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International Centre for Theoretical Physics ... „. •::^i'— IC/94/93 S r- INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS QUANTUM MECHANICS WITH SPONTANEOUS LOCALIZATION AND EXPERIMENTS F. Benatti INTERNATIONAL ATOMIC ENERGY G.C. Ghirardi AGENCY and R. Grassi UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE 1 f I \ •» !r 10/94/93 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization ABSTRACT INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS We examine from an experimental point of v'iew the recently proposed models of spontaneous reduction. We compare their implications about decoherence with those of environmental effects. We discuss the treatment, within the considered models, of the so called quantum telegraph phenomenon and we show that, contrary to what has QUANTUM MECHANICS heen recently stated, no problems are met, Finally, we review recent interesting work WITH SPONTANEOUS LOCALIZATION investigating the implications of dynamical reduction for the proton decay. AND EXPERIMENTS ' F. Heiiatti Dip.irtimento di I-'iska Tcwicii, Univorsita di Trieste. Trieste, July, G.C. (ihirardi 1. INTRODUCTION International Centre for Theoretical Physics, Trieste, Italy One of the most intriguing features of Quantum Mechanics (QM) is that it provides and too good a description of microphenomena to be considered a mere recipe to predict Dipartiniento di Fisica Tc-orica, Univorsita di Trieste, Trieste, Italy probabilities of prospective measurement outcomes from given initial conditions and, and yet, as a fundamental theory of reality, it gives rise to serious conceptual problems, among which the impossibility, in its orthodox interpretation, to think of all physical II. CSrasfli observables of the system under consideration as possessing in all instances definite Dipartiinento di Kisira, Universita di Udinc, Udrne, Italy. values. This fact can be (and actually is) accepted as far as microsystems are involved, far less so if we consider that individual macrosystems, as aggregates of microsystems, should also be thoroughly described by QM, whence they too ought to be associated with vectors \tjj > in some Hilbert space. In fact, if Q is a macroobservable with two eigenstates |± > such that: Q|± >= .t|± >, <?--=P+-P_, P± = \±><±\, (1) then, according to QM, Q does not possess a definite value unless the macrosystem is in one of the eigenstates |i: > , with the consequence that linear superpositions of M1HAMARK • TRIESTE macrostates like May 1994 IVH >=^{l+>±|-->}, (2) do conflict with macrorealism, namely with the sensible prejudice about reality that macroobservables possess definite values in all macrostates independently of any obser- vation that can be carried out on the corresponding macrosystems. This motivates the search for a theory allowing a macroobjective description of the world. 'Invited Lecture to the International Course on Advances in Quantum I'lu-noim-i Krire 10 28 February 1994. To appear in the Proceedings Plenum Press, N.-w York. In standard QM the objectification of the considered observable requires that, nonpure states like T[P±\. Indeed, the interference terms (off-diagonal elements in (4)) at the ensemble level (see, however footnote 1), the description given by the actual can be seen only by measuring observabies like R = |+ >< — | + |— >< +| which statistical operator p associated with the ensemble be equivalent to the one given by do not commute with P±. However, the assumption of the impossibility of measuring the transformed statistical operator: noncommuting macroobservables is certainly questionable in principle; among other im- plications it requires to assume that the Hamiltonian describing the system-apparatus T\p\ - P,pP, + P pP.. (3) interaction in a measurement process is not observable'1'. Just to give an example, in the case of |^t >, the statistical operators are Pf, 4. QM is a complete theory, within which one should however treat all systems, namely pure states (projection operators) corresponding to homogeneous statistical especially macrosystems, as not isolated from their environment. ensembles. If we represent them with respect to the basis |± >, the interference terms It is suggested that macrosystems (S) are unavoidably and uncontrollably coupled due to the coherent superposition of the macrostates | t > give rise to nonzero off- with their environment (E) and entangled with it by interactions. We must then consider diagonal terms: the compound systems SYSTEM + ENVIRONMENT (S+E). {•!) According to the above discussion, it should, for one: reason or another, be legiti- In order to extract the state (density matrix) ps of the system of interest out of mate to repla.ee such statistical operators by the state psm of the whole system, we have to perform a partial trace Tr^; with respect to any orthonormal basis in the Hilbert space Mj. of the environment: ><i>±\i\ t (5) ps •• Tr,.;p.s- .;. (7) 7[/'[ | are not pure states (not projection operators) and are interpreted as cor- tr responding to the inhomogeneous statistical ensemble made up by the equal weights By getting rid of the environment degrees of freedom, interference effects among mixture of homogeneous statistical ensembles associated with Ihe systems in the state (macro)states of S may be eliminated due to the fact that they are very likely to become ' I ••, respectively [ >. 7'[P+] is an incoherent mixture of the macrostates Pt without entangled with orthogonal states (vectors in M>;) of the environment. The mechanism interference between them: is quite simple: let |-]- > and |i > , i — 1,2 be orthogonal vectors in the system, IH.v, and environment Hilbert space IH;,;, respectively. If, initially, the state of the compound 'I'M '• ' 1' (6) system S I-E is Coming back to our general argument we mention that the macroobjectifscation I**(./; > - (ct.| f > +c.|- > ) ® |0 >, (8) problem, which is as old as QM itself, has been attacked in a number of different ways. 1. QM has not to be taken as a theory of physical reality, but, rather, as a means and the common dynamics is such thai, after some time, lo calculate probabilities of outcomes of prospective, measurements. This is an extreme instrumentalistic position which does not question the quantum +c. (9) mechanical scheme, but simply refuses asking what kind of reality brings about Ihe regular patterns in nature. then, taking the partial trace to single out the state of the system yields 2. QM is an incomplete theory of physical reality. Hilbert space vectors such 2 as jV'i > encompass only a partial knowledge of physical states which would be fully = Tr,,-p.v+,.; = I- >< -I I 4 \c.\ I- X - (10) characterized by specifying some set of unknown parameters (hidden variables) A. Obviously this cannot be considered as solving, in principle, the objectification problem It is suggested that QM should be completed (e.g. as in the de Broglie Bohm but simply showing that the above is not a problem in practice, because practice is not Pilot Wave Theory) so that all results of QM are reproduced, but, at the same time, accurate enough and may be never will be. it. is conceptually correct to think that a macroobservable like Q possesses a definite 5. QM is a complete theory, but its evolution equation should be changed in such a value in each of the systems (specified by the hidden variable A) that make up the way that, on a suitable time-scale, macroproperties result objectified as a consequence of inhomogeneous statistical ensemble associated with the linear superpositions \tp± >. the dynamics and not by virtue of an external assumption like the wave packet reduction 3. QM is a complete theory, but, within QM, macrosystems are singled out as postulate, while the behaviour of microsystems is disturbed as little as possible. special objects completely defined by sets of commuting observabies. In this approach it is required that the unitary evolution for statistical operators It is suggested that on macroscopic objects only classical observabies can be mea- sured, The restriction to sets of commuting macroobscrvables means that, when deal- ing with macrosystems, pure states like PJ cannot be operationally distinguished from d,p, ^ (11) 2 be changed so that microsystems keep on behaving according to (11) on any reasonable QMSL1. At random times, with mean frequency A, any system is affected by local- 3 interval of time, whereas a reduction mechanism like in (3), on a time scale suited to a ization processes in position around points x 6 1R occurring with probability P/.m(x) macrorealist position, is consistently supplied by the unique dynamics itself, and transforming their states *(q) G H according to A simple example: let the physical system be described in a two-dimensional Hilbert J space IH = C , fr1,(Ty,<Ti be the Pauli matrices and choose a. Hamiltonian H *(q) (16) 11**1! (12) (17) If the initial state is P+, then at time t, P+(t) will in general have nonvanishing off- diagonal terms, since the characteristic length a 5 and the frequency of localization being chosen to be =cos^i| + > I sin ^i (13) '- ^ 10 "'cro, A --- 10 'Sec ' . (19) Adding a term \T\pt\ to (11) (compare (3)), where X ' is some characteristic time, and a counterterm -\p, to assure that Trp, -• 1 for all t > 0, the new evolution equation The projection operator /'* --•• |i[/ >< <Ji| undergoes a process similar to the one in (5): dipt - (14) Tf;mv[P*\ dxc (20) is such that, for times much larger than A ', the off-diagonal elements of p with respect t q being the position operator of the system. to the basis | ± > will be negligible. Indeed, if we can neglect the commutator (generator of the Schrddinger evolution) in (14), the solution p,, with initial condition p(0) — p is: In spite of the fact that the conceptual relevance of QMSL, embodied in (16)-(18) resides in its yielding reduction at the individual level ', its main physical consequences —" /?«, - T\p\. (IS) can also be studied by considering what the dynamics looks like at the ensemble level.
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