17 January 2000

Ž .Physics Letters A 265 2000 153–155www.elsevier.nlrlocaterphysleta

About the notion of truth in the decoherent histories approach: areply to Griffiths

Angelo Bassi a,1, GianCarlo Ghirardi a,b,)

a Department of Theoretical Physics, UniÕersity of Trieste, 3014 Trieste, Italyb Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

Received 13 December 1999; accepted 14 December 1999Communicated by P.R. Holland

Abstract

Griffiths claims that the ‘single family rule’, a basic postulate of the decoherent histories approach, rules out ourrequirement that any decoherent history has a unique truth value, independently from the decoherent family to which it maybelong. Here we analyze the reasons which make our requirement indispensable and we discuss the consequences ofrejecting it. q 2000 Published by Elsevier Science B.V. All rights reserved.

This short Letter is a reply to Griffiths’ articleConsistent histories, quantum truth functionals, and

w xhidden Õariables 1 , in which he has raised someobjections to our paper Can the decoherent histories

w xdescription of reality be considered satisfactory? 2 .w xFor a more detailed analysis of the arguments of 2 ,

w xwe refer the reader to 3 .First of all, we would like to summarize the main

features of the DH approach, about which thereseems not to be a disagreement between Griffithsand us:

1. Within a given decoherent family everything goeslike in Classical Mechanics: the probability distri-bution assigned to the histories obeys the classicalprobability rules; it is possible to define a Booleanstructure, so it is possible to speak about the

) Corresponding author. E-mail: ghirardi@ts.infn.it1 E-mail: bassi@ts.infn.it

conjunction, disjunction of two histories and aboutthe negation of an history; moreover, one candefine the logical implication between two histo-ries, so that also reasonings of the type‘if . . . then . . . ’ are possible.

2. As Griffiths admits in the above quoted paper, itis possible to assign truth-values to all histories ofa given decoherent family. This move has animportant physical meaning: it means that, inspite of the probabilistic structure of the theory,one can speak of the properties actually possessedby the physical system under study, and not onlyof the probability that such properties be pos-sessed. In order to understand this important point,let us remember that also in Classical StatisticalMechanics one generally has only a probabilisticknowledge of the physical system; despite of this,he can claim that the system has well defined

Žphysical properties positions and momenta of itsconstituents, from which all other properties can

0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00889-0

( )A. Bassi, G. GhirardirPhysics Letters A 265 2000 153–155154

.be derived , but he doesn’t know which they aresimply because he is ignorant about the precisestate of the system. From the logical-mathemati-cal point of view, the legitimacy of consideringproperties as objectively possessed is a conse-quence of the fact that one can define a Booleanalgebra in phase space and attach truth-values toits subsets in a consistent way.In Standard Quantum Mechanics, on the otherhand, one cannot even think that systems possessphysical properties prior to measurements: mathe-matically, this is reflected in the peculiar proper-

Žties of the Hilbert space with dimension greater.than 2 : the set of projection operators cannot be

endowed with a Boolean structure, and it is notpossible to attach consistently truth-values tothem, as implied by the theorems of Gleason, Belland Kochen and Specker.Thus, giÕing a truth Õalue to the histories of agiÕen decoherent family corresponds to the asser-tion that such histories speak of specific physicalproperties that the system under study possesses

( )objectiÕely, independently from our in generalprobabilistic knowledge of the system and of anyact of measurement. This, in our opinion, is thenicest feature of the DH formalism, the one em-boding all its advantages with respect to thestandard quantum formalism.

3. When one deals with more that one decoherentfamily, things become rather problematic: if suchfamilies can be accomodated into a single deco-herent family, then all what we have said previ-

Žously remains valid. If this is not possible and.this is likely to happen most of the times , then

any reasoning, any conclusion derived by usinghistories which belong to incompatible families,are devoid of any physical meaning. Griffiths feltthe necessity to promote this fact, which we haveindicated as the ‘single family rule’, to a basicrule of the DH approach: a meaningful descrip-

( )tion of a closed quantum mechanical system,including its time deÕelopment, must employ a

Ž . w xsingle framework i.e., decoherent family 4 .This rule gives rise to some curious situations,which do not have any classical analogue, but wewill not discuss these matters now. Actually, we

agree that they do not lead to formal inconsisten-cies.Now, let us come to our argument. The formalism

of DH implies, as it is obvious and can be easilychecked, that any given decoherent history belongsin general to many different decoherent families. Aswe have argued under item a2 above, in any ofthese decoherent families, such a history has a pre-cise truth-value. As already mentioned, also Griffithsseems to agree on this. Now the relevant question is:does the truth Õalue of the considered history dependon the decoherent family to which it may belong?

ŽWe think that the answer must be ‘no’, because as.we said in item a2 truth-values refer to properties

objectiÕely possessed by the physical system understudy, and if the truth-value of a decoherent historywould change according to the decoherent family towhich it belongs, also the properties that such ahistory attaches to the physical system would changeby changing the decoherent family. We have formal-ized these considerations in the following assump-

Ž Ž . w x.tion which is assumption c of 2 :

Any given decoherent history has a unique truthŽ .value 0 or 1 , which is independent from the

decoherent family to which the history is consid-ered to belong.

w xAs mentioned in the abstract, in 1 , Griffithsclaims that such an assumption violates the ‘singlefamily rule’ and as such it cannot be considered aspart of the DH approach. With reference to this pointwe want first of all to make clear that nowhere, inthe original formulations of the ‘single family rule’,it was mentioned that a given decoherent history canŽ .or cannot have different truth values according tothe family to which it belongs; nowhere, directly orindirectly, reference was made to our assumptionŽ .since we have been the first to put it forward . Thus,it is not correct to claim that such a rule alreadyexcluded our assumption. If Griffiths claims that the

Ž .‘single family rule’ excludes c , then he is propos-ing a new, extended, interpretation of such a rule.

Having clarified this point, we are ready to acceptthat Griffiths rejects our assumption: he is perfectlyfree to do so. But we pretend that he accepts all the

Ž .consequence which we are going to analyze of

( )A. Bassi, G. GhirardirPhysics Letters A 265 2000 153–155 155

Ž .such a move. Denying c simply means to assertthat:

There are decoherent histories whose truth-valuesdepend on the decoherent family to which theyŽ .are thought or considered to belong, i.e. in somefamilies they are, for example, true, while in otherfamilies they are false.

This, in turn, means accepting that statements like‘this table is here’, ‘the Earth is moving around theSun’, ‘that electron has spin up along such a direc-tion’ are – in general – neither true nor false per se:each of them acquires a truth value only when it is

Žconsidered a member of a precise among the in-.finitely many ones which are possible decoherent

family; moreover, their truth values may changeaccording to the decoherent family to which they areassociated. In some families it may be true that ‘thistable is here’ or that ‘the Earth is moving around theSun’, while in other families it may be false that‘this table is here’ or that ‘the Earth is movingaround the Sun’. This state of affairs is the directconsequence of denying our assumption, and it shouldbe evident to anyone that if one takes such a positionthen he is spoiling the statements of the DH ap-proach of any physical meaning whatsoever.

We can then summarize the whole debate be-tween Griffiths and us in the following terms. In our

w xpapers 2,3 , we have considered the following fourassumptions:Ž . Ža Every family of decoherent histories can be nat-

.urally endowed with a Boolean structure allow-ing to recover classical reasoning,

Ž .b Within every decoherent family it is possible toassign to its histories truth values which preserve

Žthe Boolean structure i.e. they form an homo-.morphism ,

Ž .c Every decoherent history has a unique truthvalue, independently from the decoherent familyto which it may be considered to belong,

Ž .d Any decoherent family can be taken into ac-count,

and we have shown that they lead to a Kochen-and-Specker-like contradiction. This implies that at leastone of them must be rejected in order to avoidinconsistencies within the DH approach. Griffiths

Ž .rejects assumption c , while we, in accordance withthe previous analysis, believe that this move is unac-ceptable. Accordingly, in our papers we have sug-gested that one should limit, resorting to precise andphysically meaningful criteria, the set of decoherenthistories which can be taken into account. Such amove might lead to a physically acceptable andsensible new formulation of the DH approach.

References

w x1 R.B. Griffiths, Consistent histories, quantum truth functionals,Ž . Ž .and hidden variables, Phys. Lett. A 265 2000 12 this issue ,

preprint quantrph–9909049.w x Ž .2 A. Bassi, G.C. Ghirardi, Phys Lett. A 257 1999 247.w x3 A. Bassi, G.C. Ghirardi, Decoherent Histories and Realism, to

appear in J. Stat. Phys.w x Ž .4 R.B. Griffiths, Phys. Rev. A 57 1998 1604.