The Quantum Mechanics Conundrum - Interpretation and Foundations

879 Pages·2019·17.438 MB·English

by  Gennaro Auletta

Gennaro Auletta
The Quantum
Mechanics
Conundrum
Interpretation and Foundations
The Quantum Mechanics Conundrum
Gennaro Auletta
The Quantum Mechanics
Conundrum
Interpretation and Foundations
123
Gennaro Auletta
University of Cassino andSouthern Lazio
Cassino, Frosinone,Italy
Pontifical Gregorian University
Rome, Italy
ISBN978-3-030-16648-9 ISBN978-3-030-16649-6 (eBook)
https://doi.org/10.1007/978-3-030-16649-6
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Preface
The starting point of this book was my tutorship for a doctoral dissertation in the
year 2014 at the Pontifical Gregorian University in Rome. The now Ph.D. Father
LlucTorcalwroteathesistitledInSearchofanOntologythatUnderliesQuantum
Mechanics: Which Vision of the Physical World Could We Get from Quantum
Mechanics?Itwasessentiallyaphilosophicalassessmentofthemainproblemsand
solutions in interpreting quantum theory with a look at the proposals that I have
developed in my own research across more than 20 years starting from the publi-
cation of Foundations and Interpretation of Quantum Mechanics: A Critical–
Historical Analysis of the Problems and a Synthesis of the Results, published in
2000byWorldScientific.Atthattime,IspentasummerinthemonasteryofPoblet
working together with Father Torcal. Since this gave us the opportunity for
examining many crucial points in interpreting quantum mechanics, after the dis-
cussionofthementioneddissertation,bothofusenvisagedtheprojectofmakinga
book on this subject. However, when I started to work on it, I became soon aware
thattheissueatthestakewentmuchfurtherthanasimplephilosophicalassessment.
Moreover,FatherTorcal,duetohisdutiesintheCistercianorder,couldnotfollow
this project. Nevertheless, the current structure of the first three chapters partly
mirrors that of the dissertation.
Andthatwaswhymetaphysicsandsciencewere
suchcourageousenterprises,suchstartlinginventions,
biggerthanthewheel,biggerthanagriculture,
humanartifactssetrightagainstthegrainofhumannature.
Disinterestedtruth.
IanMcEwan,EnduringLove
Rome, Italy Gennaro Auletta
v
Introduction
Aswell-known,itwastheepistemologistK.Poppertospeakofaschisminphysics
determined by quantum theory.1 Retrospectively it seems opportune to ask: does
the schism consist of the introduction of the subjectivity in the mid of physics, as
Popper (following Einstein) seems inclined to think, or in accepting that events
occur randomly without any cause or even ground as stressed by Bohr? Or is it
rootedintheuncontrolledinteractionsoftheexperimenterwithquantumsystemsas
suggested by the young Heisenberg? Or even does it consist in the acceptance of
non-local correlations as pointed out by Schrӧdinger? The answer may vary
according to personal taste. These aspects essentially deal with the following
problems (keeping the same order): reality, causality, measurement, status offor-
malism. They represent a challenge to our rational understanding of Nature. They
areallfundamentalissues,butIaminclinedtothinkthatthecrucialproblemthatis
attherootofallthesepuzzlesisthenotionofreality.So,wemayrephrasePopper’s
definition of the schism in physics as the possible break down of the notion of
reality as such.
Itisthusnotbychancethatquantummechanicsstilltoday(afteracenturyfrom
itsbeginning)ispoorlyunderstoodandnotuniversallyacceptedbythecommunity
of scientists and philosophers. As a matter of fact, many physicists still today
consider it as a kind of metaphysical speculation devoid or poor of empirical
import, and when they use it in neighbouring fields (e.g. cosmology or statistical
mechanics), they limit such aconnection towhat isstrictly necessaryand often,in
these applications, forget or minimise what is really typical and fundamental of
quantumtheory.Thepresentbookisaddressedtotheproblemofmakingquantum
mechanics understandable and viable in its rational foundations to the community
of scholars.
Now, the issue of reality has in turn its roots in a categorial problem. In fact, if
our relation with reality is disturbed, it must be related somehow with the way in
which we categorise reality with our notions of physical object, law of nature,
probability,cause andso on.Moreover,if, inorder tocircumvent the problem, we
1Popper(1982).
vii
viii Introduction
denythatthereisanyrealityassuch,thenwearesayingthatourcategorialstructure
be simply the way in which we humans (and likely in different ways) build our
“story”. These are two sides of the same coin: to deny reality and to reduce cate-
gories to mere subjective constructions or games. Thus, although such a critical
philosophical problem arises in the mid of science for the first time with quantum
mechanics, it, directly or indirectly, affects the whole of our knowledge.
Thepartoftheworkthatisdevotedtointerpretationisdominatedbyfourgiants:
Bohr, Einstein, Heisenberg and Schrӧdinger. On many issues they did not agree.
Theyhadstrongdifferentscientificbackgroundsandevenphilosophicalideas:Bohr
was very much influenced by Kant and Kirkegaard and interested in epistemic
problemsofthetheory,EinsteinbySpinozaand,forwhathisrealismconcerns,bya
kind of Aristotelian background, the late Heisenberg was also influenced by
Aristotlebutwithaconstantoperationalapproach,andSchrӧdingerwasfascinated
by Schopenhauer and the Oriental philosophy. What they have in common is that
theyshowedasignificantinterestingeneralideas,acharacterthatseemstobelost
in most of the actual scientific debate, apart from very rare exceptions. In the
following, we shall see that each of these great scientists was somehow right and
thattheirviewscanbeintegrated.Infact,meanwhile,wehavecumulatedasohuge
amountofexperimentalresults(andoftheoreticaldevelopments)tobeabletosolve
most of the questions that were object of discussion at that time. However, for
doing this, a philosophical critical assessment is necessary.
It could be objected that to satisfy everybody seems too ambitious, especially
withatheorythathasraisedsomanydifferentstandpointsandevendissatisfaction.
Nevertheless, the essence of the scientific enterprise is to find common rational
explanations of our world.2 It is a permanent duty of scholars involved in these
problems, and I hope that the following examination could show at least elements
thatcouldhelpforbuildingsuchacommonviewaccordingtoquantummechanics.
The other issue is rather a foundational inquiry. We shall show that quantum
mechanicsisbasicallyaninformationtheory.However,suchareversibletheoryof
information needs to be integrated with a theory of signals when interactions are
involved. Here, the relationship with both special and general relativity is crucial.
Finally, we shall deal with Category theory and display the categorial structure of
quantum mechanics. The method of this work is not axiomatic and in general I
prefer to proceed pragmatically instead of from definitions and axioms. This
method,asweshallsee,wasanticipatedbyEinsteinwhenhesaidthatphysicsisa
science “going in the direction of increasing simplicity of the logical bases”.3 I
interpretthisstatementasmeaningthat,insteadoftryingtoleadconsequencesfrom
first truths, the right method is to start from theories and explanations that are less
general and less grounded and to proceed backwards to better foundations and
theoriesthataremoregeneral.Asweshallsee,thisisthemethodfollowedtodayby
Category theory and this explains its central role in the present work.
2As pointed out in Deutsch (1997, Chap. 1). Also C. Rovelli has insisted on the relevance of
foundationalissues(Rovelli2004,p.20).
3Einstein(1936,p.96).SeealsoAuletta(2011,Sects.5.2,2.1.2and5.4.2).
Introduction ix
Suchasubjectnecessarilyinvolvesmanyaspectsanddisciplinesandcertainlya
massive use of philosophical methods and notions. In so doing, it is mandatory to
fulfilthefivecriteriaofphilosophicaladequacy setbyAbnerShimony:coherence,
fineness of reasoning, comprehensiveness, openness to evidence, richness of con-
tents, of which the latter three require scrupulous taking into account of scientific
results.4AsSirA.Eddingtonsaid,“thecompartmentsintowhichhumanthoughtis
divided are not so water–tight that fundamental progresses in one is a matter of
indifference to the rest”.5 I completely support this point of view and have taken
inspirationfromitforthepresentbook.Inthiscontext,herecalledthatphysicswas
traditionally called natural philosophy and that a specialisation that has made
philosophy and science dumb to each other is not sane.
References
Auletta,Gennaro(incollaborationwithI.Colagè,P.D’ambrosio,andL.Torcal).2011.Integrated
CognitiveStrategiesinaChangingWorld.Rome:GandBPress.
Deutsch, David. 1997. The Fabric of Reality: The Science of Parallel Universes and Its
Implications.London:PenguinBooks.
Eddington,ArthurStanley.1939.PhilosophyofPhysicalScience.Cambridge:UniversityPress.
Einstein,Albert.1936.PhysikundRealität.JournalofFranklinInstitute221:349–382;Eng.tr.in
Einstein(1956),59–97.
Einstein,Albert.1956.OutofMyLaterYears,EstateofA.Einstein.NewYork:WingsBooks.
Popper,KarlR.1982.QuantumTheoryandtheSchisminPhysics.London:UnwinHymanLtd;
Routledge,1992,1995.
Rovelli,Carlo.2004.QuantumGravity.Cambridge:UniversityPress.
Shimony,Abner.1970.ScientificInference.InR.G.Colodny(Ed.).TheNatureandFunctionof
ScientificTheories,79–172,Pittsburgh;rep.inShimony(1993),I,183–273.
Shimony,Abner.1993.SearchforaNaturalisticPointofView.Cambridge:UniversityPress.
4Shimony(1970,p.187).
5Eddington(1939,p.8).
Contents
1 Summary of the Basic Elements of the Theory. . . . . . . . . . . . . . . . . 1
1.1 The Beginning of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 1
1.1.1 From the Classical to the Quantum World. . . . . . . . . . . . . 1
1.1.2 The Quantum Postulate and Matter’s Discontinuity . . . . . . 5
1.1.3 The Road to the Schrödinger Equation . . . . . . . . . . . . . . . 9
1.2 The Basic Principles of Quantum Mechanics . . . . . . . . . . . . . . . . 24
1.2.1 The Superposition Principle . . . . . . . . . . . . . . . . . . . . . . . 24
1.2.2 The Quantisation Principle . . . . . . . . . . . . . . . . . . . . . . . . 30
1.2.3 Physical Observables and Different Representations. . . . . . 44
1.2.4 Commutativity and Uncertainty Relations . . . . . . . . . . . . . 61
1.2.5 Unitary Transformations and Symmetries . . . . . . . . . . . . . 88
1.3 Some Further Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
1.3.1 Pauli’s Exclusion Principle and Spin. . . . . . . . . . . . . . . . . 95
1.3.2 The Correspondence Principle . . . . . . . . . . . . . . . . . . . . . 105
1.3.3 The Complementarity Principle . . . . . . . . . . . . . . . . . . . . 108
1.4 Density Operator and Compound Systems . . . . . . . . . . . . . . . . . . 110
1.4.1 Pure and Mixed States. . . . . . . . . . . . . . . . . . . . . . . . . . . 110
1.4.2 Entangled and Product States . . . . . . . . . . . . . . . . . . . . . . 116
1.4.3 Total and Marginal States . . . . . . . . . . . . . . . . . . . . . . . . 120
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2 The Main Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
2.1 Quantum-Mechanical Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 129
2.1.1 Two Pictures of Quantum Dynamics. . . . . . . . . . . . . . . . . 129
2.1.2 Probability and Interference . . . . . . . . . . . . . . . . . . . . . . . 139
2.1.3 What Does the Quantum-Mechanical Formalism
Tell Us? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
2.2 The Measurement Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
xi
xii Contents
2.2.1 It is Not a Problem for Classical Physics . . . . . . . . . . . . . 150
2.2.2 It is a Double Problem for Quantum Mechanics . . . . . . . . 152
2.2.3 von Neumann’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . 156
2.2.4 What Are the Consequences?. . . . . . . . . . . . . . . . . . . . . . 161
2.2.5 Schrödinger Cat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
2.3 The Problem of Non-locality. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
2.3.1 The Paradox Proposed by Einstein–Podolsky–Rosen . . . . . 164
2.3.2 Classical Physics and Locality . . . . . . . . . . . . . . . . . . . . . 171
2.3.3 Reality and Correlations. . . . . . . . . . . . . . . . . . . . . . . . . . 187
2.3.4 Possible Replies to EPR. . . . . . . . . . . . . . . . . . . . . . . . . . 191
2.4 The Problem of Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
2.4.1 Classically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
2.4.2 Quantum-Mechanically . . . . . . . . . . . . . . . . . . . . . . . . . . 195
2.4.3 Delayed Choice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
2.5 Summary of the Chapter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
3 The Main Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
3.1 Solutions to the Measurement Problem . . . . . . . . . . . . . . . . . . . . 216
3.1.1 Subjectivist Interpretation. . . . . . . . . . . . . . . . . . . . . . . . . 216
3.1.2 Objectivist Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 226
3.1.3 Role of the Environment . . . . . . . . . . . . . . . . . . . . . . . . . 235
3.1.4 Solution of the Cat’s Paradox. . . . . . . . . . . . . . . . . . . . . . 244
3.1.5 Summarising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
3.2 Formalism and Reality: Information. . . . . . . . . . . . . . . . . . . . . . . 247
3.2.1 Objectivism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
3.2.2 A Formalism That Denotes a Reality?. . . . . . . . . . . . . . . . 248
3.2.3 Dealing with Information . . . . . . . . . . . . . . . . . . . . . . . . . 261
3.2.4 Two Points of View on Entropy and Information . . . . . . . 271
3.3 Hidden Variables and Non-locality . . . . . . . . . . . . . . . . . . . . . . . 275
3.3.1 Logical Restatement of the EPR Argument . . . . . . . . . . . . 275
3.3.2 Bohr’s Reply to EPR. . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
3.3.3 Schrödinger’s Reply to EPR. . . . . . . . . . . . . . . . . . . . . . . 282
3.3.4 Hidden Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
3.3.5 Bell Inequalities for Testing EPR . . . . . . . . . . . . . . . . . . . 300
3.3.6 Entanglement Swapping, Teleportation, Quantum
Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
3.3.7 What Is Non-separability . . . . . . . . . . . . . . . . . . . . . . . . . 320
3.4 Back to the Problem of Causality . . . . . . . . . . . . . . . . . . . . . . . . 333
3.4.1 Form, Structure and Information. . . . . . . . . . . . . . . . . . . . 333
3.4.2 Correlations and Causal Constraints . . . . . . . . . . . . . . . . . 339
3.4.3 Potentiality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346