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Cp-Violation and Einstein's Locality Reconsidered

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Cp-Violation and Einstein's Locality Reconsidered ""'^^^ 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 space tf<'>(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<l,D) Q (11) i.e. it gives rise to a modification of the physics of system In eq.(ll) U(1,D) is the unitary operator describing the S2. It has to be stressed that the only physical meaning which evolution of Sj+D. can be reasonably given to processes like (8( is that both In such a case, as it has been proved some time ago'9', the physical results concerning S are not influenced by the systems Sx and S2 are simultaneously affected by the 2 measurement, so that the measurement itself is the unique interaction of St with D. This is obvious and follows again -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.
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