The Foundations of Quantum Mechanics: Historical Analysis And Open Questions, Cesena 2004

The Foundations of Quantum Mechanics Historical Analysis and Open Questions - Cesena 2004

editors Claudio Garola • Arcangelo Rossi • Sandro Sozzo

The Foundations of Quantum Mechanics Historical Analysis and Open Questions - CBsBna 2DD4

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The Foundations of Quantum Mechanics Historical Analysis and Open Daestions - Cesena 2D04 Cesena, Italy

4 - 9 October 2004

editors

Claudio Garola Arcangelo Rossi Sandro Sozzo University of Lecce, Italy

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THE FOUNDATIONS OF QUANTUM MECHANICS Historical Analysis and Open Questions — Cesena 2004 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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SCIENTIFIC COMMITTEE Silvio Bergia and Giorgio Dragoni (University of Bononia) Claudio Garola and Arcangelo Rossi (University of Lecce) Vincenzo Fano and Gino Tarozzi (University of Urbino) Franco Pollini (Commune of Cesena)

ORGANIZING COMMITTEE Lucio Rizzo and Sandra Sozzo (University of Lecce) Isabella Tassani (University of Urbino)

SECRETARY Maria Concetta Gerardi (University of Lecce)

SPONSORING INSTITUTIONS

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University of Bononia UNWERSITA

University of Lecce

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University of Urbino

Commune of Cesena

Physics Department of the University of Bononia

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Cesena Interuniversitary Research Center on Philosophy and Foundations of Physics

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CONTENTS Introduction C. Garola, A. Rossi and S. Sozzo

1

If Bertlmann had Three Feet A. Afriat

18

Macroscopic Interpretability of Quantum Component Systems R. Ascoli

23

Premeasurement versus Measurement: A Basic Form of Complementarity G. Auletta and G. Tarozzi

40

Remarks on Conditioning E. G. Beltrametti

48

Entangled State Preparation in Experiments on Quantum Non-Locality V. Berardi and A. Garuccio

61

The First Steps of Quantum Electrodynamics: What Is It That's Being Quantized? S. Bergia

68

On the Meaning of Element in the Science of Italic Tradition, the Question of Physical Objectivity (and/or Physical Meaning) and Quantum Mechanics G. Boscarino

80

Mathematics and Epistemology in Planck's Theoretical Work (1898-1915) P. Campogalliani

92

On the Free Motion with Noise B. Carazza and R. Tedeschi

103

Field Quantization and Wave/Particle Duality M. Cini

110

vm Parastatistics in Econophysics? D. Costantini and U. Garibaldi

118

Theory-Laden Instruments and Quantum Mechanics S. D 'Agostino

130

Quantum Non-Locality and the Mathematical Representation of Experience 142 V. Fano On the Notion of Proposition in Classical and Quantum Mechanics C. Garola and S. Sozzo The Electromagnetic Conception of Nature and the Origins of Quantum Physics E. A. Giannetto

156

178

What We Talk About When We Talk About Universe Computability S. Guccione

186

Bohm and Bohmian Mechanics G. Introzzi and M. Rossetti

197

An Objective Background for Quantum Theory Relying on Thermodynamic Concepts L.LanzandB. Vacchini

210

The Entrance of Quantum Mechanics in Italy: From Garbasso to Fermi M. Leone andN. Robotti

225

The Measure of Momentum in Quantum Mechanics F. Logiurato and C. Tarsitani

238

On the Two-Slit Interference Experiment: A Statistical Discussion M. Minozzo

248

Why the Reactivity of the Elements is a Relational Property, and Why it Matters V. Mosini

260

IX

Detecting Non Compatible Properties in Double-Slit Experiment Without Erasure G. Nistico

274

If You Can Manipulate Them, Must They Be Real? The Epistemological Role of Instruments in Nanotechnological Research A. Rebaglia

281

Mathematical Models and Physical Reality from Classical to Quantum Physics A. Rossi

293

Complex Entanglement and Quaternionic Separability G. Scolarici and L. Solombrino

301

Mach-Zehnder Interferometer and Quantitative Complementarity C. Tarsitani and F. Logiurato

311

Antonio Gramsci's Reflection on Quantum Mechanics /. Tassani

320

The Role of Logic and Mathematics in the Heisenberg Formulation of Quantum Mechanics A. Venezia

335

Space-Time at the Planck Scale: The Quantum Computer View P. A. Zizzi

345

Three-Dimensional Wave Behaviour of Light F. Logiurato, B. Danese, L. M. Gratton and S. Oss

359

INTRODUCTION

The Conference entitled The Foundations of Quantum Mechanics. Historical Analysis and Open Questions - Cesena 2004 was held in Cesena (Italy) from October 4 to October 9, 2004, and was the fourth of a series that began with a Conference in Camerino (October 31-November 3, 1988) and continued with two Conferences in Lecce (October 5-8,1993, and October 13-16, 1998). All Conferences had the same title, in order to underline their ideal continuity. Indeed, they all were conceived as interdisciplinary meetings among Italian researchers (physicists, logicians, mathematicians, historians and epistemologists) concerned with the history, the structures and the foundational problems of quantum mechanics (QM). As we have already stressed in the Introductions to the previous Conferences, the idea of grouping together scholars coming from different disciplines was suggested by the huge number of open mathematical, philosophical, historical and epistemological questions in QM, which have raised the interest of many researchers having unlike cultural backgrounds. The main aim of the Conference was then comparing the various perspectives, favouring reciprocal understanding and setting up a common language that could help crossed fertilization of ideas. Of course, this aim has been only partially achieved, but every issue of the Conference has registered some significant progress. New proposals for the solution of old and more recent problems have been forwarded, and the propensity to compare different disciplinary perspectives has become more widespread. This is witnessed also by the papers collected in the present volume, some of which clearly have an interdisciplinary character. All papers are published under the responsability of the authors, but the editors have made the effort to revise and discuss many of them with the authors in order to offer a better and more understandable final product. To this end, we also comment briefly on all articles in this Introduction, so that the reader may have an overall perspective on the content of the book. In order to facilitate readability, the papers will be grouped here, rather subjectively, into three main classes. In the first class we collect "technical" papers that propound new solutions or viewpoints within the framework

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of QM, or some generalization of it. In the second class we group those papers that have an "interdisciplinary" character, i.e., discuss QM problems introducing perspectives or notions pertaining to different disciplines. In the third class we collect the papers having a more historical, philosophical or epistemological character. Let us begin with the first class, the class of "technical" papers. The paper by A. Afriat argues against the common belief that conservation accounts for quantum correlation. The author notes that people who uphold this thesis actually have in mind conservation plus an additivity condition, and that quantum correlation has nothing to do with time, so that it should follow from the additivity condition only. Afriat recognizes that this actually occurs whenever a compound system made up by two component subsystems is considered. Yet he shows - by analyzing an entangled state of a compound system made up by three component subsystems that additivity alone is not sufficient to account for quantum correlation in the general case (we add here that the author's reasonings can be modified in order to apply to classical mixtures, attaining similar conclusions, so that they seem to be independent of the specific, non classical features of quantum correlation). The paper by R. Ascoli consists of three interconnected contributions. In the first of them the author argues that three main physical paradigms are introduced in QM whenever the interactions between quantum systems and their macroscopic environments (in particular, measurements and measuring processes) are discussed. The three paradigms have increasing complexity and the author's treatment shows that according to the third paradigm the boundary between a quantum system and its environment can be shifted by constructing a quantum model of a part of the environment. This shifting expresses the general idea of the Universality of Physics, which also guides the second and third contributions in the paper. In the former, Ascoli propounds possible definitions of semimacroscopic and macroscopic interpretability of quantum subsystems. This proposal introduces the latter contribution, which reports two theorems stated by the author some years ago and still unpublished. The theorems consider a quantum system undergoing an external process T and supply conditions for semimacroscopicity and macroscopicity of the environment of T. These conditions provide a satisfactory support to the aim at Universality of Physics quoted above, since they show that quantum models of the measurement satisfying final component system semimacroscopicity or macroscopicity can always be obtained.

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The paper by E. Beltrametti deals with the notion of conditional probabilities in statistical physical theories. To this end, the author refers to the very general notion of convexity model, which encompasses the standard classical and quantum frames, and also includes operational (or unsharp) QM. Beltrametti considers in particular a special kind of convexity model called operational probability theory (OPT). The OPT frame indeed generalizes the standard classical case and also the standard and operational quantum schemes, introducing an enlargement of the usual class of random variables in order to encompass indeterministic features. In this general frame a plurality of conditional probabilities can be introduced, which mirrors the non uniqueness of the way in which a joint measurement of two observables can be performed. The author shows that the standard choice of the conditional probability in QM can be ascribed to the so-called Luders-von Neumann recipe, which rests on a strong, not always realistic, idealization of the measurement process. The paper by V. Berardi and A. Garuccio considers the experiments that have been conceived in the last decade in order to test Einstein's locality via Bell-type inequalities by using pairs of correlated photons emitted by spontaneous parametric down conversion processes. These experiments can be divided in two classes: those in which the emerging photons have the same linear polarization (type I) and those in which the polarizations are orthogonal (type II). The authors focus their attention on type I experiments and note that photon pairs may exist in the quantum state that is introduced which travel along the same channel and reach only one of the final polarizers. This implies that the conditions that allow one to deduce the inequality obtained by Clauser and other authors, which is commonly used to test Einstein's locality, are not fulfilled. The aforesaid inequality must then be replaced by another inequality, which is explicitly written down by Berardi and Garuccio but cannot be violated by the joint transmission probabilities predicted by QM. Hence, the authors argue that type I experiments actually cannot discriminate between QM and local realism. They also show that this conclusion can be supported by further theoretical arguments and criticize some attempts at circumventing the above difficulties by introducing ad hoc assumptions. The paper by B. Carazza and R. Tedeschi analyzes the effect of a thermal background on the free motion of a classical mesoscopic particle, modeling the external noise through a longitudinal force field. The authors show in particular that, asymptotically, the position values are dispersed with a mean square deviation which increases with the time elapsed after

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the detection of the particle, while the mean square deviation of the velocity values vanishes after an initial growth. More generally, the authors' results suggest that some quantum features may be simulated by resorting to an external noise, which at first sight seems to confirm the old idea that nonrelativistic QM can be interpreted as describing a kind of Brownian motion. However, Carazza and Tedeschi point out in their conclusions some technical features that disprove this hypothesis. The paper by M. Cini provides a syntesis of some previous articles in which the author carried on a research program whose origin can be traced back to Wigner and Feynman. Cini generalizes the formalism of classical statistical mechanics in phase space, introducing an uncertainty and a discreteness postulate which imply mathematical constraints on the set of variables in terms of which any physical quantity can be expressed. These constraints entail that all variables must be represented by Dirac q-numbers and allow the author to recover the Wigner function obtained from the standard wave function of the state. By considering such a function as a pseudoprobability which can assume also negative values, one can then eliminate the problematical Schrodinger waves from QM. The author generalizes this approach to quantum field theory by applying the same procedures with simple changes and imposing Einstein's quantization to the states of a classical field. Within this generalized perspective the quantization of quantum variables is a consequence of the existence of field quanta, so that the formal rules of nonrelativistic QM follow from firstly quantizing quantum field theory. The wave-particle duality can thus be interpreted as reflecting the dual nature of the quantum field as a unique physical entity, objectively existing in ordinary three-dimensional (or fourdimensional, relativistic) space. The paper by G. Introzzi and M. Rossetti provides an overview on the de Broglie-Bohm model. The authors firstly resume the essentials of the model, in which a quantum potential plays a basic role, and briefly explain why the model can be considered empirically equivalent to standard QM. Secondly, they summarize the modern approach to the de BroglieBohm model, usually known as Bohmian mechanics. This approach avoids the introduction of a quantum potential and provides an instructive picture of physical reality that highlights the similarities and the differences between Bohmian and Newtonian mechanics. The treatment of the double slit experiment according to Bohmian mechanics is then summed up by the authors, who remind that it yields predictions about the wave function that do not differ from the predictions of standard QM. Moreover, Bohmian

5 mechanics suggests an interesting generalization of the Bohr complementarity principle, firstly propounded by Greenberger and Yasin in 1988 and successively confirmed by photon and neutron interference experiments. Finally, Introzzi and Rossetti briefly discuss some relevant features of the de Broglie-Bohm model, as causality, determinism, realism, nonlocality and holism. They point out in particular that a Lorentz-invariant formulation of the model is still lacking and that the model is realistic as long as the only measured quantities are positional ones, while all other variables are contextual and therefore not realistic. The paper by L. Lanz and B . Vacchini deals both with the foundations of QM and the quantum theory of non-equilibrium systems. Regarding the foundations of QM, Ludwig's point of view is endorsed, according to which an axiomatic approach to quantum theory must be based on the prerequisite that an objective description of statistical experiments can be given in terms of a phenomenologically established "pretheory" (not necessarily classical mechanics). The still debated problem of measurement arises if one forces, may be too naively, this objective pretheory inside quantum theory itself, taking the latter as the ultimate theory. The authors conjecture that termodynamic concepts should have a basic role in the formulation of the pretheory and consider a formalism accounting for non-equilibrium termodynamics, introducing the required objectivity elements by classical fields representing "state parameters" for local equilibrium systems. Recalling that Zubarev's approach to non-equilibrium quantum termodynamics provides a deterministic dynamics for these parameters inside quantum field theory, Lanz and Vacchini argue however that one cannot expect a general validity of such a method, just because of the evidence of the physical role of microsystems. Thus, they discuss how a breakdown of a deterministic regime for state parameters can be linked with the emergence of microsystems as the seeds of a stochastic situation: QM then appears, already in a framework which completes it, as the basic tool for the description of the stochastic regime. F. Logiurato and C. Tarsitani present two papers in this book. In the first paper (which is relevant, in particular, for a critical approach to the didactics of QM) they note that the complementarity and the uncertainty principles, though fundamental in QM, are still rather vaguely and imprecisely stated in some textbooks. The authors focus on the uncertainty principle, point out some common ambiguities in its enunciations in the literature, and observe that an operational definition of momentum is often neglected. Therefore they propose a definition based on the measurement

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of the "flight time" of the particle, and show that it allows one to deduce the de Broglie relation from the wave function instead of postulating it and to obtain the momentum distribution amplitude as the Fourier transform of the wave function at the initial time. Logiurato and Tarsitani then apply their definition to the single slit diffraction experiment and find again the above results in this special case, establishing a connection between the deduction of the uncertainty principle by means of experiments and by the method of Fourier transforms. They note however that both Heisenberg and Bohr adopted in their thought experiments definitions of Ax and Apx which do not agree with the standard definitions of these quantities as variances. Therefore they propose to take this difference into account by distinguishing two kinds of uncertainty relations in the literature. In their second paper (in which the names of the authors appear in reverse order) C. Tarsitani and F. Logiurato consider a problem that has been recently debated in the literature, i.e., the existence of doubleslit experiments that provide "which-way" information without momentum transfer to the physical objects under examination, which makes it difficult to explain the loss of interference effects. The authors consider a MachZehnder interferometer and show that also in this case one can introduce observable magnitudes that allow one to define the notions of visibility (the capability of recognizing the interference effects), predictability (the capability of predicting which path the object will choose) and distinguishability (the capability of inferring which path the object has chosen after having gone through the interferometer). By using these magnitudes one can provide a quantitative formulation of the wave-particle dualism and the Bohr complementarity principle. Moreover, the operators corresponding to them do not commute and are linked by inequalities that resemble Heisenberg's uncertainty relations. Hence, one can give a mathematical expression of the relationship between two fundamental principles of QM, which thus appear as two sides of the same coin. One can also generalize, at least in twodimensional Hilbert spaces, the complementarity relation, and explain the interference loss by means of uncertainty relations involving incompatible observables that depend on the actual experimental conditions and may not coincide with position and momentum. The paper by G. Nistico briefly reviews the ideal double-slit experiment proposed by Englert, Scully and Walther (ESW), that aims to detect which slit each particle passes through (briefly, to detect the WS property), then measuring the point of impact on the final screen. Of course, whenever repeated experiments are performed, no interference pattern ap-

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pears on the screen. Nistico then focuses his attention on the erasure phenomenon, exemplified by ESW by constructing another property of the system, incompatible with the ESW property, that can be measured without destroying interference but losing knowledge about the WS property. The author wonders indeed whether there are physical properties that are incompatible with the WS property but can be detected, for a given state of the physical system, without erasing WS knowledge. His answer to this question is positive, as proved in some previous articles. In addition, Nistico shows in his paper that, if the dimension of the Hilbert space of the physical system is suitably chosen, an ideal experiment can be contrived which makes it possible to detect a property incompatible with the WS property not only without erasure but also without correlation with this last property. The paper by G. Scolarici and L. Solombrino considers the evolution of a compound quantum system from the viewpoint of quaternionic quantum mechanics (QQM). It is well known that in complex quantum mechanics (CQM) a state of the whole system undergoes unitary evolution, while the evolution of the reduced density matrices that can be associated (via partial trace) to the component subsystems is generally not unitary. The authors show in a special case (evolution from a separable to an entangled state of a physical system made up by two s p i n - | subsystems) that the evolution of the states of the component subsystems can be described in QQM by quaternionic unitary maps, and that the non-unitary maps describing evolution in CQM can be obtained as complex projections of the corresponding quaternionic maps. Furthermore, Scolarici and Solombrino provide a description of the final state of the whole system which strongly suggests that this state should be considered separable in QQM. This establishes a relevant difference between CQM and QQM, and deserves further research (we add that such a difference could also constitute an argument in favour of QQM). Let us come now to the second class, the class of "interdisciplinary" papers. The paper by D . Costantini and U. Garibaldi propounds a general approach to equilibrium probability distributions (generalized Polya distributions) in the framework of which various kinds of uniform probability distributions on the occupation vectors can be recovered, as MaxwellBoltzmann's statistics or Gentile's parastatistics. It is well known that only limiting cases of parastatistics are needed in QM for describing equilibrium probability distributions of elementary particles, i.e., the Bose-Einstein

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and the Fermi-Dirac statistics, and these statistics can be easily obtained as particular cases within the model propounded by the authors. However, there are other fields of research in which non-uniform distributions apply (that generalize, in some sense, Gentile's parastatistics) which are also described within the Costantini and Garibaldi approach. In particular, there are economic agents in econophysics whose correlated behaviours are described by typically non-uniform probability distributions. The authors provide an example ("the ants of Kirman") and an application to stock price dynamics in order to illustrate this point, and conclude that, more generally, the behaviour of economic agents may by characterized by high correlation that can possibly be handled with an adequate probabilistic description. The paper by C. Garola and S. Sozzo inquires the notion of proposition in classical mechanics (CM) and QM. The authors note that the term proposition usually denotes an element of standard quantum logic (QL) in QM, and that the physical interpretation of such propositions is problematical since they cannot be associated with sentences of a predicate calculus, following known logical procedures. Indeed, the elementary sentences of this calculus should attribute physical properties to individual samples of physical systems, which may be meaningless because of nonobjectivity of physical properties within ortodox QM. Garola and Sozzo show that this difficulty can be removed by adopting the SR (semantic realism) interpretation of QM propounded by one of them, since physical properties are objective according to this interpretation and a unified perspective can be adopted for introducing propositions both in QM and in CM (more generally, in any physical theory T ) . One can thus construct a simple first order predicate calculus C(x) with classical (Tarskian) semantics, associating a set of physical states (called physical proposition) with every sentence of C(x). The set Vf of all physical propositions is partially ordered, and contains a subset Vy of testable physical propositions whose order structure is determined by the criteria of testability established by the theory T. In particular, V? is a Boolean lattice within CM, while it is a standard QL within QM. Hence, QL can be interpreted as a structure formalizing the properties of the notion of testability within QM, and this interpretation proves to be equivalent to a previous pragmatic interpretation of QL provided by one of the authors. Moreover, it sheds some light on the concept of quantum truth underlying standard QM, showing that it does not conflict with the classical notion of truth. Finally, the authors point out that the above results can be embodied within a more general perspective

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which considers states as first order predicates of a broader language with a Kripkean semantics. The paper by M. Minozzo propounds a new statistical discussion of the two-slit interference experiment for the ideal situation in which particles are sent sequentially (i.e., one after the other) through the interfering barrier toward the screen. The author's treatment is carried out adopting the standard axioms of Kolmogorov's probability theory, and fully exploits the sequential nature of the experimental observations. Altough a "classical" purely particle toy model is presented to explain the interference pattern and the non-additivity paradox which arises when comparing the interference pattern with the patterns obtained by closing one slit at a time, the main point of the contribution resides in the analysis of the actual experimental observations from a rigorous statistical point of view. Minozzo aims to show that, when the analysis is carried out correctly, at variance with some statistical investigations in the literature, interference experiments can be explained using standard statistical tools, without introducing waves and, in particular, without using QM. His contribution neither tries to reproduce the standard theoretical results of QM nor to provide any definite physical theory: rather, it proves, according to him, that QM is not the unique theory that may explain experimental observations and that, moreover, it has some well defined limits. Of course, it remains to establish whether the toy model mentioned above provides a convincing physical explanation of the experimental data. The paper by V. Mosini upholds that that existence of relational properties in many branches of science strongly suggests to revise the standard notion of realism, introducing a dynamic picture of reality which may change with the progress of scientific knowledge. This perspective was firstly worked out by Margenau, mainly bearing in mind QM, in his 1950 book entitled The nature of physical reality, but it was ignored or criticized by his contemporaries. Yet, some relevant features of Margenau's philosophy have been reproposed independently by several authors in the last twenty years. Mosini argues that the deep reason underlying this r e appearance is the widening of the domain of existence of relational properties in different areas of science, for new properties of this kind emerge whenever the increase of scientific knowledge leads scholars to consider more and more complex systems. As an instance, Mosini discusses in some details the case of chemical valence, showing that it is a relational property, since the same element may display different valences in its different compounds, and explaining how this may occur.

10 The paper by P. Zizzi deals with space-time at the Planck scale. As other authors, Zizzi assumes that space-time has a discrete structure at this scale, but adds the issue of quantum information propounding a Quantum Computer View (QCV) according to which each pixel of the Planck area encodes a qubit. The set of qubits forms a quantum memory register, and the information stored in the memory is processed by a network of quantum logic gates (unitary operators) that is part of quantum spacetime itself, since it describes its dynamical evolution. Zizzi claims that this model implies some interesting features of quantum space-time: reversibility of dynamical evolution, nonlocality of space-time itself at the Planck space, etc. She, however, points out some problems in her model and suggests how they can be solved. In particular, she maintains that the nonlinearity of the macroscopic level can be obtained from the linearity at the Planck scale level by considering self-organizing models. She also thinks that macroscopic irreversibility can be reconciled with the reversible dynamical evolution at the Planck scale by assuming Wheeler's picture of "space-time foam". At the end, Zizzi conjectures that micro-causality is missing at the Planck scale, but stresses that, notwithstanding its weird features, space—time seems to be able to compute its own dynamical evolution by quantum evaluating recursive functions. Finally, let us comment on the third class, the class of historical, philosophical or epistemological papers. The paper by G. Auletta and G. Tarozzi considers the three forms of duality that gave rise, historically, to the main conceptual problems in QM: (i) waves versus particles; (ii) deterministic versus stochastic dynamics; (iii) local measurements versus nonlocal correlations. The authors note that some connections between (i) and both (ii) and (iii) have already been established in the literature. Then, they provide some simple examples that lead them to argue that (ii) can be reduced to a more fundamental complementarity between "premeasurement" and "measurement". Since a further connection can be established between this complementarity and (iii), the authors conclude that all forms of duality in QM can be considered as different expressions of a unique fundamental complementarity. The paper by S. Bergia aims to recall scholars' attention on some interpretational problems in quantum electrodynamics (QED) and quantum field theory that were already discussed by Jordan and Dirac, and are yet still open. Bergia deals in particular with QED, and considers the positions of the above authors with reference to three basic issues: the difference between the quantization of the electromagnetic and the matter fields, the

11 dichotomy between waves and light quanta, the question about what quantizing a field actually means. Bergia observes that Dirac explicitly stressed the difference between "light waves" and "de Broglie or Schrodinger waves", and intended to quantize the electromagnetic field only. He was aware that no quantum mechanical wave equation exists for the photon, while QED instead introduces "complete armony between the wave and the light quantum description of the interaction". Moreover, Dirac built up the theory from the light-quantum point of view, proceeding in a direction opposite to Jordan's, who instead intended to apply QM to the Maxwell field itself. Bergia accepts the conclusion, propounded by other authors, that Jordan merely proved that the energy states of the field were quantized, which has nothing to do with the quantization of substantial Maxwell's fields. On the contrary, Dirac early succedeed in introducing a variable Nr that can be identified with the number of energy quanta in the state r arising from the quantization of a wave field. Finally, Bergia observes that Dirac also prefigured the commutation rules holding between number of photons and phase operators, and tries to reconciliate Feynman's insisting on the particle behaviour of light with Dirac's perspective. The paper by G. Boscarino considers the Einstein-Podolsky-Rosen (EPR) well known definition of "elements of physical reality" as a contradictory attempt to use an empiricist and operationalist conception of reality in order to criticize QM's claim to be a complete theory. The author maintains that Bohr's position, which affirms the completeness of QM by philosophically expliciting just the same empiricist and operationalistic conception as the most adequate to view quantum facts, is more consistent. According to Boscarino, however, an appropriate discussion of QM must be made, contrary to both Einstein and Bohr, by making explicit a different philosophical interpretation, going back to the old classical philosophicalepistemological tradition called by him "Italic" (which was diffused mainly in Magna Grecia, in ancient Italy, by Pythagorians). This interpretation was at the basis of classical physics, which maintains that one must distinguish between empirical measurements of physical quantities and physical properties in se: these have reality, even though they are not completely known. Thus, empirical measurements are only partial clues and incomplete proofs of their reality (cp. the distinction between relative and absolute space and time according to Newton). Only in this way it is possible to avoid contradictions and ambiguities of philosophical nature, not only as Bohr's complementarity, but also as EPR's "empirical" definition of elements of physical reality.

12 The paper by P. Campogalliani stresses a methodological point in science studies, opposing the neopositivistic approach: the presence of a third dimension beyond empirical and analytical knowledge, the dimension of metaphysical knowledge. This is a priori, interpretative and analogical, and also influences mathematics, which is usually identified with pure analytical knowledge, through imaginative efforts. This point of view, which enhances the role of informal thought also inside the formal one, is applied by the author to Planck's elaboration of radiation theory (1898-1915), where also the mathematical developments were influenced by conceptual models and analogies. In particular, the introduction of the statistical treatment was not justified, as in Boltzmann, by a mechanical interpretation of the second law of termodynamics, but by a microscopic hypothesis on natural radiation which only formally converged with the former, since it was conceptually very different. Planck in fact, according to his realistic metaphysics, aimed at a larger unification of physics and causal explanation, both before and after the introduction of quantum of action and discontinuity. Hence he considered the statistical approach, though elaborated in terms of phase space and sophisticated as Boltzmann's, as only temporary and provisional. The paper by S. D'Agostino starts from the known thesis that observation and experiments are theory laden in physics and studies the relationships between a primary (physical) theory and the instrumental theories that must be necessarily worked out whenever one wants to test the primary theory. Since every new physical theory includes a number of older physical theories (that can be obtained as special cases, or bringing to a limit appropriate parameters) in its so-called correspondence area, the author argues that instrumental theories associated with a given primary theory actually belong to its correspondence area. When this general viewpoint is applied to quantum measurements, the "objectification process" can be interpreted in physical terms according to the author. Indeed, D'Agostino conceives this process as a consequence of the physical process of recoursing to an objective instrumental theory as a procedure for testing the primary theory, i.e., QM. This should avoid, in particular, any need of interpreting objectification as following from the disturbance that is inherent in a measurement process. The problem remains, however, as acknowledged by the author, of reducing hard facts to instrumental theories apt to test primary theories, since the author rejects inductivist, ontological, or simply justificationist (as Popper's falsificationism) views, in favor of a more sophisticated semantic modelistic approach.

13 The paper by V. Fano proposes to compare the old question of the conflict between the qualitative character of experience and the numerical character of mathematical physics (notwithstanding the fact that the predictions of the latter refer to the former) with this violation of Bell's inequality. Indeed, also this violation breeds a conflict between the common sense locality (or factorizability principle) and the nonlocality following from the mathematical formalism of QM. Since the violation is empirically confirmed, the author tends to consider it as favouring an empiricist conception of the relation between experience and mathematical physics. In fact, Bell's inequality violation can neither be reduced to a mere statistical correlation without a common cause, nor to a mere consequence of the mathematical representation: rather, it seems a hybrid between the empirical reality and the scientific representation derived from it, in agreement with the empiricist outlook. This suggests that empiricism can give a satisfactory reply to the problem of the relation between experience and mathematics in general. Indeed it interprets the latter as a result of a process of idealization of experience, thus opposing platonism and differing from critical materialism (which makes reality beyond experience interact with supervenient physiological human structures, so creating mathematical representations) and operationalism (which reduces experience and mathematics to mere operations without any cognitive value). The paper by E. Giannetto traces back the origins of QM to the electromagnetic conception of nature, contrary to the mechanistic current view which finds out them in the extension of atomism and corpuscolar dynamics from matter to radiation (cp. Einstein). According to the author, only the electromagnetic conception, with its field view, could instead explain the peculiar statistical character of quantum phenomena: these were indeed introduced by Planck in his generalized thermodynamical approach as discontinuities of the universal electromagnetic field (cp. Larmor), expression of a "new mechanics" founded on electromagnetism both in relativity and in quantum physics (cp. Poincare). This conception agreed with Heisenberg's final formulation of QM which, because of its operational character, concentrated on electromagnetic more empirically detectable quantities, conceiving mechanical quantities, that are generally more indeterminate, as derived from the former. Giannetto's final argument is a frankly philosophical and also ethical one: the electromagnetic view favours an attitude towards nature, conceived as a dynamical and living reality, that is more respectful than the mechanistic one, which instead conceives nature as inert, dead and mechanical.

14

The paper by S. Guccione makes an effort to find out a minimal condition for the definition of Computable Universe, given that a general definition of Computable Universe is not available. The author firstly notes that it is necessary to consider a physical theory as a continuum expressed by real numbers, not limiting it to natural numbers as the computable "Turing machine" which constitutes the first model of computability. Then, Kreisel's definition of computability is adopted by Guccione, who adds to it a uniform law condition which should allow one to describe all physical processes. Notwithstanding this, it seems that theories still exist (in particular quantum gravity), which are incomputable even though they may be considered complete according to a suitable semantics. On the other side, a "theory of everything" should be computable as such, but it is not yet available (apart superstring theory, which candidates to this role). Then we must limit our discussion to universal constants in physical theories, inquiring whether they are computable or only measurable (that is, more and more approximable through different calculation recipes, not through a single rigorous one). Moreover, the physical constants seem to be not really constant, or at least, as the velocity of light, only conventionally constant. A more informative (though incomputable) theory on the universe could be quantum gravity, which, according to Hawking, may even recover, by quantum fluctuations, the information lost in black holes. Anyway, as constants may actually be slowly variable quantities and their measurability is not computability, we must content ourselves of a non perfectly uniform nature (according to Kreisel). So, the effective procedure may vary with the constant, and there may be also many procedures for a single constant. The paper by M. Leone and N. Robotti contributes to the historical reconstruction of the entry of QM in Italy. It underlines the mainly instrumentalist approach of the first Italian researchers on QM, Garbasso and Brunetti, who worked on spectroscopy (an area of research which was then very advanced in Italy), after the explicit rejection of the new theory by the authoritative physicist Corbino. The experimentalist-instrumentalist approach by Garbasso is confirmed by his subsequent attempt to interpret the results he had obtained by applying the new theory (in particular, the Stark-Lo Surdo effect) in terms of Thomson's classical theory of atom. On the contrary, Brunetti's contribution was not followed by any attempt at reinterpreting her results classically, even if her attitude was also basically instrumentalist. Anyway, the full entry of QM in Italy was due, some years later, in 1919, to Fermi. As confirmed by some documents and manuscripts in Pisa and Chicago archives, Fermi quite independently approached the

15 new physics, both from a theoretical and from an experimental viewpoint. He contributed to it in his undergraduate years in Pisa with theoretical accuracy, freedom of mind and experimental competence, well before his 1926 epoch-making contribution to quantum statistics. The paper by A. Rebaglia faces the crucial problem of the role of experiments at nanoscale (even at a single particle level), where all phenomena are, without exceptions, to be explained in quantum mechanical terms. Following Hacking's thesis, according to which we can have an evidence of reality of entities in science through their efficacy in actively manipulating with success other known realities, the author attributes a reality not only to empirical entities but also to theoretical entities, thus opposing a mere registration view of scientific truth. As an example, the wave function and the superposition of individual particle positions are real entities, since they contribute to the working, in particular, of a nanotechnological device as the scanning-tunnelling microscope, for which they are in fact theoretical starting points. Notwithstanding the appeal of this active transformative view of science, according to which we can also reproduce, by creating new devices, the creative processes of nature, the author recognizes that the problem remains of still distinguishing two different activities: first, experimenting on entities to get evidences; second, creating new technological devices by reproducing natural processes for practical use, so also giving reality to purely theoretical entities. The paper by A. Rossi underlines how the transition of the physical object from substance to function, which is to be traced back to the rise of modern physics (relativity and quanta), was made by maintaining the role of properties as constituents of any physical system, irreducible, as such, to mere measurements. In particular, the formalization of QM by von Neumann in terms of operator calculus in Hilbert space was not derived from empirical information acquired through measurements, but preceded its physical interpretation, and its application could be extended well beyond the empirical interpretation of projectors adopted by von Neumann himself. Even more, Dirac distinguished formal quantum properties from classical ones in algebraic terms, irreducible to mere measurements, though he left ambiguous the physical interpretation of his abstract formalism (particularly of his "delta-function"). The way out from the ambiguity consists in opposing the frank acknowledgement of the existence of physical properties (as Dirac's himself magnetic monopoles or non-simultaneously measurable properties in QM), which are not accessible (empirically decidable) and reducible to instrumental operations and theories, either to pragmatist re-

16

ductions or to falsificationist conceptions. The latter in particular, though avoiding to reduce physical theories and properties to mere instrumental ones, yet still consider them as quite empirically decidable (falsifiable). The paper by I. Tassani aims to analyse Gramsci's thought about microphysics as a part of a wider plan of reconstructing the cultural and philosophical European climate between the two world wars, when QM, as relativity theory (RT), was conceived and spread. Tassani stresses that QM was, contrary to RT, a difficult topic for philosophical discussion, as it was non-visualizable, complex and contrasting with common sense. In particular, these difficulties were marked in Italy, where a dominating idealistic philosophy either was not interested in new physical theories or tended to interpret them in purely immaterialistic and subjectivistic terms. Gramsci instead, though he shared the idealistic criticism of being a kind of superstiction to the positivistic exaltation of science, duly evaluated the scientific knowledge as a part of human labour and culture, thus overcoming immaterialism and subjectivism in terms of critical realism and intersubjectivity. He thus met the most advanced reflection on QM, even if he had little access to the literature since he was prisoner in fascist jails. He also underlined the relevance of linguistic considerations for avoiding paradoxical conceptions (so agreeing with some typical neopositivistic attitudes) and the impossibility of eliminating from science (inter)subjective conditions. It is then relevant to remind that Bukharin, the Sovietic exponent of historical materialism, in his contribution to 1931 London International Congress on History of Science claimed that science is only conditioned by economical-social factors, criticizing the subjectivistic intromissions of supposed religious origin. On the contrary Gramsci, stressing the role of subjective hypotheses together with experiments, anticipated, at least in part, Kuhn's historical-critical view of science. The paper by A. Venezia opposes Heinsenberg's matrix mechanics to Schrodinger's wave mechanics by assuming that the mathematical equivalence of the two formulations of QM, asserted by many authors, is contradicted by some differences existing also between the subsequent reformulations of the two perspectives (apart from the derivation of Schrodinger equation from Heisenberg's commutation rules made by Weyl using mathematical tools, as group theory, alternative to differential equations). The differences are both mathematical and logical, but the latter are more important. Indeed, the quantum logic (QL) which is more linked to Schrodinger's wave mechanics, because of its axiomatic organization and continuity, is the Birkhoff and von Neumann projectors algebra. Instead the QL which is

17

more linked to Heinseberg's matrix mechanics is an intuitionistic QL which is irreducible to the previous one, according to the author, since it rejects the law of the excluded third. In order to stress his point, Venezia proposes to derive mathematics from logic, and analyzes some of Heisenberg's reasonings on indetermination principle and commutation rules as a QL founded on the refusal of the law of excluded third. This logic is, in his opinion, more appropriate to QM than Birkhoff and von Neumann's QL. Finally, we insert at the end, out of alphabetical order, an article by F. Logiurato and other authors, which cannot be included in the above classification since it has mainly didactical purposes. Indeed, it describes a simple experimental apparatus which allows one to observe light diffraction and interference in the space and not only on a two-dimensional screen. Its aim is to illustrate the ondulatory aspects of light in the double-slit experiment used in many textbooks in order to introduce the wave-particle dualism and the Bohr complementarity principle. The pictures that have been obtained are suggestive and we have chosen them to close this book. We have thus completed the presentation of the papers collected in this volume. We would like to conclude by thanking all the Institutions that have sponsored the Conference and granted financial support to the publication of these proceedings. In particular, the Universities of Bononia, Lecce and Urbino, the Commune of Cesena, the Physics Departments of the Universities of Bononia and Lecce, the Cesena Interuniversitary Research Center on Philosophy and Foundations of Physics. Claudio Garola Arcangelo Rossi Sandro Sozzo

IF BERTLMANN HAD THREE FEET ALEXANDER AFRIAT Dipartimento di Filosofia, Universita di Urbino, via Saffl 9 1-61029 Urbino It is argued that perfect quantum correlations cannot be due to additive conservation. Dr. Bertlmann likes to wear two socks of different colours. Which colour he will have on a given foot on a given day is quite unpredictable. But when you see that the first sock is pink you can already be sure that the second sock will not be pink. Observation of the first, and experience of Bertlmann, gives immediate information about the second. [1] Most interesting features of quantum mechanics have to do with coherence (in other words with interference, with phase), which will not, however, be at issue here at all. Coherence is brought out with respect to different bases, but here the same (product) basis is adhered to throughout. It is often claimed that conservation accounts for quantum correlations (by which perfect quantum correlations will be meant). The underlying intuition is well expressed by Bertlmann's socks, or by the fact that the distribution of wine over two glasses can be worked out—provided one knows the total amount in both—by a measurement on one of them. Or consider a conservative classical Hamiltonian H =T + V(q), where T is kinetic energy and the potential V depends only on position. Conservation means that exchanges of kinetic and potential energy along a trajectory satisfy H0 = T + V, where H0 is the total energy of the motion. Kinetic energy will then be a function only of position, so that at any stage of the motion T(q) = H0 — V(q) can be deduced from the potential; the two energies are perfectly correlated. Or take two free classical particles, each one subject only to the influence of the other, with initial momenta p0 and p'0. Even if they collide their total momentum will remain TT = p0 +PQ', the momentum p' = -n — p of the primed particle can always be derived from the momentum p of the other. Such instances of additive conservation are paradigmatic. Quantum correlations are similar, especially at a given instant, and with only two subsystems; but they have nothing to do with conservation. When the contrary is claimed it seems that additive conservation is meant; but that can be broken up into two logically independent parts: 1. conservation; and 2. an

18

19 'additivity' condition, presently to be defined and denoted (A). Quantum correlations can have nothing to do with time, which has everything to do with conservation; so what is fundamentally at issue is additivity. I will argue that an additivity condition can be constructed to account for quantum correlations with two subsystems, but only with two; where there are more, quantum correlations are too strong to be explained by additivity. An explanation that only works in a restricted special case should be viewed as no explanation at all; so quantum correlations have nothing to do with additivity. Take three socks (on an equal number of feet) rather than two: once the pink sock is found on one foot, we know the remaining socks are on the other feet, but we cannot infer where the blue one is. With three glasses a measurement on one glass only tells us how much wine is in the other two together, not how much is in the third. Triorthogonal decompositions appear to go beyond the knowledge available in the above cases, and indeed to tell us where the blue sock is, or how much wine is in the third glass. Consider the triorthogonal decomposition*

W = £Utehwr->w, m

r = 1,2,3. The operator Ar establishes a one-to-one correspondence A,r

(Aj)

f~ —>

(card A j / c a r d A)

Figure 1. PARADIGM 1. The pointed downarrow expresses a transition from physical to cybernetic reality.

For the experimental physicists things are quite different: in fact they have to prepare a collection c = (CA)AGA of samples, that we label through an index set A, in the state W and to perform on each sample c\ the measurement of 77, hence the physical operations leading to decide which i has to be attributed to that sample. The activity within the physical reality finishes here. Then an activity within the cybernetic reality has to take place. First of all each experimental result has to be registered on some cybernetic support, for instance a magnetic one or a sheet of paper. Then for any i it has to be determined the subset Aj C A of the index set A of the samples for which the measurement of 77 = (77j) has led to endow the sample with that i. Finally the comparison in view of a (fraught with mystery) identification between the obtained frequencies and the probabilities of the theory (i.e. cardA e /cardA ~ pY) has to be made. In the diagrams shown here, vertical pointed arrows express interaction between physical and cybernetic realities. Thus for the experimental physicists measurement implies a transition from physical to cybernetic reality, we call it reading.

27

[We remark that the set A of the indices, in one to one correspondence with the samples of the collection c, clearly is an unnecessary structure, which is avoided in Ludwig's treatment: one can directly perform the partition on the collection c instead of performing it on the index set A. Yet one has to pay great attention to the fact that the partition concerns the original collection c and not a supposed final collection, which is not the object of Paradigm 1. Thus we prefer to introduce the index set because in this way the subject (here and in the next paradigms) can perhaps be more easily grasped. Besides, from the pragmatic point of view, any sample should always be labeled in some way, for instance according to the (irrelevant) time at which the experiment on that sample has taken place; and, after all, counting is precisely labeling within the set N of the natural numbers.]

External Process described by T — (T{) ((discrete) Davies Instrument) physicomathematical level

W

matter level cybernetic level

c WA \

Su which is one-to-one on the pure states and many-to-one on the mixtures, the counterimage of the density operator D £ Su being the family of all the convex decompositions into one dimensional projectors admitted by D. Any quantum observable A : Sy, -> M{I"(S) (if we refer to the standard quantum frame then E should be the real line) has the classical representative A o R : Mx+(fi-H) ->• M 1 + (S) which reproduces all the statistical properties of A. Clearly, not every observable on M^(VLy) need be the classical representative of a quantum observable.

3. The classical frame As already sketched in the previous section, the classical frame adopts for the set fi of pure states the structure of a measurable space, and all singletons {u>}, u> € fi, are assumed to be measurable. The set of all states, pure states and mixtures, is then identified with the convex set M+(fi) of all the cr-additive probability measures on fl. The pure states correspond

52

to the Dirac measures, namely the probability measures concentrated at a point of fi, to be denoted Su, w € ft. The set M+(ft) thus embodies the structure of a simplex whose extreme points correspond to the elements of fi: this meets the classical feature that the nonpure states have a unique convex decomposition into pure states. A random variable taking values in some measurable space 5 is usually denned as a measurable function F : fi -» S. This notion mirrors the deterministic requirement that a random variable must take definite values on pure states, namely that it has no dispersion on pure states. Notice that a measurable function F : Q, -* S extends in a natural way to the afnne map Ap : M+(fi) -> Mf(E), denned by (AFfi)(X)

:= niF-1 (X)),

X € B(E),

(1)

which will be called the distribution functional of F and corresponds to the notion of observable introduced in the previous Section. The pair (F, X) can be viewed as a two-valued experiment, which simply states whether the outcome of F does or does not fall in X. It will be called an event: the event that does (does not) occur when the outcome of F does (does not) fall in X. Clearly it is uniquely determined by the subset F~l{X) of fi: in other words the events are represented by the elements of the Boolean algebra B(U) of all the measurable subsets of fi. With some abuse of notation the subset F~X(X) will be sometimes denoted by the pair (F, X). When we refer to elements of B{Q) without special attention to the random variable they come from we will write a, b,... and say that fi{a), (J, £ A/1+(fi), is the probability of occurrence of the event o when the state of the physical system is / j . The deterministic nature of the frame here adopted means that when the state is pure, say 6U, then Su(a) = 0,1. Writing C(b\a; /x) for the conditional probability of b given a in the state (i, the standard classical recipe reads C(b\a;,) = ^

.

(2)

Let us recall a number of relevant properties of this definition. (i) If C(b\a; /x) = fi(b) then also C(a\b; fx) = n(a): in such a case the probability of occurrence of one of the two events is not affected by the occurrence of the other and we can speak of two mutually independent events. (ii) C(-|a;/i) belongs to M+ (fi) since we have C(0|a;/x) = 0, C(fi|a;/z) = 1, C(b U c\a; fj,) = C(b\a; /i) + C(c|a; /x) if b n c = 0. Hence C(-|a; fj) is a new

53

state of the physical system, to be called the conditioned state and denoted ^a\ so that we can write C(b | a;/x) = l^a\b). (iii) If we write /x = X^w» 0 < w, < 1, 53,-Wj = 1, for the convex decomposition of fj, into pure states (the countability of the decomposition is, however, unessential) the conditioned state takes the explicit form Ma) i as easily checked by noticing that ^2 Wi 6Wi (a) 6Ui (b) = Y,wi i

6

"i (o n 6) = /*(o n 6).

(4)

i

If the state \i is pure, say • M+(Ei x E2) denned on pure states by (At H A2 6u)(Xi x X2) = (AiSuXXi)

• (A2SU)(X2),

(11)

and extended to nonpure states by linearity. The conditional probability that the observable A2 takes values in X2 given that the observable A\ takes values in Xi, in the state fi £ M^~ (ft), will now be defined as C((A,,X,)|M„Jf,),rt-(^-f^').

(12)

This definition captures, we believe, the genuine statistical meaning of conditional probability. It mirrors the classical property expressed by Eq. (10) but now the nonuniqueness of the joint observable Ai,2 gives rise to a plurality of conditionals, a fact that appears not surprising in view of the nonuniqueness of the way of performing a joint measurement of two observables. Thus we have to speak of the conditional probability with respect to a specified joint observable: to outline this fact we will sometimes denote the conditional probability under discussion by C((A2,X2) \ (Ai,Xi);fi;Ai,2). Let us now examine some features of the above conditioning. As in the classical case the condition C((A2,X2) \ (Ai,Xi);fi) = (A2/J,)(X2) implies C((Ai,Xi) \ (A2,X2);(i) = (Ai(j)(Xi): this condition thus fits with the natural notion of mutual independence of the two effects. Clearly the Bayes property (see Eq. (6) of Sec. 3) C((A2,X2)

| (AuX1);fi;Al,2)

• (A1^)(X1)

=

= C((A 1 ,X 1 ) | (A2,X2);fx;Ah2)-(A2ti)(X2).

(13)

is met for every choice of the joint observable A%i2. The possibility of viewing the conditional probability of Eq. (12) as the probability of the conditioned effect (A2,X2) in some conditioned state occurs if the product A\ $3A2 is chosen as the joint observable. Indeed, writing P — Yli wi ^w; (0 < ttTj < 1, £ \ wt — 1) for the convex decomposition of fj, into pure states, we have C((A2,X2) | (AuX^nAiBAi) where

= (A2^'X^)(X2),

(14)

56

To prove this statement just notice that

(A2^X^)(X2)

E

= (AJ){Xi)Ylwi , A K7I A i

• W.*)(*i) • (A2SUI)(X2) =

\fv

v

(yli/iXXO Y

,

(Al ®A2 fl)(Xl

X X2)

(^M(*i)

(16) The map /J, i-+ ^(^i' x i) specified above is nondisturbing since it leaves fixed the pure states: if /x is pure, say Su, then 6^' 1 = Su. Thus we see that the properties of the standard classical case quoted in items (ii) and (iii) of Sec. 3 are preserved in the OPT frame when we refer to the product Ai G3 A2 as the joint observable. Notice that the equality dl *' l } = 8W implies (see Eq. (14)) C((A2,X2) | (AuX^S^A^At)

= (A26U)(X2).

(17)

which expresses the fact that any two effects (Ai,Xi), (A2,X2) become mutually independent in a pure state with respect to the choice Ai K A2 for the joint observable. The standard classical property (see Eq. (10)) of no correlations in pure states is thus recovered in the OPT frame when we refer to the joint observable Ai E3 A2. However, two effects need not be independent in a pure state when we refer to different choices of the joint observable: in this way the typical quantum phenomenon of correlations in a pure state manifests itself in the OPT frame through the existence of joint observables that do not have the product form, hence thanks to the nonuniqueness of the joint observable. The so-called quantum entanglement here appears connected to the way in which two observables are paired together to form a joint observable: in this sense the entanglement cannot be viewed as a property pertaining only to a state [10,11]. Let us stress that the occurrence of a conditioned state having the nondisturbing nature expressed by Eq. (15) rests crucially on the simplex nature of the convex set of states Mj + (Q). We have indeed the following theorem. Theorem 4.1. If S is the convex hull of fi := {uii,u2, ••••} then a map of the form ^2wi ciui

S 3 a = ^2wiC0i i-> — i

Wi

^i

Ci

>

0 < Wi ,ct < 1, y^Wj = 1

i

(18) exists only if S is a simplex.

57

Proof. Suppose S is not a simplex, so that the minimal cardinality of the set Cl of its extreme elements is 4. If there are just 4 extreme elements then they have to be " coplanar", and by a proper ordering of them, the segment (wi, 0J2) will intersect the segment (W3,CJ 4 ) in one point a which will admit the two convex combinations a = Wi U1+W2 W2 and a = W3 U3+W4 W4. A nondisturbing state transformation would move the first convex combination along the segment ( ^ i , ^ ) and the second convex combination along the segment (^3,604) thus producing two different elements of S. If the cardinality of fi is greater than 4 then it is always possible to pick up 5 extreme elements, say wi,a;2,W3,a;4,W5 in such a way that the segment (wi,W2) intersects the triangle {u)3,U4,u)§) in one point a which will admit the two convex combinations a = W\ a>i + u>2 ^2 and a = W3 0J3 +W4UJ4+W5 UJ5. A nondisturbing state transformation would move the first convex combination along the the segment (u>i,w2) and the second convex combination along the triangle (LJ3,W4,W5) thus producing again two distinct elements of S. • We finally remark that in the OPT frame the conditional probability defined by Eq. (12) does not fulfill a property analogous to the one expressed by Eq. (7). Indeed, the condition (A2,X2) < (Ai,Xi), which explicitly reads (A2IJ,)(X2) < (Aifjb)(Xi) for every // £ M 1 f (fi), does not imply C((A2,X2) I (A1,X1)^;Ah2) = { ^ j j f ^

(19)

nor (A2,X2) > {A1,X1) implies C((A2,X2) | {A^X^n;Ah2) = 1- In fact, the property of Eq. (7) rests on the deterministic nature of the standard classical case. Notice that also the repeatability condition expressed by Eq. (8) is not preserved in the OPT frame.

5. Quantum conditioning As in Sec. 2, we write A for the self-adjoint operator of 7i associated to the quantum observable A : Su ->• M^"(R), and P j x for the projection operator defined by the pair (A,X); when explicit reference to this pair is not needed we simply write P (or P\,P2,.. if different projectors are called into play). These projection operators are representatives of two-valued observables, hence of quantum events. The traditional recipe expressing the quantum conditional probability

58 C(P2 | Pi; D) of P 2 given P x in the state D £ Su reads C(P2|Pi,£)-

^(£>A)

,

(20)

which can be rewritten as

C(ft|J\;2>)='&(fl W ft),

D(A) =

J^L

(21)

hence as the probability of "occurrence" of P 2 in the conditioned state D(Pl\ This is usually referred to as the Liiders-von Neumann rule for the quantum conditioning, a rule based on the sequential picture. Notice that the condition C(P2 \ P\]D) = Ti(DP2), which states that the probability of occurrence of (the event associated to) P 2 is not affected by the occurrence of Pi, does not imply C(Pi | P2;D) — Tr(DPi); only in case Pi and P 2 commute does the above condition recover the symmetry property. Thus the conditional probability expressed by Eq. (20) fails to give rise to a natural notion of mutual independence of two events. The map of Su into itself defined by D \-> D^p^ does not leave the pure states (namely the one-dimensional projectors) fixed: such a map does not belong to the family of nondisturbing state transformations previously discussed. Actually, a nondisturbing map on S-u would be prevented by Theorem 4.1 of Sec. 4. This implies that the conditional probability C(P2 \ Pi; D) need not collapse into the probability of occurrence of P 2 in the state D when the latter is a pure state: in other words, correlations can appear in pure states. The Bayes property, that now would read Tr(£) Pi P 2 Pi) = Tr(Z)P 2 PiP2)), does not hold true, except the case in which Pi and P 2 commute. We have however the analogue of the classical property expressed by Eq. (7): C(P2 | P i ; D ) = ^ ~ \ if P 2 < Pi (andC(P 2 | PX;D) = 1 if P x < P 2 ). ir(.L'Pi) (22) Actually, this property is sufficient to imply Eq. (20), as shown in [12]. In particular we have the repeatability property C(Pi | Pi; D) = 1 which can be read by saying that the occurrence of Pi is certain in the conditioned state £)( P l ). The conditional probability of Eq. (20) can be connected to a notion of joint probability distribution only in the case of commuting observables.

59 Coming back to the more detailed notation PA. X- f° r t n e projection operator associated to the observable Ai and to Xi €