A Quantum Mechanical Theory of Local Observables and Local

V o l . 1 , 1N, o . 3 , M a r c h1 9 8 4 Printed in Belgium

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

A QuantumMechanicalTheory of Local and Local Operations Observables Willem M. de Muynck' ReceiuedMay 2, 1983 Local operators are characterized mathematically b1' means oJ projection operators on the Banach space of bounded operators. The idea of microlocalitl'. as opposedto macrolocality, is implemented into the theory 56 as to enable us to deJ'ineoperations that are strictly local. Necessary and sufJicient conditions are intestigated in order that the interaction of a local measurementinstrument tt'ilh 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 issueof 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 the microphy'sicalworld has receivedmuch attention,mainly in the context of experimentssuch as thoseconsideredfor the first time by Einstein,Podolsky, and Rosen.(')The study of this problem has picked up considerableimpetus These inequalities,which are by the discovery of the Bell inequalities.(2) derivedon the basis of a theory of so-calledlocql hidden variables,express the fact that such theoriesdo not allow correlationsbetweendistant particles to exceed a certain value. From the experimentalviolation of the Bell inequalities,(r'1)itis often inferredthat the microphysicalworld is nonlocal, and that there can exist influences(')or even signals(o)thatpropagatefaster than light. Since quantum mechanicsgives a good descriptionof the excess correlation revealed by the experiments, and hence seems to provide an t

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

r99 0Ol5'9018/84/03000199$03.50/0c 1984PlenumPublishingCorporation

200

de Muynck

of a nonlocal world. it is often consideredto be itself adequaterepresentation a nonlocal, or even noncausal.theory (see.e.g., Ref. 7 and referencescited 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 mechanicalanalysisof the measuringpocess,(n'it was concluded that the measurementresults of a measurementperformed on one subsystemare completely independentof which measurementis performedsimultaneouslyon another subsystemthat is located at a big distance.In these anaiysesquantum nonlocality, if it exists.does not manifestitself on the level of the measurementresults. l'he idea that quantum mechanicsis basically local is also transparent ) h e r el o c a l o b s e r v a b l easr e p o s t u l a t e d i n t h e t h e o r y o f q u a n t u mf i e l d s . ( r 0 ' ' rw to commute if they pertain to regions that are causally disconnectedlthe postulateof local commutativity), This postulateexpressesthe expectation will not that, becauseof relativisticcausality,mutually distant measurements disturb each other. In accordancewith this expectationa reiation between was local commutativityand mutual nondisturbanceof distant measurements demonstratedbv means of a quantum mechanical analysis of the joint m e a s u r e m e notf t w o o b s e r v a b l e s . ( rH 2e ) r e , l o c a l c o m m u t a t i v i t yw a s d e r i v e d of mutual nondisturbanceof trvo distant measurements. as a consequence which nondisturbancewas requiredto hold for the measuringprocess. Since quantum nonlocality can be demonstratedin a direct way only if by the presupit manifestsitself as a disturbanceof a distant measurement. positiott of nondisturbancewe restricted ourselves in Ref. 12 to those measurementproceduresin which the quantum nonlocality, if it exists,has no 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 different point of view in the presentarticle,the conversequestionis considered,viz. whether local commutativity is sfficient for nondisturbance. It is investigatedwhether a quantum mechanicaldescriptioncan be given of a joint measurementprocedure in which two distant measuring instruments interact locally with one and the same object system.in such a way that the interactionprocessestake place in a mutuall,vnondisturbingway. ln this investigation both object and measuring instruments are describedby quantum fields,thus avoiding the seriouslimitation inherentin the treatmentsof Refs.8 and 9, that each of the two measuringinstruments is taken to be sensitiveto one of the subsystemsonly. It seemsthat by this limitation the possibilityof obtaining experimentalevidenceof nonlocality is excludedbeforehand.As a matter of fact. since there is no referenceto the distancebetweenthe measuringinstruments,the reasoningsof Refs.8 and 9 give the same results if the measuringinstrumentsare not far apart. This

Quantum Mechanical Theory of Local Observablesand Local Operations

201

indicatesthat these analyseshave no bearing on the problem of quantum nonlocality. A genuine proof of nondisturbance of mutually distant measurementsshould demonstratein which way the disjunction between measuringinstrumentand distant subsystemmakes the interactionbetween thesetwo ineffective.It is the purposeof the presentarticle to study quantum (non)localityfrom this point of view. ln order to be able to perform such an investigation.we will first have to clear up the nature of the locality involved in the interactionbetweenthe q u a n t u mf i e l d s .I n t h e t h e o r yo f l o c a l q u a n t u mf i e l d s , ( r 0 ' r r ) l o c aolb s e r v a b l e s . or. more generally.local operators,are defined to belong to some reglon as representingphysical either of space,timeor of tilr, and are interpreted(r0) and ry(x) field operators,y*i-r) Thus. the perlormed in region. this operations operators hence are particle and in x, respectively, a and annihilate, create pertainingto 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 deducethat local commutati[i/.f is not necessaryfor an interpretationof field operatorsas representinglocal 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 representedmore faithfully'.Such a criterion is developedin Sections2-4. Operatorsobeying this criterion will be calledmicrolocal operators.becauseit derivesfrom the tnicroscopicpropertiesof the quantum mechanicalstates.By way of contrast. local commutativity, being derivable from the nondisturbance of measurements. essentiallyis a mqcroscopicproperty. which probably has referenceonl\ to operatorsthat can play the roles of observables.If mutual n o n d i s t u r h a n coef d i s t a n t m e a s u r e m e n tiss i n t e r p r e t e da s a c o n s e q u e n coef macrolocalitvor macrocausality.then local commutativity of observablesis the reflectionof this macrocausalityin the quantum mechanicalformalism. It is the purposeof this article to investigatewhether it is possibleto devise.u'ith the help of microlocal fields,joint measuringprocessesthat are mutually' nondisturbing.In doing so we are able to study the interrelation betweenthe conceptsof microlocality and macrolocality,and to implement t h e s u g g e s t i o n ( rr'6r ) t h a t q u a n t u m n o n l o c a l i t y m e r e l y i s a v i o l a t i o n o f microlocalitl' or microcausality.without macrocausalitybeing violated. To this end in Sections 5 and 6 definitions of the four notions micro- and macrolocality, and micro- and macrocausalityare proposed,in order to make these notions more precise.After defining in Section7 what is to be understoodby a microlocal interactionbetweenquantum fields, in Section8 necessaryand sufficientconditionsare investigatedfor this interactionto be microcausal. ln Section9 the theory that is developedin the preceding

202

de Muynck

sectionsis applied,in a rather provisionalway! to the interactionof quantum electrodynamics.Finally, in Sectionl0 implications which our theory may have on the issue of quantum nonlocality as studied in relation to the Bell inequalitiesare discussed. We close this Introduction by noting that, in developingthe notions mentionedabove and in proving the theorems,we did not pursue maximal mathematicalrigor, in order not to obscure the conceptual issues.Thus, theorems are proven only for bounded operators with a discrete spectrum, whereasin Section2 use is made of a basisfor Fock spacewhich only exists in the improper Dirac sense.(In AppendixA it is shown that the analysisof Section2 can also be carried throush without referenceto such a basis.)

2. LOCAL ALGEBRAS: EXPLICIT REPRESENTATION FOR A LOCAL SCALAR FIELD The presentsectionwill have mainly a heuristiccharacter.We consider the simplestmodel of a quantum field, viz. the scalarfield obeyingcanonical commutation relations,in order to develop a mathemaliccl criterion for an operator to be a local operator pertaining to some region C of conhguration space ]fi3. In this way we can obtain a better insight into the nature of the quasilocal algebra generatedby these operators.By exhibiting a concrete representationit is, moreover.shown that the more generalaxiomatictheory to be developedin Section3, is not an empty structure.

2.1. Localized Quanta The Hilbert space.7 of the systemis taken to be the usual Fock space. A generalstate of the field is given as(")

''' p " ( r , , . . . , r ,vr r) * ( r ,.). . y t ( r n ) lyl): \' (N!) | t u , . . . ' l1i r l d r . n l0) (l) 'llll N:0

I n ( l ) P . ( r , , . . . ,r . ) i s a function which is symmetric under permutationof the coordinates.The field operators 14*1r)and r4(r) satisfy the canonical commutationrelations

I v $ ) ,v r ( r ' ) ) : d ( r- r ' )

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

(2)

Quantum Mechanical Theory of Local Observablesand Local Operations

203

T h e N : 0 t e r m i n ( l ) s h o u l d b e i n t e r p r e t e da s Y o l 0 ) , Y o a c o n s t a n t ,l 0 ) The set of being the vacuum state of the field, defined by y(r)10):0. vectors t''rlt'(:r,)...rti*(.,u)10) (3) ]r,,...,r"): (N!) can be consideredas a complete,orthonormal set of improper vectors in Fock space.(r8) Our heuristicsstarts from the observationthat the vectors (3) might be interpreted as describing localized states of the field. More generally, a state r, in which all quanta are localizedin some region C of lF, might be obtained a n a l o g o u s l yt o ( 1 ) a s f -

y)c:

\' \:0

(Nl) "'I

a r r . . . l a . r P , ( r ' , . . . , r r ) y * ( r , ) . ' . , i r t ( t r ) 1 0 )( 4 ) '('

Since the creation ofsuch a localized state in C from the vacuum ]0) is a physical operationperformedlocally in region C, it seemsnatural to require that the operatorsU'(r), r € C be local operators,pertainingto region C (see alsoRef. 19). Although the interpretationof a local operator as representinga local ph-vsicaloperationis analogousto the one adoptedby 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 of Minkowski space,we take here IRr as the conhguration space (in Ref. ll these two interpretationsare consideredside by side). lt is clear that this choice restricts our treatment to a nonrelativisticone. For this reason we shall not requireour local operatorsto obey Lorentz covarianceas is done in Ref. 11. Also local commutativity, being defined here as commutativity of operatorspertainingto disjoint regionsof Rr. is not requiredfrom the outset Ialthough eventuallylocal commutativity will turn out to be necessarylor the existenceof certain microlocal interactions(cf. Section8)1. Instead,in the following we shall try to give a mathematicalcharacterizationof a local operatorpertainingto some region C of lRr. which is in line with the idea of a physical operationperformedin C. To this end we denote the states (3) in the following according to I n t h i s n o t a t i o n n c a n b e i n t e r p r e t e da s t h e liti' lnl). E(n+n):Iy'. occupation number of the single-particlestate d(r - ro), ro € C, and {n} as the set of these numbers if ro ranges over C. For {li} an analogousinterp r e t a t i o no b t a i n sw i t h r e s p e c t o C . t h e c o m p l e m e not f C . l t i s i n t e r e s t i n tgo note here that, although in the following the heuristicsis basedon the delta function representation,the notions to be developedare independentof this special representation.For this reason our theory encompassesboth the

de Muynck

204

notions of strict locality and of essential locality (Ref. 20; see also Section4), the latter corresponding to a representationin which the deltafunctionsare replacedby a completeorthogonal set of nearlt, localized single-particlestates. 2.2, Local Operators We shall now give a characterizationof which operatorson the Hilbert space,,7',having l1nl, 1ri)) as a completeorthonormal set of states.are to be consideredas local operatorspertinent to region C of Pr. Such operators should obey the following two requirements: l.

They should not affect the number of quanta outsideC.

2.

The effect of the operator inside C should be independentof the presenceof quanta outsideC.

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

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

(s)

Operatorsl. obeying (5)can be characterizedas follows. Considerthe set .d(f) of boundedoperatorsA on .7. Define a mapping PI ol ,19(7) into itself, accordingto

PtA:

\(n)\illml

( { n l .1 0 }A I \ m l , 1 0 l 1 ) r f ,{ i 7 1 2 ( l n{ n r f-,}

(6)

Sincethe operatorP|, definedbV (6).is easily seento be idempotent,

(PI)' : PI

(7)

and is continuous in the uniform topology of ,4(f), it is a projection operator on.7(7). The rangeof Pf follows from (5) and (6) to be precisely the set of local operators,4. pertinentto C. So, we can characterizethis set by meansof the equality

PIAr': tr,

(8)

It is easily seenthat PII : I,

for all C

(9)

that is, the unit operator1 is a local operator,perinentto any region C. Let 7(7) denote the Banach space of traceclassoperatorsB on,,{. Then Tr AB, A e 3(f) is a linear functional on 7(.7).In fact, taking the

Quantum Mechanical Theory of Local Observablesand Local Operations

usual operator norm as a norm in ,*(7), this Banach spaceis the dual of 7(f) (see Refs. 21,22).It is now straightforwardto show that P/ is the adjoint with respectto the linear functional TrAB, of an operatorP(- acting on / ( 7) and defined by

PcB:

\'

( 1 n f{, , r iBl \ m \ , 1 , 4 1 ) l 1l 0nf)f ,( l r r f .{ 0 i

lnll nllml

Be v(r)

(10)

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

(ll)

F r o m ( 7 ) a n d ( l l ) i t i s c l e a r t h a t a l s o P , , i s a p r o j e c t i o no p e r a t o r .

Pi _ P, M o r e o v e r .t a k i n gI : 1

(12)

i n ( l l ) . i t f o l l o w sf r o m ( 9 ) t h a t TrPr.B:TrB.

R e s t r i c t i n gB t o t h e p o s i t i v ec o n e{ ( ; t ' ) , P , B b e l o n g st o 7 ( - Z ) * . T h u s . B € 7(7)

t

Be;(tf)

(13)

o f 7 ' ( 7 ) ^ ( 1 0 ) i m p l i e st h a t a l s o

- . : ,P r . B e ' d ( 7 ) t

(l4)

't-(f)* with TrA: l. T h e m a p p i n g( 1 4 ) , i f r e s t r i c t e dt o t h e o p e r a t o r s . Be i s a n e x a m p l eo f a m a p p i n g ,s o m e t i m e sc a l l e da m i m o r p h i s m , ( 2 rw) h i c h i s a linear mapping of the base of a base norm space into the base of another b a s en o r m s p a c e .I n d e e d .l r o m ( 1 3 ) a n d ( 1 4 ) w e s e et h a t . i l B i s a d e n s i t y operator.P, B is also a density operator.Since TrA.P:TrAr.PtP

(15)

it is clear that Prp embodies the same information as p concerningthe e x p e c t a t i o no s f o p e r a t o r s1 , . p e r t i n e n tt o C . F r o m ( 6 ) a n d ( 1 0 ) i t i s a l s o d i r e c t l vs h o w nt h a t

TrA7,Prp: (0 ,4. 0)

(l6)

0 ) b e i n g t h e v a c u u m s t a t e( j 0 ) : ] { 0 1 ,{ 0 f ) ) . S o t h e i n f o r m a t i o nc o n t e n to f P,.p. relating to operators 16" pertinent to the complement C of C, is equivalentto that of the Fock spacevacuum. This suggeststhat the density operatorP..p describesa state in which all particlesare localizedin C. We shall now show that this is indeedthe case.

de Muynck

2.3. Localized States In order to give a definition of a localized state we introduce the projection operator P.' operating on 6 (7) according to Pip:

PrpPs,

pc: \l { n } ,{ 0 } ) ( { n } ,{ 0 } ,

p e rQn

(17)

We shall say that p representsu'1,*. localizedin C, if and only if Pip:

p

(18)

that is, p belongsto the range ofP/, From (10) and (17) it can directly be proven that PIP]:

(19)

nr-

Since (19) implies that Pr.p is in the range of P,l, we concludethat for an arbitrary density operatorp the projectedstatedescribedby Pcp is localized in C. From the definitions(10) and (17) we can, equivalently,prove that PrP;:

(20)

P/

The two equalities(19) and (20) imply that the projectionoperatorsP,. and P| have equal ranges.As a matter of fact, in Hilbert-Schmidt space(with inner product Tr A+B) the Hermitian projection Pt- would be related to the non-Hermitian projection P. as orthogonal and nonorthogonalprojections, respectively,onto the same subspace.For the characterizationof a statep.. as a state localizedin C, i.e.,having only quanta inside C. the projectionP,. is as suitableas P. is. For this reason,the relation P,'9: P

Ql)

can be used as an alternativeto (18) as a definition of a state which is localizedin C. By choosingthe definition (21)we can take advantageof the fact that not Ptp but Prp representsthe information contained in region C: c o n t r a r y t o ( 1 5 ) w e h a v e i n g e n e r a lT r A r p * T r A r P ; p . From (10) it follows directly that Pc 10)(01: 10)(01,

for all C

(22)

that is, the Fock space vacuum is localized in any region of lFi. This somewhat metaphorical though not inappropriate result stems from our definition (6) of a local observable.It is shown in AppendixB that an alternative dehnition is possibleon which the vacuum is localizednowhere.Since

Quantum Mechanical Theory of Local Observablesand Local Operations

207

by this alternative definition certain observablesare excluded that are physically relevant,we stick to the definition presentedin this section. 2.4. Local Observablesand Local Algebras From the definition (6) of Pf it is easily seenthat the local operators , e shallindicate 1 , . s a t i s f y i n g( 8 ) c o n s t i t u t ea * a l g e b r a . ( r 0 ' r rH) e n c e f o r t hw this algebraas the local *-algebra (/, of local operatorspertinentto region s n a l g e b r aw i t h r e g i o nC . C. Note that also (21)associatea The selfadjoint operators of (1c.are the local obseruablespertinent to C. If QrP^

(23)

is the spectralrepresentation, it is easily shown that P^ : \

(24)

P^ . r a t Inl

in which P,,,u, is the projection operator of a subspaceof Fock space spannedby vectorshaving definitenumbers {li} of quanta in C. From (24) it f o l l o w st h a t

(2s)

PIP^- P^

It is seenlrom (24) that each eigenvaluea,, of the local observableis highly d e g e n e r a tsei n c ee i g e n v e c t o rhsa v et h e f o r m L , , C , , , l 1 n i . l n l ) , w h i c h b e l o n g t o t h e s a m e e i g e n v a l u ea * f o r a l l { t r f . B e c a u s eo f ( 1 5 ) a n d ( 1 6 ) i t s e e m s reasonable to associate local observables1,. with local measurements performedin region C. Ii D is anotherregion of llrr, the projectionsPf and P|.,, are defined analogouslyto (6). By specifyingthe states(3) accordingto whetherquanta are locatedeither in C\C a D, D\C ^ D, C o D, or C l) D, it can be shown that

Ptno: PIPfi: P|PT

(.26)

From (26) it followsthat the local operators satisfythe properties t.

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

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

(isotony)

(21) (28)

de Muynck

Thesepropertiesmake it possible('0'tr)todefinethe quasilocalalgebra r1 as the C*-inductive limit of the union of the local algebras17. pertinentto ail 3. boundedregionsC of rl Then, t1 is a C*-algebra.From the definition (6) it is straightforwardlydemonstratedthat all operatorspertinentto disjoint regions are mutually commutative. This verifies the property of local commutativity of the local observablesof f 1, which, here, is a direct consequenceof the model. It is possibleto construct in an analogousway a quasilocal algebra based on the local x-algebrasgeneratedby the projectionsP,. Although from a mathematicalpoint of view both algebras are equally interesting' becauseof the physical interpretationwe shail restict our attention to the quasilocal algebra generatedby PI.The first reason for doing so is that generallythe product of two density operatorsis not a density operator.So. there cannot exist a local algebraof densitl'operalors. 2.5. The SchliederCondition A s e c o n dr e a s o ni s d e r i v e df r o m t h e S c h l i e d e cr o n d i t i o n . l r t ) w h i c hf.o r our definition of a local algebra,reads A(..Ar):0,

C^D:A.>Ac:0

or

Ar,:0

(29)

This condition which, in its relativistically generalized form. can be demonstratedto hold for field theoriesof the Wightman type.(tt'25)plays a prominent role in recent researchon the causality propertiesol local field theories(for a review, see Ref. l2). It is easily seenthat (29) is satisfiedby the local operatorsdefinedbV (8). That the Schlieder condition (29) is not automatically fulfilled for arbitrary quasilocalalgebrascan be seenfrom the fact that. contrary to the one based on P$. the quasilocal algebra based on P,. does nol obey the Schliedercondition. This follows most easily from a considerationof the

o p e r a t o {r n s f .{ 0 f) ( { n i .{ 0 i 1 {, n l + 1 0 1a n dl { 0 11. t l ) ( 1 0 1f .t l , 1 t l + { 0 1 , which belongto the rangesof P.: snd P.=,respectively.Multiplication of these operatorsgives zero. without vanishingof either of the two operators. This example answers an observationmade in Ref. 26 regarding the question of the universal validity of the Schliedercondition for quasilocal algebras.Evidently, it is possibleto conceiveof such a definition of local operatorsthat the Schliedercondition is not satisfied.ln order to obey this condition. it is not sufficient that the operators are merely pertinent to disjoint regions. It is suggestedby these considerationsthat the Schlieder condition could be used as a requirementto be fulfllled by a quasilocal algebra in order that the operators of this algebra representstrictly local operations.

Quantum Mechanical Theory of Local Observablesand Local Operations

2Og

Such operators should be equivalentto the unit operator ,l outside their d o m a i n o f o p e r a t i o n .S o , . 4 . ' , 4 , s h o u l d b e i n D e q u i v a l e n tt o A , , i l s h o w i n gt h a t A r . ' A n c a n n o t v a n i s h u n l e s sl 1 ) v a n i s h e sS. o , CaD:g, point of view, the Schliedercondition is compulsory for physical from a strictly local operations. representing local operators Notwithstanding the importance of the Schliedercondition as exem plified by the above considerations,in the generaltreatment 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 operatorscorrespondto local operations.As a matter of flact,with.4.'. also ( / t . l t i s t h e n c l e a rt h a t i n . e l o n g st o t h e l o c a l a l g e b r a uAr.. a a constantb general a local operator ,4(. also changes the state outside C, thus invalidatingthe heuristicargumentwhich was basedon the interpretationol a local operator as representinga local operation.The secondreasonis that local commutativity'.which is often taken as one of the defining characr e r i s t i c so f a q u a s i l o c a la l g e b r a( a n d w h i c h w i l l b e s h o w n t o b e n e c e s s a r y a l s o i n o u r t h e o r y ) ,i s q u i t e i n d e p e n d e notf t h e S c h l i e d e cr o n d i t i o n .T h i s c a n b e s e e na s f o l l o w s . T h e q u a s i l o c a la l g e b r ab a s e d o n P ( ( 1 0 ) . n o t o b e y i n g t h e S c h l i e d e r c o n d i t i o n .t u r n s o u t t o b e a l s o n o t l o c a l l y c o m m u t a t i v eT. h a t i s . i n g e n e r a l

lP(.A.PDBl+0.

C.D:A

(30)

A d e t a i l e d i n s p e c t i o n .h o w e v e r ,o f t h e c o m m u t a t o r ( 3 0 ) s h o u ' s t h a t t h e t a i l u r c t o o b e - vb o t h t h e S c h l i e d e rc o n d i t i o na n d l o c a l c o m m u t a t i v i t yd o e s of thesetwo propertles not hale a common origin. The relative independence quasilocalalgebra modified slightly the by considering can be demonstrated s e e n t h a t t h i s a l g e b r ai s ( 1 0 ' ) B . I t i s e a s i l y A p p e n d i x l o r m u l a o f d e l l n e db l c o n d ition. S c h l i e d e r n o t o b e y t h e b u t d o e s i o c a l l vc o m m u t a t i v e respect.and too in one too strong to be seeming The Schliedercondition u.eakin another.in order to provide a characterizationof local operators.we s h a l l l o o k i n t h e n e x t s e c t i o n sf o r a b e t t e rc r i t e r i o n .

2.6. Theorems F o r l a t e rc o m p a r i s o w n e c l o s et h i s s e c t i o nb y p r o v i n gt h e f o l l o w i n gt w o theorems. Theorem 2.1. The projectionP,. can be definedby

p(B:\- A,u,BAlr,, B € r(r) lxl

(31)

210

de Muynck

in which the operators A1n1 and A,f,-, are partial isometries which are pertinentto C, i.e., Pf A,7,: A 171 PtA[,t:

(32)

A[,t

and satisfy the equality Y A{r,A,r,: t

(33)

t n-]

Proof.

Taking

(34)

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

(31) and (32) follow by direct inspection. Since A [ r , A , u , : ) ' i { n } ,{ i a } ) ( { n, f] i z } l

(35)

is a projection operator,it i, ,..'n thatAlny andAf^are partial isometries. Equation (33) is directly entailedby (35). I Theorem 2.2. If I

is an operator on Fock space.obeying I

0):

a,

then PtgPFA):o Proof.

f o r a r b i t r a r yB

(36)

Since

PtA:

I

( { 0 i I, r i p * r i { 0 t . \ 4 1 ) \ p l , i f l ) (pll , \ q } l

:

( \ m l ,\ m l l B \ n t ,{ t i ) ( 1 0 i \. p l p { A i 0 } ,i r i )

IpilFlts)

we get

B P t A-

I m l I m lpI l tnllnl

. l \ m | ,\ m f) ( { n } 1, , 'lt - f and

PI(BPFA): :

tmllmllplln)

( 1 " 2i10,l lB l \ n l ,i t i ) ( { 0 i ,\ F l P t A 1 1 01} 0 , 1)

. l \ m l \, n l ) ( \ n l\,n f l Since,4 0):0

impliesPtAIO>:0, thetheoremfollows.

(37) I

Quantum Mechanical Theory of Local Observablesand Local Operations

Corollary.

If ,4 l0) : 0, then also PtA ' PcP:O

(38)

Proof. Equation (38) is the adjoint of (36) with respect to the functional Tr AB. I

3. LOCAL OBSERVABLES AND LOCAL OPERATORS: GENERAL THEORY 3.1. Definitions In the foregoingsection we consideredpropertiesof the local algebras defined bV (6) or (10). In the sequelof this article we shall relinquish the special representationpresentedthere. The theory of Section2 will be generalizedso as to be valid for more general fields. although it will maintain its nonrelativisticcharacter. Taking Fock space as the Hilbert space.7 of the field states,the vacuum state 0) is defined as the state without freld quanta. If 1/ is the observablemeasuringthe number of quanta,then

1 il 0 ) : 0

(3e)

As to the operatorsworking on ,j/' we restrict ourselves.as before,to the set l(7) of bounded operators oD.7, which can be consideredas a Banach spacewith respectto the operator norm. As an algebra,.V(7) is a C'f -algebra.(rr)Also F(.f) is defined,as in Section2, as the Banach space of traceclassoperatorscontainingthe density operatorsin its positive cone z-(;f)*. Thenthe expectation values

(A'):TrAp,

A e.4(z),

p e 7(7)

(40)

are bilinear functionals.The operator A is completely determinedby the restrictionof the functional to g(f)*. Also, p is determinedcompletelyif (l ) is given lor all A € .'tGf). In characterizinghow the locality of an operator has to be specifiedin the generaltheory, we first turn our attentionto self-adjointoperators.If C is some (bounded)region of []r. then a (bounded)self-adjointoperator,4,-will be taken to be pertinentto C, if the measurement of the correspondingobservable is a local measurementin C. This meansthat the measuringinstrument for A, draws its information entirely from C. In Section2 it was seenthat the information content of C can be representedby the density operatorP(.p,

de MuYnck

212

i n w h i c h P . o b e y s( 1 2 ) t h r o u g h( 1 5 ) . T h e s er e l a t i o n sa r e i n d e p e n d e notf t h e special representationused in Section2. and hence can be generalized. Taking, as before,Pf as the adjoint operator to P. with respectto the linear functional (40), we give the following definition. Definition 3. l. (Local Operators). If P, projectsrQf ) into itself.that is.

is a mimorphism which

(4i)

B€r(7)*..>PCBe{(7)r Tr P.B:

Tr B.

B e 7(f

)

(42) ( 4 3)

P'r: P. then .4. is a local operator pertinentto C il

PIA,:A,

(44)

From (44) the equality (15) for generalfields directly follows, showingthat, if ,4. is a local observablepertinentto C. the density operatorsp and Prp give the same expectationvalues. Generalizinganalogouslyrelation (2 1), we arrive at the following. Definition 3.2 (LocalizedStates). If the mimorphism P, is definedas in (a1)-(a3), then p.. is the density operator of a state localizedin C if Ptpc: pr

(45)

For stateslocalizedin C we have. for arbitrary A € .'t(,f), Tr A Pr :lr

Pt A ' Pr

(46)

So. as far as region C is concerned,I and P.*,4 representthe same information about the system.If I is an observable.then PI A can be interpreted as the restrictionof this observableto C. Thus. if l{ is the total number of quanta,PIN is the number of quanta containedin C (in Theorem3.1 it is shown that Pf ,.{ is self-adjointill isr seealso Theorems3.8 and 3.9).

3.2. Postulateswith Respect to Locality We now formulatethe basic assumptionof the presentarticle. 3 Postulate 3.1. With every (bounded)region C of Fl is associateda mimorphism P. obeying

Quantum Mechanical Theory of Local Observablesand Local Operations

( i ) ( 4 1f ( 4 3 ) r on l(7); (ii) P7r,: I, / beingtheunitoperator (iii) if C and D aretwo regions,then Pt-atr: PrP,r: PnPr.

(41)

(48)

r. Equation (47) expresses[cf. (45)] that on1' state is localized in F The equality (48), which is equivalentto the adjoint relation Icf. (26)l

P r . , : P tP t : P I P [

( 4e)

which that the informationrvith respectto an observableI € l(7) expresses O D is region C way the of the a D is independent region C in is contained propertlr region' of this a unique information making this singledout. thus i nl c l u s i o n( 2 7 ) a n d i s o t o n y ( 2 8 ) F r o m ( 4 8 ) t h e p r o p e r t i e so f s e t - t h e o r e t i c a performedin C o D. general measurements for local case. in the f ollou'. also g e n e r a l i z a t i o onf ( 1 5 ) : ( 2 8 ) f o l l o w i n g w e f i n d t h e From directly TrAr.P:Tr Ac.PrP.

Cc D

(s 0 )

F r o m t h e p h y ' s i c ailn t e r p r e t a t i oint s e e m sr e a s o n a b l teh a t ( 5 0 ) r e m a i n st r u e i f . 1, i s r e p l a c e db y a p r o d u c tA r B r . o l o p e r a t o r sp e r t a i n i n gt o C . T h i s a s k sf o r , h i c h i s c o m m o n l yr e q u i r e dl o r l o c a l o p e r a t o r s . ( r 0 ' r r ) a s e c o n dp o s t u l a t ew

Postulate3.2. For any region C the operators pertaining to C c o n s t i t u t ea n a l g e b r a t, h a t i s ,

Pt(Ar*8,.): A( + B( P[(A, . Br): A, . B,

( sl )

Actually. it will be shown in Theorem3.2 that this algebra is a x algebra. W e s h a l l r e f e r t o t h i s a l g e b r aa s t h e l o c a l x - a l g e b r a/ / c . . S i n c e P f i s a proJectionoperator on 4(7), it should be continuous in the uniform topolog-v.From this it follows that tlc includesall of its limit points with respectto this topology. Then we also have P t - f@ r . ) : " f ( A r ) '

/ a n a r b i t r a r yb o u n d e df u n c t i o n

(52)

of the local B.v meansof (52)the equality (25) for the spectralrepresentation observable(23) can be shown to hold also in the generalcase. Then, the e q u a l i t y( 9 ) .

PII:I

(53)

214

de Muvnck

which follows directly from (42), can also be derived from the representation I : l,^ P.' As in Section2, the quasilocalCx-algebra (1 can be definedas the C*inductivelimit of the union of the local algebrast'7c. As will be discussedin more detail in Section4, the present definition of local operators does not imply local commutativity for operatorspertainingto disjoint regionsof lRr. So, according to this definition, local commutativity may be fulfilled, or it may not be so. As a matter of fact, up until now the meaning of locality is not uniquely fixed. Different notions of locality may be accommodated by our definition (cf. Section2.l). By ascribing additional properties to the projections P. and PF, the class of localization definitions may be further restricted. One such restriction,which is encounteredalso in Ref. 20, is basedon the premisethat a statePrp,being localizedin C, shouldbe equivalentto the vacuum state l0) outsideC. We take this premiseas the following. Postulate3.3. If C is the complementof C in tr r. then

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

(54)

As will be seenin Section4, also with this additionalpostulatelocality is not specifiedso as to warrant local commutativity.

3.3. Theorems We shall now derive some usefultheorems. T h e o r e m3 . 1 . I f A : l

+. t h e nP t 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, becauseof (4 l), also Tr A prp is real, and . hence Tr PI A p is real for all p e 6 (-T) -. I

Theorem3.2. lf PI A: l, thenPI Ar : Ar.

(56)

Proof. SinceTr 13 : (Tr B'A'1*, we have Tr PIA' ' p:T, Theorem 3.3.

A'Prp:

(Tr(p.p)t .A)* : (TrAp)* :Tr Atp

I

If p is a density operator, then

P : 10)(01 "p

(57)

Quantum Mechanical Theory of Local Observablesand Local Operations

g i v e sC :

Proof. Taking in (54) C:A

P#A: A,

215

l R r . B e c a u s eo f ( 4 7 ) w e h a v e

A e."(7)

Then (54) may be written accordingto

Tr AP.p:rr A l0)(0,

A e 1J(r)

This directly entails (57).

(s8) I

Theorem 3.4.

P;A: (0lr l0)1

(se)

Proof. This follows directly from (58) srnce Tr AP.p:Tr

PIA . p:

( 0 A O ) T rp

a

U s i n g ( 5 9 ) w e c a n g e n e r a l i z e( 5 7 ) t o a r b i t r a r yB € { ( 7 ) : Theorem 3.5. PtB:(rrB)10)(01,

B€r(7)

(60)

Proof.

T r A P , B - T r P $ A . B : ( 0 A O ' ) TBr : ( T r z l 0 ) ( 0 1 ) ( rr )r

I

Theorem 3.6. Pc0)(0:10)(01,

CcFr

(61)

Proof. From (48) and (57) it follows that for arbitrary A: T r A P r . 0 ) ( 0 : T r A P r . P n p : T r A P a p : T r , . 1l 0 ) ( 0 1

I

Corollary.

\o Pt A 0) : (01.40),

A € ?Qn

62)

CoD:A

(63)

Theorem 3.7. TrArPr.p:(0 lDl0), Proof. This generalization of and (57).

(54) directly follows from

(48) I

de Muynck

Theorem 3.8. If the operator ,4 is given as t _ l' dr A(r),

PI A(r): A(r),

r€ C

(64)

J lltl

then

P TA :

l a r a 6 + iJ a dr(01,l(r) l0)

Jc

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

(6s) and

(se).

I

T h e o r e m 3 . 9 . I f N i s a s e l f - a d j o i not p e r a t o rN > 0 . N 0 ) : 0 .

PJN>0.

PrNl0):0

then

( 6 6)

Proof. By (55) Pf N is self-adjoint.Tr l,{p) 0V, > Tr NPr.p: TrPIN .p>OV PI N)0. E q u a t i o(n6 1 ) i m p l i e s ( 0 l P d 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 Section3 the problem of local operatorswas tackled on the basis of the notion of local measurements. A complementaryline of approachingthis problem presentsitself if we start from the idea of local statepreparationor locai changeof state.A local operationin region C, then, should changethe density operator of the system only in C. It will turn out that this latter requirementadds a new element to the notion of locality, which is not embodiedin the idea of local measuremenr. Since this new elementis essentially of a microscopic character,we shall refer to that notion of locality which includesthis new elementas microlocality. If by some physical operationthe stateof a physicalsystemis changed. in general,both initial and final statesof the systemshouldbe describableby density operators.For this reasonan operationshould be a trace-preserving mapping of (a subset of) 6(.7)* into 6(.7)*: p-+T(p). If the operation correspondswith a linear mapping, it is mimorphism. However, in the sequel we shall encounteralso onerationswhich are nonlinear.

Quantum Mechanical Theory of Local Observablesand Local Operations

217

In the presentsectionwe shall ignore the fact that physical operations generallyare not instantaneous but need sometime in order to be completed. Time-dependentoperations will be dealt with in later sections,in which operations are studied which are brought about by the interaction with another system. We shall give now a definition of a microlocaloperationin region C. Definition 4.1 (Microlocal Operation). A microlocal operation 7,.: p - Tc@), in region C of F 3 is a trace-preserving mapping of (a subsetof) 'a(7'). into 7(,7)*, obeying,on its domain, the two requirements

(i) | r.,P..l: 0. (ii) PrTr:Po.

(61) (68)

C.D:O

We shall first discuss the two requirements(67) and (68). Equality (6i) seemsa reasonablerequirementbecausea microlocal operation T, should have to do only with that part of the state which is localized in C. The equality is equivalentto the two relations T(.P(.: PcTcP(

(6ea)

PcTc: Pc T(.P(

(6eb)

Then. (69a) signifies that a microlocal operation I,.. operating on a localized statePcp, doesnot spoil the localizationof this state.Also (69b) is plausibleif we assumethat a microlocal operation 2. is constitutedout of local operatorspertinentto C. Then, for any B € 4(f ). Tt B PrTr(.PrP): fr PI B ' Tr(PrP)

: rr Pf (r](Pf 8)) . p :rr r{QI B) . p - Tr B PrTr(p) s i n c et h e l o c a l o p e r a t o r sP $ B a r e i n t h e s a m el o c a l * - a l g e b r a/ 1 r . a st h o s eo f

rf lci.(sl ) l. The requirement(68) is equivalentto

T r P I B. r r ( p ) : T r P t B . p ,

C.D:O

or

TtetrBt: PtB.

C.\ D -- a

(70)

expressingthat the microlocal operation in C does not influencemeasurement outcomes of measurements which are performed simultaneouslvin a disjoint region D.

218

de Muynck

The two properties(67) and (68) are not completelyindependent. From (69a) we get, becauseof (57), C a D : a = >P o T r ( P r p ) : P o P c T c ( P . p ) : l 0 ) ( 0 1: P o P c p which is obtained also if (68) is applied to stateslocalizedin C. However, this does not imply that either of the two requirementswould be superfluous. On the one hand, (68) cannot be derived from (67) for states that are not localized 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 seenfrom a simple counterexample,constructedfor the local scaiar field discussedin Section2. If we define an operation Z by

r ( l { p }1, 0 1 ) ( { p{ 0} ,} l ) :l { 0 }1, 0 } ) ( { 0{ }0,1 r ( 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 not fulfilled, since,for \ pl + O,

r @ , 1 \ p l{,p - } X { p{}t ,} l ): l { 0 1{ ,0 1 ) ( 1 {001},l but

P , r ( l \ p l {, 1 } ) ( { p }{ ,t } l ) : l l p l ,{ o } ) ( { p {l ,0 l l We closethis sectionby remarkingthat, by virtue of (a8), the equalities(67) and (68) remain valid if T. is replaced by P6, thus showing that the projection P7 is a microlocal operation performed in C.

4.2. The Kraus Representation;Microlocal Operators A very common kind of operationsis given by the mimorphismsp r @ ) : A p A r , A u n i t a r y . M o r e g e n e r a l l y ,i t w a s s h o w n b y K r a u s ( 2 7 ) t h a t operationsas definedby us (which correspondto the nonselectiveoperations of Ref. 27) can generally be representedaccording to

A u p A l , , Y , q I , 4 o t:

r@:\-

(71)

kk

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

r*(B):ltlat

o

(72)

Quantum Mechanical Theory of Local Observablesand Local Operations

219

is a transition map as definedby Mercer(28)(seealso Ref. 12, Section3.1). The projection operator P. of the scalar field representationwas shown in Theorem2. I to have preciselythis form. Moreover, from (3 1) and (32) we see that in this_specialcase the operatorsAo of (7 1) are local operators pertaining to C, This seems to reflect the fact that P. is a microlocal operation in C-. It is tempting to extend this to microlocal operations in general,by defining a microlocal operation by the expression(71) in which now all I o andA! are requiredto pertainto the sameregion,that is, -Twt ,n \ :

\'pt,a

_-(,-k

. r n- (P, .\4a +

(73)

With this possibility in mind we prove the following two theorems. Theorem 4.1. Local commutativity of the operators pertaining to regions C and D (with C )D: O) is necessaryand sufficientin order that all operationsof the form (73) obey (68). Proof. ( i ) I n s e r t i n gi n ( 7 0 ) t h e s p e c i a lo p e r a t i o n sr @ ) : A r p A [ . , A [ . A r . : 1 , we immediatelyobtain

AtrPtB.Ac:PiB, C^D:a Hence11..,PtB):0, C n D : A for arbitraryunitaryoperators 1. . (ii) If local commutativityis assumed,we get for any B (with A*:PtA*):

T r B P r r @ l : T r B P o5 - O r r e l \ \Tt : ''

' A 1 ,P . ATPTB

}

:Tr

Ai.Ao.PtB.p k

:Tr B png which directly entails(68). In view of (70) and (72) this rheoremcan be seen to coincide with Proposition3.1 of Ref.28, as far as applied to transition maps. I Theorem 4.2. The operations(73) satisfy equality (69b).

de MuYnck

220

Proof. Since the operators pertinent to C constitute a x-algebra, it follows that for any B and any operator .4. pertinent to C

' Ac . A,): AtrPtB Pt@trPtB So.

T r B P , . ( A r P ., pA l ) : T r r [ @ [ r t B ' A r ) ' P :Tr ALptB. Arp:Tr BPr.(ArpA[)

Hence Pr ( A r P ,p . A [ ) : r r @ , p A [ ) which implies (69b) also for the more generaloperations(73).

I

Becausein the generaltheory, up until now, we did not specify any property of local operatorscharacterizingtheseas entitieswhich are related to microlocal operations,it is not possibleto prove that the operations(73) also satisfyequality (69a) (althoughit is easily proven that they do so in the scalar field theory of Section2). In order that the operations(7-l) will fulfill all requirementsof a microlocal operation,we shall now restrict somewhat further the notion of locality as embodied by the projectionsP. and Pf . This will be done by generalizingthe requirement(69a), viz. that microlocal operationsdo not spoil the localizationof a state,to operators.To this end we first give the following definition. Definition 4.2 (Microlocal Operators). The local operators defined in Section3 will be called microlocal operators if the projections P. are such that AcPcB: Pc(AcPcB) Qaa) ano PcB.Ac:Pr(PrB.A)

(74b)

The property (14a), (74b) will be called the miuolocality property. We now can prove the following. Theorem 4.3. If the operatorsPf,4* of the operation (73) have the microlocality property,then Z(p) satisfies(69a). Proof.

Since by Qaa), Qab)

. Atr) ArPrp . AI: Pr(ArPrp). AI: Pc(AcPcp the theorem follows by inspection.

Quantum Mechanical Theory of Local Observablesand Local Operations

221

In the theory to be developedin the sequelof this article we shall only consider fields that satisfy the microlocality property. Since this property obtainsfor the fields introducedin Section2, such microlocal fields evidently exist. In consideringmicrolocal operationswe shall, however,not rely on the Kraus representation(73), becauseit is not clear whether this is the most general representationof a microlocal operation. As a matter of fact, the results obtained in the following set of theoremssuggestthat the properties (54) and Oaa), QaV, which specifythe more characteristicpropertiesof our microlocal operators,are not sufficientto derive local commutativity. This suggeststhat it might be possibleto devisemicrolocal operations,which are constituted out of microlocal operators that need not obey local commutativity, and which, becauseof Theorem4'1, should have a more generalform than (73). So, we shall stick in the following to the definitionof microlocal operationsas given in Definition4.l. This also impliesthat we do not take local commutativity as an 4 priori property of (micro)local operators.Instead,the properties(54) and Qaa), Qab) will be taken as the starting point of the presentinvestigation. 4.3. Propertiesof Microlocal Operators We shall now derive some propertiesof microlocal operators. Theorem 4.4.

P t @ A c ) :P r B . A ( P t @ c B ) : A c '. P t B

(1s)

.B arbitrary. Proof. From (74a) it follows that Tr BA rP r.P : Tr BP c(AAr:g

or

Be:0

or

(0,4c10):(0lBel0):0

de Muynck

224

Tr ArBTp:0 Proof. Since,by assumption,

for all p' Ve

Tt ArPr(B7P):Tr B7PfuAr):0

p: Pcp or p: P6p,this gives'by (79)' Takingin eitherof theseexpressions (01,4c10).TrB7p:Q

ve

and ( 0 l8 6 l 0 ) . T r A r p : Q to the desiredresult. This is equivalent

5. MICRO. AND MACROLOCALITY ln this sectionwe draw a distinction betweenthe idea of microlocality as introducedin Section4, and a notion of macrolocality,which is definedin the following as the implementationof microlocality on the macroscopic scale of measurementresults.This distinction is inspired by the difficulties which aboundin relativisticquantumtheory as regardsa covariantdefinition of a position observable,which seemto put the idea of locality into sertous doubt. Sincethere is no direct evidencewhatsoeverof a violation of locaiity on the scale of (macroscopic) experimentation, it is sometimes difficultiesare characteristicfor that the above-mentioned conjectured(r3-r6) by replacingthe requirementof be solved and can level only, microscopic the microlocality (or -causality)by the weaker requirementof macrolocality (or -causality). In order to implementthe distinctionbetweenmicro- and macrolocality, we resort to the theory of quantum measurementas developedin Ref. 12. In this theory propertiesof the object system (which are representedby selfare mapped onto properties of the adjoint operators A:1,^a^P^) Eq' (7), (8) and (10) of Ref. 12) (cf. to measuringinstrumentaccording

pP^ T . r U ^ S rO p o S * : T u r

(87)

in which p and po are the initial statesof object o and measuringinstrument c, respectively,Sp@poS'is the final stateof the combinedsystem,and Em is the macroscopicproperty of the measuringinstrument correspondingto P ^ . B y m e a n so f ( 8 7 ) i t i s p o s s i b l et o t i e t h e D e f i n i t i o n4 . 1 o f a m i c r o l o c a l operationto the macroscopiclevel:

Local Operations Quantum Mechanical Theory of Local Observablesand

225

Definition 5.1 (Macrolocal Operation). An operation T: p-+ 7(p) is a macrolocaloperationin region C of R' if, up to experimentalerror, (i)

T r oE ^ S T ( P r p ) A p " S +: T . r E ^ S P r T ( p ) @ p . S '

(88)

T . rE u .ST(p)O post :

(89)

lor all .8.. (ii)

T.,

Z,S, @PoSo

f o r a l l E - c o r r e s p o n d i n g t oa P ^ w h i c h i s p e r t i n e n t o D , w i t h D ' C : A ' Sincefrom (67) and (68) we get, for arbitrary P^'

Tr P^Tr(Pr rl : Tr P^P cTc(P)

(e0)

and

T,,r PIP^. Tr(P):TorPfiP^' P'

c.D:Q

(91)

it follows irom (87) that, accordingto our definition,microlocal operatlons are also macrolocal.Conversely,since (67) and (68) follow from (90) and ( 9 1 ) i f i n t h e l a t t e re x p e s s i o nPs^ c a n b e t a k e nt o b e a r b i t r a r y ,m a c r o l o c a l i t y would also seemto entail microlocality.This, however,neednot be the case, becausethe arbitrarinessof E. in (88) and (89) does not necessarilyimply that the correspondingset of P.'s separatesthe statesol the object system. Yet. it is clear that a distinctionbetweenmicrolocality and macrolocality,as is made here. makes sense only by virtue of the assumptionthat it is impossible to determine the quantum state completely by means of macroscopicmeasurements.Since all measurementis subject to a certain So, inaccuracy.this assumption,however,does not seemto be unreasonable. macrolocalityis reconciledwith a possibleviolation of microlocality because there exist such limitations on the possibilityof measuringlocal observables does that the restrictionof (88) and (89) to actually possiblemeasurements not entail (67) and (68). Yet, theselimitations seemsto have rather a practical than a fundamentalorigin. The distinction of micro- and macrolocality as given above could be implementedby an idea encounteredin the relativistic theory of quantum fields.(2e'r0) t:iz. that the notion of strict localization, as defined bv (45)' should be replacedby the weakernotion of essentiallocalizationof the state of the quantum fie1d.That is to say, there do not exist local operations preparing the initial statesp and po of object and measuring instrument, to be containedexactly in some boundedregion of beforethe measurement, Fl3. Then the equalities (88) and (89) obtain, because the essentially

226

de Muynck

localized state is supposed to yield for all local measurements,up to experimentalerror, the same results as can be calculated for the strictly localizedstate of which it is an approximation. Although, for reasonsnot to be discussedhere,it seemsvery reasonable to assumethat the initial statesof the quantum measuringprocesscannot be strictly localizedbut are at bestessentiallylocalized,we shall,for the sakeof clarity, in this article stick to the notion of strict localization as definedby (45). As a matter of fact, the mathematicalpossibility of strictly localized statescan be demonstr&ted,(zo':01 though the physical interpretationof such statesremains somewhatobscure.So, although it seemspossibleto have a theory of macrolocal operationswhich are not microlocal, in the following we shall aim at a quantum measurementtheory in which the (instantaneous) influence of the localized state of the measuring instrument on the object systemis a microlocaloperation.If it is possibleto devisesuch a theory, the measurementinteraction will then automatically obey also macrolocality, and simultaneousmeasurements in disjoint regionswill be nondisturbing(if thesemeasurements are instantaneous). In Sections7 and 8 it will be shown that such a theory can be constructedon the basisof the assumptionthat the field operators constitutingthe interaction hamiltonian betweenobject and measuringinstrumentare microlocal operatorsthat commute with operarors pertaining to disjoint regions of R3; that is, both local commutativitv and microlocality are assumed. As we saw in Section4, microlocality and local commutativity seemro be more or less independentpropertiesof operators.Hence.we evidentlydo not completelysucceedin what we set out to do. viz. derive nondisturbance from local commutativity. As a matter of fact, local commutativity is only sufficient for nondisturbance,in our derivation, if it is supplementedwith other, possibly independentrequirements.This state of aflairs may be-and hopefully will be-improved on, to the effect that nondisturbanceis derived from local commutativity without supplementaryrequirements.yet, our presentderivationappearsto be illuminating.By constructinga theory which is not only macrolocalbut which is also microlocal,we get at our disposala model theory to which we can compareour physicaltheoriesin order to see how far in thesetheoriesmicrolocalityis satisfiedor violated.Thus, quanrum electrodynamicswill be seen, in Section9, to be a theory which, ln our definition,is nol microlocal. From our analysisit is also possibleto infer in which way the measuringprocesshas to be restrictedin order that quantum electrodynamics can be a macrolocal theory, although violating microlocality.The possibilityof obtaining theseresultsseemsto be suffrcient justification to considertheoriesin which the idea of strict localizationplays a role. The replacementof strict localizationby essentiallocalizationwould introduce an additional source of nonlocality by which the violation of

Quantum Mechanical Theory of Local Observablesand Local Operations

))1

microlocality is enhanced.In the presentanalysisthe possibilityof this kind of contributionto nonlocalitv is left out of consideration.

6. MICRO- AND MACROCAUSALITY Up until now we did not take into accountpossibletime dependence of the operations.Thus, the transformationp . T(p) could be interpretedas an instantaneouschangeof the density operator. However, if such a transfor, mation is the resultof a causalphysicalprocess,the correspondingchangeof the density operator will take some time. We shall discussin this section such time-dependentoperations, with special attention directed toward causalityproperties. Let T(p(t),t2 t,)-p(tr). tz)tt (e2) describe a general time-dependentoperation transforming the density operatorat /, into p(tr). lt is noted that the representation (7 1) of Kraus also accommodatesthis kind of operations,but we shall not make any use of it. A very common operationof the type (92) is given by T'(p,t):

Tr s(r)p, 6;prs(r)*, s(l):

e

tt(IIt+IIt+IItl)

( e3)

describingthe influenceon systeml, causedby the interactionwith some other system2. This exampleprovidesan excellentopportunityto discussthe relation between the notions of microlocality and microcausality. In Section4 microlocality was introducedas a property of operationson the density operatorof the object system.such operationscan be brought about either through the free evolution of the state ol the object system, or by means of an interactionwith a secondsystem.The propertiesof these are governed by the free field hamiltonian H, and the total hamiltonian H t + H 2 1 - H r z , r e s p e c t i v e l yI n. b o t h c a s e st h e p r o p a g a t i v i t yo f t h e s o l u t i o n s of the field equationswill induce a violation of the microlocality conditions ( 6 7 ) a n d ( 6 8 ) , s i n c ea s y s t e mw h i c h , a t t : 0 , i s l o c a l i z e di n a r e g i o nC w i l l occupy a different region at a later time. Relativistic (Einstein) causality demandsthat this region should be containedin the domain of influencec, of C, defined by C,: Q(r, ct) (e4) r€C

9(r, ct) being a sphericalneighborhoodof r with radius c/. Implementingthis idea of relativisticcausalityinto the notion of a microlocal operationwe get the followins.

de Muynck

228

Definition 6.1 (Propagative Microlocal Operation). A propagatiue microlocal operation Tr(',t) is an operationsatisfying (i)

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

(ii)

PrTr(Pr,p. tl: PcTr(p.tl

(iii) PDTc(P,t):PoU(t)P'

DAC,:g

( esa) (esb) (e6)

U(r) representingthe free evolution of the system. The equalities(95a), (95b) and (96) reduce to (67) and (68)' respectively, in case there is no propagativity.More specifically,(95a) and (95b) are the generalizationsof (69a) and (69b), respectively,to the propagative situation.If all operationsthat can be applied to a system are propagative microlocal operations,we shall say that the systemis microcausal. If T. is a nonpropagativemicrolocal 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), I J ( t )P r T c :

P c , U ( t )P c . T c

which, for 7. :1, implies the equalitY

lU(t). Pr,lPc : 0

( 9 7a )

can be derivedfrom (95b),viz. An analogous expression Pclu(t),p.,]:0

(e7b)

In Theorem 6.2 it will be shown that, for a convex set C' (97b) is also a consequenceof (96). Equalities (97a), (97b) express the miuocausal evolutionof localizedstatesof the free field. Although (97a), (97b) is a necessarycondition for microcausalityof a system,it is not sufficient.If the system interactswith another system,the locality propertiesof the operation (93) describingthis interaction are a of both the free evolutionof the two systemsand the propertles consequence of the interactionoperatot Htz.If I/r, does not have such propertiesthat it can be characterizedas a local interaction betweensystems I and 2, the whole idea of the possibility of a microlocal operationevaporates.In order that the operation(93) be a propagativemicrolocal operation,it is sufficient, as will be demonstratedin the next sections,that (i) Hrrbe a microlocal interaction hamiltonian (cf. Section 7 for the definition), and (ii) the free field propagationbe microcausal.Then, the system is microcausalif it has only microlocal interactionswith other systems.

Local Operations Quantum Mechanical Theory of Local Observablesand

229

As is well known, microcausalityis one of the outstandingproblematic features of relativistic quantum theory. Especially the condition (97a) is often judged to be impossible, because of the problems encountered in constructingphysically relevant solutions of the wave equationswhich do not spreadwith a velocity exceedingthe velocity of light.(3t)In the context of local observablesthis problem is demonstratedespecially lucidly by viz. that for on the basis of a theorem proved by Borchers,G3) Schlieder(32) zero in any be cannot any local projection operator [cf. (25)] Tr P^U(t)p in a region p is localized open subsetof the t-axis, even if the initial state measurement. the C of which is locatedarbitrarily far away from the region data the requirementof microcausaiity Becauseof the above-mentioned is sometimesdropped,and replacedby a requirementof macrocausality,on the basis of argumentationssimilar to those given by us in Section5 on account of the locality problem.(r3-r5)At this moment, however, it is not completelyclear whether this weakeningof the generaldemandsto be met by the theory is really necessary.As a matter of fact, both the infinitevelocity wave packet spreading and the Schlieder-Borchers theorem are of the spectrumof the one-sidedboundedness of a presupposed consequences from below, it is often free-fieldhamiltonian.If the spectrumis not bounded fronts and behaving wave possibleto constructwave packetshaving sharp m i c r o c a u s a l l y . ( r 3 ' 3Irn) d e e d , i t i s s o m e t i m e st h o u g h t t r s ' : ' t )t h a t c e r t a i n paradoxeswith respectto causalitycan be solvedby allowing the spectrum of the hamiltonian to be unbounded also from below. considerationsshow that it is not possible,as yet! The above-mentioned to disqualify microcausalityas a property of relativisticquantum fields, on the basis of its flree-fieldbehavior.For this reasonwe feel free to postulate (97a), (97b), thus assumingmicrocausalityof the free field. By doing this we are able to disentanglepossibleviolationsof microcausalityoriginatingfrom its two possiblesources,viz. free-fieldevolution and interactions,the latter being the proper subjectof this article. Of course,this procedureis only of pragmatic value, since both sourcespossibly contribute to the violation of microcausality.In a more "realistic" treatmentwe have to take into account also the violation of microcausalityby the free-fieldevolution.The analysis of the measuringprocesswill have to explain, then, how it is possiblethat this latter violation of microcausalitydoes 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 calculatedto be too small to be detectedunder presentlaboratoryconditions. So, it seemsthat a possible violation of microcausality by the free-field evolution will not spoil our derivation of nondisturbance of joint local in causallydisjoint regions' measurements We concludethis sectionby deriving the following theorems,

2Jo

de Muvnck

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

Pt,BcQ): 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 operationsof the form U(t)Tc,I. a microlocal operation, C convex, are propagative microlocal operations, then (97b) follows from (96). Proof. The projectionP., is a microlocal operationin 1-,,). Since,for convex C, (e ,),: C-,it follows irom (96) that PDU(t)Pc,:PoU(t), Then (97b) follows by taking D:

(ee)

D-C

t

C.

7. LOCAL INTERACTIONS In order to be able to handle compoundsystems,we have to extendthe formalism of local operatorsin the usual way to encompassdirect products of operatorsof the differentsystems.Thus, if At, and Bi, are local operators of systemsI and 2, respectively,we have ( P t c@ P ' z ; * @ i @ B i l :

P l * g P , , * ( A L@ B r , )

: P [ * A ' ,@P l , *B ' r : A i @B ' ,

(loo)

This allows a straightforwardgeneralizationtoward the tensorproduct of the Banach spacesof the operators of systems I and,2, which contains the interaction operators of these systems. we shall take such interaction operatorsto have the form

grr:

At$) 8lB'(r) .l*,dr

Moreover,^I1,,will be called a local interactionif l'(r) operators, satisfying P[* Ar (r) : At (r),

r€ C

P?* B'1G): B'(r),

r€ C

(l0l )

and B2(r) are local

(102)

Quantum Mechanical Theory of Local Observablesand Local Operations

2Jl

In the following it is also assume that At(r) and B2(r) satisfy the microlocality property (7 4a), (7 4b). If we restrict our attention, as we will do in the following sections,to a specialkind of microlocal interactions,namely interactionsdescribingonly the scatteringof two field quanta, we may assumethat the operators,4'(r) and B2(r) of (l0l) both consistof an orderedproduct ofa creation and an annihilationoDerator.In this casewe have

l ' ( r )l 0 ): n ' z Gl 0) ) : 0

(103a)

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

(l03b)

For the scalarfield theory of Section2, Theorem2.2 and its corollary can be generalized to compound systems such that from (103a), (103b) the following equalitiescan be derived: A t ( r ) P t r p : f ' ? ( rP) ' r p : 0 P ' r p. , q ' t ) :

P ' r p . - 8 2 ( r ;: I

, d C

(104)

In the generalcase it is possibleto derive from (103a), (103b), (79), and (8la) that Pi(l'G) p'rp):PNAft)p'rp):0, r € C e t c . A n a l o g o u s l yt o the remark made after Theorem4.7, it does not seempossibleto derive the more generalrelations(104) also here. In the following it will be convenient to restrict the class of local interactionswhich will be consideredto those H r r i n w h i c h t h e o p e r a t o r sl t ( r ) a n d B ' z ( r ) s a t i s f y( 1 0 4 ) . It is noted here that the restrictionof local interactionsto interacuons (101) satisfying(104) is, strictly speaking,too severe,becausethis restriction makes it difficult to apply the theory to, for instance, the interaction of quantum electrodynamicswhich does not obey (104). Since,however,in the presentinvestigation it is our main intention to find sulficient conditions for a field theory to be microcausal,this is not a seriousdrawback. For the interaction hamiltonian (101) obeying (104), the following theoremsare now derived. Theorem 7.1. If the systems I and 2 arelocalized in regions C and D, respectively,which do not intersect,then TrHrrP[@Pip:0,

C^D+A

or, equivalently,

P[*e;P]o*H,r:9.

CaD:a

(105)

de Muynck

232

Proof.

H,r: I a, Ptr*A'trl Pl* @ P'o* O P3*PZ-x82(r) * J|D a, Pl"Pl,*A'1r;5l rl* 4':111

+ Jl'

coD

drP!*A' 1r1 6 ri * r'1r;

: l, ar,4'G)(o |o) IB'z(r) ( r )B' z( r ) + { a' 1o1,arlo) +

[-

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

b e c a u s eo f ( 4 9 ) , ( 5 9 ) , a n d ( 1 0 3 a ) ,( 1 0 3 b ) . Theorem 7.2. Pl*@P2D*Hi2:0,

C.D*A,n)2

(106)

Proof. f,

H i r : l l t ur [ * , t ' ( . ) O P i * B ' ( r+)' |' d rP( l x A \' '(v. ) O p l * r ' t r t l I L'c So, Pf.* @ P'o*Hi, is a multiple integral of a sum of terms of the form

l ' ( r ' ) ' " r l - *n 1 6 , 1 1 P i . * e t cAf ' ( t , ) . ' . P [ : A ' ( . , ) )@P l , * ( P lB w i t h 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 if not all Cr: C. So, '(r")) : 0, P L * ( P L fA ' ( t , ) " . P L :A

u n l e s sV ' C ' : C

SincePj*(Pt * B' (r r) ..' P2r*82 (1)) : 0 if C a D : g, the theoremfollows.

T

Quantum Mechanical Theory of Local Observablesand Local Operations

233

From (105) and (106) we get the equality pl*@p2D*eiHt2t-1,

(107)

C^D:A

These results can easily be generalizedto interactionhamiltoniansthat are sums of terms of the form (101) with microlocal operatorsobeying (104). Theorem 7.3. The equality (107) is equivaientto v"Pl @ [email protected]"P t La p"p):pi :PL@P'op,

e P L ( P ' ,@ P L p ' e i ' ' 1 t ) (lo8)

CaD:Q

Proof. Equation (108) follows directly from a generalizationof (77) to tensorproduct space,glvlng piil: PL @ Pl)@tH "r Pl- E

P:* I

P2o* einrtt ' pi @ pLp

:PL@PLp,

C^D:a

Conversely.from (108) we get the adjoint relation @ p ' o * B. e i I I ' , t ) : p l * @ P ? , * 8 .

vrpl*@p:*(pl*

CaD :a

(r 0 e ) I

T a k i n g - B : 1 , t h i s e n t a i l s( 1 0 7 ) . Theorem 7.4. l p tcT2r (eiH,,tp[ @ pL p) : rf Tr (pf @ p'" p . eiH'. ) :PtrTrrp, Proof.

CaD:@

(110)

Since

A', Atr:P[*@P?,* it follows, by a generalizationof (75) to tensor products.that pl* g prr*(AtreiHnt)- Atttr* @p2* eiHtlt So,if C ) D:

e , w e h a v eb y ( 1 0 7 )

Pl.*@P"*(ALuinttt):At6; c^D:A

(l1l)

Since ( I I I ) obtains for AI: Ptr*At with arbitrary A', (l10) follows I as the adjointof (11l). straightforwardly

de Muynck

The equality (110) clearly admits the interpretationthat, if systems1 and 2 are both localized in nonintersectingregions,the interaction is not effective, at least not as far as local measurementsin C on system I are concerned.In the next section we shall relax the restrictionof the statesof system I to localized states,and keep only system 2localized. We shall then demonstratethe more generalproperty of a local interaction,viz. that the influenceof system2 on systemI obeys all the propertiesof a microlocal operationas definedby Definition 4.l.lf a local interactionoperatorIl,, has this property, we shall term it a microlocal interaction operator. 8. MICROLOCAL INTERACTIONS AND MICROLOCAL OPERATIONS T h e o r e m 8 . 1 . I f 1 1 , , i s a l o c a l i n t e r a c r i o nd e f i n e db y ( l 0 l ) , ( 1 0 2 ) ,a n d ( 1 0 4 ) ,t h e n givrlt pi

Proof.

O

,ilr12t2: pt.(giHrttt pirp . ei?rttrl,

j : r,2

(l 12)

Writing

H rr: H,r^l H,rI

: I d, P'r*A'(r)Q Pf xB'z(r)+ p[*AtF) @pa*B'z(r) '( .l_dr c it follows from (104) that

H , r . P [p : P I p . H , , - : 0

( rr 3 )

So, H,rPLp:Hrr,PLp:f[(n,r,r'rp)

(lt4)

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

P L ![*A' . PLD") :