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Quantum Reality
As probably the most successful scientific theory ever created, quantum theory has profoundly
changed our view of the world and extended the limits of our knowledge, impacting both the theo-
retical interpretation of a tremendous range of phenomena and the practical development of a host
of technological breakthroughs. Yet for all its success, quantum theory remains utterly baffling.
Quantum Reality: Theory and Philosophy, Second Edition cuts through much of the confu-
sion to provide readers with an exploration of quantum theory that is as authoritatively comprehen-
sive as it is intriguingly comprehensible. The book has been fully updated throughout to include the
latest results in quantum entanglement, the theory and practical applications of quantum computing,
quantum cosmology and quantum gravity. Needing little more than a school level physics and math-
ematics background, this volume requires only an interest in understanding how quantum theory
came to be and the myriad ways it both explains how our universe functions and extends the reach
of human knowledge.
Written by well-known physics author and teacher Dr. Jonathan Allday, this highly engaging
work:
• Presents a thorough grounding in the theoretical machinery of quantum physics

• Offers a whistle-stop tour through the early part of the 20th century when the founding
fathers of quantum theory forever altered the frontiers of human thought
• Provides an example-filled interpretation of the theory, its applications, and its pinnacle in
quantum field theory (QFT), so crucial in shaping ideas about the nature of reality
• Separates fact from speculation regarding quantum physics’ ability to provide a starting
point for philosophical queries into ultimate understanding and the limits of science
The world beneath the one that we experience with our senses is profoundly mysterious, and while
we may never completely unravel that mystery, quantum theory allows us to come closer than ever
to understanding where the science leaves off and the mystery begins. Quantum Reality: Theory
and Philosophy, Second Edition makes that understanding accessible to anyone possessing a quest
for knowledge and a sense of awe.
Quantum Reality
Theory and Philosophy
Second Edition
Jonathan Allday
Cover Image: Harris & Bush, MIT
Second edition published 2023

by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press

4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2023 Jonathan Allday
First edition published by CRC Press 2009
CRC Press is an imprint of Taylor & Francis Group, LLC
Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been
acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or
utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including
photocopying, microfilming, and recording, or in any information storage or retrieval system, without written
permission from the publishers.
For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the
Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are
not available on CCC please contact mpkbookspermissions@tandf.co.uk
Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for
identification and explanation without intent to infringe.
Library of Congress Cataloging‑in‑Publication Data

Names: Allday, Jonathan, author.
Title: Quantum reality : theory and philosophy / Jonathan Allday.
Description: Second edition. | Abingdon, Oxon ; Boca Raton, FL : CRC
Press, 2022. | Includes bibliographical references and index. |
Identifiers: LCCN 2022014186 | ISBN 9781032127347 (hbk) |
ISBN 9781032122380 (pbk) | ISBN 9781003225997 (ebk)
Subjects: LCSH: Quantum theory. | Quantum theory—Philosophy.
Classification: LCC QC174.12 .A45 2022 | DDC 530.12—dc23/eng20220820
LC record available at https://lccn.loc.gov/2022014186
ISBN: 978-1-032-12734-7 (hbk)

ISBN: 978-1-032-12238-0 (pbk)
ISBN: 978-1-003-22599-7 (ebk)
DOI: 10.1201/9781003225997
Typeset in Times
by codeMantra
To my parents
My point, which you’ll hear me rant about again, is that at both
the conceptual and the mathematical level, quantum mechanics
is not just a funny-looking reformulation of classical physics. The
two physical theories are fundamentally, physically different.
Michael A. Morrison
The average quantum mechanic is no more philosophical
than the average motor mechanic.
Rev. Dr. John Polkinghorne KBE FRS
Of two alternative futures which we conceive, both may now be really possible;
and the one become impossible only at the very moment when the other excludes it
by becoming real itself. Indeterminism thus denies the world to be one unbending
unit of fact. It says there is a certain ultimate pluralism in it; and, so saying, it
corroborates our ordinary unsophisticated view of things. To that view, actualities
seem to float in a wider sea of possibilities from out of which they are chosen; and,
somewhere, indeterminism says, such possibilities exist, and form a part of truth.
William James
Physics may reveal the mind of God, but only if
he happens to be thinking about dirt.
Ken Wilber
Science… proceeds by elucidation, so that feats of genius
can become ordinary learning for beginners.
Roland Omnès
Pictogram Credit: Emrys

Contents
Forward...........................................................................................................................................xvii
Preface.............................................................................................................................................xix
About the Author.............................................................................................................................xxi
Introduction.......................................................................................................................................1
I.1 Physics................................................................................................................1
I.2 Philosophy.......................................................................................................... 2
Realists..................................................................................................3
Instrumentalists.....................................................................................3
PART 1
Chapter 1 Our First Encounter with the Quantum World: Light................................................... 7

1.1 Some Opening Thoughts.................................................................................... 7
1.2 A little Light Reading......................................................................................... 7
1.3 Lasers and Video Cameras................................................................................. 9
1.4 Photons............................................................................................................. 10
1.5 An Interference Experiment............................................................................. 11
1.5.1 Interference as a Wave Effect.............................................................. 12
1.5.2 Mach–Zehnder with Photons.............................................................. 16
1.5.3 Delayed Choice................................................................................... 18
1.6 Summary..........................................................................................................20
Notes��20
Chapter 2 Particles....................................................................................................................... 23
2.1 Particles and Waves.......................................................................................... 23
2.1.1 Electrons and Electron Guns............................................................... 23
2.2 The Stern-Gerlach Experiment........................................................................24
2.2.1 Turning Things Around...................................................................... 27
2.2.2 Things Get More Puzzling..................................................................28
2.2.3 So, Where Did It Go?.......................................................................... 29
2.2.4 What Does It All Mean?..................................................................... 31
2.3 Summary..........................................................................................................34
Notes�� 34
Chapter 3 Quantum States........................................................................................................... 35

3.1 Where Are We Now?........................................................................................ 35
3.2 Describing Classical Systems........................................................................... 35
3.2.1 Chaos................................................................................................... 37
3.3 Describing Quantum Systems.......................................................................... 38
3.3.1 Specific Example: Mach–Zehnder Again...........................................40
3.3.2 Probability Amplitudes.......................................................................44
vii
viii Contents
3.3.3 Relating Amplitudes to Probabilities..................................................44

3.3.4 Amplitudes, Complex Numbers and Phase......................................... 45
3.3.5 States in Stern–Gerlach Experiment................................................... 48
3.3.6 General Stern–Gerlach States............................................................. 49
3.3.7 Some Further Thoughts....................................................................... 50
3.4 What Are Quantum States?.............................................................................. 51
Notes��52
Chapter 4 Amplitudes.................................................................................................................. 55
4.1 More on Amplitudes......................................................................................... 55
4.1.1 Change of Basis................................................................................... 58
4.2 Dirac Notation.................................................................................................. 59
4.2.1 Orthonormal Bases..............................................................................60
4.2.2 New Light Through…......................................................................... 61
4.2.3 Going the Other Way.......................................................................... 63
Notes��63
Chapter 5 Measurement............................................................................................................... 65
5.1 Embracing Change........................................................................................... 65
5.2 Types of States.................................................................................................. 65
5.2.1 Eigenstates........................................................................................... 65
5.2.2 Mixed States........................................................................................66
5.3 Expectation Values...........................................................................................66
5.4 Operators.......................................................................................................... 67
5.4.1 Operators and Physical Quantities...................................................... 69
5.4.2 Classical and Quantum....................................................................... 69
5.5 How States Evolve............................................................................................ 70
5.5.1 Why Is State Collapse Necessary?...................................................... 73
5.5.2 Behind the Veil.................................................................................... 74
5.5.3 Determinism and Free Will................................................................ 74
Notes��75
Chapter 6 Interference................................................................................................................. 77
6.1 How Science Works?........................................................................................ 77
6.2 The Double-Slit Experiment............................................................................ 77
6.2.1 The Double Slit with Electrons........................................................... 79
6.2.2 Wave/Particle Duality......................................................................... 82
6.2.3 Wave Nature of Electrons.................................................................... 82
6.3 Double-Slit Amplitudes.................................................................................... 83
6.3.1 Phase and Physics................................................................................84
6.3.2 An Experiment with Phase.................................................................. 86
6.3.3 The Interference Term......................................................................... 87
6.3.4 Amplitudes and Electron Strikes........................................................ 87
6.4 Last Thoughts................................................................................................... 88
Notes��89
Chapter 7 Free Particles............................................................................................................... 91

7.1 The Position Basis............................................................................................ 91
7.2 The Amplitude for a Free Particle.................................................................... 91
Contents ix
7.2.1 Classical Waves...................................................................................92

7.2.2 The Complex Wave of the Amplitude.................................................94
7.2.3 Frequency............................................................................................ 95
7.2.4 What Does the Amplitude Tell Us about the Motion of
a Free Particle?....................................................................................96
7.2.5 Amplitudes, Energy, and Momentum.................................................97
7.3 Where Next?..................................................................................................... 98
Notes�� 98
Chapter 8 Identical Particles........................................................................................................99

8.1 Some Opening Thoughts..................................................................................99
8.2 Particle Dodgems.............................................................................................99
8.2.1 Scattering Amplitudes....................................................................... 101
8.2.2 The Moral of the Story...................................................................... 103
8.3 States of More Than One Particle.................................................................. 104
8.3.1 Identical Particles.............................................................................. 106
8.3.2 States in Real World.......................................................................... 109
8.3.3 Overall States.................................................................................... 111
8.3.4 More Than Two Particles.................................................................. 111
8.3.5 More General States.......................................................................... 112
8.3.6 A More Elegant Approach................................................................ 112
8.4 Final Thoughts................................................................................................ 113
Notes�� 113
Chapter 9 Scattering Identical Bosons....................................................................................... 115

9.1 Scattering........................................................................................................ 115
9.2 The Same, but Different: Identical Particles.................................................. 117
9.2.1 Using the Whole Detector................................................................. 118
9.2.2 And Another Way.............................................................................. 119
9.3 Transitions Away from States......................................................................... 120
9.3.1 Spontaneous vs Stimulated............................................................... 122
9.3.2 Lasers................................................................................................ 122
9.4 Bose–Einstein Condensates........................................................................... 123
9.4.1 Einstein’s Argument.......................................................................... 125
Notes�� 126
Chapter 10 Spin............................................................................................................................ 127

10.1 Fermions, Bosons, and Stern–Gerlach Magnets............................................ 127
10.2 Angular Momentum....................................................................................... 127
10.2.1 Angular Momentum in Quantum Theory......................................... 129
10.2.2 Eigenstates of Angular Momentum.................................................. 129
10.2.3 Magnetic Moments............................................................................ 131
10.2.4 The Magnetic Moment of an Electron.............................................. 132
10.2.5 Intrinsic Angular Momentum........................................................... 133
10.3 Spin Operators................................................................................................ 134
10.3.1 Spin Matrices.................................................................................... 135
10.3.2 Fermions and Bosons........................................................................ 139
10.4 Quantum Scale, Spin, Spinors and Twistors.................................................. 140
Notes�� 140
x Contents
Chapter 11 Fermion States........................................................................................................... 143

11.1 States, Normalization, and Phase................................................................... 143
11.2 Exchange and Rotation................................................................................... 144
11.3 Rotational Symmetry of States....................................................................... 145
11.3.1 Reversing the Polarity of the Neutron Flow...................................... 146
11.3.2 Coffee Mugs and Quantum States.................................................... 149
11.3.3 Spin, Symmetry, and Exchanges....................................................... 150
11.4 Time................................................................................................................ 151
11.4.1 Spinning Things Round.................................................................... 153
11.4.2 Rotation for More Fun and Profit...................................................... 155
11.4.3 So Spin Is?......................................................................................... 156
11.5 Boson Spin States........................................................................................... 157
11.5.1 More on Time Reversal..................................................................... 158
11.5.2 Time-Reversed Boson States............................................................. 159
11.6 Deep Waters................................................................................................... 160
Notes�� 160
Chapter 12 Continuous Bases...................................................................................................... 163

12.1 Representations............................................................................................... 163
12.2 Two Issues....................................................................................................... 165
12.2.1 Probability Density............................................................................ 165
12.2.2 Infinite State Expansions................................................................... 167
12.2.3 The Identity Operator........................................................................ 168
12.2.4 A Short Aside: Projection Operators................................................. 169
12.3 State Functions and Wave Functions.............................................................. 171
12.4 Observables.................................................................................................... 172
12.4.1 The Problem of Momentum.............................................................. 172
12.4.2 Momentum in Quantum Theory....................................................... 173
12.4.3 Operators and Representations.......................................................... 176
12.4.4 Expectation Values Again................................................................. 177
12.4.5 Operators and Variables.................................................................... 178
Notes�� 179
Chapter 13 Uncertainty................................................................................................................ 181

13.1 Expectation Is Not Enough............................................................................. 181
13.1.1 Developing Uncertainty.................................................................... 183
13.2 Heisenberg’s Principle.................................................................................... 186
13.2.1 So What?........................................................................................... 186
13.2.2 I’m Not Sure What You Mean by Uncertainty….............................. 187
13.3 Yet More Uncertainty..................................................................................... 188
13.3.1 The Generalized Uncertainty Principle............................................ 189
Notes�� 190
Chapter 14 The Equations of Quantum Theory.......................................................................... 191

14.1 The Schrödinger Equations............................................................................ 191
14.1.1 Ê and Ĥ.............................................................................................. 193
14.1.2 Stationary States................................................................................ 193
Contents xi
14.2 Ehrenfest’s Theorem....................................................................................... 194

14.2.1 The Classical Limit........................................................................... 196
14.2.2 Constants of Motion.......................................................................... 197
14.3 The Energy-Time Inequality.......................................................................... 197
14.3.1 I Really Don’t Have the Time…........................................................200
14.3.2 Energy/Time Uncertainty.................................................................. 201
14.4 Time Evolution............................................................................................... 201
14.5 Conclusions.....................................................................................................202
Notes��203
Chapter 15 Constrained Particles................................................................................................205

15.1 A Particle in a Box......................................................................................... 205
15.1.1 Another Brick in the Wall.................................................................206
15.1.2 Normalization....................................................................................208
15.1.3 Energy within the Box......................................................................209
15.1.4 Momentum in the Box....................................................................... 210
15.1.5 Spatial Distribution........................................................................... 211
15.1.6 Wave Packets..................................................................................... 212
15.1.7 Two-Dimensional and Three-Dimensional Boxes............................ 216
15.2 The Hydrogen Atom....................................................................................... 216
15.2.1 Quantum Numbers for Hydrogen...................................................... 220
15.2.2 Visualising Hydrogen State Functions.............................................. 221
15.3 A Box Containing More Than One Electron.................................................224
15.3.1 Temperature and the Fermi Gas........................................................ 225
15.3.2 White Dwarf Stars............................................................................ 226
Notes��229
PART 2
Chapter 16 Genealogy................................................................................................................. 233

16.1 The Scientific Community............................................................................. 233
16.2 “It Was the Best of Times, It Was the Worst of Times”................................. 234
Notes�� 235
Chapter 17 Planck and Einstein................................................................................................... 237

17.1 Where to Start?............................................................................................... 237
17.2 Planck’s Life................................................................................................... 237
17.3 Planck Enters Research.................................................................................. 237
17.3.1 Planck’s Formula for Black Body Spectra........................................ 239
17.4 Einstein...........................................................................................................240
17.4.1 Quantization of Light........................................................................ 241
17.4.2 The Photoelectric Effect.................................................................... 242
17.4.3 Enter the Photon................................................................................ 242
17.4.4 Bosons............................................................................................... 243
17.5 Final Thoughts................................................................................................ 243
Notes�� 244
xii Contents
Chapter 18 Bohr........................................................................................................................... 247

18.1 The Godfather................................................................................................ 247
18.2 Early Life........................................................................................................ 247
18.3 Atomic Theory...............................................................................................248
18.3.1 Atomic Spectra..................................................................................248
18.3.2 Bohr’s Atom...................................................................................... 249
18.3.3 Developments.................................................................................... 252
18.4 Complementarity............................................................................................ 255
18.4.1 Extensions......................................................................................... 256
18.5 Later Life........................................................................................................ 257
Notes�� 258
Chapter 19 Heisenberg................................................................................................................. 259

19.1 Early Days...................................................................................................... 259
19.2 The Development of Quantum Theory.......................................................... 259
19.2.1 Cloud Chamber Tracks...................................................................... 261
19.2.2 The Uncertainty Principle................................................................. 261
19.2.3 Quantum Concepts............................................................................ 263
19.3 Later Life........................................................................................................264
Notes�� 265
Chapter 20 De Broglie & Schrödinger........................................................................................ 267

20.1 Beginnings...................................................................................................... 267
20.1.1 Electron Diffraction.......................................................................... 268
20.2 Enter the Wave Equation................................................................................ 269
20.2.1 Matter Waves..................................................................................... 270
20.2.2 So What Is ψ?.................................................................................... 271
20.2.3 Nobel Prizes...................................................................................... 272
20.3 Schrödinger’s Philosophy............................................................................... 272
Notes�� 273
Chapter 21 Dirac.......................................................................................................................... 275

21.1 Dirac’s Influence on Quantum Physics........................................................... 275
21.2 Dirac, the Person............................................................................................ 277
21.3 Dirac’s Views on the Meaning of Quantum Theory...................................... 278
Notes�� 281
Chapter 22 Conclusions............................................................................................................... 283

Notes�� 284
PART 3
Chapter 23 Quantum Correlations............................................................................................... 287

23.1 Two Threads................................................................................................... 287
23.2 Is Quantum Theory Complete?...................................................................... 287
Contents xiii
23.2.1 The EPR Argument........................................................................... 288

23.2.2 Follow-Up by David Bohm............................................................... 291
23.2.3 Bohr’s Reply to the EPR Argument.................................................. 293
23.2.4 Einstein and Bohr.............................................................................. 294
23.3 Schrödinger Introduces Entanglement........................................................... 295
23.3.1 Entanglement and Measurement....................................................... 295
23.3.2 The Sorry Tail of Schrödinger’s Cat................................................. 297
23.4 John Bell and Bohm’s EPR............................................................................. 299
23.4.1 Bell’s Argument................................................................................300
23.4.2 A Toy Model...................................................................................... 301
23.4.3 Bell’s Formula...................................................................................302
Experimental Correlations, Se..............................................302
Local Hidden Variable Correlations, Sh............................... 303
Quantum Mechanical Correlations, Sq................................. 305
23.4.4 Aspect’s Experiment.........................................................................306
23.5 Implications....................................................................................................308
Notes�� 308
Chapter 24 Quantum Computing................................................................................................. 311

24.1 Historical Perspective..................................................................................... 311
24.2 The Fundamentals of Digital Computing....................................................... 311
24.2.1 A Bit More Information.................................................................... 312
24.2.2 Logic Gates....................................................................................... 312
24.3 Quantum Analogues....................................................................................... 313
24.3.1 Qubits................................................................................................ 313
24.3.2 Quantum Gates.................................................................................. 315
24.3.3 The No-Cloning Theorem................................................................. 317
24.3.4 What Makes a Quantum Computer Quantum?................................. 318
24.4 Quantum Teleportation................................................................................... 319
24.4.1 Experimental Implementation........................................................... 322
24.5 Practical Quantum Computers....................................................................... 322
Notes��323
Chapter 25 Density Operators...................................................................................................... 325

25.1 Great Expectations......................................................................................... 325
25.2 Why Bother?................................................................................................... 327
25.3 The Density Operator and EPR/Bohm-Type Experiments............................ 329
25.3.1 Representing a State.......................................................................... 330
25.3.2 The Density Operator and Entangled States..................................... 331
25.4 The Density Matrix and the Measurement Problem...................................... 332
Notes�� 334
Chapter 26 Interpretations........................................................................................................... 335

26.1 What is An Interpretation?............................................................................. 335
26.2 A Collection of Problems............................................................................... 336
26.2.1 The Nature of Probability................................................................. 336
26.2.2 Reduction of the State Vector............................................................340
26.2.3 Entanglement..................................................................................... 342
xiv Contents
26.2.4 Measurement..................................................................................... 343

26.3 Important Theorems....................................................................................... 343
26.3.1 Bell’s Inequality................................................................................ 343
26.3.2 The Kochen-Specker Theorem.........................................................344
26.3.3 Proving the Kochen-Specker Theorem.............................................344
Opening Moves....................................................................344
Development........................................................................346
Endgame�� 348
26.3.4 Consequences.................................................................................... 352
26.4 Carnegie Hall................................................................................................. 352
Notes�� 352
Chapter 27 The Copenhagen Interpretation................................................................................ 353

27.1 Bohr’s Influence.............................................................................................. 353
27.2 Bohr’s View of Quantum Theory................................................................... 354
27.2.1 Classical Concepts Must Be Used to Describe the Results
of Any Experiment............................................................................ 354
27.2.2 During a Measurement It Is Impossible to Separate a
Quantum Object from the Apparatus................................................ 355
27.2.3 The Results of One Experimental Arrangement Cannot
Necessarily Be Related to Another................................................... 358
27.2.4 Classical Explanations....................................................................... 359
27.2.5 Drawing the Threads Together..........................................................360
27.3 Heisenberg and Potentia................................................................................. 361
27.4 Von Neumann and Measurement................................................................... 363
27.4.1 The Mind of an Observer..................................................................364
27.5 The Deep End…............................................................................................. 365
27.6 Criticisms of the Copenhagen View............................................................... 367
27.6.1 The Problem of the Cut..................................................................... 367
27.6.2 Problem of Collapse.......................................................................... 370
Notes�� 370
Chapter 28 The Many Worlds Interpretation............................................................................... 373

28.1 Everett, Wheeler, Bohr & DeWitt.................................................................. 373
28.2 The Relative State Formulation...................................................................... 374
28.3 Measurement Records.................................................................................... 376
28.3.1 And the Next One….......................................................................... 378
28.4 The Ontological Step...................................................................................... 379
28.5 Many Worlds Arrives..................................................................................... 380
28.6 Many Worlds Matures.................................................................................... 381
28.6.1 The Nature of Probability................................................................. 381
Everett’s Solution................................................................. 382
Other Approaches................................................................ 383
Decision Theory Enters the Argument................................ 383
28.6.2 State Reduction................................................................................. 387
28.6.3 Entanglement..................................................................................... 387
28.6.4 Measurement..................................................................................... 387
28.6.5 Bell’s Inequality and the K-S Theorem............................................. 387
Contents xv
28.7 Criticisms of the Many Worlds View............................................................. 388

28.8 Time Thoughts............................................................................................... 390
Notes��390
Chapter 29 Assorted Alternatives................................................................................................ 393

29.1 Being in Two Minds about Something…....................................................... 393
29.1.1 Mindless Hulks….............................................................................. 394
29.1.2 The Advantages of Having More Than One Mind........................... 395
29.2 Objective Collapse.......................................................................................... 396
29.2.1 The Penrose Interpretation................................................................ 396
Notes��399
Chapter 30 Consistent Histories................................................................................................... 401

30.1 Frameworks.................................................................................................... 401
30.2 Quantum Reasoning.......................................................................................405
30.2.1 Moggies and Sample Spaces.............................................................405
30.2.2 Meaningless Statements....................................................................405
30.2.3 Contextuality.....................................................................................407
30.2.4 Non-Locality.....................................................................................407
30.3 Histories..........................................................................................................407
30.3.1 Combining Histories.........................................................................408
30.3.2 Probabilities.......................................................................................409
30.3.3 Consistent Histories........................................................................... 411
30.3.4 Histories and Mach-Zehnder............................................................. 412
30.3.5 Measurement..................................................................................... 415
30.3.6 Decoherence and the Classical World............................................... 417
30.3.7 Histories in Cosmology..................................................................... 419
30.4 Ontology......................................................................................................... 420
30.4.1 Pre-Probabilities................................................................................ 421
30.4.2 Unicity............................................................................................... 421
30.4.3 Probability (Again…)........................................................................ 422
30.4.4 Other Issues....................................................................................... 422
Notes��423
Chapter 31 The Ontological Interpretation.................................................................................. 425

31.1 Physics and Philosophy.................................................................................. 425
31.2 Wave and Particle........................................................................................... 426
31.2.1 Bohm’s Version of the Schrödinger Equation................................... 426
31.2.2 The Quantum Potential Energy......................................................... 429
31.3 Probability...................................................................................................... 432
31.4 Quantum Potential Energy in Action............................................................. 433
31.4.1 Quantum Potential Energy and the Double Slit Experiment............ 433
31.4.2 Quantum Potential Energy and the Particle in a Box....................... 434
31.4.3 Spin.................................................................................................... 435
31.4.4 Entanglement..................................................................................... 436
31.5 Information and Wave Function Collapse...................................................... 436
31.6 Deeper Waters................................................................................................ 439
31.7 Reactions to Bohm’s Theory..........................................................................440
Notes��441
xvi Contents
Chapter 32 Quantum Field Theory.............................................................................................. 443

32.1 Why Are We Doing This?.............................................................................. 443
32.2 Taking Identical Particles Seriously............................................................... 443
32.2.1 Particle Labels...................................................................................444
32.2.2 Substance Abuse............................................................................... 445
32.3 States in Quantum Field Theory.................................................................... 445
32.3.1 Fock States........................................................................................446
32.3.2 The Vacuum...................................................................................... 447
32.3.3 Up and Down We Go….................................................................... 447
32.3.4 Change of Basis................................................................................. 447
32.3.5 Orderly Matters.................................................................................448
32.3.6 Fermions and Bosons........................................................................449
32.3.7 The Number Is Up............................................................................. 450
32.3.8 Normalization.................................................................................... 451
32.3.9 Round and Round We Go….............................................................. 452
32.3.10 Multiparticle Operators Representing Observables.......................... 453
32.4 Basis for Progress........................................................................................... 454
32.4.1 So Why Is It Called Quantum Field Theory?................................... 455
32.4.2 Wave-Particle Duality....................................................................... 457
32.5 Interactions in Quantum Field Theory........................................................... 458
32.5.1 Interaction Operators......................................................................... 459
32.5.2 Interaction Potentials......................................................................... 461
32.6 Vacuum Fluctuations...................................................................................... 463
32.6.1 Fields and Numbers...........................................................................464
32.7 Quantum Gravity............................................................................................465
32.7.1 Loop Quantum Gravity (LQG).........................................................466
32.7.2 String Theory....................................................................................466
32.7.3 Prospects...........................................................................................466
Notes��467
Chapter 33 Personal Conclusions.................................................................................................469

33.1 Popular Opinion.............................................................................................469
33.2 Quantum Reality............................................................................................469
33.2.1 Critical Realism................................................................................. 470
33.2.2 Copenhagenism & Consistent Histories............................................ 471
33.2.3 Many Worlds and Many Minds......................................................... 472
33.2.4 The Ontological Interpretation.......................................................... 473
33.2.5 Objective Collapse............................................................................. 473
33.3 Conclusions..................................................................................................... 473
Notes��474
Appendix List of Important Rules.............................................................................................. 475
Index............................................................................................................................................... 477
Forward
Quantum mechanics is “at first glance and at least in part, a mathematical machine for predicting
the behaviors of microscopic particles — or, at least, of the measuring instruments we use to explore
those behaviors [Ismael, 2020].” Why does the machine work? This is the question of how to ‘inter-
pret’ quantum mechanics. It turns out to be intensely controversial. Carroll recalls,
At a workshop attended by expert researchers in quantum mechanics… Max Tegmark took an...unsci-
entific poll of the participants’ favored interpretation ….The Copenhagen interpretation came in first
with thirteen votes, while the many-worlds interpretation came in second with eight. Another nine
votes were scattered among other alternatives. Most interesting, eighteen votes were cast for “none of
the above/undecided.” And these are the experts [2010b, 402, n. 199].
There are two reasons why this is not really disagreement over the interpretation of quantum
mechanics, in any ordinary sense of ‘interpretation’. First, at stake is not what people happen to
mean by technical terms, like ‘state vector’, ‘collapse’, and so on. This would be a question of (pre-
sumably empirical) natural language semantics, and would tell us nothing about the physical world.
Second, the ‘interpretations’ do not even all agree on the machine. For example, Bohmian mechan-
ics (Chapter 31) amends the equations, and makes subtly different predictions.
Physics has made impressive progress without addressing the interpretational question. But
there is a growing sense that progress on the deepest mysteries, like how to reconcile quantum
theory and General Relativity, may require its resolution [Hossenfelder & Palmer, 2020]. Philosophy
is becoming harder to avoid. The situation resembles the one in the early 20th century, when philo-
sophical reflection inspired some of the most penetrating arguments in the history of physics, such
as the EPR argument and the SchrÖdinger Cat thought experiment.
This book is unique in the physics landscape. It is not a textbook, a guide to solving the
SchrÖdinger equation. It is also not a philosophy text, assuming familiarity with metaphysics and
epistemology. It is a serious survey for non-specialists of what the mathematics could mean. It
offers, all in one place, an accessible introduction to the theory (up through a sketch of quantum
field theory), an overview of the ‘no-go’ results, and a careful discussion of some important interpre-
tations. Philosophers will appreciate the self-contained introduction to the theory, while physicists
will learn from the philosophical analysis. Newcomers will delight in all of it.
We need more books like this. Understanding the nature of value, consciousness, mathematical
truth, and possibility and necessity, will also require insights from both philosophy and science.
Deep interaction between philosophy and mathematics has already born plentiful fruits outside of
philosophy proper, like proof theory, model theory, and theoretical computer science. We may hope
that a meaningful exchange between philosophy and physics will be comparably fecund.
Justin Clarke-Doane
Columbia University
IAS, Princeton
xvii
Preface
The world is not what it seems. Behind the apparent solidity of everyday objects lies a seething
shadow world of potentiality which defies easy description, as it is so different from our everyday
experience. In some manner, familiar objects such as solid tables, cricket balls, stars, and galaxies
arise from what transpires underneath. We do not know precisely how this comes about.
There is a theory that describes the underlying world: quantum theory. It is one of the most suc-
cessful scientific theories of all time and it has profoundly changed our view of the world.
Quantum understanding is vital to our current science and technology; its application is not
restricted to esoteric experiments in high-energy physics. The theory certainly helps us under-
stand the inner mechanisms of neutron stars, superconducting materials, and possibly even the
early moments of the Big Bang, but without it we would have no appreciation of why the table on
which this laptop sits is solid. The LED bulb on the table next to me is generating light (which is a
quantum phenomenon) as electrical charge in the form of tiny particles called electrons (which we
need quantum theory to understand) are passing through a material and transferring energy. The
material in the wires leading to the bulb has a property called resistance, which can only be fully
understood by applying quantum laws.
Yet for all its success, aspects of quantum theory remain utterly baffling. While the mathematics
is clear (albeit occasionally hard to deal with), interpreting what it is saying about the world remains
a profound challenge.
In the 100-odd years since quantum theory was born, there have been many books written that
attempt to explain quantum physics to the interested amateur. This is an important endeavor. The
world beneath the one that we experience with our senses is profoundly mysterious, and there are
some important philosophical messages about the nature of reality and the limits of science that
need to be put across. I hope that this book can contribute to that effort.
THANKS TO THE FOLLOWING (FIRST EDITION)

The Master and Fellows of Gonville and Caius College, Cambridge, and especially Dr. Jimmy
Altham for access to the college library and the time and space to work during the summer of 2002
while I was writing the first edition.
Rev. Dr. John Polkinghorne KBE FRS (1930–2021) for an inspiring lunchtime discussion and
his encouraging support.
Dr. Lewis Ryder (1941–2018) for reading parts of the manuscript.
Dr. David Wallace for reading the section on the Many Worlds interpretation.
Dr. Grahem Farmello for reading the section on Dirac.
Greg Manson, Md, for help with uncertainty.
THANKS TO THE FOLLOWING (SECOND EDITION)

Many thanks to Mrs Rebecca Hodges-Davies, Dr Kirsten Barr and latterly Dr Danny Kielty all of
CRC Press, Taylor & Francis Group for guiding me and this manuscript through the production
process. Also the team at codemantra.
Special thanks to Scott Hayek, Bruce Newhall, John Sweeney, Neal Brower (all of the Johns
Hopkins University Applied Physics Laboratory, retired) and John R. Moore (retired) who reached
out to me with regard to General Relativity and then (undaunted) ventured into quantum reality.
They read and commented on sections of the revision with sympathy and in detail. Somehow, they
managed to embody my ideal reader and still be willing to talk to me. I’m looking forward to some-
time meeting you guys in person!
xix
xx Preface
I am grateful to Dr Philip Davies of Bournemouth University for also contacting me, offering the
very flattering, and daunting, prospect of being interviewed on YouTube and then managing to edit
me into sounding reasonably coherent. His willingness to then read and comment on some of this
second edition was a welcome bonus.
Many thanks to my school friend Professor Simon Hands, University of Liverpool, who read
many of these chapters and challenged various bits of wonky physics. Of course, any mistakes
remain my responsibility. Here’s to the classes of 79/80.
None of this could be done without the continual support of my family and friends. Carolyn has
had to carry a lot over the last 18 months never mind the additional burden of a whiney author. My
love and thanks to her. Unfortunately, I have already started again…
And finally, many thanks to Emrys who has shown me that the world is exactly as strange as I
suspected it to be.
Jonathan Allday
2nd Edition
16th January 2020, Yorkshire
13th February 2022, Worcestershire
Jallday40r@me.com
About the Author
For 30 years, Dr. Jonathan Allday taught physics at a range of schools in the UK. After taking his
first degree in Natural Sciences at Cambridge, he moved to Liverpool University where he gained
a PhD in particle physics in 1989. While carrying out his research, Dr. Allday joined a group of
academics and teachers working on an optional syllabus to be incorporated into A-level physics.
This new option was designed to bring students up to date with advances in particle physics and
cosmology. An examining board accepted the syllabus in 1993, and now similar components appear
on many advanced courses.
Shortly after this, Dr. Allday started work on Quarks Leptons and the Big Bang, now published
by Taylor & Francis and available in its third edition, which was intended as a rigorous but acces-
sible introduction to these topics. Since then, he has also written Apollo in Perspective, Quantum
Reality and Space-time, co-authored a successful textbook, and contributed to an encyclopedia for
young scientists.
Dr. Allday’s interest in the physics and philosophy of the quantum world dates back to his school
days, where he remembers reading an autobiography of Einstein. As an undergraduate, he special-
ized in relativistic quantum mechanics and field theory, writing his third-year project on Bell’s
inequality, as well as taking a minor course in the history and philosophy of science. The idea
for this book occurred during a summer placement at Cambridge, hosted by Gonville and Caius
College.
Other than physics, Dr. Allday has a keen interest in cricket and Formula 1.
xxi
Introduction
Suppose for example that quantum mechanics were found to resist precise formulation.
Suppose that when formulation beyond FAPP [for all practical purposes] is attempted, we
find an unmovable finger pointing outside the subject, to the mind of the observer, to the
Hindu scriptures, to God, or even only Gravitation? Would that not be very, very interesting?
J S Bell1
[my addition]
I.1
PHYSICS
Quantum theory has passed through three distinct stages during its evolution. The first is generally
referred to as ‘old quantum theory’. During a period roughly between 1900 and 1925 Planck, Bohr
and Einstein, among others, grappled with various experimental and theoretical crises by patching
up classical (Newtonian) mechanics with some new quantum ideas applied as band-aids. This is not
to disparage the work; it was vitally important, and you can only do what is possible at the time.
Gradually, however, Heisenberg, Jordan, Schrödinger, Born, Bose, de Broglie and Dirac developed
a coherent structure that would be recognized as quantum theory as it exists now. This broadly took
place in the mid-1920, but progress did not stop there. Born, Heisenberg, Jordan, Dirac and Pauli
pieced together quantum field theory between 1926 and 1929. Other physicists, such as Fermi and
Fock later added refinements to the basic structure. In some ways, this evolution can be seen as a
progressive acceptance of quantum concepts that were increasingly distant from classical thinking.
This book is divided into three parts to somewhat mirror this evolution. Part 1 covers the basic
conceptual and mathematical machinery of mid-1920s quantum theory. In Part 2 we take a whistle-
stop tour through old quantum theory and the development of its successor by focussing on the work
and views of some of the central figures involved2. Finally, in Part 3 we survey some important
interpretations and tackle an outline of quantum field theory. At various points, we will draw atten-
tion to the key interpretive issues and discuss what is at stake in our view of reality.
The quantum world is very different to the picture painted by classical physics. That being the
case and given the extreme nature of some of these differences, we might question the wisdom of
accepting the quantum view. The answer is that quantum theory is unsurpassed in its record of
explaining different aspects of nature, some of which we will touch upon later. A few significant
areas include:
• The otherwise mysterious aspects of radioactivity which found a natural explanation within
quantum theory.
• The quantum theory of electrons in atoms allowed us to understand and categorize the
spectral lines of atoms and molecules.
• As a parallel development, chemistry, the nature of the chemical bond, the periodic table
and the structure of molecules all gained a secure theoretical basis.
• Applying quantum theory to matter allowed an understanding of some anomalous aspects
of heat capacities, the conductive (thermal and electrical) properties of materials, and
the development of the band theory of solids. Without this key theoretical advance, vital
aspects of modern technology, including the development of semi-conductors, would not
have been possible.
• Equally, the magnetic properties of materials are a quantum issue. This aspect of theory
has led to the development of important magnetic technologies, such as MRI imaging.
DOI: 10.1201/9781003225997-1 1
2 Quantum Reality
• Quantum theory has also proven to be flexible enough to tackle some surprising discover-
ies, such as superconductivity, the nature of stellar interiors, nuclear fission and fusion and
aspects of Big Bang theory.
Given this stellar record from a broad range of different aspects of physical theory, we are bound to
take the concepts of quantum theory seriously despite their counter-intuitive aspects.
I.2
PHILOSOPHY
Scientists are very ambitious. They’re very competitive. If they really thought philosophy would help
them, they’d learn it and use it. They don’t.
L. Wolpert3
Now I need to say something about philosophy. Don’t put the book down, it will all be over in a
minute.
Scientists can be quite disparaging about philosophy, sometimes while in the process of espous-
ing, quite stridently, a philosophical position of their own. The fact is all scientific theories point
beyond themselves to some degree. At the very least, they open up questions such as: ‘what does this
theory tell us about reality?’, ‘what aspects of this theory, if any, directly relate to real elements of
the universe?’, ‘how do we know that any scientific theory is true, and what does truth mean in this
context?’, ‘how does science work as a reliable tool for scrutinizing the world?’. Such issues cannot
be fully addressed from within science.
Scientific advances come from the joint application of experiment and theory. While it is difficult
to maintain that these are separate disciplines, when you analyse the situation in detail, the world is
too strange and surprising to be understood in all its aspects purely by philosophical reflection. Who
would have thought that quantum theory was likely? We need the constant nudge and corrective of
experiment to direct the focus of theory.
However, not all the important questions about the world are directly addressable by science.
Nor is it sufficient to simply define important questions as being those that are open to scientific
techniques, not least because that is in itself a philosophical position. An argument along the lines
of ‘our subject makes progress and yours doesn’t, so sucks boo to your subject’ is hardly a mature
analysis. Perhaps the apparent lack of progress in philosophy is not a reflection of any lack of rigour,
but more to do with the difficulty of the problems it seeks to address.
Take the example of quantum theory. The mathematical structure of the theory has been in place
for over 100 years, albeit subject to the occasional and useful refinement in that time4. However,
the interpretation of the theory is still an open question. The fact that there are several competing
views is a signal point. Establishing the correct interpretation is not itself a scientific matter. If
each approach agrees on the physical content, they have the same predictive power hence they are
experimentally indiscernible5. Nevertheless, they project radically different outlooks on the nature
of reality (contrast the Many Worlds interpretation with Many Minds, just for one example). The
choice that each individual makes is made, consciously, or not, on its consonance with their overall
worldview. Philosophy naturally enters the discussion, even if tacitly.
Philosophers have a variety of views on the nature of knowledge. At issue are matters relating
to how we know something, how reliable our knowledge is, whether all our knowledge comes from
the world via our senses or are there some things that we just ‘know’ etc. Discussions of this kind
are covered by a branch of philosophy called epistemology. A closely connected, but distinct, area
is ontology. This is the inquiry into what is actually out there for us to know. As a rough example,
the existence of electrons is a matter for ontology; how we know about them and their properties
falls to epistemology.
Introduction 3
Epistemologically there are two approaches to how science works, or rather what it is that science
sets out to do.
If you are a realist, then you believe that science is an accurate map of what is really out there.
The various ideas and pictures that we come up with (such as electrons, black holes, the Big Bang,
and DNA) are elements of reality and we are discovering true information about the world. From
this perspective, the purpose of science is clear: to find out as much as possible about what is going
on in the world. To a realist, a good theory is one that convinces us that the things it speaks about
are not just figments of our scientific imaginations.
However, you might be an instrumentalist, in which case you are not too bothered about the
accuracy or reality of your ideas, as long as they fit the data and allow us to make accurate pre-
dictions. An instrumentalist may not believe that electrons are real. They will agree that various
experiments produce clumps of data that can be gathered under the heading “that’s an electron” and
will use this data to predict another set of experimental readings under slightly different circum-
stances. However, they will draw short of committing to the objective existence of electrons. You do
not have to believe that Colonel Mustard is a real person to have fun finding out if he is a murderer
in the game Cluedo. To an instrumentalist, a good theory is one that allows us to play the game well.
Various scientists have embraced and promoted one approach or another over the years:
Realists
Physicists believe that there exist real material things independent of our minds and our theories. We
construct theories and invent words (such as electron, positron etc.) in an attempt to explain to ourselves
what we know about our external world … we expect a satisfactory theory, as a good image of objective
reality, to contain a counterpoint for every element of the physical world.
B. Podolsky
A complete, consistent, unified theory is only the first step: our goal is a complete understanding of the
events around us, and of our own existence.
S. Hawking
The great wonder in the progress of science is that it has revealed to us a certain agreement between
our thoughts and things …
L. de Broglie
Instrumentalists
I don’t demand that a theory correspond to reality because I don’t know what it is. Reality is not a qual-
ity you can test with litmus paper. All I’m concerned with is that the theory should predict the results
of measurements
S. Hawking
In science we study the linkage of pointer readings with pointer readings.

A. Eddington
There are arguments on both sides. A realist would say that the only satisfactory way of explaining
the success of science is by believing that are talking about reality. An instrumentalist would coun-
ter by saying that in Newton’s age we believed that time was the same for everyone, then Einstein
comes along and declares that time is different depending on our state of motion, or if we happen
4 Quantum Reality
to be in a gravity field. What next? Often our ideas of what is ‘out there’ change radically, so why
believe any of it? If our ideas let us fly to the moon, cure diseases, and make good plastics, who
cares?
Many scientists6 go about earning their daily bread without being bothered about the philosophi-
cal niceties; “shut up and calculate” would be their motto. Unfortunately, tackling quantum physics
raises questions that are difficult to put aside. It is all to do with the state of a quantum system. A
realist has some trouble believing that a quantum state is an ontologically real thing, as it seems,
at least in part, to depend on our knowledge about a system. An instrumentalist would have no
problem believing that states are nothing more than a concise expression of our information about a
system. More of a challenge would be explaining why the objects that we study behave in radically
different ways if their state changes, which suggests that they have some ontological relevance.
Throughout this book, I am going to try and remain as neutral as possible and point out where
realism and instrumentalism have their strengths when applied to quantum theory. You may find
that exposure to these ideas forces you to refine your own thinking.
NOTES
1 J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press; 2nd edi-
tion, 2004.
2 As I am not aware of a collective noun for these individuals, I have coined the term Founding Fathers
to use in this book. I am amused by the whimsical implied connection to the framers. The unfortunate
gender specificity in this phrase is a matter of historical fact. Mme Curie, for example, did very impor-
tant work that should not be ignored, but it is not directly related to our story.
3 L. Wolpert (1929–2021), University College London, Round Table Debate: Science versus Philosophy?
https://philosophynow.org/issues/27/Round_Table_Debate_Science_versus_Philosophy.
4 When we get to the chapter on Consistent Histories, we will see something of that nature.
5 Objective collapse is a clear exception to this. As we will see later, this entails some modification to the
theory as it stands and hence has different predictions which are, just about, accessible by experiment.
However, objective collapse addresses one, critically important, aspect of interpretation, but not the
whole philosophical ballpark.
6 Perhaps the majority.
Part 1
1 Our First Encounter with
the Quantum World
Light
1.1 SOME OPENING THOUGHTS
The first draft of this chapter was written while sitting in a college garden under a cloudless sky,
with the bright sunlight flooding over some particularly well-manicured lawns (Figure 1.1).1 I clearly
remember struggling to see what I was typing over the reflected glare from my laptop screen.
I still find it hard to reconcile the beauty of such scenes with what I know about the nature of
light. This is part of the mystery that shrouds quantum reality.
Large-scale (macroscopic) objects, such as trees, bushes, and cricket balls, are made up of
small-scale (microscopic) things such as protons, neutrons, and electrons. The laws of physics that
describe the large-scale world have been broadly understood since the 1700s. Our first tentative
exploration of the physics of the small-scale started in the 1900s. As we rapidly came to realize, the
laws governing the small-scale world describe behaviour that, judged by the standards of everyday
experience, is utterly bizarre. It is very difficult to see how all the funny business going on at the
atomic scale can underpin the regular, reliable world we spend our lives in.
This contrast between the microscopic world (‘seen’ via experiment) and the macroscopic world
(experienced via our senses) is a theme that will recur throughout this book.
1.2 A LITTLE LIGHT READING

When you set out to understand some new phenomenon, it’s a good idea to start by looking for
similarities with something that you have already figured out. In the case of light, there appears to
be two possible comparisons. Light might be a wave (a spread out, rather flappy thing that varies in
both space and time, like the ripples on a pond) or a stream of particles (localized hard lumps, like
cricket balls, that simply change their position with time). Up until the 1800s, there was no experi-
mental way of settling the issue, so most people took sides on philosophical or theoretical grounds.
Two big-name physicists squared up in opposite corners: Thomas Young favoured a wave view of
light with Isaac Newton (see Figure 1.2) championing a particle interpretation.
In 1665 Newton made some fundamental discoveries about light while taking leave from his stud-
ies at Cambridge, which was under threat from the plague. In one of his classic experiments, he
allowed a thin shaft of sunlight to fall on a glass prism, producing a spectrum of colours. Newton
explained this by assuming that the colours of light corresponded to distinct types of particle.
In this view, white light was a stream of all the different particles mixed together, rather than a
distinct colour of itself. As they passed through the prism, the various types of particle interacted
with the glass differently, which caused them to emerge along separate paths, separating to give
the observed spectrum.
With Newton as a public supporter, 2 the particle view was bound to hold a certain sway, but the
existence of sharp shadows near opaque objects did not hurt the argument.
To the casual glance, shadows have well-defined edges. This is tricky to explain if light is a type
of wave. Other examples, such as water waves, clearly bend around objects that get in their way. So,
if light is a wave, shadows should have rather fuzzy outlines, or so it was thought in Newton’s time.
DOI: 10.1201/9781003225997-3 7
8 Quantum Reality
FIGURE 1.1 The opening of this chapter was written while sitting outside one of the windows of Harvey
Court, part of Gonville and Caius College in Cambridge.
FIGURE 1.2 An engraving of Isaac Newton (1642–1727)—his pioneering experiments with light led him to
propose that light was composed of a stream of particles.
However, not all were convinced, and in 1801, Young carried out an experiment that was sensi-
tive enough to reveal the wave aspects of light. The key to Young’s discovery was the use of two
linked sources of light to produce an interference pattern. We will go into the details of how inter-
ference works in Section 1.5.1, but for a simple illustration imagine dropping two pebbles into an
otherwise smooth surface of water. Ripples spread out from each impact point and inevitably over-
lap somewhere. The result is a complex pattern of motion on the surface of water: an interference
Our First Encounter with the Quantum World: Light 9
FIGURE 1.3 Thomas Young’s original diagram explaining his interference experiment. A and B represent
light sources that send out waves of light that spread out in circular patterns centred on each source. These
waves look rather like the ripples that would spread out on a lake if pebbles were dropped into the water at A
and B. Complex patterns are formed where the waves overlap. C, D, E, and F are places where light and dark
bands would appear on a screen.
pattern. In Young’s version, specially prepared light from two sources was directed to overlap on a
screen. In the region of overlap, instead of a patch of illumination, a series of bands were seen. The
natural explanation was that the waves from the two sources were combining, like the ripples on
water, causing bright patches where they reinforced each other and dark regions where they got in
each other’s way (Figure 1.3).3
Young was able to use these observations to estimate the wavelength of light. For water waves
the wavelength would be the distance between two neighbouring peaks (high points) on the surface.
The wavelength of light is a little harder to interpret, as it is related to the electric and magnetic
fields that comprise the light wave (Section 1.5.1). However, if we take a wave view of light, the
colour is related to the wavelength, with red light being long wavelength compared to blue. Light’s
wavelength is incredibly tiny, in the region of one-tenth of a millionth of a meter. This explains why
we observe sharp shadows. Waves will only bend round objects that are about the same size as their
wavelength. The objects that we see casting shadows are much bigger than the wavelength of light,
hence the light does not leak around them to blur the edges of the shows.
1.3 LASERS AND VIDEO CAMERAS

In the twentieth century, the study of light was revolutionized by the ability to produce precisely
controlled beams that are tightly collimated and made of light that is all the same colour: laser
beams. These days, lasers are among the most ubiquitous of devices. They are found Blue-ray play-
ers, laser pointers, measuring devices and other common pieces of equipment. Lasers are used in
schools to carry out experiments similar to that done by Young, but with considerably more ease.
They have transformed the teaching of optics.
Another remarkable technological development has been that of the CCD camera. CCD stands
for Charge-Coupled Device. They are very sensitive detectors of light. Relatively cheap ones are at
the centre of digital cameras, converting the light falling on them into electrical signals that can be
processed and stored. CCDs are also used in infrared detectors such as those in spy cameras and
security alarms. Even more interestingly, they have helped to transform astronomy by making it
possible to detect very faint objects in the night sky. Such highly sensitive CCDs have to be cooled
to low temperatures so that thermal noise does not mask the image.
10 Quantum Reality
By employing controlled beams from a laser and CCDs to detect very faint amounts of light, we
are able to carry out experiments similar to Young’s basic design, but in ways that he could not have
imagined. The results of these new experiments are so radical that they call into question everything
that we have said so far.
1.4 PHOTONS
Figure 1.4 shows a very simple experiment where a laser beam is aimed directly at a CCD detector
(from now on we will just call them ‘detectors’) the output of which is transferred to a computer and
displayed graphically on a screen.
At moderate intensities, the light seems to be spread equally over the sensitive surface of the
detector. However, as we further reduce the intensity of the beam, the image starts to break up into
a sequence of tiny speckles (Figure 1.4). Reducing the intensity further makes these speckles occur
less frequently, and consequently, they seem to be scattered randomly across the screen. With a
suitable laser, the intensity of the beam can be reduced to the point at which only a single speckle
occurs at any one time with a notable interval between it and the next one.
A natural way of interpreting these results, aside from thinking that the detector is broken, would
be to suggest that the light is a stream of particles. When a particle strikes the detector, it off-loads
its energy and produces a single speckle on the screen. At high intensities, there are millions of
particles arriving within tiny intervals of time, and the detector records a uniform illumination.
Nowadays, we refer to these particles as photons.
Now we have two contradictory experiments: one suggests that light is a wave (Young’s interfer-
ence) whereas the other points to the existence of photons. One simple fix would be to suppose that
lasers produce photons whereas other sources of light produce waves. Unfortunately, this is not the
case. As mentioned earlier, the use of CCDs in astronomy has enabled us to study objects that are
so faint that the light is recorded (with the aid of a telescope) a single photon at a time. Clearly, stars
and galaxies also produce photons.
Although modern day laser/CCD combinations enable us to perform a simple demonstration
that reveals the existence of photons, historically they were detected well before the invention of
the laser. Arthur Holly Compton carried out a crucial experiment4 in 1923 while investigating the
scattering of X-rays5 by atoms.
By 1923 physicists had already successfully produced interference patterns from X-rays, so their
wave nature seemed settled. Given this, Compton expected to find that a beam of X-rays would be
scattered by electrons inside atoms. The electrons would absorb the energy in an X-ray and then
rebroadcast it as a new X-ray sent out in a random direction, but with the same wavelength.
He actually discovered that the X-rays coming off the electrons were of a lower wavelength than
those in the incoming beam. Furthermore, the electron struck by the X-ray recoiled as if hit by a
physical lump of matter. A detailed examination of Compton’s results showed that the energy of the
FIGURE 1.4 Using a laser beam to experiment with light.

FIGURE 1.5 In Compton’s experiment it seemed as if X-ray photons were colliding with electrons like
physical lumps of matter. In the process, they transferred some of their energy to the electron, which recoiled
from the collision.
incoming X-ray had been passed on to the electron in exactly the same fashion that one snooker
ball passes energy onto another when they strike (Figure 1.5). This was completely contrary to the
wave picture of light. Compton could explain these results by replacing the wave picture by one that
had the X-rays as a stream of photons, but nobody could reconcile this with the interference results.
So, in the mid-1920s physicists found themselves in a bit of a mess. The issue of the wave/particle
nature of light, which seemed settled a hundred years before, was now opened up again. However,
this time it was worse. Earlier there had been two competing views of the situation waiting for a
decisive experiment to declare which one was right. Now there were two contrary experiments,
revealing light as a wave in one instance and as a particle in another.
In theoretical terms, a complete resolution to this problem was not to come until the develop-
ment of quantum field theory and its daughter quantum electrodynamics (a continuous development
between the 1930s and the 1950s), both of which are subjects for later. For the moment, we will ‘ride
the paradox’—thinking on the one hand that light is a particle (photon) and on the other hand that it
is a wave - and move on to explore some experiments that demonstrate the split personality of light
even more effectively.
1.5 AN INTERFERENCE EXPERIMENT

A key aspect to the next series of experiments is the use of beam splitters. An ordinary mirror con-
sists of a shiny or silvered surface designed to reflect all incident light, although in practice, no mir-
ror is ever 100% efficient and some energy is lost. With a beam splitter, 50% of the light is reflected
while 50% is transmitted. There are many different varieties of beam splitter, which affect the light
as it is transmitted and reflected in slightly different ways. In this context, I have in mind a dielectric
50:50 beam splitter, although the details of how it works are not necessary for our analysis.
If you direct a high-intensity laser at such a beam splitter, two beams emerge, each one having
half the intensity of the incoming beam.
The next step is to place two ordinary mirrors, one in each path from the first beam splitter,
arranged to divert the beams toward a second beam splitter (see Figure 1.6), where the same
thing happens to each beam: half of the light arriving passes straight through and the other half is
reflected. Finally. two detectors, X and Y, are placed in the beam paths.
Detector X picks up light that has been reflected by the first beam splitter (and so travelled
the ‘top’ path), then transmitted by the second beam splitter. It will also collect light that was
12 Quantum Reality
FIGURE 1.6 Using beam splitters to divide a laser beam that is then recombined at a detector.
FIGURE 1.7 Light waves, like water waves, have peaks (P) and troughs (T). The wavelength of the wave is
the distance between two successive peaks or troughs.
transmitted by the first beam splitter (along the ‘bottom’ path) and reflected by the second one.
Any light reaching detector Y must have been either reflected by both beam splitters (top path) or
transmitted by both (bottom path).
This arrangement of beam splitters, mirrors and detectors is called a Mach–Zehnder interfer-
ometer and similar instruments (without modern electronics) have been used for sensitive opti-
cal experiments since 1891. Once a Mach–Zehnder interferometer is set up using a standard light
source or laser, it is easy to confirm that the intensity reaching each detector depends on the relative
distances along the top and bottom paths. If the equipment is very finely adjusted so that these two
paths are of exactly the same length, detector Y records no light at all, whereas detector X gets all
of the intensity entering the experiment. Without this very critical adjustment, X and Y collect light
in varying relative amounts: if more light arrives at X, then less will reach Y (and vice versa).
In classical (pre-quantum theory) physics this effect is explained by calling on the idea that light
is a wave.
1.5.1 Interference as a Wave Effect

Consider some ripples6 crossing the surface of a lake. In some places, the water level is higher than
normal (these are peaks) and in others, it has dropped below normal (troughs). The wavelength of
the ripple is the distance between two successive peaks, which is the same as the distance between
successive troughs. The frequency of the wave is the rate at which complete cycles (from peak to
trough to peak again) pass a fixed point, and the period is the time taken for one cycle (Figure 1.7).
The motion of any particle in the surface of the water will track through a repetitive cycle from
its starting point to the local peak, then back down to a local trough and up again to the peak. At any
moment, the particle is at a certain phase of its motion. (This is a term borrowed from lunar observations,
where the phase of the moon at any stage in its monthly cycle is an indication of the specific section of
that cycle that the moon is currently displaying.) We can quantify the stage of the cycle by comparing
the motion of particle or point on the wave with a similar point rotating around a circle, as in Figure 1.8.
The phase is then represented by an angle, ϕ , measured in radians. On this measure, two points
on the wave separated by a wavelength will have a phase difference of 2π. The phase difference
between a peak and a trough is π and that between a peak (or trough) and the central undisplaced
line of the wave is π/2.
As light is composed of electric and magnetic fields,7 its wave nature is rather more complicated
than a simple ripple. The peaks and troughs in a light wave are not physical distances, as in the
height of water, instead they are variations in the strength of the field. As this is quite a tricky con-
cept to imagine, we can continue to think of a light as being somewhat like a ripple, provided we
don’t take the analogy too seriously.
Typically ripples on a lake have wavelengths that are comfortably measured in centimetres. Light
waves, on the other hand, have wavelengths better measured in nanometres (10−9 m), which makes them
very sensitive measures of distance. Thinking back to the interference experiment in Figure 1.6, imagine
dividing the distance travelled by a light wave into chunks that are equal to the wavelength of the wave.
Almost certainly, the distances involved will not be a whole number of such chunks. Equally, the device
would have to be very finely calibrated for the two path lengths to be exactly the same number of chunks.
If the distances are not precisely the same, the light travelling along each route will have gone
through a different number of complete waves by the time it gets to the detector. As the light has a
common source at the first beam splitter, the two beams will set off on their different routes in phase
(i.e., in step) with each other (see Figure 1.9). If we could see their peaks and troughs directly, they
FIGURE 1.8 Comparing the phase in a wave motion with a point rotating around a circle. The phase can
then be characterized by an angle, ϕ , in radians.
FIGURE 1.9 The waves labelled A and B are in phase with each other (peak to peak and trough to trough);
waves B and C are exactly out of phase with each other (peak to trough).
14 Quantum Reality
would be marching along peak for peak and trough for trough. However, by the time they get to the
detector, the two beams may no longer be in phase, due to the different distance travelled. One could
be reaching a peak in its cycle as it arrives and the other a trough (like B and C in Figure 1.9). If this
happens, the waves will cancel each other out and there will be little energy entering the detector.
Exact cancellation would only happen if the waves met precisely peak to trough (π phase differ-
ence), which is not possible for any length of time due to small variations in distance (the mirrors
will be shaking slightly) and fluctuations in the laser.
To complete a detailed analysis of our experiment, we also need to take into account any phase
changes that happen to the light at the various mirrors. Generally, when light bounces off a mirror, the
reflected wave is out of phase with the incoming wave by half a wavelength (π phase difference). Things
are slightly different with a dielectric beam splitter. Its specially prepared surface, which is bonded to
a glass block, can reflect light from either side (the dashed line in Figure 1.6 indicates the reflecting
surface). If the reflection takes place from the surface without the light having to pass through the glass
block, then the ordinary π phase shift takes place. However, any light that has to pass through the block
before reaching the reflecting surface is not phase shifted on reflection. However, there is some phase
shift as the wave passes through glass, even if it is not reflected, as illustrated in Figure 1.10.
If we now track the progress of a light wave through the upper arm of the interferometer, we can
see that the cumulative phase shift of the wave by the time that it arrives at detector Y is 2π + φ + φ ′
(Figure 1.11).
FIGURE 1.10 Phase shifts on passing through the glass of a beam splitter. The reflecting surface of the
beam splitter is shown by the dotted line. On the left, a light wave that passes horizontally through the splitter
without reflecting, undergoes a phase shift of φ . On the right, a wave passing vertically through the glass has
its phase shifted by φ ′ .
FIGURE 1.11 A wave passing through the top arm of a Mach-Zehnder interferometer on the way to detector
Y will undergo a series of phase shifts on reflection and passing through the glass of a beam splitter.
FIGURE 1.12 A wave passing through the lower arm of a Mach-Zehnder interferometer on the way to detec-
tor Y will also undergo a series of phase shifts on reflection and passing through the glass of a beam splitter.
In this case the cumulative phase shift is different by π compared with that of the top arm.
FIGURE 1.13 Light waves arriving at detector X via the top and bottom arms of the instrument arrive
exactly in phase with each other, and so constructively interfere.
Applying the same logic to the lower arm of the produces a cumulative phase shift of π + φ + φ ′
(Figure 1.12).
In this analysis, we are assuming that the path lengths through the top and bottom arms are
identical, so the only phase shifts are due to the reflections and the passage through the glass. Given
this fine adjustment of the instrument, the waves arriving at detector Y have a phase difference of
( 2π + φ + φ ′ ) − (π + φ + φ ′ ) = π , so they destructively interfere, and no light is seen by the detector.
On the other hand, detector X will see some illumination as the waves arriving there are exactly
in phase and so constructively interfere (Figure 1.13).
In most experimental setups, the paths through the interferometer are not equal in length. As we
shall see in more detail in Chapters 6 & 7, this also has an impact on the relative phases of the beams,
something that has not been incorporated into the argument thus far, on the assumption that both paths
were exactly the same length. Given the ability to move one of the fully silvered mirrors, so that the
relative path lengths were changed, the experiment could be developed to study the variation of bright-
ness in X and Y as the relative path length varied. In essence, this would be an interference pattern.
Young did not have access to a Mach–Zehnder interferometer (on the very reasonable grounds
that they hadn’t been invented), but he combined light from two sources to produce an interference
pattern on a screen. The results of his experiments could also only be explained by using a wave
theory of light.
16 Quantum Reality
1.5.2 Mach–Zehnder with Photons

With an appropriate laser as a light source, we can turn down the intensity of light entering the
Mach–Zehnder experiment so that the beam resolves into a stream of photons. If we reduce the
laser’s intensity sufficiently, the time between photons emerging from the laser will be more than
the time it takes a photon to completely traverse the apparatus. Then there is only one photon in the
interferometer at any one time. However, we can’t determine exactly when a photon will be emitted
by the laser as that is a random event. Sometimes a few come reasonably close together, sometimes
quite a long time passes between them. We have control over the average rate, but nothing more.
One might expect that the photons arriving at the first beam splitter have a 50:50 chance of pass-
ing through or reflecting off. Another perfectly logical possibility would be that two reduced energy
photons emerge from the mirror, one heading in each direction. It is easy enough to determine
experimentally exactly what happens. All we have to do is place some photon detectors just after
the beam splitter in the path of each beam.
This simple experiment produces an interesting result: half the time the photon is reflected, and
half the time it is transmitted; you never get two photons at the same time. However, there seems to
be no inherent difference between the photons that get through and those that reflect. For example,
the sequence is not so regular as one reflects and the next goes through, and then the next reflects
etc. In fact, there is no pattern to the sequence, except that overall half reflect and half get through.
This is our first encounter with an important facet of the quantum world. Some aspects of nature
lie beyond our predictive ability e.g., which way the photon will go.
This inability to predict with certainty, even in principle, is a specific feature of quantum theory.
Sometimes we lack an advanced grasp of the situation and have created simple models that do not
catch every facet. Sometimes, the situation is so sensitive, we would need an unrealistically precise
grasp of initial conditions in order to make suitable predictions. In either case, it comes down to
ignorance or inability on our part. Quantum reality appears to confront us with a third possibility:
genuinely random events that have no causal handle to give us leverage. These are very deep philo-
sophical and physical waters. Unfortunately, for the moment we must simply note this as something
for further discussion and move on.
Having established that a photon reaching the first beam splitter in a Mach–Zehnder interferom-
eter will either reflect and travel the top path through the device or transmit and follow the bottom
path, our interest must now turn to what happens at the detector end.
For ease, we assume that the detectors are 100% efficient and that no photons get ‘lost’ while
crossing the apparatus.8 If we add together the number of detections at X and Y over a time period,
the answer will correspond to the number of photons emitted by the laser during the same period.
However, the relative number of photons arriving at the two detectors depends on the path lengths:
• If they are exactly equal, then no photons ever arrive at Y.

• If the paths are not exactly equal, then we find that the detection rate at each detector reflects
the intensity of the interference pattern observed when we had the intensity turned up.
To clarify, let’s imagine that during a high-intensity experiment, I had arranged for the path lengths
to be adjusted until 70% of the total light intensity entering the experiment arrived at X and 30% at
Y. Once we turned the intensity down so that we could resolve individual photons, we would find
that 70% of the time a photon is detected at X and 30% of the time at Y. There is never a ‘double
firing’ with photons arriving at X and Y together (as long as we have the laser turned down so that
there is only one photon in the system at any time). This experiment has been done under extremely
well-controlled conditions, and there is no doubt that the photon arrival rate directly reflects the
interference pattern in the way described.
Stated rather quickly in this manner, it doesn’t sound like there is much of a problem here.
Yet there is.
If a photon is a small particle of light, then how can the different paths have any effect on one
single photon?
We confirmed that photons randomly ‘pick’ reflection or transmission at a beam splitter. After that
they proceed along one path or the other to a detector. It is hard to imagine a single photon going along
both paths at the same time. Even if we could sustain that idea for a particle, it is not supported by the
experimental evidence9. Recall that when we put two detectors directly after the beam splitter, they
only picked up one photon at a time down one or the other path. There was no sign that the photon
traversed both paths simultaneously…
Now a wave can do this. It can spread throughout the experiment (think of the ripples formed
when you toss a pebble into a lake) so that parts of the wave travel along each path at the same time.
When the two parts of the wave combine at the far side of the experiment, the information about
both paths is being compared, which leads to the interference pattern.
A single photon must surely have information about only one path, so how can single photon
experiments produce interference patterns?
It transpires that there is a flaw in our argument. It is extremely subtle and cuts to another of the
primary issues that physicists have to face when dealing with the quantum world.
We confirmed that the photons randomly divert at the beam splitter by placing detectors in the
two paths. However, this eliminates any chance of picking up the interference pattern. If the detec-
tors have stopped the photons, then they have not travelled the paths. In principle, this does not tell
us anything about what might happen when no detectors are present.
Of course, it is simply ‘common sense’ to assume that the photons do the same thing with or
without these detectors in the experiment, but we have already seen that the interference pattern for
photons hardly seems to be a matter of common sense.
There is a way to investigate this further. All one has to do is place just one photon detector, Z, after
the beam splitter, say in the path of the reflected beam. If we detect a photon there, then we certainly
won’t get one at the far side of the experiment, at X or Y. On the other hand, if we don’t pick a photon at
Z, we can assume that it has passed through the splitter, rather than reflecting, and so we can expect to see
it at the far end. The experiment is easily done, given the equipment, and confirms that for every photon
leaving the laser we pick one up either at the far end (X or Y) or in the reflected beam (at Z) (Figure 1.14).
FIGURE 1.14 Mach–Zehnder with photons. A photon arriving at the first beam splitter has a 50:50 chance of
being reflected and picked up at the detector. In which case, nothing is seen at X or Y. However, if the photon
is transmitted then there is a 50:50 chance of it arriving at X or Y, no matter what the length of the path is.
18 Quantum Reality
What we find for the transmitted photons is that half of them arrive at Y and the other half at X,
no matter what the length of the path is. In other words, there is no interference pattern. Removing
detector Z opens up the top route to the far side of the experiment. At the same time, it removes any
direct knowledge that we might have about the behaviour of the photons at the beam splitter.
It does, however, restore the interference pattern…
Gathering our conclusions:
• with a standard Mach-Zehnder experiment, adjusting the path lengths by moving one or
more of the standard mirrors, produces an interference pattern of light at the detectors;
• when we turn the intensity of the laser down, the light beam resolves into a stream of
photons;
• the rate of photons emitted by the laser is related to the light intensity;
• reducing the intensity of the beam does not affect the interference pattern—now it’s the
arrival rate of the photons that depends on relative path lengths10;
• if we adjust the experiment, so that we can tell which path was taken by the photon (directly
or indirectly) at the first beam splitter, then the interference pattern is destroyed;
• if we are unable to tell the path of the photon, then there is an interference pattern, which
seems to imply that the photons arriving have information about both routes through the
experiment;
• opening up the top path (by removing detector Z) can actually reduce the number of pho-
tons arriving at Y;
• in the extreme case, if the paths’ lengths are the same, opening up the top path means that
you never get any photons at Y.
1.5.3 Delayed Choice
It is possible to develop the experiment so that the results are even more puzzling.
To do this we introduce a device called a Pockels cell (PC) into one of the routes (in Figure 1.15
it can be seen in the reflected route). PCs are crystals that change their optical properties when an
FIGURE 1.15 In this experiment a PC is used. Such a device is capable of passing photons or diverting them
to a detector. Passing an electrical current through the cell rapidly changes its setting.
electrical current is applied to them. Without a current, the cell allows photons to pass. Applying a
current changes the cell so that it diverts photons, which can then be picked up by another detector.
Consider the scenario shown in Figure 1.15. The PC is initially set to divert photons. A photon
leaves the laser and arrives at the first beam splitter. If it is reflected, then the setting of the PC will
divert it to Z, and we don’t see it at X or Y. However, if the photon is transmitted by the first beam
splitter, it misses the PC, and it turns up at either X or Y (50:50). In either case there is no interfer-
ence pattern.
If instead we set the PC to pass photons, we get an interference pattern. We will stipulate that
the experiment has been finely calibrated so that the paths lengths are an exact match, so that there
is no detection rate at Y11.
So:
• if the PC is set to transmit, we get no photons at Y and all of them at X;

• if the PC is set to divert, then only half of the photons reach the far side of the apparatus,
but they then have an equal chance of being picked up at either X or Y.
This result alone is enough to give us pause. If the photon takes the lower route with the PC set to
divert, then it can get to X or Y. If it takes the lower route with the PC set to pass, then the photon
never arrives at Y. But if it takes the lower route it doesn’t go anywhere near the PC, so how can the
setting of that device affect things? Is this a further hint that somehow or other the photon travels
both routes at the same time?
Now we get devious: we initially set the PC to divert photons, but while the photon is in flight,
switch the cell over. As the cell responds quickly to a current, we can make the change after the pho-
ton has interacted with the beam splitter. One way to do this would be to leave the PC on divert and
establish the timing rhythm from a run of photons at a certain laser intensity. That tells us the rate at
which photons are being emitted at that intensity. We can easily measure the distance from the laser
to the beam splitter, so we know how long it takes a photon to reach that first point in the apparatus.
Provided we switch the PC over after this time, but before the photon has had time to reach detec-
tors X and Y, we will achieve the desired effect. We can then set a switching frequency for the PC.
After the experiment has run for a while, we can use a computer to wade through the data. It
will find some photons arriving at Z, which always happens when the detector is set to divert, and
some at X and Y. We program the computer to ignore the photons at Z and sort the X & Y group
into those that arrived when the cell was set to divert, and those that made it through when it was
set to pass (Table 1.1). Remarkably, when the data are separated out in this manner, the photons that
arrived at the far side with the PC set to pass show an interference pattern. The other photons that
arrived with the PC set to divert (but obviously were committed to the other path and so missed it)
show no interference pattern at all.
Recall that in every case the PC was initially set to divert photons and was only switched over
after they left the beam splitter. With the PC set to divert, we have seen that the photons follow one
TABLE 1.1
The Pattern of Detections for Different Settings of the Pockels Cell
Detectors
Cell Setting Z X Y
Divert Yes, if reflected at first beam Yes, if transmitted at first beam Yes, if transmitted at first beam
splitter (50%) splitter (25%) splitter (25%)
Pass Never Yes, all photons arrive here due to Never, due to the interference
path lengths being equal (100%) pattern
20 Quantum Reality
path or another (top route, via the Pockels cell to Z, or bottom route via the beam splitter to X or Y).
Whilst they were in flight, we sometimes switched the PC, removing our ability to know which path
the photons travelled, and producing an interference pattern. The presence of an interference pattern
suggests that the photon travelled both paths.
It’s hard to believe that changing the setting of the PC can have an influence that travels backward
in time to affect the photon at the first beam splitter. What we can say is that the ability to deduce
the path of the photon (PC set to divert) results in no interference. If we can’t directly or indirectly
determine the path of a photon (PC set to transmit), then we do get an interference pattern.
The mathematical machinery of quantum theory describes the photon leaving the mirror as
being in a combination (superposition) of two distinct sates, one for travelling each path. In a stan-
dard interpretation, when the photon arrives at the PC, this combined state randomly collapses into
one or other of the distinct states (corresponding to the photon being on one path or the other). An
alternative interpretation talks of parallel worlds, with a photon always travelling along one path in
each world, but the two worlds being able to influence each other to some small degree, resulting in
an interference pattern.
1.6 SUMMARY
Although this chapter has only been a starting point, we have already come across some fundamen-
tal issues. We have seen that a description of light must somehow encompass both wave-like and
particle-like natures, depending on the circumstances. The underlying randomness that can appear
in the quantum world has made itself known via our inability to tell which way a photon will travel
at a beam splitter (in an experiment set up to detect its path). Finally, and in my view most impor-
tantly, we have indications that quantum mechanics is going to be a contextual theory: an adequate
description of the behaviour of a quantum object (light in this case) will require an understanding
of the whole experimental setup; the behaviour depends on the context.
NOTES
1 My thanks to the Master and fellows of Gonville and Caius College, Cambridge, for the opportunity to
spend some time in College writing the first edition of this book.
2 Newton’s book, Optiks, was published in 1704 and put forward a strong case for the particle view.
3 When ripples are supporting each other, this can cause a patch that is either deeper or higher than nor-
mal. In light, bright bands can be caused by deep and high patches. Dark bands are formed when the
light waves oppose one another.
4 For which he earned the 1927 Nobel Prize in physics.
5 X-rays had been discovered in 1895 by Röntgen (for which he was awarded the first Nobel Prize in phys-
ics). By this time, their properties had been well established and their nature, as part of the electromag-
netic spectrum, confirmed.
6 What people normally think of as water waves (the things you see on the beach) are not really waves in
the strict sense of physics. Beach waves are a mixture of ripples and tidal movement of water.
7 In a non-photon model, that is.
8 This is an exaggeration, I’m afraid. No detector has 100% efficiency. What I am trying to suggest here
is that nothing “odd” happens to the photons in flight. Each one gets through the experiment and is, in
principle, detectable at the far end.
9 We will discover that the experimental evidence from one setup is not always directly related to a dif-
ferent setup. Hence it is not quite true to say at this stage that the evidence does not support a photon
travelling both paths. Here we have an example of the contextuality of quantum mechanics.
10 Some readers may be worrying about what I mean by the arrival rate of the photons. One picture in your
mind might be that, somehow, we are slowing the photons down in the experiment, so it takes longer
time for them to get to the far side. In fact, it is a rather more complicated situation. First, we cannot be
exactly sure of the moment that a photon leaves the laser (this is due to the uncertainly principle that we
will discuss later). Second, the different positions of the detector mean that there will be different travel
times, but light is quite quick, so this is not a major factor. You have to take a more overall view of the
experiment. If the detector is at a position that corresponds to a dim part of the pattern, then when we
reduce the intensity to the single photon level we have to consider the experiment as a whole—in which
case the position of the detector is influencing the probability that a photon leaves the laser. The whole
thing from leaving the laser to travelling through the experiment to arriving at the detector at the far side
is an interlocking process and each stage has an influence on the others.
11 Note that this is still an interference pattern.
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— Sa, il professore è venuto — mi disse la Gegia, entrando in
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Avevo le palpebre gravi, l’ossa peste dalla notte insonne. Mi posi a
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La Gegia prese la chicchera del caffè dalle mie mani che tremavano,
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Licenziai la Gegia e saltai giù dal letto. Avrei voluto esser vestita in
un attimo, e invece la mia toilette mi occupò una mezza oretta
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Rammento un nodo dovuto rinnovare tre volte, un bottone passato e
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Quando fui pronta, esitai ad uscir dalla stanza; perchè, sebbene
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Gegia nella sua zucca vuota? Chi sa se Verdani si sognava neanche
di attendermi?
Ma la Gegia aveva colto nel segno, e il professore mi attendeva
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— Desideravo — egli principiò alquanto impacciato, e guardando il
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Egli m’introdusse nel salottino ove la signora Celeste stava una
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ricamo o con un libro. Adesso la signora Celeste non c’era; eravamo
soli, il professore ed io.
Verdani mi pregò di sedere. Egli si mise a camminare in su e in giù,
come aveva camminato nella propria stanza quelle sere in cui io
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mattina.... Se devo imbarcarmi venerdì....
— Ed è necessario, assolutamente necessario che s’imbarchi
questa settimana?
— Mio fratello mi scrisse di prendere il vapore del 18 o del 25....
Quello del 18 non l’ho preso; dunque....
— E se domandasse una proroga?...
— Al punto in cui siamo?... Dopo aver fatto tutti i preparativi, dopo
essermi accommiatata da quasi tutti i miei conoscenti?... No, no,
nemmen per sogno....
— Se poi ha tanta premura di lasciarci! — egli interruppe con
amarezza.
— O professore — esclamai, e sentivo un nodo alla gola — non sia
ingiusto.... Crede che me ne vada fino al Caucaso per un
capriccio?... Avrò avuto torto ad accettar con tanta precipitazione
l’offerta di mio fratello, ma si metta al mio posto.... al posto d’una
ragazza che non è coraggiosa, che non è forte, che non ha spirito
d’iniziativa.... Vedevo non lontana la miseria, l’umiliazione di ricorrere
alla carità degli estranei, e afferrai la prima tavola di salute che mi fu
gettata.... Ormai....
— E non c’è nulla, nulla che potrebbe trattenerla? — seguitò Verdani
con calore.
Mi sforzai a dissimulare con una facezia la mia crescente emozione.
— Vuol che speri in una lotteria guadagnata senza biglietti, in un
impiego ottenuto senza le cognizioni occorrenti per esercitarlo?
La fisonomia di Verdani ebbe una contrazione dolorosa. — Non c’è
altro, non c’è proprio altro?
Dio mio! Che cos’è questo riserbo che c’impone di reprimere i nostri
slanci, di nascondere i nostri sentimenti? È una virtù o è un vizio? Io
lo vedevo soffrire; potevo forse con una parola dissipar le sue
sofferenze, infranger l’ultima tenue barriera che si ergeva fra noi e la
felicità, e non osavo dir quella parola, non osavo neanche guardarlo
in viso.
— Ebbene — ripigliò Verdani mutando posizione e venendo a
sedermisi accanto — scriverò a mia madre che m’ero ingannato.
— Sua madre? Come c’entra la sua mamma?
— Oh se c’entra!... Avevo affrettato la mia gita a Bologna per questo.
Volevo consultarla, lei che è tanto savia e buona; volevo comunicarle
un mio disegno.... S’ella lo disapprovava avrei chinato il capo in
silenzio, perchè non oserei far cosa di cui mia madre avesse a
dolersi.... Ma ell’approvò tutto; ella mi disse con la sua solita, cieca
fede in me: Ciò che tu fai è ben fatto; le persone che tu ami io le
amo; c’è sempre posto per esse nel mio cuore e nella mia casa....
Io tremavo come una foglia.
— Professore.... — balbettai confusa.
— Non mi chiami così — egli proruppe con impeto abbandonando la
mano ch’io avevo lasciata nella sua. E seguitò con voce raddolcita:
— I miei amici mi chiamano Verdani, mi chiamano Gustavo. — Egli
scosse tristamente il capo e soggiunse: — È vero ch’ella mi conosce
appena. Le son vissuto accanto parecchie settimane senza
occuparmi di lei, sfuggendola quasi.... Però, quando il caso ci
avvicinò, quando ci scambiammo le prime confidenze, quando la
seppi sul punto di prendere la via dell’esilio, provai dentro di me
qualche cosa che non avevo provato mai.... La mia scuola, i miei
studi aridi e gelati non mi bastavano più; sospiravo il momento
d’incontrarla, sospiravo l’ora del pranzo.... Mi pareva che ci fosse
una certa analogia fra i nostri caratteri; anch’ella era timida, era
riservata come sono timido e riservato io, e la semplicità de’ suoi
modi spiccava maggiormente per l’affettazione di altri.... sa bene a
chi alludo.... di altri che s’era pur fitto in capo di piacermi.... O
signorina, se fossi stato ricco, avrei ben vinto prima la mia ritrosia....
Ma come non esitare se non potevo offrirle, per ora almeno, che un
nome oscuro, una vita modesta, fatta di privazioni e di sacrifizio? Ciò
non ostante, lo vede, il coraggio lo avevo trovato; ma capisco ch’era
un sogno.... un bel sogno....
Ah, in quell’istante trovai io pure il coraggio di dire a Verdani che il
suo sogno era stato il mio sogno, che quello ch’egli mi offriva
superava di molto ciò ch’io avessi osato chiedere alla fortuna, che lo
amavo....
Egli mi strinse sul petto bisbigliando con accento ineffabile: — Elena,
anima mia....
Allorchè mi sciolsi dalle sue braccia, mi sovvenne di Odoardo. — E
mio fratello che m’aspetta, che mi ha mandato il denaro pel viaggio?
— Tuo fratello? — disse Gustavo. — Gli telegraferai che non puoi
partire. Il resto glielo spiegheremo per lettera.... Ha vissuto tanti anni
senza di te; si adatterà a vivere ancora.... In quanto al danaro, se
non vorrà lasciarlo alla sorella come regalo di nozze, ho qualche
risparmio, glielo restituirò io.... Sarà il dono che farò alla mia
fidanzata.
Gustavo mi presentò come tale alla signora Celeste, la quale mi
abbracciò con trasporto, vantandosi d’aver contribuito a questo lieto
avvenimento.... Mai, mai le passò pel capo di far sposare ad un
uomo come il professore quella caricatura della Giulia.... Sarà....
Quel giorno stesso, dopo pranzo, mi parve che una nuvola
oscurasse la fronte di Gustavo, e gliene chiesi la ragione.
Egli mi rispose con un’altra domanda: — Sei ben sicura di non
pentirti?
— O Gustavo....
— Fosti colta così di sorpresa!... Talvolta il cuore umano inganna sè
medesimo.... Amandomi oggi, t’è parso d’avermi amato anche
prima.... Se fosse un’illusione?
Non gli risposi; gli feci segno d’attendere, entrai nella mia camera e
ne presi questo libro, che deposi sul tavolino davanti a lui.
Egli m’interrogò con lo sguardo.
— È un libro — io spiegai — da leggere questa notte.... in quiete....
Non subito.... no.
A malgrado del mio divieto. Gustavo aveva sollevato la coperta
dell’album, e ne andava sfogliando le pagine.
— Una specie di diario?
— Appunto.
— Di tuo pugno?
— Di mio pugno.... Ma leggerai dopo... te ne prego.
Gustavo ubbidì a malincuore.
La mattina seguente lo vidi raggiante di contentezza. — O cara, cara
— egli mi disse. — Ora non dubito più.... Non puoi immaginarti che
gioia sia il sapere d’essere stati amati quando non s’era detto ancora
che si amava.
Io sorrisi. — Sì che me l’immagino, poichè è quello che è toccato a
me.
— Hai ragione — egli soggiunse abbracciandomi teneramente. —
Adesso però convien scrivere l’epilogo.
Mi strinsi nelle spalle.
— Che importa? Questi sfoghi dell’anima s’addicono più ai giorni
tristi che ai lieti.
— No, no — insistè Gustavo. — È una storia intima che non può
rimanere incompiuta. Devi promettermi di finirla.
Glielo promisi. Ma non trovavo mai il verso di accingermi all’opera.
Ieri egli me ne rimproverò con dolcezza. — Se tardi troppo scriverai
di maniera. Scommetto che a quest’ora hai dimenticato molti
particolari del colloquio che decise della nostra sorte.
— Non scommettere — replicai. — Perderesti.
Fra poco darò da leggere queste pagine a Gustavo, ed egli, leale
com’è, sarà costretto a riconoscere che avrebbe perduto. Sono certa
di non aver nulla dimenticato e nulla inventato; dalla prima all’ultima
pagina la mia semplice cronaca non ha che un pregio, la sincerità.
FUORI DI TEMPO E FUORI DI POSTO.
I.
L’Università di X è da qualche tempo un po’ scaduta di credito; ma

dieci anni or sono essa era certo tra le più riputate del Regno, e vi si
contavano a dozzine i professori aventi un nome celebre nella
scienza. Nella facoltà giuridica il Bertioli, il Soreni, il Mereghini, nella
fisico-matematica il de Ziani e il Luserta, nella medico-chirurgica
l’Astigiano e il Barelli, in quella di filosofia e lettere il Meravigli, il
Dalla Volpe, il Frusti, il Teofoli, il Canavese, il Pontevecchi, ch’era
anche rettore. È verissimo che molti di questi uomini insigni
appartenevano alla classe dei professori che chiameremmo
decorativi, perchè le loro relazioni con l’Università si limitavano a
qualche lettera scritta al segretario economo per farsi mandar lo
stipendio. Il Bertioli, per esempio, era senatore e i suoi doveri di
cittadino lo costringevano a frequentare le sedute della Camera
vitalizia; il Sereni e il Mereghini erano tutti e due deputati e avevano
obblighi uguali verso la Camera elettiva; anzi il Mereghini, nel cui
cranio capace alloggiavano comodamente le legislazioni di tutti i
paesi del mondo, poteva considerarsi un’appendice del Ministero di
grazia o giustizia, ove i successivi titolari dei portafogli si servivano di
lui per l’eterno rimaneggiamento dei codici. Ciò non gl’impediva del
resto di fare all’Università una lezione ogni dicembre annunziando la
materia che avrebbe trattato e che naturalmente non trattava nel
corso dell’anno. Il de Ziani e il Luserta, onore della facoltà
matematica, ambidue senatori in pectore, erano anch’essi pieni di
cariche, membri dell’Accademia dei Lincei, membri del Consiglio
superiore dell’istruzione pubblica, ecc., ecc., autori di relazioni e di
programmi di studi in perfetta contraddizione fra loro. Dell’Astigiano
e del Barelli non si parla. Erano medici di fama europea e non
potevano rifiutare l’opera loro a chi li chiamasse a consulto in Italia e
fuori d’Italia. Spesso li si chiamava tutti e due in una volta, giacchè
essendo l’Astigiano profondo nella diagnosi e il Barelli nella
terapeutica poteva accadere che il primo, infallibile nel determinare
la natura del morbo, sbagliasse nel suggerire la cura, e il secondo,
senza rivali nella cura, prendesse in iscambio un male per l’altro.
Del rimanente questo stato di cose conciliava le vedute delle famiglie
degli scolari con quelle degli scolari medesimi. Le famiglie si
riempivano la bocca coi gran nomi dei professori dei loro figliuoli; i
figliuoli esultavano delle continue assenze dei professori e
mancavano regolarmente alle lezioni dei sostituti.
Il rettore Pontevecchi, celebre orientalista ma non energico uomo, si
consolava pensando che nella facoltà di filosofia e lettere, ch’era
proprio la sua, le cose procedevano alquanto diversamente. In tanti
professori non c’era che un unico deputato, il Meravigli, e anche
quello andava di rado alla Camera perchè l’aria di Roma non gli era
propizia. Gli altri erano puramente uomini di studio e non volevano
saperne della vita pubblica.
Primeggiava tra questi il Teofoli, professore di filosofia, spirito largo
ed acuto, parlatore limpido ed efficacissimo, ammirato dalla
scolaresca, stimato e rispettato da tutti i colleghi. Due di essi, il Dalla
Volpe e il Frusti, lo seguivano come la sua ombra, e la gente, a forza
di vedere quei tre sempre insieme, aveva preso a chiamarli per celia
i tre anabattisti. Il Dalla Volpe aveva moglie, una moglie terribile fino
a trentacinqu’anni per la sua galanteria, da trentacinqu’anni in poi
per la sua devozione: il Frusti era vedovo e grande odiatore delle
donne; il Teofoli pareva deliberato a rimaner scapolo, e sebbene non
partecipasse ai pregiudizi del suo amico Frusti contro il bel sesso,
preferiva tenersene alla larga e frequentava soltanto il salotto della
contessa Ermansi, ch’era una signora matura.
Ben provveduto di mezzi di fortuna, il professore Clemente Teofoli
aveva un bel quartierino, una magnifica biblioteca e un’ottima tavola
a cui egli invitava spesso qualche collega, e, nelle grandi occasioni,
anche qualche discepolo preferito. Pegli altri due anabattisti, non c’è
bisogno di dirlo, c’era sempre un posto e una posata disponibile. Il
Dalla Volpe in particolare si rifugiava dall’amico il venerdì e le altre
vigilie, per evitare la cucina di magro che la sua degna consorte gli
avrebbe inflitta inevitabilmente.
Quei pranzetti, che la signora Pasqua, governante del professore
Teofoli, una virago baffuta e contro le tentazioni, sapeva ammannire
con arte sopraffina, erano rallegrati da discussioni dottissime fra i tre
inseparabili. Il Teofoli parlava volentieri dell’opera ch’egli stava
maturando da più anni sul tema già trattato alla fine del secolo
scorso dal Dupuis, L’origine delle religioni; il Frusti e il Dalla Volpe
facevano il possibile per tirare il discorso l’uno sulla storia antica e
l’altro sulla moderna o a meglio dire su quel periodo di storia antica e
moderna ch’essi prediligevano. Poichè, a voler essere sinceri, i due
amici brillavano piuttosto per la profondità che per la varietà delle
ricerche. Il Frusti non si occupava volentieri, nella storia moderna,
che della rivalità tra Carlo V e Francesco I, e il Dalla Volpe, nella
storia antica, non aveva occhi che per le gesta della 19ª dinastia
tebana le cui glorie cominciano con Setti I, soprannominato
Merenaphtha o Menaphtha (caro a Phtah), le cui imprese però,
come sanno anche gli studenti di ginnasio, furono confuse con
quelle di Ramesse II, suo figlio. Una volta preso l’aire, il dotto uomo
non si fermava più, salvo che qualcheduno non trovasse il modo di
richiamarlo alla memoria delle sue tribolazioni coniugali. Allora egli
dimenticava Menaphtha e Ramesse e sfoggiava una facondia
mordace che agli spiriti frivoli poteva parer preferibile alla grave e
ponderata eloquenza con la quale egli esponeva le vicende
memorabili dell’Egitto.
— Ero un bel somaro a pigliarmi tanti fastidi in gioventù per le
scappatelle della mia signora consorte, — egli diceva sovente. —
Quelli eran tempi beati in confronto d’adesso. C’erano, sì, delle
chiacchiere in paese; c’erano spesso tra i piedi dei seccatori; ma
almeno la Luisa era d’un umore gaio, piacevole, ed era bellina, ciò
che non guasta. Le vere calamità, son principiate dopo quel fatale
vaiuolo che la lasciò tutta butterata. Non vedendosi più un cane
intorno, le son spuntati i rimorsi, l’è venuto il bisogno imperioso di
espiare le sue colpe e di rimettersi in grazia di Domeneddio. E vigilie,
e digiuni, e ogni momento in chiesa, alla messa, ai vesperi, alla
benedizione, al confessionale, e preti, e frati e monache in casa.... e,
s’io arrischio una parola, mi sento a rispondere: — Se ho commesso
dei falli non puoi dire ch’io non ne faccia penitenza. — Così ho il
gusto di aver la confessione esplicita di mia moglie, e quello di far
penitenza insieme con lei.... Ah le donne!
Il nostro Teofoli notava che quando si ha avuto la sfortuna d’incappar
male non è lecito giudicar tutte le donne alla stregua di quelle che ci
hanno fatto soffrire.
Ma questa ragionevole osservazione dava sui nervi al terzo
commensale, il professore Frusti. — È falso. Anzi è precisamente
l’opposto. I soli che possono esser indulgenti con le femmine sono
quelli che incapparono male. A loro almeno è permesso di credere
che ce ne siano d’una pasta diversa dalle poco di buono che
conoscono. Chi ha conosciuto le migliori non ha più illusioni possibili.
E la mia era una delle migliori. Tutti lo dicevano, tutti continuavano a
dirlo.... anche quando non c’era più un dubbio al mondo ch’ella mi
menasse pel naso. E io sono intimamente convinto che avessero
ragione.... Ma era donna e faceva la sua parte di animale nocivo.
Dopo queste dichiarazioni ripetute ogni tanto su per giù con le
stesse parole e la cui amarezza lasciava sospettare una ferita
ancora sanguinante, il professor Frusti aveva l’abitudine di
tracannare un bicchiere di vino. Qualche volta, se la signora Pasqua
era presente (ed ella usava dar di quando in quando una capatina in
salotto da pranzo per sentir lodare i suoi manicaretti), egli si
appellava al giudizio di lei ch’era uno spirito assennato e non aveva
mai voluto esser confusa con le persone del suo sesso.
E la signora Pasqua approvava energicamente. — Parole d’oro —
ella diceva con la sua voce grossa. — Son tutte tagliate sul
medesimo stampo.
Le dispute fra i tre amici si prolungavano sovente durante la
passeggiata e s’inacerbivano nelle sere in cui Teofoli, invece di
andare in birreria coi colleghi, si recava dalla contessa Ermansi.
Poichè Frusti e Dalla Volpe non gli potevano perdonare questa sua
debolezza. Com’essi non avevano mai accettato gl’inviti di quel bas
bleu ch’era la Ermansi, così avrebbero preteso che non li accettasse
lui e che non si prestasse gentilmente a far la parte di bestia rara nel
serraglio della contessa.
II.
La conoscenza di Teofoli con la contessa Susanna Ermansi datava

dal giorno ch’egli aveva tenuto all’Università una prolusione a cui
assisteva il fiore della cittadinanza e nella quale erano adombrate le
idee fondamentali dell’opera sull’origine delle religioni. Non si
ricordava all’Università un trionfo simile. Che il Teofoli avesse
ingegno e dottrina all’altezza del tema lo sapevano tutti, ma non tutti
presumevano che insieme col filosofo non rifuggente da nessuna
audacia dell’intelletto ci fosse in lui un poeta atto ad intendere ogni
aspirazione dell’anima, ogni inquietudine della coscienza. Nulla nel
suo discorso che ricordasse la critica superficiale, beffarda del
secolo XVIII, ma una larga tolleranza, ma una simpatia schietta per
tutti gli sforzi con cui l’umanità tenta di penetrare il mistero che ne
avvolge, per tutte le ipotesi pie che il sentimento tramuta volentieri in
certezze. Così, mentre gli uni applaudivano l’erudito, gli altri
battevano le mani all’artista, che vestiva di forme elettissime gli
astrusi concetti, e l’eleganti donnine, alle quali tra la messa, il magro
e il confessionale non dispiace qualche spruzzo di libero pensiero,
erano le più entusiaste ammiratrici del facondo professore che si
faceva perdonare l’ardito razionalismo con un caldo soffio d’idealità.
In quel dì memorabile Teofoli non potè esimersi dall’esser presentato
a una ventina di contesse, marchese, baronesse, eccetera eccetera,
che andarono a gara per colmarlo d’elogi e per sollecitarlo a tener
presto una serie di conferenze a cui esse si sarebbero fatte una
festa d’intervenire.
Non c’è dubbio che la vanità dell’uomo era lusingata da questo
incenso; tuttavia, egli non perdette il suo sangue freddo e non si
lasciò prendere negli ingranaggi fatali del cosidetto bel mondo. Si
schermì molto cortesemente dagl’inviti che gli piovevano da ogni
parte, si schermì dal tener le conferenze che gli si domandavano, e
di tante nuove relazioni che avrebbe potuto iniziare non ne accettò
che una sola, quella della Ermansi, il cui salotto era frequentato
anche da parecchi colleghi dell’Università e della quale egli
conosceva da un pezzo il marito. Superba di questa preferenza, la
contessa colmava il professore d’attenzioni e di regalucci; lo sapeva
appassionato dei fiori e gli mandava le più belle rose del suo
giardino; lo sapeva ghiotto delle frutta e gli mandava le primizie del
suo orto; e quando il conte marito tornava dalla caccia il professor
Teofoli era sicuro di ricevere dal palazzo Ermansi o un invito a
desinare o il dono d’un capo di selvaggina, che, dopo esser stato
oggetto delle cure più amorose da parte della signora Pasqua, era
servito in tavola a uno dei soliti pranzetti con l’intervento di Dalla
Volpe e di Frusti. In queste occasioni Teofoli diceva scherzosamente
ai suoi due commensali: — Dovete pur convenire che la mia amicizia
con la Ermansi ha il suo lato buono.
— Sì, sì, — borbottavano gli altri; — se tutto si limitasse a ricever dei
regali di frutta e di selvaggina. Ma presto o tardi la Ermansi ti farà
qualche brutto tiro.
— O che tiro volete che mi faccia? — esclamava Teofoli. — Farsi
sposare no sicuramente. È maritata.
— Le donne maritate possono restar vedove.
— Il conte Antonio gode una salute di ferro. E in ogni caso la
contessa è fuori di combattimento.
— Non si sa mai.... Del resto in casa sua ci vanno anche delle
signore giovani.
— Oh che uccelli di malaugurio! — replicava Teofoli infastidito. —
Per le giovani son vecchio io.... E sul serio, avete paura ch’io mi
metta a fare il galante?
I due amici tentennavano la testa con aria lugubre, e Frusti
sentenziava con la sua voce cavernosa: — Tutto è possibile.
In verità non era facile rappresentarsi il nostro Teofoli sotto l’aspetto
d’uomo galante. In primo luogo gli mancava quello che i francesi
chiamano le physique de l’emploi. Tozzo della persona, con una
fisonomia espressiva ma irregolare, con certi movimenti bruschi e
nervosi, egli non era mai stato l’Apollo del Belvedere. Nell’età critica
in cui noi l’incontriamo, cioè a cinquant’anni sonati, egli aveva già la
vista indebolita dalle lunghe veglie sui libri, aveva sull’ampia fronte i
segni dell’intensa applicazione mentale, e i capelli radi e grigi non
lasciavano nemmeno sospettare la chioma folta e ricciuta ch’era
stata forse l’unica bellezza della sua infanzia. Vestiva con proprietà
ma senza la minima ricerca d’eleganza; soprabito nero di taglio
professorale, cravatta pur nera, calzoni e guanti scuri, cappello a
tuba, occhiali fissi, mazza d’ebano col pomo d’avorio. Certo che a
sentirlo discorrere si dimenticava la sua apparenza infelice. Non lo si
poteva confondere coi Dalla Volpe, i Frusti e similia, che portavano
la cattedra dovunque andassero. Egli era piacevole, arguto, alieno
da qualunque pedanteria, e aveva uno spirito così largo e una
cultura così varia che nessun argomento grave o leggero lo coglieva
alla sprovveduta. E anche con le signore era amabile e disinvolto più
che non si sarebbe supposto in un uomo tanto dedito agli studi. Non
che di tratto in tratto non gli accadesse di commettere qualche
goffaggine, di toccare qualche tasto falso, di dir qualche madrigale
che sentiva di rancido e di stantìo, ma eran peccatucci veniali che gli
si perdonavano volentieri, in grazia delle molte sue qualità.
Anzi alla contessa Susanna non bastava averlo frequentatore
assiduo del suo salotto; ell’avrebbe voluto accaparrarselo per la sua
villeggiatura. — Venga a passare un mesetto con noi.... due
settimane almeno.... nel nostro romitorio di Sant’Eufemia, a tre ore
dalla città, in luogo tranquillo, con aria salubre e vista incantevole....
Venga, venga. Farà un vero piacere a me e a mio marito.... E sarà in
libertà piena.... Potrà portarsi i suoi libri, le sue carte, potrà
studiare.... Da noi non ci sono cerimonie, non ci sono etichette....
Ospiti, o nessuno, o pochissimi, e gente alla buona.... Venga, venga.
Il conte Antonio faceva eco alla moglie. E pigliando a parte il
professore, soggiungeva in segreto: — Se ci onora della sua visita le
mostrerò la mia collezione di edizioni rare del 1600. La tengo in
campagna per godermela nelle giornate di brutto tempo.... Qui ho
altre occupazioni.... Ma in campagna quando non posso andare alla
caccia non trovo divertimento maggiore che quello di starmene fra i
miei vecchi libri.
Notiamo fra parentesi che chi avesse argomentato da ciò che il
conte Antonio Ermansi fosse una persona colta avrebbe pigliato un
bel granchio. Il conte Ermansi era un bibliomane; nulla più e nulla
meno. Egli non amava i libri per sè, ma per le loro curiosità
tipografiche. E anche le sue ricerche in proposito si limitavano al
secolo XVII. La più preziosa opera stampata nell’anno 1599 non
valeva per lui quanto la più stupida stampata nel 1601. D’altra parte,
nello stesso secolo XVII egli non si curava affatto degli autori celebri,
noti, i cui scritti erano stati pubblicati e ripubblicati; a’ suoi occhi non
avevano pregio che gli oscuri, quelli che nessuno conosceva, quelli
che forse in tutta la loro vita non avevano dato alla luce che un
misero opuscolo di venti pagine. Già il conte Ermansi non leggeva
nè i volumi grandi, nè i piccoli; una volta sicuro che del libercolo da
lui scovato fuori su un muricciuolo non c’erano che cinque o sei
esemplari in Europa, egli era contento come una Pasqua. Del resto,
non era più noioso degli altri della sua specie.
Comunque sia, è probabile che la collezione del conte Ermansi
esercitasse una scarsa attrattiva sul professore Teofoli e contribuisse
a fargli rimandar da un autunno all’altro l’accettazione dell’invito. Egli
si scusava adducendo la sua antica abitudine d’intraprender nelle
vacanze un lungo viaggio fuori d’Italia, a Parigi, a Vienna, a Berlino,
a Londra, a Edimburgo, allo scopo di rovistar biblioteche, di
annodare o di rinfrescar conoscenze coi confratelli di studio sparsi
pel mondo. Guai per lui se cedeva alla tentazione d’impigrirsi negli
ozi campestri.
Ma gli Ermansi non si davano per vinti. No, no, badasse a loro. Un
po’ di quiete è indispensabile sopratutto agli uomini che affaticano
molto il cervello. Avrebbe lavorato meglio dopo. In ogni modo, non si
pretendeva ch’egli rinunziasso al suo viaggio. Avrebbe fatto un
viaggio più breve, ecco tutto.... Anzi, se si fosse trovato male,
sarebbe ripartito il giorno dopo il suo arrivo, senza che nè lei nè suo
marito se ne adontassero.... Ma s’immagini. Con un vecchio
amico!...
Alla lunga Teofoli si lasciò carpire una mezza promessa per
l’autunno 187.... Non voleva impegnarsi, ma insomma, se gli era
possibile, al ritorno dalla Germania sarebbe passato a fare una
visitina a Sant’Eufemia.
E avvenne proprio così.
III.
Dalla Volpe e Frusti non seppero nulla di questa visita. Nelle

vacanze i tre indivisibili si dividevano. Quell’originale di Dalla Volpe,
appena finiti gli esami, partiva per ignota destinazione, guardandosi
bene di dare a chicchessia il suo indirizzo. Non voleva che la moglie
potesse raggiungerlo nè con la persona nè con le lettere. — Il mio
matrimonio — egli diceva — non mi accorda ormai altro benefizio
che questo; di poter viver tre mesi lontano dalla mia dolce metà, di
starmene pacificamente in qualche angolo remoto del mondo
cullandomi nella beata illusione d’esser scapolo o vedovo, o
pensando almeno che la cara Luisa urla, strepita, sbuffa ed espia i
suoi vecchi peccati senza di me.
Fedele al suo programma, durante le sue assenze non scriveva a
nessuno. Un anno lo si era visto in una dello stazioni alpine più
romite e solitarie; l’anno dopo si seppe ch’egli era in Egitto alle
rovine di Tebe dove corso il rischio di morire da un colpo di sole
pigliato nel decifrar geroglifici.... Ma neanche la paura dei colpi di
sole l’avrebbe indotto a rinunziare a quello ch’egli chiamava il suo
bagno nel celibato.
In quanto a Frusti, egli rimaneva sepolto dal luglio all’ottobre d’ogni
anno in qualche biblioteca d’Europa a ricercar documenti relativi a
Francesco I e a Carlo V. E ogni nuova scoperta era per lui una
grandissima gioia; non però una gioia senza mistura d’amaro,
accadendogli spesso di trovare un documento favorevole a
Francesco I quand’egli stava per mostrar le sue simpatie a Carlo V e
uno favorevole a Carlo V quand’era sul punto di giungere a una
conclusione opposta.
Per solito Frusti e Dalla Volpe erano di ritorno dalle loro
peregrinazioni soltanto dopo l’amico Teofoli, il quale nel suo zelo per
l’Università non voleva mancare nemmeno alla prima seduta del
Consiglio accademico. Si pensi quindi che maraviglia fosse la loro
quando, arrivati a X la mattina stessa dell’apertura dei corsi, seppero
che Teofoli non sarebbe giunto che fra due o tre giorni. Peggio poi
quando udirono il resto dalla signora Pasqua scandalizzata. Il
professore era stato in Germania sino alla metà di ottobre; poi s’era
fermato un paio di giorni nella villa dei conti Ermansi; di là era venuto
a casa per poche ore, tanto da comperarsi alla sartoria della Ville de
Rome un vestito completo e da far qualche altra spesuccia; e la sera
stessa, senza dire nè ai nè bai, senza voler dare una spiegazione
soddisfacente a lei, la signora Pasqua, che pur ne aveva diritto,
aveva ripreso il treno per Sant’Eufemia. Ah c’era del buio, molto
buio. Un uomo come il professore Teofoli, un uomo ch’era stato
sempre così savio, così costumato!...
Frusti e Dalla Volpe si guardarono tentennando il capo. L’avevano
sempre detto che la relazione degli Ermansi doveva esser fatale al
loro amico.
La condotta del nostro Teofoli al suo ritorno non tardò a giustificare
le maggiori apprensioni. Già bastava vederlo per capire che non era
più quello di prima. C’era nella sua toilette, nella sua andatura,
nell’espressione della sua fisonomia qualcosa di civettuolo che lo
rendeva irriconoscibile. Dal rettore al bidello, dai professori agli
studenti tutta l’Università era commossa da questa trasformazione.
Ogni giorno se ne sentiva una di nuova. Teofoli s’era abbuonato dal
parrucchiere, e aveva il fazzoletto impregnato d’acqua di Colonia!
Teofoli aveva ordinato al confettiere Grandi di spedire a
Sant’Eufemia (ove gli Ermansi si trovavano ancora) una colossale
scatola di dolci! Teofoli s’era comperato due cravatte di raso color
crema e un paio di lenti da sostituirsi in certi casi agli occhiali, troppo
solenni e cattedratici! Teofoli, invece della sua mazza d’ebano col
pomo d’avorio, aveva un leggero bastoncello di canna d’India!
Teofoli aveva minacciato di licenziare la signora Pasqua s’ella si
permetteva di seccarlo con le sue querimonie!
Nè le osservazioni dei due indivisibili erano accolte meglio. Egli si
meravigliava delle loro meraviglie. S’era forse impegnato a vestir
sempre ad un modo? O che un professore non potrà mettersi una
cravatta di raso chiaro e farsi ravviare dal parrucchiere i pochi capelli
che gli restano? Credevano di giovare alla scienza con simili
pedanterie? No, no, egli era persuaso che quell’abisso voluto
scavare fra gli studiosi ed i semplici mortali era un ostacolo alla
diffusione del sapere. In quanto a lui era risoluto a esser un uomo
come tutti gli altri, e non trovava necessario di andar a pescare dei
motivi misteriosi a una determinazione così naturale.
— Teofoli, non ce la dai ad intendere — dicevano sarcasticamente
Frusti e Dalla Volpe. — Tu non ti profumi d’acqua di Colonia per
agevolar la diffusione del sapere. Qui sotto c’è una femmina.
Il professore alzava le spalle in atto stizzoso. — Che femmina, che
femmina?
Ma ogni volta che gli toccavano questo tasto, diveniva rosso come
un papavero.
Che la femmina ci fosse non c’era dubbio. Restava a sapere chi
fosse.
Era evidente che Teofoli doveva averla incontrata in villeggiatura
dagli Ermansi ove quell’autunno c’era stata più gente del solito, e
ove con una magnanimità degna di lode la contessa Susanna,
riconoscendo la propria insufficienza fisica, aveva invitato anche
cinque o sei signore giovani e belle. La più bella, la più giovine era la
contessa Giorgina Serlati, sposa da due anni di un lontano parente
degli Ermansi, vissuta fino allora tra Roma e Parigi e rassegnata
adesso, per riguardi di economia, al soggiorno meno costoso di X....
Questa Giorgina non s’era vista a X che di passaggio subito dopo il
suo matrimonio, e aveva prodotto una notevole impressione per la
singolare avvenenza dell’aspetto e per la festività un po’ rumorosa e
bizzarra del carattere. La dicevano adesso ancora più seducente,
ancora più originale; insomma una di quelle che paiono nate apposta
per corbellare gli uomini. Aggiungasi un marito melenso,
insignificante, persuaso da un pezzo della vanità d’ogni suo tentativo
d’invigilar la moglie, e disposto a chiuder un occhio pur di esser
libero d’occuparsi de’ suoi cavalli e delle sue galanterie di bassa
lega.
Che fosse mai questa la donna che faceva girar la testa al
professore Teofoli? È ben vero ch’egli poteva esser suo padre; ma
non importa. In amore, le bestialità più grosse sono le più probabili, e
non c’era da stupirsi se Teofoli a cinquant’anni sonati aveva preso
una cotta per una donna di ventidue o ventitrè. In ogni caso, la
faccenda si sarebbe chiarita appena gli Ermansi avessero
abbandonato la villeggiatura, tirandosi dietro gli ospiti che
rimanevano ancora presso di loro. E i Serlati erano appunto tra
questi.
Ora il 25 novembre di quell’anno il professor Teofoli finì la sua
lezione dieci minuti prima che il bidello suonasse la campana, e,
congedandosi nell’atrio da tre o quattro studenti che avevano
l’abitudine di accompagnarlo a casa, entrò in un fiacre appostato
presso il portone dell’Università.
— O dove andrà il professore? — chiesero due di quei bravi
giovinotti.
— Ve lo saprò dire più tardi — soggiunse un terzo che non aveva
fretta di far colazione. E senza por tempo in mezzo montò in un altro
fiacre che passava di là ed era vuoto.
Teofoli non si recava in nessun luogo illecito e misterioso. I due
fiacre si fermarono alla stazione. Il professore discese dal suo e lo
studente fece lo stesso; il professore si mise a passeggiare su e giù
in atto d’uomo che aspetta, lo studente andò a sedere al caffè.
Circa dieci minuti dopo giunse una corsa, e Teofoli ch’era riuscito a
spingersi fin sotto la tettoia ricomparve in mezzo a una folla di
persone tra le quali lo studente riconobbe i coniugi Ermansi. Ma più
dei coniugi Ermansi lo colpì una signora giovine, alta, bellissima, dai
grandi occhi bruni che lampeggiavano sotto la veletta, dal corpo
svelto e flessuoso, dalla voce argentina, squillante. La seguiva a
pochi passi di distanza un uomo pur giovine, in soprabito grigio,
dall’aria annoiata, certo il marito. Al fianco di lei c’era Teofoli e le
parlava animatamente, e teneva sul braccio un suo impermeabile, e
si tirava dietro col cordino una cagnetta pinch alla quale la bella
signora slanciava degli sguardi teneri chiamandola a nome: Darling,
Darling. Facevano parte della brigata altri tre o quattro signori, senza
tener conto d’un codazzo di servi d’ambo i sessi, carichi di valigie, di
sacchi da viaggio, di panieri, d’ombrelli e perfino di gabbie di
canarini.
Fuori c’erano le carrozze, e la comitiva si divise con gran
dimostrazioni di cordialità. Gli Ermansi salirono in un landau chiuso,
l’altra coppia prese posto in un legno scoperto insieme con la
cagnetta. Però nel momento che il cocchiere stava per allentar le
redini sul collo dei cavalli la signora disse una parolina a Teofoli, e
questi ch’era ancora ritto davanti allo sportello mise il piede sul
montatoio e con una prestezza di movimenti di cui non lo si sarebbe
creduto capace fu in un attimo nella carrozza seduto accanto alla
bella persona che lo aveva invitato.
Rinvenuto appena dalla meraviglia di veder il suo professore
dileguarsi in quell’equipaggio signorile e al fianco di quella splendida
fata, lo studente colse a volo alcune frasi d’un colloquio fra due
zerbinotti ch’erano arrivati anch’essi in compagnia degli Ermansi e
che s’avviavano in città a piedi seguiti da un fattorino a cui avevano
consegnato il loro piccolo bagaglio.
Uno di questi zerbinotti che lo studente conosceva di nome, il
marchese di Montalto, diceva dispettosamente all’amico: — Alla
lunga quel balordo di Teofoli dà sui nervi.
— Non crederai mica che la Serlati lo prenda sul serio?
— Lo so anch’io che non lo prende sul serio. È però una gran noia
l’averlo sempre tra i piedi.
— Speriamo che quando ella lo avrà reso completamente ridicolo lo
getterà da parte.
— Sì, sì.... intanto si rende ridicola anche lei.
— Oh — notò l’interlocutore che prendeva le cose con maggior
calma — una donna bella come la contessa non si rende mai
ridicola.
Lo studente non intese più di così, ma quello che aveva inteso, unito
con quello che aveva visto, gli bastò per riferire ai suoi condiscepoli
che la donna alla quale il professore Teofoli prestava i suoi omaggi
era la contessa Serlati, una creatura deliziosa, nel primo fiore degli
anni, un bocconcino insomma più adattato agli scolari che ai
professori. E quei bravi ragazzi che pur volevano un gran bene a
Teofoli, che lo consideravano un luminare della scienza, che
l’avrebbero difeso accanitamente contro i suoi detrattori, provavano
in quell’occasione una specie d’animosità contro di lui e si sentivano
disposti a far eco a quel mezzo cretino del marchese di Montalto che
con tanta disinvoltura gli aveva dato del balordo. Gli è che se non
capita mai il momento in cui il balordo paia un uomo di spirito, ci
sono anche troppi momenti nella vita in cui l’uomo di spirito pare, ed
è davvero, un balordo.

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Quantum Reality: Theory and

Philosophy, 2nd Edition Jonathan

Quantum Reality 2nd Edition Jonathan Allday

An Introduction to Moral Philosophy 2nd Edition

Quantum Field Theory 2nd Edition Michael V. Sadovskii

Temporal Logic From Philosophy and Proof Theory to

Primary Mathematics 3A Hoerst

An Introduction to Indian Philosophy Perspectives on

QUANTUM INFORMATION PROCESSING: Theory and

Statistical Approach to Quantum Field Theory 2nd

• Presents a thorough grounding in the theoretical machinery of quantum physics

Second edition published 2023

and by CRC Press

© 2023 Jonathan Allday

First edition published by CRC Press 2009

CRC Press is an imprint of Taylor & Francis Group, LLC

Library of Congress Cataloging‑in‑Publication Data

ISBN: 978-1-032-12734-7 (hbk)

Pictogram Credit: Emrys

Chapter 1 Our First Encounter with the Quantum World: Light................................................... 7

Chapter 3 Quantum States........................................................................................................... 35

3.3.3 Relating Amplitudes to Probabilities..................................................44

Chapter 7 Free Particles............................................................................................................... 91

7.2.1 Classical Waves...................................................................................92

Chapter 8 Identical Particles........................................................................................................99

Chapter 9 Scattering Identical Bosons....................................................................................... 115

Chapter 10 Spin............................................................................................................................ 127

Chapter 11 Fermion States........................................................................................................... 143

Chapter 12 Continuous Bases...................................................................................................... 163

Chapter 13 Uncertainty................................................................................................................ 181

Chapter 14 The Equations of Quantum Theory.......................................................................... 191

14.2 Ehrenfest’s Theorem....................................................................................... 194

Chapter 15 Constrained Particles................................................................................................205

Chapter 16 Genealogy................................................................................................................. 233

Chapter 17 Planck and Einstein................................................................................................... 237

Chapter 18 Bohr........................................................................................................................... 247

Chapter 19 Heisenberg................................................................................................................. 259

Chapter 20 De Broglie & Schrödinger........................................................................................ 267

Chapter 21 Dirac.......................................................................................................................... 275

Chapter 22 Conclusions............................................................................................................... 283

Chapter 23 Quantum Correlations............................................................................................... 287

23.2.1 The EPR Argument........................................................................... 288

Chapter 24 Quantum Computing................................................................................................. 311

Chapter 25 Density Operators...................................................................................................... 325

Chapter 26 Interpretations........................................................................................................... 335

26.2.4 Measurement..................................................................................... 343

Chapter 27 The Copenhagen Interpretation................................................................................ 353

Chapter 28 The Many Worlds Interpretation............................................................................... 373

28.7 Criticisms of the Many Worlds View............................................................. 388

Chapter 29 Assorted Alternatives................................................................................................ 393

Chapter 30 Consistent Histories................................................................................................... 401

Chapter 31 The Ontological Interpretation.................................................................................. 425

Chapter 32 Quantum Field Theory.............................................................................................. 443

Chapter 33 Personal Conclusions.................................................................................................469

Appendix List of Important Rules.............................................................................................. 475

THANKS TO THE FOLLOWING (FIRST EDITION)

THANKS TO THE FOLLOWING (SECOND EDITION)

In science we study the linkage of pointer readings with pointer readings.

1.2 A LITTLE LIGHT READING

1.2 A LITTLE LIGHT READING

1.3 LASERS AND VIDEO CAMERAS

1.5 AN INTERFERENCE EXPERIMENT

1.5.1 Interference as a Wave Effect

1.5.2 Mach–Zehnder with Photons