""'^^^ IC/88/208

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

CP-VIOLATION AND EINSTEIN'S LOCALITY RECONSIDERED

GianCarlo Ghirardi

INTERNATIONAL ATOMIC ENERGY AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1988 MIRAMARE-TRIESTE

IC/88/208

International Atomic Energy Agency 1.INTRODUCTION and The idea, which has been repeatedly put forward in the United Nations Educational Scientific and Cultural Organization past^1), that the collapse of the wave function in experiments of the EPR-type should allow f aster-than-light communication, INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS has recently given rise to a renewed debate in connection with a proposal, whose novelty, in the authors' opinion^2', derives from considering CP-violating interactions which imply the occurrence of linear superpositions of non-orthogonal states. In spite of the fact that sound criticisms about the CP-VIOLATION AND EINSTEIN'S LOCALITY RECONSIDERED arguments discussed in ref.(2) have been presented'3) and that the conclusions drawn by the authors have been proved to be unappropriate by a rigorous discussion of the physical aspects GianCarlo Ghirardi of the considered process in two recent papers'4'5', the International Centre for Theoretical Physics, Trieste, Italy, authors of ref,(2) have stressed once again'**) their previous and claims. Ref.(6) pretends to give an answer to the above quoted Dipartimento di Fistca Teorica, Universita di Trieste, Trieste, Italy. criticisms to ref.{2),and strongly suggests that the problem is still open and deserves, at least, further investigation. In our opinion, ref.(6) contains various statements and exhibits a way of reasoning which could give rise to serious ABSTRACT misunderstandings. As we have stated some years ago'7), in discussing analogous problems: "any critical reconsideration A critical reconsideration of the arguments put forward by A. Datta, D. Home of the conceptual foundations of generally accepted and A. Raychaudhuri in two recent papers is presented. All crucial points of their analysis theoretical schemes for the decryption of natural phenomena, are examined and it is shown that their conclusions are not appropriate. to be useful, must be carried out with great logical rigour". It seems therefore appropriate to reconsider the arguments of refs. (2) and (6), to make more precise the exact terms of the problem under discussion.

MIRAMARE - TRIESTE August 1988

Submitted for publication. -2- correspond to situations in which one can claim that the 2. THE ROLE OF CP-VIOLATION particle has "spin up" in a given direction (x or y, Refs.<2) and (6) make continuous reference to the peculiar respectively). character of CP-violating interactions, giving rise to linear ii. <*|T> | 0, so eq.(3) represents a linear superposition of superpositions of factorized states of a composite system in non-orthogonal states. which the states of the component subsystems are iii.|»> and |T> are the two linearly independent non-orthogonal. (non-orthogonal) eigenstates of the non-hermitian It has to be stressed that, as it stands, this statement operator is meaningless since any state of a composite system can be written as a linear combination of orthogonal or non-orthogonal states of the constituents systems, provided ^1 the two set of states span the same manifolds. As a simple belonging to the eigenvalues +1 and -1, respectively. example one could consider the singlet spin state, which arises naturally in various decay processes: The arguments of refs.(2) and (6) are based precisely on the consideration of an expression of type (3). It is evident \S> =( 12,-z that if one chooses such an expression and assumes that a measurement procedure, involving only particle 1 and allowing If one introduces the two states j *> and I T> , which are the to detect whether this particle is in state |*> or IT> , can be eigenstates, belonging to the eigenvalue +1, of ax and o-y, devised and, moreover, one postulates that this measurement respectively: reduces state (3) either to |(t>1>|TJ> or to IT1^ I »2>. one would be facing the same problem and could draw the same conclusions as those in refs.(2) and (6). The fact that ref.(2) deals essentially with non-hermitian operators has been nicely illustrated and criticized in a recent paper'e^. one immediately sees that state (1) can be written as: Our conclusion of this sketchy analysis is then: The occurrence of CP-violating interactions is totally irrelevant \S> = -[(l+i)//2] {l*^ & |T2> - |T»> Q |»2>), (3) for the subsequent arguments developed in refs.(2) and (6).

where the upper indices denote the particle to which the 3. THE ROLE OF THE ORTHOGONALITY OF THE PROJECTIONS states refer. Another point which is repeatedly stressed by the authors We stress the following facts: is that all previous proofs of the impossibility of i. I*> and IT> have a clear physical meaning, since they superluminal communication rely heavily on the assumption of a reduction mechanism based on the use of orthogonal projections. This statement, too, can give rise to misunderstandings if it is not made precise. In fact, as can be easily seen, it is not the orthogonal nature of the projection operators used in the standard formalism which plays the key role in the impossibility proofs, but the -3-

-4- fundamental fact that they are operators acting only in the responsibJe of the apparent non-local effect. Hilbert space of one (the measured one) of the two systems of the composite system. In fact, quite in general'4>s) even if 4. POSSIBLE PHYSICAL PROCESSES AFFECTING A CONSTITUENT OF one considers a completely arbitrary, trace-preserving map of A COMPOSITE SYSTEM the statistical operator p(l,2) of the composite system At this stage, one is naturally led to raise a fundamental question: what do the authors of ref.{6) want to prove? They seem to think that they have made plausible that the basic

=> P(l,2)= (5) postulates of quantum mechanics, i.e., Schrodinger's evolution and wave packet reduction, could conflict with Einstein's locality. where the bounded operators Ak and Bk act only on the Hilbert space H^1' of system Sj, and are such that The real issue is whether one can, by exerting some physical action on system S,, influence the results of EtBiJ1) Ai,**) = 1, (6) perspective measurements on the far apart system S2, Such a question to have any physical meaning and to allow some then, due to the possibility of cycling operators acting on rigorous conclusions to be drawn, must be tackled by making the Hilbert space on which one is taking the partial trace: precise what kind of action is actually taking place on system St. Different attitudes can be taken. We list below all the natural ones. Tr< = Tr< (7}

A. Suppose that system Sj is made to interact with some This means that no physical effect is induced on system S2 by the process ( 5) . device D, the Sj-D interaction being described as a standard On the other hand, even if use is made of orthogonal Schrodinger evolution governed by a Hamiltonian H(1,D) acting projections acting not only on H'1) but on ti(*,2\= Hl'lx H'2' in the Hilbert spacetf<'>(x) ff(D), Then , starting from a state then any mapping (10)

P(l,2) =» P(l,2)=Ek Pk(l,2) Pk(l,2) (8) where the states with apices denote arbitrary states of the implies in general that subsystems Si and S2, arid |D> describes the initial state of the device D, one gets by evolution:

Tr< O P( 1 ,2) if Trl * > P<1 ,2) (9) |Y\t>=a|x2>[U

-5- -6- from the possibility of cycling operators under the trace Ak P Ak + (121 taken on the Hilbert space on which they act.

with EkAk+Ak=l> *s not' as stated in ref.(6) "an abstract B. Suppose that one wants to explore also the consequences argument based on the first representation theorem", but a of assuming that the interaction of S and D is not accounted t precise and rigorous general result. As proved in refs.(4) and by the Hamiltonian evolution, but is described by the (5), in the case of a composite system, eq.(12) entails again postulate of wave packet reduction, in terms of orthogonal that, if the operators A^ act only on the Hilbert space of projection operators acting only on system 5t. This case system St, then no detectable effects are induced on S2 and, deserves a separate consideration in view of the difficulties conversely, if the physics of S2 is changed by the process, of accounting for wave packet reduction by the linear then the operators A^ must necessarily act on tf(''@ tf'2'. evolution of the theory, so that one can take the wave packet Anybody willing to give a physical meaning whatsoever to reduction as a basic independent postulate of quantum process (12), on the basis of the argument presented now, mechanics. As well known 19>10), and explicitly admitted by would be compelled to conclude that, if the physical the authors of ref.(6), also in this case no instantaneous predictions for S2 are changed, then one is actually dealing effect at a distance occurs. with a process which simultaneously affects both systems Sj

and S2, as shown by the explicit dependence of the operators At this point, from a basic epistemological point of view, describing the process from the variables of both systems. one could already state that quantum mechanics, with all its Concluding: even if the only logical and rigorous basic assumptions, does not conflict with Einstein's locality. generalization of the reduction postulate, and therefore of the quantum formalism, is considered, no conflict with However, in refs.(2) and (61 it is claimed that quantum Einstein's locality emerges. Any apparent non-local effect is mechanics is not able to describe measurements of the kind induced by the simultaneous action exerted on the two far that the authors have in mind, i.e., devised to detect apart systems. non-orthogonal states. This statement seems to us extremely vague and obscure, but we can try to follow the authors on 5. ANALYSIS OF MORE GENERAL DYNAMICAL MECHANISMS this line. We consider then another possibility. Recently there have been several proposals, aimed to find a solution for various practical or conceptual problems, of C. As it has been clearly illustrated in refs.(4) and (5), modifying the standard quantum dynamics by introducing, e.g., one can actually generalize the standard reduction postulate. stochastic non-hamiltonian terms in the Schro'dinger's This has been done long ago, not for abstract mathematics' evolution equation. Since, as we have seen, the proposal of sake, but for precise physical reasons, i.e. to account for ref.(6) does not fit within the standard quantum rules, it realistic measurement procedures. The generalization of the constitutes a real modification of quantum mechanics. To be as standard reduction postulate has to respect some basic general as possible in analyzing the ideas put forward by the physical requirements. The fundamental result by Kraus 'l1', authors, it is appropriate to see whether, within the that the most general map between statistical operators which framework of such enlarged dynamics, one can give a precise satisfies these essential requirements has the form: meaning to their arguments.

-8- -7- Before going on, however, it is necessary to stress that, measurements on S2. However to assert, on the basis of this from the epistemological and conceptual points of view, the vague model, that the change (13) can be induced (and this is terms of the problem are now completely different front those the crucial point under discussion) by a physical action discussed in the previuos sections. In fact, we are no more exerted only on system St, is clearly meaningless. discussing whether the standard quantum mechanical scheme We can now come back to consider the dynamical models conflicts with Einstein's locality but simply whether the which assume a basic modification of standard quantum specific models which will be considered exhibit such a formalism. Various models of this type have recently been feature. proposed and extensively investigated. Without trying to be exhaustive we mention refs.(12-1 7 ) . As a preliminary step of the new attitude in looking at It is not appropriate to comment here on these works and ref.(6)i it is appropriate to point out that, even though the on the results that have been achieved in them. However, there authors have tried to make the reduction mechanism introduced is a point which emerged from these investigations and which in ref.(2) more definite, by taking explicitly into account has some relevance to the problem we are discussing. some necessary consistency requirements, they have not given a When considering, e.g., stochastic generalizations of the 1 precise description of what they think happens when Sl is Schrbdinger s evolution, one faces the problem that the subjected to a measurement, so one has no indication helping resulting dynamics can induce faster-than-light effects. As to understand the physical details of the process. It should stressed by Gisin'13', this occurs when the evolution equation be clear that, if one accepts to invent new mechanism* for the statistical operator has not a closed form. According inducing changes of the statistical operator, the simple to this author the possibility of inducing superluminal imposition of general conditions does not eliminate the effects is taken as a criterion to abandon the models which enormous arbitrariness about the details of the process. Since exhibit this feature. different specific processes have different meanings and allow Thus, even for those physicists who are willing to different physical interpretations, one cannot, in absence of consider possible modifications of quantum mechanics, the more precise specifications, draw any significant conclusion requirement of avoiding any conflict with Einstein's locality about the process under discussion. is a basic guiding principle. The reasons for taking this To make the above statements clearer it can be useful to attitude should be obvious: whenever one tries to build up a discuss a trivial case. Suppose one assumes that a composite new theoretical framework, if one wants to attribute to it system, initially in the pure factorized state |*1>|^2> is some basic significance, one has to require that it could be subjected to an action which transforms it into another considered as a non-relativistic approximation of a factorized state, let us say |T1>|M2>, with arbitrarily chosen relativistic theory. In this context, it is illuminating to normalized states |T1> and \u2>. In the language of the remark that, in discussing the model proposed in ref.(14), the statistical operator one has first problem that Bell (181 has tackled and solved is to test that the model possesses "a residue, or at least an analogue |*1XJX**x2l=P(l,2)*P( 1 ,2)= (13) of Lorentz invariance" i.e., that no instantaneous action at a distance is involved. The process (13) obviously preserves probability and also An analogous attitude has been taken in ref.(8) in obviously changes the physical predictions about perspective connection with generalizations of quantum mechanics which

-9- -10- would allow to consider observables associated to 12, P. Pearle, Phys. Rev. D, 13, 857 (1976); Found. Phys. 12, 249 (1982); Phys. Rev. D, 29, 235 (1984), ibid. non-herraitian operators. We think that the same attitude would D 33, 2240 (1986); Preprint IC/88/99, Trieste, be taken by most of the members of the scientific community. 13. N. Gisin, Phys. Rev. Lett. 52, 1657 (1984); ibid. 53, Concluding: the modifications of quantum mechanics which 1776 (1984); Preprint: Stochastic Quantum Dynamics and allow faster-than-light effects (and therefore, in particular, Relativity, Alphatronix Lab. SA, 1988. the proposals made in refs.{2) and (6)) should be disregarded, 14. G.C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D, 34, just on the basis of the fact that they exhibit such a 470 (1986). feature. 15. T. Banks, L. Susskind and M.E. Peskin, Nucl. Phys. B, 244, 125 (1984); J. Ellis, J.S. Hagelin, D.V. Nanopoulos and M. Srednicki, ibid. B, 241, 381 (1984). REFERENCES 16. A. Frenkel, The Reduction of the Schrodinger Have Function 1. K.R. Popper., Quantum Theory and the Schism in Physics, and the Emergence of Classical Behaviour,Preprint,Budapest Vol.3 of Poscript to the Logic of Scientific Discovery, KFKI-1988-17/A. edited by W.W. Bartley (Hutchinson, London) 1983; N. Cufaro-Petroni, A. Garuccio, F. Selleri and J.P.Vigier, 17. L. Diosi, Phys. Lett. A, 122, 221 (1987). C.R. Acad.Sci. Ser.B, 290, 111 (1980); F. Selleri, in Dynamical Systems and Hicrophysics, edited by A. Blaquiere 18. J.S. Bell, in: Schrodinger - Centenary Celebration of a et al. (Springer-Verlag, New York.N.Y.) 1980; N.H.Herbert, Polymath, edited by C.W. Kilmister (Cambridge Univ. Press, Found.Phys., 12, 1171 (1982). Cambridge) 1987, p.41. 2. A. Datta, D. Home and A. Raychaudhuri, Phys. Lett. A, 123, 4 (1987). 3. G. Lindblad, Phys. Lett. A, 126, 71 (1987); E. Squires and D. Siegwart, Phys. Lett. A, 126, 73 (1987); J. Finkelstein and H. Stapp, Phys. Lett. A, 126, 159 (1987). 4. M.J.W. Hall, Phys. Lett. A, 125, 89 (1987). 5. G.C. Ghirardi, R. Grassi, A. Rimini and T. Weber,Europhys. Lett. 6, 95 (1988). 6. A. Datta, D. Home and A. Raychaudhuri, Phys. Lett. A, 130, 187 (1988). 7. G.C. Ghirardi and T. Weber, Nouvo Ciu. B, 78, 9 (1983). 8. E.J. Squires, Phys. Lett. A, 130, 192 (1988). 9. G.C. Ghirardi, A. Rimini and T. Weber, Lett. Nuovo Cim. 27, 293 (1980). 10. P.M. Eberhard, Nuovo Cim. B, 46, 392 (1978); A. Shimony, in Proceedings of the International Symposium on the Foundations of Quantum Mechanics, edited by S. Kamefuchi et al., (Physical Society of Japan, Tokyo) 1983, p.225. 11. K. Kraus, States Effects and Operations, (Springer,Berlin) 1983. -12- -U- 7

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