El camino a la realidad

Dave Langford, SF&F critic and reviewer, in his long-since defunct column for White Dwarf magazine, once said that, "There is a tendency to over-praise big books simply because one has got through them." I agree that this tendency exists but note that Langford gave no reason for it. I think the reason is more or less macho intellectual pride; look at me! I read this honking great saga! It must be great or I'd have to admit wasting my time! And I need to show off my intellectual credentials! Now imagine that the book is not only huge, but really difficult because, say, it's dense with obscure references (e.g. Ulysses) or full of mathematics and not kiddie maths, either...the temptation must be even worse.

Hence I'm going to start my review with a couple of gripes: This book, which is full of maths, much of which would make your average undergrad scientist grunt with the strain at the very least, as well as physics to post-grad level in places, has no glossary of technical terms. There is ample cross-referencing and an index but these are no substitute. When you want to know, Clifford Algebra, which one was that again? (Because you've met ordinary algebra, complex algebra, Clifford Algebras, Lie Algebras and Grassman Algebras...), going back and reading through an entire section again to find out, is a bit annoying - a list of definitions at the back would have helped enormously. Admittedly this would have made a big book even bigger but it would have made it much more user-friendly.

Gripe number two is in a similar vein; Penrose fails to supply a list of the upper and lower case Greek alphabet symbols and their names or a similar list for obscure mathematical symbols, such as del and scri. Given that nobody without training in Greek or in science is going to know these and such a list would only take up one page, its omission is egregious.

This leads neatly into a topic that has been dicussed quite a bit here on Goodreads - namely, who is this book aimed at? What is it's purpose? Firstly I would point out that the subtitle "A Complete Guide to the Laws of the Universe" is not really true: Classical Thermodynamics is barely seen as we rush straight into the statistical view of the Second Law. Of the other three Laws of Thermodynamics, Zero is never mentioned and the others barely name-checked. I doubt many physicists would consider that all the basic theories have been covered in such a circumstance. But I think this is a marketing problem; I don't believe Penrose ever intended to write such a "Complete Guide."

In the preface Penrose talks about wanting, with this book, to make cutting edge physics available to people who struggle to understand fractions. Now, this can only be taken as a joke, considering what one is up against only in chapter 2, but I would guess that Penrose genuinely wants to have the widest possible audience for his book whilst not compromising his aims.

What are those aims? In my view he wants to give his personal views on the state of cosmology and fundamental physics but to be able to do it at an advanced technical and mathematical level and additionally to give his own philosophies regarding the nature of thought, science, maths and...Nature! This means that he wanted to deliver Chapters 27 - 33 on the physics/cosmology, bracketed by Chapters 1 and 34 of philosophising. The entirety of the rest of the book is simply there in order to equip readers to understand what he says in those six technical chapters! This requires 15 Chapters of maths and ten Chapters of physics/cosmology...

Looked at this way, the book begins to reflect the genius and madness of the author: Many of the explanations in earlier stages of the book left me thinking, why do it that way? That's not the easiest way to understand this if you've never come across it before! He also goes straight to very general mathematical principles, missing out intermediate levels of abstraction that might make what comes later easier. He chooses to emphasise the geometrical/topological view of everything, which, it might surprise one to know, is not always the easiest way to understand things. Many of the choices of what to emphasise and what to ignore seem odd...that is until one gets to the late stages of the book.

Upon arrival at Chapters 27 - 33 (i.e. what I think Penrose really wants to talk about) one can see that everything that has gone before has been put together in order to provide the most efficient route to understanding - hardly a page has been wasted. All those strange choices of what to emphasise, all the peculiar, non-standard explanations when easier explanations exist, all the leaping to the most general mathematical ideas, all the things missed out, all these things are done so that the points he wants to discuss can be followed without wasting time or space in what is a 1000+p book as it stands. The necessary skill, thought and effort required to do this impress me enormously.

Inevitably this means that most of what is covered in the book of "standard" physics has been explained better (by which I mean more readily comprehensibly), even at a mathematical level, elsewhere - but not in one volume! The consequence of this is that Penrose's widest possible audience may not be all that wide: although he suggests one could read the book and ignore every equation in it, (something I often do when reading technical literature!) I suspect one would rapidly become bored and disenchanted. The unavoidable fact is that the greater your mathematical capabilities, the more you will get from this book and additionally, the more maths and physics you know before starting, the more you will gain from this book.

Further, the more you are willing to study the book the more you will gain. Manny approached it by reading 3 hours per night until done. I would suggest that the nearer to that approach you can get the better off you will be, even though I failed miserably to do so. There are numerous excercises scattered through-out, which I did not attempt, but I would suggest that if you are determined to attempt them, you should read the remainder of each chapter as soon as you hit a hard problem, then go back and look at the problems again. (And note the solutions web address given in the preface!)

So what did I gain from the unavoidable slog of this book?

The general philosophising of Chapters 1 and 34 struck me as a waste of time; I either thought what was being espoused was obviously clap-trap or obviously true - and for me the questions he raises mostly aren't interesting to me anymore. (They were back when I hadn't reached my own conclusions yet.) Others, may feel very differently, however - and many would not agree about which parts are claptrap! The remainder offered me quite a bit, however.

For instance, a frankly embarressing mis-understanding of the EPR paradox I was labouring under was corrected! (Something of a body-blow to me as it is undergrad physics!) On the other hand, Penrose makes an astounding mistake at one point, where he gets himself horribly messed up with basic (high school) probability theory and time-reversal. (Pretty good combination to the head from me!) This is a good reminder that there is no argument from authority in science: just 'cos Penrose says it, doesn't make it right! This wrong argument is then used to go on to explain a completely freaky (and I suspect wrong) prediction about basic quantum theory. I am not clear that the example, which is definitely wrong, invalidates his whole line of reasoning, though; it may be that other examples show the general argument to be correct.

Then Penrose delivers the knock-out punch: Conservation of energy/momentum/angular momentum in General Relativity is non-local! Not only that but it has only been proved to be true at all in a subset of cases! Seriously, how could I have never known this before?! (Non-physicists may well have no clue why I am so thunder-struck by this revelation, but it is not far short of learning that there's a whole continent you'd never heard of before.) It's completely gob-smacking. And I can't see how I didn't get told as an undergrad.

Further, Penrose's main purpose was achieved; I have a much better understanding of the main approaches to tackling the outstanding problems in cosmology/fundamental physics than I did before and along the way I gained some insights I previously lacked. Two examples are the Higgs boson explanation of the origin of mass and spinors. The Higgs boson theory is barely touched upon and is one of the rare examples of something being included that is not strictly necessary later. I wish there had been more about it, whilst recognising precisely why there is not. What material there is made the theory seem much less arbitrary than it had previously.

Spinors are a mathematical concept that feature heavily in the book, mainly because they feature extremely strongly in Penrose's Twistor Theory of quantum gravitation. Penrose gives an assessment of his own theory that I respect enormously and cannot praise highly enough; he expresses clearly what it it can acheive and equally clearly and forthrightly what it cannot. Every weakness and limitation is mentioned and explained. The only time I have previously come across a scientist giving such an honest and complete assessment of the weaknesses of his own theories in a popular account is when I read Charles Darwin's Origin of Species. I cannot express how much respect Penrose earns from me by doing this. Suffice to say that most popular science books will make out that the author's ideas are obviously and unassailably correct. Further, many technical papers fail to match this level of dispassionate critical assessment.

But back to the spinors; they feature in the now well established Dirac Equation for a relativistic electron but the (non-standard) way Penrose shows this and explains their connection with the left-handedness of the Weak nuclear force and how they link to the Higgs boson ideas are fascinating. However I am not clear about them in one (crucial) regard: are they real? Penrose says they are. I am not sure (because my understanding is still muddy) and I find (somewhat to my horror!) that even though I've read all 1050p of the main text, all of Chapters 2-17 twice and many individual sections several more times, I am still not done with this book! I have to go back and see if I can make sense out of these spinors. Also, I owe Manny a discussion of Inflationary Cosmology: I'm going to have to read the relevant chapter again in order to provide it.

Wish me luck as I delve back into the very deep waters of this book!

Cosmology, Early Universe Symmetry Breaking and Inflation
So, Manny requested my views on the above topics: blame him!
I must say at the outset that I am no expert in this field and in fact Manny has read much more about modern cosmology than I have, so I’m not sure how much value should be placed on the following; it’s a pretty naïve collection of speculations and intuitions.

Early Universe Electro-weak Symmetry Breaking (EUSB)
The current theory of the weak nuclear force and the electromagnetic force relies on a “broken” symmetry. That is to say all the relevant particles and their interactions were more symmetrical when their temperature was higher; so high that one has to look back to shortly after the Big Bang to find anything with a high enough temperature. Penrose gives the (standard) analogy of a lump or iron cooling down; at some critical temperature, the atoms go through a phase change and instead of having randomly aligned magnetic fields, these fields all line up in one direction. This creates the macroscopic magnetic field but in the process reduces the symmetry of the iron. It used to look the same in every direction but now it has an obviously different look, depending on the direction the magnetic field is pointing. So the idea is something similar happened when the universe cooled down below a critical value and the weak nuclear force and the electromagnetic force now look different because of the reduction in symmetry. But the lump of iron in fact won’t spontaneously have all its atoms line up in exactly the same direction unless it is cooled very slowly. Instead, “domains” develop. Inside a domain all the atoms are lined up the same direction but each domain has its own direction, which is why any old lump of iron is in fact not a macroscopic magnet. All the fields from the microscopic domains, pointing in different directions tend to cancel each other out. Which leads to weirdness when talking about the particles and forces of the universe doing the same thing: there are equivalents of the directions of the magnetic domains that the cooling particles could drop into that are different from what we observe. So the fundamental interactions would look different in a different domain. And the universe did not cool slowly, so it is much more likely than not that such “domains” did form if the theory is true. Now the boundaries of these domains would look and behave very strangely. In fact one type of boundary predicted by the theory, Cosmic Strings, can lead to something really bizarre: time travel! That is, technically, space-like movement into the past.

Well, I just don’t believe time travel into the past is actually possible, which means I don’t believe cosmic strings exist which means I think there’s something wrong with current electro-weak theory. However all the “low” energy density tests done show electro-weak theory to do very well indeed, thanks! So I’m in a bind; the high-energy prediction of a low energy theory that works really well predicts something I don’t believe. What to do? Well, it is often possible to write a theory in more than one way, mathematically, so I would search for a different mathematical description of the low energy theory that did not rely on the EUSB idea, thus getting rid of the unwanted cosmic strings and parts of the universe that are radically different from ours altogether. I don’t know if this is possible. There has been one claim that a cosmic string has been observed but I don’t know if any corroboration of the claim exists. If they do, then electro-weak theory as it stands gets an extra-ordinary boost.

Inflation
Penrose states that the initial motivation for the idea of cosmological inflation was to “explain” why magnetic monopoles are not observed but exist anyway. Magnetic monopoles are neat as they would explain why electrical charge comes only as integer multiples of a fundamental value (though not what the value is). The trouble is, nobody has ever seen one and if they were formed at all it would have been with such concentration that they would have been easily spotted by now. Unless inflation had reduced their concentration radically by expanding the universe at a ridiculous rate…
Well, this seems to me to be only half the problem; the other half is demonstrating that monopoles must have formed and that they did so prior to inflation and not afterward.
Later, people suggested that inflation could explain homogeneity and flatness. Homogeneity is the fact that wherever we look the in the universe the matter seems to be distributed in a similar way (i.e. stars, galaxies, clusters, super-clusters…). Flatness is the idea that the universe is, over-all, expanding at a rate just high enough to prevent it collapsing again because of the gravity of all the stuff in it.

Penrose presents cogent arguments as to why inflation actually cannot explain homogeneity. They seem indisputable to me. That the universe started off in an extremely low entropy state seems an unavoidable fact. Why was it like that? I don’t know. It’s a big mystery.
Inflation cannot solve the problem of EUSB either, but hey! it’s a hypothesis and it can make predictions, so astronomers should try to see if they can prove it wrong or not. As far as I have gathered, the observational status of inflation is ambiguous. My feeling is that the whole idea is very arbitrary: a field occupying the whole universe must exist in order to provide the opportunity for inflation to occur. No hint of a carrier-boson for said field has ever been found. I don’t know if theory can predict anything at all about such a particle, apart from it must have integer spin. There is talk of a second inflation occurring because it seems that some astronomical objects are not only moving away from us but accelerating away from us. This suggests to me that whilst there may in fact have been inflation in the early universe, current theory is nowhere near adequate to explain it: why then? Why again now? What causes the transition? It could be a purely random event where the field transitions from one state to another quantum mechanically but to do so the new state must allow an immediate drop of energy in the field. How can one decide what the possible energy levels of the field are? Or if we are in the ground state now or not? The theory also has implications for the nature of the vacuum i.e. it is different before and after an inflation period starts. The vacuum in the quantum mechanical sense seems to me not to be understood at all well. I think that if inflation has ever happened it indicates that a deep theory explaining the nature of the vacuum is required.

The Anthropic Principle
This gets dragged up a lot in discussions of cosmology. It’s annoyingly persistent. It comes in a weak and a strong form. Starting with the weak form (WAP): Many adherents claim that the WAP has predictive power and is therefore in some sense “correct” i.e. some deep law of the universe. What it states is that there is sentient life therefore…X. X is a prediction of some phenomenon. The most famous example is a certain energy level of the Carbon nucleus. The argument went, we have life because we have heavy atoms so there must be a way of fusing lighter atoms to make heavier ones but to make any atom heavier than carbon, carbon itself must have this specific energy level…and it does! Triumph for the WAP! Except that is rubbish, because the argument doesn’t rely on the existence of life at all; it is easy to imaging a lot of heavy atoms floating about but no life. The real argument, stripped of inessential guff, is simply: there are atoms heavier than carbon – there must be this energy level of the carbon nucleus for that to happen. Life, let alone human sentience doesn’t feature at all. All WAP arguments fail in this manner: it turns out that life is inessential to the argument.
Then there is the strong anthropic principle (SAP): This states either the universe was fine-tuned for the existence of human sentience to be possible or there are in fact a huge number of universes that are somehow different from each other so ours is just a statistical freak. This assumes that the only possible way that sentience could occur is if physics follows the laws we see. I don’t buy this at all: are people seriously claiming that they know what all the possible emergent ramifications of some grand set of all possible sets of physical laws are and only a tiny fraction of them could sustain sentience? It seems to me we don’t even know all the ramifications of the laws of the observed universe yet, let alone any other one. But there is some sort of prediction here; there might be other universes. Maybe that is a testable proposition. I don’t know.

Spinors and Spin
I re-read the material on the Dirac Equation and spinors a while ago but I've only just got round to discussing it. The discussion is moved to the comments due to the character limit on reviews!