck

Corollary. The operation T'(p,.), defined by (125), satisfies

P L T ' ( p , p r ) : P L p , c o D : a , p t : T r r p ( 1 3 5 )

Proof. This follows directly by taking the adjoint of (127). I

Summarizing, it is seen from Theorems 8.3 and 8.4 that the operationT' (p, . ) , def ined by (125) , sat is f ies the condi t ions (67) and (68) of a

microlocal operation i l and only if the constituent operators ,4'(r) of theinteraction hamiltonian -F1,, obey the local commutativity relation (134).

Consequently, this is also a necessary and sufficient condition that our localinteraction (101), satisfying (74a), (74b) and (104), is a microlocalinteraction. In the following this result is extended to the more physical

operations (93), presupposing microcausality for the free-field evolutions ofsystems I and 2.

Theorem 8.5. Defining the operation Tt(p... f) according to

T t ( p , p r , / ) : : T r U ( t ) P ' 1 r p , p t : T { p( 1 3 6 )

U ( t ) p : e - i ( H o + H t ) t p g i ( H o * I I r t t t , H o : H r l H ,

and assuming microcausality for the free-field evolutions, the operationT'(p,.,/) is a propagative microlocal operation in C, if the operationT' (p, . ) , def ined by (125) , is micro local . That is (c f . Def in i t ion 6.1) ,

f '@i.p, P'rp, t ) : PI,7 '(P'cp, PIp, t )

f t r f t ( f t r ,p, PL,p, t ) : PLT'(p, p, , t )

PLT ' (p , ' , t ) : PL L I , ( t ) , D o C, : g

U t ( t ) P : s - i n ' t P e ' ' "

( 1 3 7 a )

( 1 3 7 b )

( 1 3 8 )

Proof. In the derivation we make use of Trotter's product formulaG5)

e i (H j+H t ) t : l im (7 iH tQ /n t e iH t2 ( / n t ) n ( 139 )n 4 @

Application of this formula in U(t) Pfp yields a double sequence, which hasthe same l imi t as the sequence of i ts d iagonal terms. So, for j :1 ,2,

u(t)Prrp: ( g - i r t oQ /n ) e iH12Q/n ) )n P I , . 1 t , ,

t r , , , n t e iHo \ I t n t ) n ( 140 )l imn 4 6


By successive applications of (97a) and (l 12) to each term of this sequence,we get

U ( I ) P L P : P t r . U Q ) r t r P . i : 1 , 2 ( 1 4 1 )

Then,

T' (Pt.. P, Pt.. p, t) : Tr U(t) Pi- e P| pI

: P" ,T: U(DPIC A P"P

: r t r , r t@|p , p r rp . t )

thus proving (137a).In order to derive (137b) we first prove the following lemma.

Lemma. lf H n is a local interaction hamiltonian, satisfying (74a),(74b), (104), and the condition of local commutativity as given inTheorem 8.4. then

u{r(t)p[* @ pl* At2 - p[* €) p:.*(ufr(r)pl* e pl.*A") (t42)

Proof. By (75) and (l 19) it follows that

pl* g Pzc* @fr(t) Pt * g pl.*trt2)

:P I * @ P \ . r , ( e iH ' / P l * I P ! *A " . e ' ' ' , ' )

: p l * g p 7 * e i H t 2 t . p l * @ p ? * A ' r . p l * @ p \ . x e i H ' 2 t

: e i t l 1 7 ( t P l * I P l * A t , . , - i H 1 1 r l

H,r, being defined as in the proof of Theorem 8.1.On the other hand, it follows from the local commutativity of the

operators l '(r) and 82(r) which constitute H,, that

e i H t 2 c t P r * @ P 7 * A t 2 . , i l r 1 2 r - t

: g i I I l r t e iHna r P r * @ p?*A t2 . e i t l t 2a t , iH r4 t

: s iHp t p l * @ p l . *A t , . e iHn t

: U(t)* PLx @ Pt'*A"

thus proving the lemma.

240 de Muynck

Returning to the derivation of (137b), we apply Trotter's product

fbrmula (139) , analogously to (140) , according to

u(t)* PI*A ' : l im (It fr(t ln) utQln)) ' Pl.*A\ ( 143)

in which

U t \ t )A t2 - e iHo t A t2 , - i l l o t , Ho : H t t Hz

U f r ( t ) A t 2 - e t H t z t A t 2 e i H t l t

We now show by means of (143) that

u(t1* p\*o ' : ; ,1.,* ,s p:,- u(.t)* pt *A ' 1t++1

This is proven as follows. First consider

{J ' ; r ( t ln) U{( t ln)P[*A ' : Uf r ( t ln) UtQln)PI* t P i*A '

: p[: '@ pl* ' , ,uf,(t lr)p2i,,3) p,[, ,u((.t ln1pl.* I pl, 'A'

which fo l lows by successive appl icat ion of (98) and (142) . Repeat ing th is^ , ^ ^ - , 1 , , , . r l a f t a rP r v L L u u r ! t

(U{ , ( t I n) Ut Q I n) )^ P i . * A '

: Pl.: , , ,6; P2!,, ,(I t ' f ,( t ln) L){(t ln)) ' ' P(\*A1, I ( n ( rr

wh ich . by ( l a3 ) . d i r ec t l y imp l i es (144 ) .From (144) we get . by apply ing P,1*. th . equal i ty

pll s p2.* uQ)* p|.*At : pi* LrQ)* pl ' .A' (145 )

which is seen to be the adjo int o l (137b).We f inal ly prove (138) . This equal i ty , which is equivalent to

, , : r ; *o ' . u6r ' rp :T: P;*A, . (1 . , ( t )T. r p . D n C, : g (146)

is proven by once more applying Trotter's product formula and making

successive use of the equal i t ies (97a) and ( l l2) , to get

u( l r 'z rp: l i ry l I (p .1, , , ,%( t ln)Pl , , , , , ( / , r ( t ln) )Pl .p (141)n a . t ; ' ,


in which Co: C, and the product is an ordered product, T increasing from

right to left. Inserting (147) into the left-hand side of (146) gives

,, lr 'r*o' . t-tg1rl.p

: l im ry np?* l l1ui ,1t1n1Pl. l , , , , , ̂U{( t ln)Pl* , , , , , , , )P},*A'n + 6 . l 2

J I

The product in this expression is successively evaluated, Since z{' only refers

to system l . we have

P'1c* P;*At : PL*A'

For the same reason, and because ol (98), we subsequently get

Pl : , , , , , ,u{ ( t I n ) P i * , t ' : P L! ' ^u { ( t I n ) P 'u" A '

whence

Pi: ' , , , , , ,Lr{rQln) Pl! , , , , , , ( / {( t ln)P),xA1: P2 t , , , , , , , ,u f r ( t I n) P ) , , ! , ,u Y 1t 1 n1 r ; ,* ,1 '- Lrl(t ln)P],*A'

the la t ter equal i ty fo l lowing f iom (127).

C r , , r , n i D , ' r : a

Successive appl icat ion of the same ru les( 138) . in a d i rect rvar ' .

slnce

i f C a D , : g

f inal ly enta i ls (146) . and henceT

QUANTUM ELECTRODYNAMICS

The interaction hamiltonian of quantum electrodynamics.

H , , : I H , " , ( r ) d r : i e I r l r N ( r r r ( r l 7 , A ' ( r l u t ( r \ l ( r 48 )

in which N denotes normal order ing, has the form ( l0 l ) o f a localinteraction operator. Since. however, each of the operators l(r), ry(r). andl'(r) is a sum of a creation and an annihilation operator. it is lairly obviousthat (148) does not sat is fy (103a), (103b) and (104) : tak ing systems I and 2

to represent the photon system and the electron-positron system. respec-

242 de Muynck

t ive ly , we may have, for r€ C, that A ' ( r )Ptrp and Pt .p.1 ' ' ( . ) are nonzero,and ana logous l y f o r N ( r z ( r ) VF ) ) .P ' ) cp and P2 rp .N ( l ( r ) r z ( r ) ) . Th i s makesit impossible to apply the method of the foregoing sections in a direct way,in order to see whether the interaction of quantum electrodynamics ismicrolocal. Notwithstanding this, it is interesting to have an idea about theway this difference interferes with the derivations of the preceding sections,so as to get an impression as to what extent microlocality could be violatedby quantum electrodynamics. In the present section we shall give a short,rather qualitative discussion of the reasons why quantum electrodynamicsmay be expected to be a theory violating microlocality.

Starting with Theorems 7.1 and 7.2 it is easily seen that lor theinteraction hamiltonian (148) Theorem 7. I holds as before. Theorem 7.2.however, is no longer derivable. This is most easily seen by calculating

(01 H,",(r,) r1,n,(.') I0) : - #t# (:2!fuT9 - r)

( p o : E p Q r : E ) ( 1 4 9 )

which is nonzero, the nonvanishing contribution stemming from the vacuumdiagram depicted in Fig. 1.

If we calculate the more general expectation value Tr H,n,(.,) H,n,(.r)pL@p'"p, CaD:9, assuming that the equal i t ies ( l0a) are val id for a l linteraction operators which are not excluded above, then there are severalmore nonvanishing contributions. These contributions stem from processes inwhich the interaction of an electron with a photon is made possible by firstcreating a photon in D or an electron-positron pair in C, which eventually isannihilated again. Such processes are generally interpreted as interactionsinvolving virtual quanta, which interactions only contribute to the self-energies of the electron and photon system, respectively (Fig. 2).

In most applications of quantum electrodynamics vacuum diagrams areomitted completely because they cannot contribute to any real transition.'ro'

Fig. l . Vacuum diagram.


Fig. 2. Sel f -energy diagrams.

since no quanta are involved originating from the incoming field. Also theself-energy terms are taken into account only to a certain extent. As is wellknown, higher-order terms of this kind are lumped together in a process ofmass and charge renormalization, which is interpreted physically as a tran-sit ion in the theory from "bare" particles to "physical" or "dressed"particles: the cloud of virtual quanta surrounding a real (incoming) particleis included in this new theoretical entity, which. henceforth, is thought of asa more or less pointl ike particle.

It is important to note here that, by neglecting both the vacuumdiagrams and these self-energy terms, precisely such processes are neglectedby which quantum electrodynamics violates microlocality. Microscopicallyspeaking there is a nonvanishing probabil ity of detecting a virtual particle inthe physical ground state (the physical "vacuum") of the quantum elec-trodynamical system, as well as in the cloud of a real-particle state.Experimental detection of such virtual particles would mean that quantumelectrodynamics not only would violate microlocality but also macrolocality.This, however, does not seem to be observed, thus i l lustrating the distinctionbetween micro- and macrolocality that was drawn in Section 5. As a marrerof fact, in order that there be no macroscopic violation of locality, ourparticle detectors should not be sensitive to virtual particles but only react to"physical" particles. Although this seems to be a rather trivial conclusion, itactually amounts to a rather severe restriction of the possibil i t ies of probingquantum reality. These restrictions, howeverr seem to be of a practical nature

243

244 de Muynck

only and cannot be seen to have a fundamental origin' If our measuring

instruments are subtle enough, we might get experimental evidence of the

violation of locality. It seems that the Lamb shift. which depends on self

energy terms analogous to those depicted in Fig. 2, can be seen as an indirect

evidence of this effect: certain values of the energy observable (which is not a

/occl observable pertaining to a bounded region of [1.1 would be different if

the interaction terms violating microlocality are absent in the interaction

hamiltonian of quantum electrodynamics. So, even if there are fundamental

objections against a direct observation of nonlocality by means of the

detection of a virtual particle (for instance. because then the detector would

have to supply an infinite self-energy to transform a virtual particle into a

real one), it seems possible to obtain some experimental information about

the way the nonlocal virtual-particle cloud changes in a real-particle tran-

s i t ion.By considering the possibil i ty of proving Theorem 8. I for quantum elec

trodynamics, we meet a second source of violation of microlocalit l ' which

has to do with real quanta only. If we start. as before, with a state in which

the photon system is localized in C and the electron-positron system in D.

with C o D : A, then there is also a contribution to Tr H'1t2Pl ?: Pl,p trom

that part of II ,, which describes electron-positron annihilation. By this term.

for instance, in D a photon can be created which, because of the symmetrl '

of the wave function, is instantaneously correlated with the photons in C.

This can be interpreted as a violation of microlocality/causality. because the

newly created photon is not distinguished from the photons which were

originally present in C. This implies that the photon in D constitutes one

single system together with the photons in C, thus entail ing a "' iolation

of( 1 1 2 ) f o r T : L

It seems that microcausality can only be implemented into a theorf in

which the photon created in D can be considered as a system which is

dilferent from the photons in C. As we showed elsewhere.'rt rn' such

theories, in which the identical particles are treated as distinguishableparticles. can be constructed. So. the violation of microcausalit l discussedhere seems to be a consequence of the usual f ield theoretic description. rathcr

than caused by actual physical processes. and is essentially spurious. Since.however. it is a feature of the usual f ield theory, it could have consequenceson the macroscopic level of measurement. That this is not the case is oncemore a consequence of a restriction of the class of measurements: onll ' such

observables are compatible with the usual f ield theory, which do not discern

between, on the one hand, a product state of the photons in C and those in

D, and, on the other hand, the state which is obtained by symmetrizing thisproduct state. Such measurements are not sensitive to the violation ofmicrocausality discussed here. So, a real test requires a measurement which


is outside the class of measurements described bv usual quantum fieldtheory.

From the examples discussed here it is seen that quantum elec-trodynamics cannot be considered to be a microlocal theory, even if the free-field evolution is thought to be microcausal. Of course, contributions to theviolation of microcausality due to the free-field evolution wil l add to the onesdiscussed here. A detailed discussion of the whole problem is beyond thescope of the present article.

We close this discussion of quantum electrodynamics by noting thatmeasuring instruments consist of the same kinds of particles the objectsystem is composed of. This implies that due to the interaction of object andmeasuring instrument we may expect nonlocal effects analogous to the Lambshift. caused by a nonlocal influence exerted by the measuring instrument onthe virtual'particle cloud of the object system. It seems not improbable thatthis nonlocal interaction between measuring instrument and object systemmay be viewed as the physical basis of the fundamental inseparabil ity ofthese systems in the measurement act.(r ') On the other hand. we may expectthe nonlocal interaction to be far less effective if the distance between trvosystems is great. Consequently, this kind of inseparabil ity presumably isrestricted to systems which are in close interaction.

IO. DISCUSSION AND CONCLUSIONS

ln the present article the problem of (non)locality of the microscopicworld is studied on the basis of the ideas of local operation and localmeasurement. Since we can. at least in principle, make these local operationshappen at wil l, the theory seems to fulf i i l the criteria of what is called byd'Espagnatls) an entailment theory of causation. Nonlocality would beproved if a local operation would have observable effects in a region which iscausally disjoint.

It was demonstrated that the interaction between object system andmeasuring instrument can be, under certain specified conditions, a localoperation performed on the object system. This result would endorse theconclusions drawn in Refs. 8 and 9 (cf. Introduction) that in the case of theEPR experiment the two measurements are completely independent.However, from our discussion of quantum electrodynamics it is also clearthat realistic physical interaction hamiltonians may have terms violatinglocality. Apart from these interaction effects, the free-held evolution alsointroduces nonlocal/noncausal features into quantum mechanics.

Before the invention of the Bell inequalit ies, virtually no one realizedthat quantum mechanics might be an essentially nonlocal theory. The

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postulate of local commutativity was accepted as a natural warrant of thenondisturbance of local measurements performed in causally disjoint regionsof space-time.

We have not been able to demonstrate that local commutativity alonewould be suffrcient for nondisturbance. So. there are still the twopossibil i t ies, viz. that either the violation of locality is merely a microscopiceffect that has no influence on macroscopic measurement results, or that theclass of local measurements considered up unti l now is too small toencompass such measurements which are sensitive to the quantumnonlocalities. The first alternative seems to be favored by the experimentaldata which show the measurement results of local measurements to be inde-pendent of the presence of a distant measuring instrument. However, as wesaw in Section 9, a careful observation of energy (which, as a local obser-vable, pertains to the whole of trr) may reveal certain consequences ofnonlocality. This points in the direction of the second aiternative. Perhapsthe correlatiol? measurements involved in tests of the Bell inequalities, inwhich a fast switching of the measurement setup is performed (as proposedby Aspect(40)), wil l be sensitive enough to test quantum nonlocality (thepreliminary experiment(ot is not sich a testl). It wil l be extremely diff icult,however, to interpret the results of these experiments, if they deviate fromquantum mechanics.

Several authors(a'-ai) have expressed an expectation that theseexperiments wil l also have results in accordance with quantum mechanics.This expectation may be based on the supposition that, if a localmeasurement is not disturbed by another, distant, local measurement, thiswil l also be the case if that distant measurement is switched as in the Aspectexperiment. It would indeed be rather surprising if the nonlocal effects wereso strong that the measurement results of one stationary local measurementcould be influenced by switching the arrangement of a distant localmeasurement, since this would entail the possibil i ty of faster-than-lightsignall ing. In fact, the main part of this article is devoted to a demonstrationof the feasibil i ty of this very position within quantum mechanics, by showingthe possibil i ty of a quantum mechanical theory of local operations. theinfluences of which behave at least macrocausally.

It is, however, less clear whether, under switching conditions, we have toexpect quantum mechanics to hold also for correlation data. Such data canonly be derived from measurements performed on both particles in the EPRexperiment. So, in the switching experiment the particles interact withnonstationary measuring instruments. Now, if there is a nonloral interactionof a measuring instrument with the particle neorest to it, the switching of theinstrument may cause observable effects which are analogous to the Lambshift.


Stated differently. by changing the measurement arrangement, the stateof the object system is also changed (see also Section4 of Ref. l2). If thechange is performed slowly, this need not have observable consequencesexceeding those described by quantum mechanics. However, if the change isso fast that the interaction of incoming particle and measuring instrumentcannot be treated adiabatically, deviations from the quantum mechanicalpredictions can be expected. Since, presumably, the commutation frequencyof 250 MHz, as proposed in Ref. 40, is too low to reach the nonadiabaticregime, the expectations of Shimony(ar)and Vigier et al.G2'43) may cometrue. Indeed a switching experiment at 50 MHz shows excellent agreementwith quantum mechanics,(oo) The Aspect experiment, restricted to thesefrequencies, seems to probe only the inseparability of the two subsystems inthe EPR experiment.

Deviations from quantum mechanical predictions are to be expectedonly if the measurement conditions apply to a situation which is outside thedomain of application of quantum mechanics. The boundaries of this domainare unknown, and measurements l ike Aspect's switching experiment aremeant to explore its extension. The leading idea of this experiment is that, byswitching the measurement arrangement very fast, it is possible to break theinseparabil ity of the distant parts of the object system. For this it is thoughtnecessary that the switching time be smaller than the time-of-fl ight of a l ightsignal between the two measuring instruments of the EPR setup. If ourconjecture (cf. Section 9) is correct that the nonlocal interaction between thedistant subsystems cannot, because of the great distance, be seen as animportant source of inseparabil ity having observable macroscopic conse-quences, we cannot expect much effect from an effort to break it.

The situation is different if we consider the interactions of each of thesubsystems with the measuring instrument it enters. Here the nonlocalinteraction is expected to induce inseparabil ity of subsystem and measuringinstrument, so as to form, what is called by Bohr, an indivisible whole ofobject and measuring instrument. Breaking this kind of inseparabil ity wouldseem to be possible only if the switching time is shorter thandfc, dbeing thedimension of the commutator. So, even though Aspect's switchingexperiment yields the correlations predicted by quantum mechanics, this doesnot seem to be interpretable as a consequence of a fundamental nonlocalityof the world, but it follows because the experiment has remained inside thedomain of validity of quantum mechanics. For deviations from quantummechanics the switching frequency should presumably be one or two ordershigher.

Our discussion of quantum nonlocality is far from final. Much work hasto be done in order to clarify whether quantum nonlocality is of afundamental nature, or can be derived from an underlying local theory. It

de Muynck

was noted by d'Espagnat(5) that a study of the existence of faster-than-light

influences cannot confine itself to the level of mere quantum mechanical

description of measurement results, but should rely on the notion of attribute.

or property, of the microsystem. It was demonstrated in the present article

that a quantum mechanical treatment of measurement makes it possible to

improve our understanding of nonlocality also on the quantum level. We

agree with d'Espagnat that an understanding of quantum nonlocality on the

basis of a more fundamental /ocal subquantum theory wil l need notions

which transcend those of quantum mechanics. It is questionabie, however.

whether such theories wil l predict, for EPR-like experiments, deviations from

quantum mechanics as large as represented by the Bell inequalit ies, if the

interaction of object and measuring instrument is duly taken into account'

ACKNOWLEDGMENT

The author would l ike to express his sincere gratitude to Professors Jan

Hilgevoord and Boudewijn Verhaar for careful reading of the manuscript and

many valuable suggestions for improvement.

APPENDIX A. EXPLICIT EXPRESSIONS FOR PC AND PE

In this Appendix the projections P( and P| of Section2 are exhibited ina representation that makes no use of the existence of the improper Fockspace vectors (3). Instead, all derivations can be performed on the basis ofthe eoualitv

(0 i v ( * , ) . . . vG) r l r+ (y , ) . . ' v r ' ( y , ) i 0 )- , , ,

; d ( " , - i l y , ) . ' ' 6 ( x , - I I y , )

which follows directly from the canonical commutation relations (2) (\-r,

ind icates summat ion over a l l permutat ions of (y1, . . . ,y , ) ) . Using (A. 1) . i t canbe demonstrated that the operator on Fock space defined by

tt

Pr : \ ' Pc (N)

ar, . ' . ) rdr"

rTrr(r ,) . ' . rz+(r") 0)(0 ,r [ r) . ' . vt(r r) (A.2 )q t

- \ ' ' l,rrt I

. \ : { , i r " (

( A . l )

(A .3)

is a projection operator, transforming the state (1) into (4). that is.

I Y ) . : P , lY )


In (A.2) P.(N) is also a projection operator, selecting the states with

precisely N_bosons in C.With C the complement ol C in iF3, the operator P.(1/) is one of a

family of orthonormal projection operators on Fock space, defined by

l ' , r r lP , ( , r y . S ) : r u | d f t - . . I l T r s I d r s . , . . . l _ d t , u / * ( r r ) . . . i / * ( r r )

. S : ( l / - S ) l . c . ( - . . . c

. l 0x0 l y ( r " ) . . . y ( r , )

S : 0, 1,.. . , 1y' (A.4)

Pc(l/, S) corresponds to the ly'-boson states of which S bosons are in C andN- .S i n C . C lea r l y ,

P(.(N) : P. (1/. 1/) (A .s )

l t is easilv demonstrated that

Pc(N, .S) :1 ( A . 6 )

and

P. (N, S) PrlN' , S') : d,r,^, d.., P(.(1/, S) ( A . 7 )

Using (A.2) in the def in i t ion (17) of the or thogonal pro ject ion P. 'on v(7) .

the subspace of states localized in C can be defined by (18). The represen-tation of the nonorthogonal projection P. on 7(7). obeying (19) and (20).is given by

p . , : ( ' i ' I I Iio r-o \-o s: r| .\ !

' ) ,d ' , " ' I d* , . l : , or , ' ) ) ,or , , l i t - ' t ' , " ' l - " '

( A . 8 )

' (0 v(" , ) " ' v (xs) v@). . . v@) pv+(v ) . " rzr (v . ) v ' (2 , ) " ' v / - (2 , ) i0). r / ' (x , ) . . . y ' (xs) l0) (0 y0, ) . . . v0 r )

Analogously, it is shown that the adjoint P/ of P., defined according to( I 1) , is g iven by

250

6 :

PtA :.:. ), I

" 7r N:

),d*, J '*. I ,or, " ' l ,ar, l .az, " ' i1-dz,

. (01 , r0,) . . . y(y.) Av' \ ) . " y ' (* ' ) l0)

. , t t ( y , ) " ' v ' ( y r ) , y ' ( r , ) . . . , t t ' ( r r l | 0 ) (0 | r y ( x ' ) " ' , f ( " r ) v@ ) " ' v@ r )

and that the projections P$ satisfy the properties (26f(28) characterizing aquasilocal algebra. From (A.9) it is easily derived that on this definit ion the

field operators are local operators:

Pt vG): v$), x € c

PF,z* ( * ) :v /+ (x ) , x€C

P l r T ( x ) : a . x € C

P f r z * 1 x ) : 0 . x € C

Then, defining

v ( f ) : ) , d x f ( x ) v $ )

with/(x) a scalar function, (A.10) yields

pt v(.f): l ,a* -f(x) v6) ( A . l l )

and analogously for y(.f) ' . So, VU) is a local operator in C if /(x) has itssupport in C. Then, tak ing in (A.11) / (x) : -Vd(x-y) , i t fo l lows that

Pf vra(x): I oto:'' x € interior C

x € interior C

de Muynck

(A.e)

( A . 1 0 )

(A . r 2 )

and analogously for Vy'(x).It is also possible to demonstrate that any polynomial of f ield operators

represented by the right-hand side of (A.11) is again a local operator in C.

APPENDIX B. AN ALTERNATIVE DEFINITION OFLOCALIZATION

In order to avoid the consequence (22) stemming from the definition oflocalization as expressed by (21), viz. that the vacuum is localized in an1'


region of i l i3, we can change (17) slightly, to the effect that the vacuum stateis no longer included in the summation. Thus, defining

0 . : , , ) 0 , { n } { 0 f ) ( { n } { 0 } 1 , Q i p : Q , , p Q , ( 1 7 ' )

and

QcP: I ( l n l i r r l i p l \m l \ n l ) l { n f { o f ) ( { r r } { o fl n l ln l ln l

( { z l . { u } + { 0 } )

we have

Qi : Q,

Q[Q, : Q ,

Q, Q t : Q t

and

QtA : ) ' ( { " r f { 0 l l l { r f { 0 i ) l \m l \ n l ) ( 1n l1 i r } ll n l l m i l n l

( { r l . l u l r { 0 1 )

QT. : QT

Obviously, the operators pfl constitute an algebra.From the equality

rr QIA . Qep : <ol QtA ]0) : 0

which obtains for all A and all C, it now follows that

0( 10) (01 :0 v (

(12 ' , )

( 1 9 ' , )

( 2 0 ' )

( 1 0 ' )

( 1 6 ' )

(22 ' , )

(6 ' , )

( 1 ' , )

So, in this definit ion, the vacuum state is not localized anywhere in i ' .r.

Since the operators defined by (6') constitute a subset of the set ofoperators (6) defined by Pf, both the Schlieder condition and localcomrrutativity remain valid properties.

Notwithstanding the attractive property (22') of the localization basedon Q,-, this definit ion does not seem fully appropriate yet. The reason forthis is that the algebra of operators (6?) does not contain certain operatorswhich, from a physicist's point of view, deserve the qualif ication of a localoperator. For instance, out of the observables

\n l \n l ) (n l \n lt m l

t { t de Muynck

representing the probabil it ies of f inding {n} quanta in C. the observable with

l rz i :10f is not conta ined in th is a lgebra, contrary to the observables wi th

l t ' i + i 0 f .Directly related to this fact, we have

e I1 : \ - l r r l l r 7 lX {n l \ n l i + I ( 9 ' )

, , i,l 'l 1,1,,u'hich implies that, if the density operator has the vacuum as one of its

components. then

T r Q , p + T r p ( 1 3 ' )

So Qt is not a mirnorphism, contrary to P, . Clear ly . the requi rement that

the localization definit ion corresponds to a mimorphism guarantees that the

class of local observables is rich enough in order to encompass all physicallt '

acceptable observables.

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a quantum mechanical theory of local observables and local operations

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