Fashion, Faith, and Fantasy in the New Physics of the Universe

Fashion, Faith, and Fantasy in the New Physics of the Universe

by Roger Penrose

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Product Details

ISBN-13: 9780691119793
Publisher: Princeton University Press
Publication date: 09/27/2016
Pages: 520
Sales rank: 493,743
Product dimensions: 5.90(w) x 8.70(h) x 1.50(d)

About the Author

Roger Penrose, one the world's foremost theoretical physicists, has won numerous prizes, including the Albert Einstein Medal, for his fundamental contributions to general relativity and cosmology. He is the bestselling author, with Stephen Hawking, of The Nature of Space and Time (Princeton). Penrose's other books include Cycles of Time: An Extraordinary New View of the Universe and The Road to Reality: A Complete Guide to the Laws of the Universe (both Vintage). He is the Rouse Ball Professor of Mathematics Emeritus at the University of Oxford and lives in Oxford, England.

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Fashion Faith and Fantasy in the New Physics of the Universe


By Roger Penrose

PRINCETON UNIVERSITY PRESS

Copyright © 2016 Roger Penrose
All rights reserved.
ISBN: 978-1-4008-8028-7



CHAPTER 1

Fashion

* * *


1.1. MATHEMATICAL ELEGANCE AS A DRIVING FORCE

As mentioned in the preface, the issues discussed in this book were developed from three lectures given, by invitation of the Princeton University Press, at Princeton University in October 2003. My nervousness, with these lectures, in addressing such a knowledgeable audience as the Princeton scientific community, was perhaps at its greatest when it came to the topic of fashion, because the illustrative area that I had elected to discuss, namely string theory and some of its various descendants, had been developed to its heights in Princeton probably more than anywhere else in the world. Moreover, that subject is a distinctly technical one, and I cannot claim competence over many of its important ingredients, my familiarity with these technicalities being somewhat limited, particularly in view of my status as an outsider. Yet, it seemed to me, I should not allow myself to be too daunted by this shortcoming, for if only the insiders are considered competent to make critical comments about the subject, then the criticisms are likely to be limited to relatively technical issues, some of the broader aspects of criticism being, no doubt, significantly neglected.

Since these lectures were given, there have been three highly critical accounts of string theory: Not Even Wrong by Peter Woit, The Trouble with Physics by Lee Smolin, and Farewell to Reality: How Fairytale Physics Betrays the Search for Scientific Truth by Jim Baggott. Certainly, Woit and Smolin have had more direct experience than I have of the string-theory community and its over-fashionable status. My own criticisms of string theory in The Road to Reality, in chapter 31 and parts of chapter 34, have also appeared in the meantime (predating these three works), but my own critical remarks were perhaps somewhat more favourably disposed towards a physical role for string theory than were these others. Most of my comments will indeed be of a general nature, and are relatively insensitive to issues of great technicality.

Let me first make what surely ought to be a general (and perhaps obvious) point. We take note of the fact that the hugely impressive progress that physical theory has indeed made over several centuries has depended upon extremely precise and sophisticated mathematical schemes. It is evident, therefore, that any further significant progress must again depend crucially upon some distinctive mathematical framework. In order that any proposed new physical theory can improve upon what has been achieved up until now, making precise and unambiguous predictions that go beyond what had been possible before, it must also be based on some clear-cut mathematical scheme. Moreover, one would think, to be a proper mathematical theory it surely ought to make mathematical sense – which means, in effect, that it ought to be mathematically consistent. From a self-inconsistent scheme, one could, in principle, deduce any answer one pleased.

Yet, self-consistency is actually a rather strong criterion and it turns out that not many proposals for physical theories – even among the very successful ones of the past – are in fact fully self-consistent. Often some strong elements of physical judgement must be invoked in order that the theory can be appropriately applied in an unambiguous way. Experiments are, of course, also central to physical theory, and the testing of a theory by experiment is very different from checking it for logical consistency. Both are important, but in practice one often finds that physicists do not care so much about achieving full mathematical self-consistency if the theory appears to fit the physical facts. This has been the case, to some considerable degree, even with the extraordinarily successful theory of quantum mechanics, as we shall seeing in chapter 2 (and §1.3). The very first work in that subject, namely Max Planck's epoch-making proposal to explain the frequency spectrum of electromagnetic radiation in equilibrium with matter at a fixed temperature (the black-body spectrum; see §§2.2 and 2.11) required something of a hybrid picture which was not really fully self-consistent [Pais 2005]. Nor can it be said that the old quantum theory of the atom, as brilliantly proposed by Niels Bohr in 1913, was a fully self-consistent scheme. In the subsequent developments of quantum theory, a mathematical edifice of great sophistication has been constructed, in which a desire for mathematical consistency had been a powerful driving force. Yet, there remain issues of consistency that are still not properly addressed in current theory, as we shall see later, particularly in §2.13. But it is the experimental support, over a vast range of different kinds of physical phenomena, which is quantum theory's bedrock. Physicists tend not to be over-worried by detailed matters of mathematical or ontological inconsistency if the theory, when applied with appropriate judgement and careful calculation, continues to provide answers that are in excellent agreement with the results of observation – often with extraordinary precision – through delicate and precise experiment.

The situation with string theory is completely different from this. Here there appear to be no results whatever that provide it with experimental support. It is often argued that this is not surprising, since string theory, as it is now formulated as largely a quantum gravity theory, is fundamentally concerned with what is called the Planck scale of very tiny distances (or at least close to such distances), some 10-15 or 10-16 times smaller (10-15 meaning, of course, down by a factor of a tenth of a thousandth of a millionth of a millionth) and hence with energies some 1015 or 1016 times larger than those that are accessible to current experimentation. (It should be noted that, according to basic principles of relativity, a small distance is essentially equivalent to a small time, via the speed of light, and, according to basic principles of quantum mechanics, a small time is essentially equivalent to a large energy, via Planck's constant; see §§2.2 and 2.11.) One must certainly face the evident fact that, powerful as our present-day particle accelerators may be, their currently foreseeable achievable energies fall enormously short of those that appear to have direct relevance to theories such as modern string theory that attempt to apply the principles of quantum mechanics to gravitational phenomena. Yet this situation can hardly be regarded as satisfactory for a physical theory, as experimental support is the ultimate criterion whereby it stands or falls.

Of course, it might be the case that we are entering a new phase of basic research into fundamental physics, where requirements of mathematical consistency become paramount, and in those situations where such requirements (together with a coherence with previously established principles) prove insufficient, additional criteria of mathematical elegance and simplicity must be invoked. While it may seem unscientific to appeal to such aesthetic desiderata in a fully objective search for the physical principles underlying the workings of the universe, it is remarkable how fruitful – indeed essential – such aesthetic judgements seem to have frequently proved to be. We have come across many examples in physics where beautiful mathematical ideas have turned out to underlie fundamental advances in understanding. The great theoretical physicist Paul Dirac [1963] was very explicit about the importance of aesthetic judgement in his discovery of the equation for the electron, and also in his prediction of anti-particles. Certainly, the Dirac equation has turned out to be absolutely fundamental to basic physics, and the aesthetic appeal of this equation is very widely appreciated. This is also the case with the idea of anti-particles, which resulted from Dirac's deep analysis of his own equation for the electron.

However, this role of aesthetic judgement is a very difficult issue to be objective about. It is often the case that some physicist might think that a particular scheme is very beautiful whereas another might emphatically not share that view! Elements of fashion can often assume unreasonable proportions when it comes to aesthetic judgements – in the world of theoretical physics, just as in the case of art or the design of clothing.

It should be made clear that the question of aesthetic judgment in physics is more subtle than just what is often referred to as Occam's razor – the removal of unnecessary complication. Indeed, a judgement as to which of two opposing theories is actually the "simpler", and perhaps therefore more elegant, need by no means be a straightforward matter. For example, is Einstein's general relativity a simple theory or not? Is it simpler or more complicated than Newton's theory of gravity? Or is Einstein's theory simpler or more complicated than a theory, put forward in 1894 by Aspeth Hall (some 21 years before Einstein proposed his general theory of relativity), which is just like Newton's but where the inverse square law of gravitation is replaced by one in which the gravitational force between a mass M and a mass m is GmMr,-2.00000016 rather than Newton's GmMr-2. Hall's theory was proposed in order to explain the observed slight deviation from the predictions of Newton's theory with regard to the advance of the perihelion of the planet Mercury that had been known since about 1843. (The perihelion is the closest point to the Sun that a planet reaches while tracing its orbit [Roseveare 1982].) This theory also gave a very slightly better agreement with Venus's motion than did Newton's. In a certain sense, Hall's theory is only marginally more complicated than Newton's, although it depends on how much additional "complication" one considers to be involved in replacing the nice simple number "2" by "2.00000016". Undoubtedly, there is a loss of mathematical elegance in this replacement, but as noted above, a strong element of subjectivity comes into such judgements. Perhaps more to the point is that there are certain elegant mathematical properties that follow from the inverse square law (basically, expressing a conservation of "flux lines" of gravitational force, which would not be exactly true in Hall's theory). But again, one might consider this an aesthetic matter whose physical significance should not be overrated.

But what about Einstein's general relativity? There is certainly an enormous increase in the difficulty of applying Einstein's theory to specific physical systems, beyond the difficulty of applying Newton's theory (or even Hall's), when it comes to examining the implications of this theory in detail. The equations, when written out explicitly, are immensely more complicated in Einstein's theory, and they are difficult even to write down in full detail. Moreover, they are immensely harder to solve, and there are many nonlinearities in Einstein's theory which do not appear in Newton's (these tending to invalidate the simple flux-law arguments that must already be abandoned in Hall's theory). (See §§A.4 and A.11 for the meaning of linearity, and for its special role in quantum mechanics see §2.4.) Even more serious is the fact that the physical interpretation of Einstein's theory depends upon eliminating spurious coordinate effects that arise from the making of particular choices of coordinates, such choices being supposed to have no physical relevance in Einstein's theory. In practical terms, there is no doubt that Einstein's theory is usually immensely more difficult to handle than is Newton's (or even Hall's) gravitational theory.

Yet, there is still an important sense in which Einstein's theory is actually a very simple one – even possibly simpler (or more "natural") than Newton's. Einstein's theory depends upon the mathematical theory of Riemannian (or, more strictly, as we shall be seeing in §1.7, pseudo-Riemannian) geometry, of arbitrarily curved 4-manifolds (see also §A.5). This is not an altogether easy body of mathematical technique to master, for we need to understand what a tensor is and what the purpose of such quantities is, and how to construct the particular tensor object R, called the Riemann curvature tensor, from the metric tensorg which defines the geometry. Then by means of a contraction and a tracereversal we find how to construct the Einstein tensorG. Nevertheless, the general geometrical ideas behind the formalism are reasonably simple to grasp, and once the ingredients of this type of curved geometry are indeed understood, one finds that there is a very restricted family of possible (or plausible) equations that can be written down, which are consistent with the proposed general physical and geometrical requirements. Among these possibilities, the very simplest gives us Einstein's famous field equation G = 8πγT of general relativity (where T is the mass–energy tensor of matter and [??] is Newton's gravitational constant – given according to Newton's particular definition, so that even the "8π" is not really a complication, but merely a matter of how we wish to define γ).

There is just one minor, and still very simple, modification of the Einstein field equation that can be made, which leaves the essential requirements of the scheme intact, namely the inclusion of a constant number [??], referred to as the cosmological constant (which Einstein introduced in 1917 for reasons that he later discarded) so that Einstein's equations with [??] now become G = 8πγT + [??]g. The quantity [??] is now frequently referred to as dark energy, presumably to allow for a possibility of generalizing Einstein's theory so that [??] might vary. There are, however, strong mathematical constraints obstructing such considerations, and in §§3.1, 3.7, 3.8, and 4.3, where [??] will be playing a significant role for us, I shall restrict attention to situations where [??] is indeed non-varying. The cosmological constant will have considerable relevance in chapter 3 (and also §1.15). Indeed, relatively recent observations point strongly to the actual physical presence of [??] having a tiny (apparently constant) positive value. This evidence for [??] > 0 – or possibly for some more general form of "dark energy" – is now very impressive, and has been growing since the initial observations of Perlmutter et al. [1999], Riess et al. [1998], and their collaborators, leading to the award of the 2011 Nobel Prize in physics to Saul Perlmutter, Brian P. Schmidt, and Adam G. Riess. This [??] > 0 has immediate relevance only to the very distant cosmological scales, and observations concerning celestial motions at a more local scale can be adequately treated according to Einstein's original and simpler G = 8πγT. This equation is now found to have an unprecedented precision in modelling the behaviour, under gravity, of celestial bodies, the observed [??] value having no significant impact on such local dynamics.

Historically of most importance, in this regard, is the double-neutron-star system PSR1913+16, one component of which is a pulsar, sending very precisely timed electromagnetic signals that are received at the Earth. The motion of each star about the other, being very cleanly a purely gravitational effect, is modelled by general relativity to an extraordinary precision that can be argued to be of about one part in 10 overall, accumulated over a period of about 40 years. The period 40 years is roughly 10 seconds, so a precision of one in 10 means an agreement between observation and theory to about 10 (one hundred thousandth) of a second over that period – which is, very remarkably, indeed just what is found. More recently, other systems [Kramer et al. 2006] involving one or even a pair of pulsars, have the potential to increase this precision considerably, when the systems have been observed for a comparable length of time as has PSR19+16.

To call this figure of 10 a measure of the observed precision of general relativity is open to some question, however. Indeed, the particular masses and orbital parameters have to be calculated from the observed motions, rather than being numbers coming from theory or independent observation. Moreover, much of this extraordinary precision is already in Newton's gravitational theory.


(Continues...)

Excerpted from Fashion Faith and Fantasy in the New Physics of the Universe by Roger Penrose. Copyright © 2016 Roger Penrose. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

Acknowledgements ix

Preface xi

Are fashion, faith, or fantasy relevant to fundamental science? xi

1 Fashion 1

1.1 Mathematical elegance as a driving force 1

1.2 Some fashionable physics of the past 10

1.3 Particle-physics background to string theory 17

1.4 The superposition principle in QFT 20

1.5 The power of Feynman diagrams 25

1.6 The original key ideas of string theory 32

1.7 Time in Einstein’s general relativity 42

1.8 Weyl’s gauge theory of electromagnetism 52

1.9 Functional freedom in Kaluza–Klein and string models 59

1.10 Quantum obstructions to functional freedom? 69

1.11 Classical instability of higher-dimensional string theory 77

1.12 The fashionable status of string theory 82

1.13 M-theory 90

1.14 Supersymmetry 95

1.15 AdS/CFT 104

1.16 Brane-worlds and the landscape 117

2 Faith 121

2.1 The quantum revelation 121

2.2 Max Planck’s E = 126

2.3 The wave–particle paradox 133

2.4 Quantum and classical levels: C, U, and R 138

2.5 Wave function of a point-like particle 145

2.6 Wave function of a photon 153

2.7 Quantum linearity 158

2.8 Quantum measurement 164

2.9 The geometry of quantum spin 174

2.10 Quantum entanglement and EPR effects 182

2.11 Quantum functional freedom 188

2.12 Quantum reality 198

2.13 Objective quantum state reduction: a limit to the quantum faith? 204

3 Fantasy 216

3.1 The Big Bang and FLRW cosmologies 216

3.2 Black holes and local irregularities 230

3.3 The second law of thermodynamics 241

3.4 The Big Bang paradox 250

3.5 Horizons, comoving volumes, and conformal diagrams 258

3.6 The phenomenal precision in the Big Bang 270

3.7 Cosmological entropy? 275

3.8 Vacuum energy 285

3.9 Inflationary cosmology 294

3.10 The anthropic principle 310

3.11 Some more fantastical cosmologies 323

4 A New Physics for the Universe? 334

4.1 Twistor theory: an alternative to strings? 334

4.2 Whither quantum foundations? 353

4.3 Conformal crazy cosmology? 371

4.4 A personal coda 391

Appendix A Mathematical Appendix 397

A.1 Iterated exponents 397

A.2 Functional freedom of fields 401

A.3 Vector spaces 407

A.4 Vector bases, coordinates, and duals 413

A.5 Mathematics of manifolds 417

A.6 Manifolds in physics 425

A.7 Bundles 431

A.8 Functional freedom via bundles 439

A.9 Complex numbers 445

A.10 Complex geometry 448

A.11 Harmonic analysis 458

References 469

Index 491

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