Power of the Invisible

Front matter

er of als Invisibl

Amsterdam University Press

Power of the Invisible

Introduction

When it comes to atoms, language can be used only as in poetry. The poet, too, is not nearly so concerned with describing facts as with creating images.

Niels Bohr, ‘Atomic Physics and the Description of Nature’ (1934)

In this introduction we show how the book is structured and give some advice on how to read it.

Mathematics as a language of Nature. Quantum theory is known to be a difficult subject and becomes completely unfathomable if you have to rely entirely on our feeble natural language to describe it. Therefore | hope that you will not be scared away by the book’s rather mathematical appearance, particularly the second volume which looks as if itis full of equations. Don’t put the book aside just because of its intimidating appearance. Natural language is not the optimal means precisely because in quantum theory we enter realms of reality that are quite remote from our everyday experiences and preconceptions. Our cherished ‘common sense’ appeared to be of limited use and easily led us astray. Some call the quantum world mysterious or alien, while others see iut as elusive or unfathomable; indeed one may easily get drowned if the message is communicated to you in words only.

Mathematics is here to rescue us; it allows us to construct smart and elegant notions that perfectly fit nature’s needs it comes with a beautifully efficient notation. The lengthy descriptions one would need in natural language to convey the essentials of quantum reality would too easily clut-

The quantum leap. This art work called ‘The running knot’ is located in the city park of Kanazawa, Japan. (Source: Eryn Vorn, FLICKR)

ter the mind and lead to the utmost confusion, as | have seen happening in quite a few ‘no formula’ expositions of the quantum world to the layperson. So there are ample reasons to be courageous and go ‘symbolic.’

Great narratives choose their own language. The heart of music is in the sound and a verbal substitute would not do. And as we all know, it takes guidance to learn how to hear what it expresses. The same is true for the visual arts. It is hard to imagine a book about Picasso without pictures. And this is what Sagredo in the Dialogos of Galileo confided: ‘If | were again beginning my studies, | would follow the advice of Plato and start with mathexviii

INTRODUCTION matics. Yes, the narrative of Nature expresses itself most eloquently in mathematics. So, we take Sagredo’s advice to heart and will gently introduce some of the quantessential mathematical concepts along the way, but always in a rather pedestrian way’. Math, as a language of nature, as a means for understanding, but not as a purpose on its own. To that end | have included several so-called Math Excursions at the end of the third volume. These excursions explain in a user-friendly way what the math in the main text is about and will tell you all you need — but maybe never wanted — to know about matters like functions, complex numbers, matrices, algebras or vectors. Checking out these excursions will help you to get more out of this book.

The best part of climbing a mountain is often the splendid view from the top. In a similar way we work our way up to some of the quantessential equations, not in praise of rigor, but in praise of clarity and beauty. | tell my students that equations love people and they better do because they owe their existence to them. Bearing that in mind, isn’t it amazing that this man-made language of mathematics turns out to be the most ‘natural’ after all? This fascinating fact inspired the famous mathematical physicist Eugene Wigner to write an interesting essay about this paradox titled: ‘The unreasonable effectiveness of mathematics in the natural sciences.’ And as | intend to remain your travel ing companion all along the winding road to the quantum world, | hope that you will be patient with some of the math that we will encounter along the way. Think of it as the poetry of reality: a sublime shorthand endowed with a built-in integrity. A minimal yet powerful representation of reality. There is some truth in what John von Neumann, as keynote speaker at the first national meeting of the Association for Computing Machinery in 1947, quipped: ‘If people do not believe that mathematics is simple, it is only

? As we will indeed encounter many of the fundamental equations of physics along the way, the interested reader who is not at all versed in these equations may want to look at my popular science book entitled The Equations: icons of knowledge (Harvard University Press, 2005).

Figure 1: Adinkra symbol. Adinkras are symbols of the people of the Ashanti Kingdom in West Africa (Ghana) that represent concepts or wise sayings (aphorisms). This adinkra is called ‘nea onnim no sua a, ohu, which translates as ‘he/she who does not know can become knowledgeable through learning.’ | happen to see many interlocked copies of the letter ‘E’ , from Education, a striking coincidence!

S. James Gates, Complex ideas, complex shapes (2012)

because they do not realize how complicated life is.’

To whom am | talking? One of the first questions a potential publisher will throw at you as potential author is about who your perceived audience is. Who is going to read (or rather, buy) this book? So many pages, so many topics, so many equations, who the hell do you think..../f you cannot kill your darlings they will kill you! My answer is encrypted in the symbolic aphorism depicted in Figure 1 saying: ‘he/she who does not know can become knowledgeable through learning’. Keeping in mind that this holds true for basically everybody, it stands for the notion of education permanente, which advocates a broader spectrum of conceivable audiences for books on knowledge. There is the questionable dichotomy that books about science have, for some reason, to belong to either the categories ‘popular’ or

NATURE IS QUANTIZED xix

‘textbook’, with basically nothing in between. From my experience of teaching science to all kinds of people, | know that there are many diverse audiences between those of laypeople and Harvard graduates. And these present us with a need for books that try to bridge the intellectual pseudo gap | just mentioned. And with the availability of internet sources like Wikipedia and Youtube there is still a clear need for in-between books that give a broad coherent account with some theoretical depth. My hope is that this book provides an example thereof. So who are the would-be members in this perceived audience: students of various backgrounds and disciplines, from motivated high school whizzkids to multidisciplinary college students, as well as their teachers. | think of students in the disciplines neighbouring physics in the natural sciences, as well as engineering, mathematics and information science. | think of journalists and of the growing group of seniors who finally have time to get to grips with some of the deep scientific subjects that over the last century through technological developments have so radically transformed the world around them. | dedicate this work to the bright young people throughout the world who share that insatiable hunger for true knowledge and | hope that it will inspire their honorable quest. Students tend to be overwhelmed by the ‘how to’ questions, which means that the ‘why’ and ’what does it mean’ questions are neglected. Let me close this little pitch with a quote from the early muslim polymath Al Kindi!®, who lived around 850 AD:

We should not be ashamed of recognizing truth and assimilating it from whatever source it may reach us, even though it might come from earlier generations or foreign peoples. For him who seeks truth there is nothing of more value than truth itself. It never cheapens or abases him who searches for it, but ennobles and honors him.

WAI Kindi wrote more then 250 books. His Manuscript on Deciphering Cryptographic Messages, in which he laid the foundation of crypto-analysis using statistical interference and frequency analysis, is remarkable.

Nature is quantized

Quantum theory is not a theory of one particular system like the atom; it is a set of universal principles that applies to all of nature.

We present an overview of how this elaborate field is structured as a whole and thereby motivate the lay out of the book.

Quantum theory is based on a set of fundamental principles that nature appears to obey at basically all scales and therefore underlies all of physics, and more indirectly also all of chemistry and biology. The dictum is ‘One Nature, One Science’. Deep down all physical theories have to behave according to the quantum rules and therefore all our theories have to be ‘quantized, somewhat like kids have to be potty-trained, and dogs have to go to obedience school to learn not to bark. The quantum postulates forced us to reinvent the whole of fundamental physics from a new conceptual basis. We have quite successfully quantized particles and mechanics, electrodynamics including optics, and liquids, solids and other condensed forms of matter. But also unified theories describing subnuclear physics have been successfully quantized and led to the celebrated Standard Model. And finally, not so long ago, we realized that even information should be quantized. This Ongoing quantization process has led to a much deeper understanding of the fundamental structure of nature, but also to a huge number of breathtaking applications and quantum technologies that have only just started to take off. Indeed, technologies involving quantum information processing are expected to generate a highly disruptive transition with a huge socio-economic impact. Yet, this having been said, there are still many fundamental challenges, like the quantum interpretation of gravity, the oldest known force, which are required to be tackled in order to understand the origins of the universe or how black holes work.

XX INTRODUCTION

Chemistry Volume Ill: Condensed matter Hierarchies:

t

Molecular k Volume Il:

the emergence of diversity

Quantessence: how quantum theory works

Atomic Quantum Physics

Nuclear

Volume l: Elementary The journey: from Particles classical to quantum worlds

Figure 2: Three volumes. Quantum theory was introduced to physics at the atomic level. From there it started spreading into the other levels of physics, at both larger and smaller scales.

PHYSICS, MATHEMATICS AND CONCEPTS xxi

Three volumes. Quantum theory basically originated at the level of the atom, and by modern standards that is an intermediate length scale. From there the applications of the basic theory developed in two opposite directions, as indicated in the left column of Figure 2. On the one hand to ever smaller distances, all the way down to modern particle physics, and on the other hand to ever larger distances, moving up all the way to modern (bio)chemistry and condensed matter physics. The arrows pointing upward underscore the basic fact that quantum effects are by no means restricted to the microscopic domain. There are many research fields devoted to the study of quantum principles on macroscopic scales, which amounts to applying quantum theory to collective phenomena. In that sense every cell phone is full of quantum.

Even though | will restrict myself to the ‘quantessentials’, the subject is so vast that the book is divided into three parts, i.e. volumes, which — as also indicated in Figure 2 — can be characterized as follows.

The first volume of the book, called The journey: from classical to quantum worlds, starts with the highlights of classical physics and informatics after which it descends into the quantum world. It is the narrative guided by man’s passionate quest for the most basic building blocks of nature and their interactions. We start with marbles and end up with quarks and even superstrings.

In the second volume of the book, called Quantessence: how quantum theory works, we delve deeper into the structure of the theory and present some of its mathematical representations. And we will talk about the conceptual issues concerning quantum states, observables and measurements that we encounter along the way. There we will be concerned extensively with mind-boggling notions like entanglement, particle interference and quantum teleportation.

In the third and final volume called Hierarchies: the emergence of diversity, we discuss quantum theory as it applies to the structural hierarchy of matter from the atomic level to chemistry and the quantum physics of condensed states of matter. We not only consider the hierarchy in a spatial sense but also how that hierarchy arose in a temporal sense during the early stages of cosmic evolution. It closes with a chapter on scaling, discussing notions such as self-similarity, scale invariance and renormalization of theories in order to understand their asymptotic behavior if one imagines the behavior of theories as models of nature, at ever smaller or larger scales. We conclude this quantum trilogy by offering a concluding chapter with a more general science-driven perspective.

Physics, mathematics and concepts

If you look long enough, anything becomes abstract Diane Arbus

This section presents a meta-perspective on how to read this quantessential book. The quantum world can be traversed in many ways, all pertaining to a certain ‘logic’. Taking a single path will enlighten certain aspects but may obscure others. Therefore it is better to combine different paths to get an optimal feeling for the quantum landscape. To get to the quantessence, one would have to add up the contributions of all the different paths”! .

Once a field of science (like physics) has matured sufficiently, one can learn something interesting about the structure of scientific knowledge in general. This is indicated in the layered structure of quantum knowledge in the scheme of Figure 3, in which the three columns refer to the three layers of knowledge that | like to distinguish between and which will be explained shortly.

Hin a symbolic — if not ironic — sense, you could call this a ‘path integral approach to the understanding of quantum theory.

xxii INTRODUCTION

Layer A Layer B Layer C

Quantum Mathematical Quantum Phenomenology Representations Concepts

Indistinguishability Exclusion principle

Chemistry uantum strings Condensed matter Q 8

T

Quantum fields

=

Quantum particles Wavefunctions

T

=

Teleportation Computation Measurements Interference Quantum information Observables Qubits Incompatibilit |

Elementary Quantum State vectors Particles syntax/logic Entanglement

Figure 3: Three layers. In quantum physics one may distinguish three layers displayed here as columns. From left to right, (A) is about the phenomenology of systems in which quantum theory manifests itself, (6) is the layer of mathematical representations or realizations and (C) is the layer of quantum concepts and principles. Note that the layers are coupled together as a whole, not via their individual components.

PHYSICS, MATHEMATICS AND CONCEPTS xxiii

Theoretical physics is basically about constructing optimal mathematical models of reality. It usually starts by effectively describing certain regularities apparent in some observed physical phenomena. The next step — if possible — is to relate different phenomena through the model. This amounts to reducing the number of independent parameters in the models. Finally, one hopes that it will make predictions and suggestions as to where to look for unique signatures of new phenomena. Over time this modelling has been done in an ever more sophisticated way, exploiting existing as well as developing new mathematical and computational tools.

A first step in modelling a physical system is to just identify which degrees of freedom are relevant to the phenomena one wants to study and understand. A second crucial step is to identify what the relevant interactions between these basic degrees of freedom are. For the moment these are just words referring to basic notions, which have to find their way into some symbolic representation or mathematical framework. We may, in the end, have to extend our set of basic concepts and rules, our grammar so to speak, in order to accommodate new phenomena and new underlying principles.

In the development of quantum theory over the past century, this is exactly what happened. It turned out that we needed new mathematical realizations and ever more sophisticated representations of the material world. It is a multitude of unfolding insights intertwined with the dramatic growth of our experimental means to probe physical reality that marked the advances in theory over the last century. And finally, once the mathematical, maybe somewhat pragmatic modeling has advanced sufficiently, one should try to come to a more fundamental insight as to what these models imply for the logical structure of the underlying physical reality. Here we enter a realm with philosophical ramifications, where we move from the syntax anchored in the mathematical consistency of the model, to its semantics and interpretation. We can pose ontological questions about what is ‘to be and/or not to be’, as well as questions about the epistemology and about what is ’knowable’. We enter the territories of beables and knowables: in short, the realm of meaning.

Three layers. Quantum theory at large comprises a huge body of knowledge | like to think of as consisting of three layers as depicted in the columns of Figure 3. The A-layer comprises the physical realizations and manifestations of quantum matter, the -layer is about mathematical representations and realizations, and finally the C-layer concerns underlying concepts and principles, and their logical structure and interpretation. Indeed it is only after one has a mathematically consistent formulation of the theory that conceptual questions force themselves on us in a way that we can make sense out of them. Yet, one cannot avoid switching between the layers if one is to give a coherent account of the subject as a whole.

The first layer A refers to quantum phenomenology, the body of observational evidence concerning the broad spectrum of quantum phenomena that we will consider in this book. It is in fact the same as the first column in Figure 2, but note that the other columns of the two Figures refer to qualitatively entirely different things.

The second layer B refers to mathematical representations or models. This is already more abstract, as we ascend to the mathematical modeling of the observed phenomena. One might for example think of quantum states being elements of some vector space referred to as the Hilbert space, or of the mathematics of a qubit, or of a wave function. Or consider physical observables as represented by operators that act on that Hilbert space, like matrices or differential operators. And we may think of the dynamics of the quantum system described by famous differential equations, such as the Schrödinger, Heisenberg and Dirac equations.

And indeed, in the middle column from bottom to top we

XXxiv

INTRODUCTION see increasingly complex realizations of the same quantum principles, which are stated in the first step at the bottom. It is a hierarchy of degrees of freedom. We start with the discrete case of qubits and ‘qubit mechanics’, and move one step up to the simple continuous case of a single quantum particle. In the next step we face the problems of many particles of one single type or species, and the interactions between these species, which leads us to the theory of quantum fields. This level includes multi-particle states, and the creation and annihilation of particles; furthermore the forces are included and quantized. We finally end up with theories (and so far only theories) that attempt to combine all types of fields (or particle species) in the spectrum of a unique quantum (super)string. At this level space-time is included and quantized. So what we have indicated in the second column is the idea that states representing the physics at one given level form a small subspace of the set of states in the next step. It represents a modelling hierarchy.

We have mapped the system onto a mathematical model that allows us to make calculations and predictions, but models also pose new challenges for finding out what the essential concepts are that underlie all those quantum phenomena. We like to understand what the generic features are that set the quantum world apart from what we were used to in classical physics. That is what the next layer is about.

The third layer C is concerned with the conceptual implications of the mathematical framework, where we are required to interpret the basic mathematical entities back into physical terms. You may compare this to coming home from an exciting journey to some unknown country, and being forced to describe to your colleagues what the exquisite, extremely exotic food tasted like. You may think of mathematical models that manage to successfully describe and predict measurement outcomes, but at the same time force us to reinterpret what the very nature of physical reality is. There is the saying cherished by many theorists that ‘equations speak for themselves’, but that is often not the case. For example, you may know that the mathematics of special relativity is surprisingly simple, but its physical ramifications are certainly not; it forced us to fundamentally redefine our concepts of space and time. Something similar happened in the realm of quantum theory with respect to the true nature of what we, for convenience, call ‘matter’, or ‘radiation’, or ‘energy’, or ‘information’. Here we encounter the necessary consequences of the Hilbert space formalism, such as the existence of quantum entanglement and quantum interference. And we have to cope with non-commuting observables leading to fundamental uncertainties in measurement outcomes. These unambiguous consequences of the mathematical formalism, which by itself is clear cut, will, as we will show, pose quite formidable epistemological and philosophical questions. It suffices to refer to the infamous Einstein, Podolsky, Rosen (EPR) paradox, which lies at the heart of the well-known Einstein—Bohr debate about how quantum theory defines what we call reality. This debate has been going on for three quarters of a century and only now appears about to be settled.

Going from left to right in Figure 3 is, in some sense, a perspective marked by experimental discoveries and as such a rather historical perspective. Going from left to right is therefore hard because it is erratic, and it moves slowly except for sudden jumps. It is highly unpredictable because it basically lacks internal logic: there is no strictly logical path from classical to quantum physics. The path from left to right is the historic one, and therefore bumpy, but also paved with would-be miracles and intriguing misconceptions, which indeed make a wonderful narrative with ample heroism and drama.

But once the subject has matured, there is the other possibility, namely to start on the right with the concepts and a logical, abstract framework, and from there move back to the left. A theorist like myself naturally prefers a presentation from right to left, which in a sense is highly anti-

PHYSICS, MATHEMATICS AND CONCEPTS

XXV chronological, but would be more comprehensive because it has an internal logic and systematics. | believe that once things are understood, going from right to left is easy. Moreover, it would give the author the freedom to limit himself to the quantessence of a coherent body of knowledge.

Yet, in spite of this argument, it would be a bad idea to really treat the three layers sequentially from right to left, because you need the stuff on the left to appreciate the content of the right column. This suggests the option of a left-right compromise, or left-right coalition, just like that is often the case in the politics of healthy democracies.

Combining parts and layers: the outline. After some reflections on the general structure of the book, let me now just give a more detailed description of the layout of the chapters. As mentioned, | have divided the book in three parts or volumes, as indicated in Figure 2. Volumes | and IIl are primarily descriptive and do not require much math, since they are phenomenologically-oriented. So in the context of the layers, Volumes | and Ill mainly deals with A with some attention to layer B. Volume Il, with a title that refers to the ‘quantessence’, focusses more on the mathematical and conceptual structure of the theory, and covers the layers 6 and C. As a matter of fact, quantum lovers with an outspoken fear of formulas may prefer to read only Volumes | and III as a single coherent descriptive account of what quantum theory has achieved. The following preview may help you to make up your mind.

Volume | talks about The journey, where we follow a path starting at the level of atoms, and descending deeper into matter to the worlds of nuclei and elementary particles and their interactions. This part is so to speak inward bound. But before we embark on this descent in Chapter 1.4, we give a review of what classical physics is about in Chapterl.1. Chapter 1.2 deals with the very breakdown of classical physics, from which crises the theories of relativity and quantum emerged. Here we also included a section on the physics of geometry and a section on the notions of information and computation, highlighting another fundamental turning point in twentieth century science and technology. In Chapter 1.3, on units, scales and universal constants, we obtain surprisingly deep insights in the domains of validity of our cherished theories by applying what we call ‘dimensional analysis. It provides us with a heuristic quantitative sense of what the characteristic scales in nature are, and why. In Chapter 1.4 we describe the quest for the basic building blocks of matter all the way from atoms down to the most fundamental constituents of matter and radiation.

In Volume II — called Quantessence: how quantum theory works — we give an accessible introduction to the mathematical modelling tools and representations that comprise quantum theory, including those which led to a number of remarkable conceptual and semantic puzzles. This part emphasizes the two deeper layers | alluded to before, i.e. the layers 6 and C of Figure 3.

This second part also leads us deeper into the subjects of quantum information and computing. To that end we first have to contrast the setting of quantum theory with its classical counterpart. In Chapter ll.1, the first of Volume Il, we start by introducing quantum states as vectors in Hilbert space. | discuss the structure of Hilbert space for qubits and quantum information in quite a lot of detail. In the second Chapter II.2, | discuss the quantessence of observables, why we think of them as operators acting on Hilbert space, and what it means to make a quantum measurement. In this chapter the Heisenberg uncertainty relations are also introduced. In Chapter II.3 | talk about the measurement process more extensively with a particular focus on quantum interference phenomena. Chapter Il.4 examines quantum entanglement and some of the modern experiments addressing the profound questions of cloning, Schrédinger’s cat, hidden variables, as well as quantum teleportation and computation. In Chapter ll.5 | explain the concepts of quantum particles, fields xxvi

INTRODUCTION and strings. There, the famous equations of Schrödinger, Heisenberg and Dirac that describe the time evolution of states and observables, will be introduced. | also explain properties like quantum spin, quantum statistics and their relationships. In Chapter 11.6, the final chapter of Volume Il, we introduce the notions of symmetry and symmetry breaking which play a central role in all of modern physics. The notion of symmetry served as a powerful guiding principle in our quest to understand nature.

In the third and final Volume of the series we return to model physical reality but we now move upwards from the atomic scale. In Chapter Ill.1 we discuss how matter sequentially evolved in the very early universe, from quarks, to nucleons, to atoms, and from simple molecules to the basic (bio)chemistry concerning the molecules of life.

Chapter Ill.2 and III.3 are devoted to the splendid diversity of quantum phenomena in the physics of many body systems that are manifest in gaseous, liquid as well as solid phases. Where in Chapter IIl.2 we consider the atomic and nuclear lattices and to what extent these are ordered, the focus in chapter III.3 is on the electronic behavior in solids and the quantum phenomena they display.

In Chapter IIl.4, we touch upon the quite advanced notions of scale dependence and renormalization. Part III could well be called outward bound, certainly if reasoned from the atomic scale where quantum theory made its first appearance. The criteria of inward and outward bound refer to the arrows in the left column of Figure 2.

In the concluding Chapter III.5, we zoom out and look at the meaning and impact of quantum in the broader context of science, technology and society.

After the concluding chapter you find a set of Math Excursions, appendices in which we offer rather minimal but tailor-made introductions to the mathematics used throughout the book.

Choosing the structure of a three-volume book means that we couple together the layers of Figure 3 so as to enable a coherent presentation of the quantessence as a whole, which is accessible without being too superficial. What you see is that quantum theory, even when you restrict yourself to the quantessence, is a huge field. and that is why | divided the book up in three volumes.

As you may be aware, an impressive number of Nobel prizes have been awarded in the course of the past century to quantum discoveries in physics and chemistry. We list most of them in an appendix (on page 644) at the end of the final volume. There we also provide some of the chronology, and list the names of many of the influential thinkers who made seminal contributions to the field. It may also help you to follow up specific topics that have caught your interest while reading.

| like to think of the three parts of the book as a kind of a triptych, where the central panel covers the deeper quantum scenery, while the side panels are more descriptive and discuss lots of real physics, from quarks all the way up to bio-chemistry and the splendid diversity we encounter in the condensed states of matter.