Trends in theoretical physics Book

I am finished, finally. All the 1050+ pages of this ambitious behemoth - including many exercises. What a ride!...
Finished? Well you are never finished with such a book, titled "The road to reality" but actually providing more than that: providing nothing less than a "road-map" to reality, and opening to the reader new beautiful vistas in modern mathematics and physics. I am sure that I will come back to this book in the future, as a source of inspiration and for future reference.

Before I start, I must confess that it feels slightly ridiculous for me to critique such a monumental work by a crazy genius like Penrose, who clearly lives in a higher realm of consciousness than the large majority of us common mortals. Never mind - the best I can do is to describe my personal, deeply humbling experience in reading this amazing piece of intellectual prowess. Penrose is clearly the Mozart of mathematical physics: genius, craziness, wild imagination and pure technical virtuosity all combined into a unique intellect.

Let me start by highlighting some of the peculiarities of this book (due, in my personal opinion, much to editorial/commercial choices/pressures, and only partially to personal choices by the author himself). This is also important to the proper positioning of this book within the overcrowded (and of hugely varying quality) world of popular science books:

A) this book is sold as a science "popularization" (probably with the objective to attract as wide an audience as possible): in reality this book requires a significant amount of background knowledge in mathematical physics; in fact there are some sections where the equivalent of a full maths/physics degree would be required to fully appreciate all the subtleties of some of the subjects being treated. I would personally recommend, as a prerequisite, the following minimum:
- calculus (including multivariate)
- ordinary and partial differential equations (at least at rudimentary level)
- complex analysis
- linear algebra
- basics of topology
- ideally, differential geometry at basic level
- in the physics area, some prior exposure to Maxwell's equations, the Lagrangian and Hamiltonian formalisms, the basics of quantum mechanics and relativity (at least special) would also be a great help.
If a reader tried this book with no prior exposure to any such topics, I am afraid that it all might prove a very steep learning curve indeed. I had all the above pre-reqs, and still it was, for me, an occasionally demanding if not challenging book in terms of required focus, time and intellectual stamina. But this is also a strength of this book - one of its greatly rewarding aspects is that the author does not refrain from exposing the reader to real maths (some of the exercises can be pretty hardcore), and to get into the heart of modern physical theories such as QFT and GR, if you want the real thing, you simply can't avoid getting into some real maths.

B) this book is not really a textbook, or at least not an ordinary textbook: it lacks the necessary structure and flow, and (too) many important derivations are left as exercises. The depth of mathematical analysis and rigour are uneven, and they seems also highly dependent on the occasionally whimsical personal level of the author's interest in the subject. Moreover, the formal treatment of many subjects is highly original and something that, unless you have had some previous exposure to them, might be confusing and certainly not easily reconcilable with what standard textbooks present. But this is part of the beauty of this masterpiece.

C) this book, contrarily to what stated in the title, is not a "complete" guide to the laws of the Universe. While managing to concentrate so many sophisticated and fascinating subjects into a single book, and at a serious level of detail (which is no mean feat, and something quite unique), I think that Penrose should have written two books (one on the mathematical underpinnings, and a separate one on the physical aspects) rather than trying to concentrate such a massive amount of information in one single big book.
This approach (admittedly less palatable from a commercial standpoint) would have allowed him to expand some important areas that he has unfortunately neglected in his otherwise magnificent book (such as finiteness and re-normalization issues in QFT) and which would have also allowed him to include other subject matters too - for example, the coverage of thermodynamics focuses mainly on the statistical view of the Second Law, neglecting all the other elements.
But make no mistake - this book feels like a "War and Peace" of mathematical physics - a colossal enterprise that just keeps giving and giving, a treasure trove of original insights, beautiful hidden connections, amazing stories and revelations. A fantastic and detailed exploration of quantum physics, relativity, of the current trends in the attempts to unify the two, and of cosmology. An exhilarating intellectual adventure in the company of a crazy genius.


Let me now make a few comments about the author's peculiar personal style and approach:
- Penrose's prose is beautiful, even if considered from a purely literary perspective; often very clear even if sometimes a bit too concise; it usually flows very smoothly, and it occasionally even acquires some poetical undertones in sections where aspects of philosophy of science and philosophy of mathematics are treated with insightful passion.
- Penrose relies heavily on a diagrammatical/pictorial/visual/topological representation of the concepts being treated. In doing so, many beautiful, eye-catching and highly informative hand-drawn diagrams/pictures are presented to the user. This approach generally works very well, but there are instances where (in my personal opinion) the most intuitive and simplest way to learn/teach a new concept is to present it analytically (in mathematical format), and where the purely visual approach might be quite limiting if not outright confusing. For example, the visual explanation of the concept of "covariant derivative": while important and helpful, is in my opinion by itself not sufficient to get a real appreciation of the mathematical features of this entity: its analytical derivation, and its expression using the Christoffel symbols, would have actually better clarified its tensorial nature.
- In order to condense as many subjects as possible within the constraints of the space allowed by one single book (even one of 1050+ pages) Penrose has left the derivation of many important results as "exercises". The exercises are therefore very important, they are a must in order to get a full appreciation of the underlying theory, and some of them are beautiful and rewarding (like exercise 22.32, which entails a beautifully elegant derivation of the Laplacian in spherical coordinates, using the curved metric and covariant derivatives). But some others should not have been left as exercises, and they seem no more than an editorial cop-out: there is one exercise that is even asking the reader to complete a significant derivation step of the GR field equations! (thankfully there is a website where many exercises have been solved by other readers (the majority of them clearly with a professional background in mathematical physics) - see ).
- I love Penrose's great intellectual honesty in not just debunking much of the hype behind String Theory and Multiverse Hypotheses in general, and in destroying the so-called "Strong Anthropic Principle", but also in making very clear the speculative character of some of his own positions and theories. I also love his great originality and independence of mind, and his nuanced, multi-disciplinarian approach which includes aspects of philosophy of science and considerations of pure mathematical nature.
- Penrose has reached such a higher, rarefied level of proficiency in mathematical physics that he must have completely forgotten how common humans think in relation to mathematical formalism. Just as an example: after getting into pretty advanced stuff such as hyperfunctions, and treating it like if it is the simplest thing on Earth, the author then refuses to get into a detailed definition of differentiability of a many variable function (which is quite simple, really) because "it is too technical"!! So his ideas of what is complex can occasionally be significantly at variance with what the common mortal might think.


Finally, let me add here some miscellaneous notes about selected individual chapters of this majestic book (it is a very incomplete list, only a severely reduced sample, as there is simply too much stuff in this book for me to be able to analyse in just one review):
- In the first few chapters, after a fascinating introduction of overall philosophical character, Penrose briefly addresses some of the basic fundamentals of mathematics, including a short but intensely interesting discussion about the Axiom of Choice and its relationship to the Zermelo-Fraenkel set theory. A totally fascinating subject that unfortunately Penrose only touches, without developing into more detail.
There is also a fascinating treatment of hyperbolic geometry, and of the number system/s. However, when dealing with the relationship between real numbers and reality, the author does not make any reference to important aspects such as computability and irreducible complexity, as for example addressed by the excellent work done by G.Chaitin.
- The author then gets into one of the most beautiful and fascinating realms of mathematics - complex analysis. The treatment is concise but done well, and I completely share Penrose's enthusiasm and love for the aesthetically as well as functionally beautiful world of complex numbers, which he calls "magical" with good reason.
- Another subject of interest treated are the fascinating quaternions, which extend the complex numbers and provide the uncommon and interesting features of a non-commutative division algebra. There are in the book a couple of minor missing things in relation to the algebraic structure of quaternions: an algebra is a vector space that must also be equipped with a bilinear product, and a ring (being also an abelian group under addition) must be provided with an additive inverse.
Moreover, Penrose is dismissive of the utility of the quaternions in the development of physical theories - this is correct, but it must also be said that quaternions find important uses in Information technology applications, in particular for calculations involving three-dimensional rotations, computer graphics and computer vision.
- Penrose could not have forgotten the extremely important Clifford/Grassmann algebras, which are foundational to the entire architectural structure of mathematics, and he didn't; they are treated well, but a bit too succinctly in my opinion, considering their importance.
- Symmetry groups are treated really well, clearly and concisely. Very nice.
- Manifolds and calculus on manifolds: all is treated quite well, but I would have found quite helpful some more analytical detail rather than a focus almost exclusively on the visual/topological approach. I confess that here, in order to get a proper detailed handle of tensor calculus and exterior calculus, I had to consult other more traditional, "textbook-type" sources.
- Chapter 16 (Cantor's infinities, continuum hypothesis, Godel's incompleteness, Turing computability and similar) is OK and well written, and it would be utterly fascinating to a neophyte - but of course it could not have been detailed nor exhaustive, considering the complexity and width of such subjects.

- The "Physics" part proper starts with Chapter 17. Chapter 17 on spacetime, and chapter 18 on Minkowskian geometry (and special relativity of course) are succinct, but riveting and beautifully written. Just one small note: Penrose has a fetishism for non-standard, or uncommon, notational or representational choices, not always justified: for example, by using a non-standard metric tensor in page 434, Penrose gets himself into an error at the end of page 435 (c^4 should have been used rather than c^2) and I think that he would not have fallen into this typo, had he used the more standard and simple-to-use metric.
- Chapter 19 (Maxwell and Einstein) are beautiful; Maxwell equations and Einstein field equations in tensorial/differential form are mind-blowing for their mathematical conciseness and beauty, and this is where the reader can start to see the value of the mathematical apparatus described in the previous chapters: things such as manifolds, tensors, exterior derivatives and bundle connections. Einstein's fields equations of general relativity are beautiful. Very rewarding. It all seems so neat and perfect, but then, the bombshell (at least to me, who was never told of such a thing!): the energy/momentum stress tensor does not account for the energy density of the gravitational field itself, and it seems that conservation of energy/momentum is non-local !
- Chapter 20: Lagrangians and Hamiltonians: I must say that they are too hastily discussed - the derivation of the Euler-Lagrange equation (one of the most classic proofs in mathematics) is not treated, the Legrende transform to derive the Hamiltonian from the Lagrangian is not treated, the derivation of Hamilton's evolution equations is not treated either. Not very happy with this chapter.
- The subsequent chapters on quantum mechanics are beautiful, and the description of the EPR issue is really nice. The derivation of Dirac's equations is equally beautiful. I also love Penrose's discussion and perspectives about the measurement paradox. Beautiful stuff indeed. An exception to the overall masterful treatment of Quantum Physics is chapter 26 (Quantum Field Theory) which is is a bit too qualitative - more like a "popularization" rather than a treatment at the same good level as done in the other chapters of this book - and unfortunately the important subject of re-normalization is treated only briefly and qualitatively.
- The chapters on cosmology are very interesting and nicely written.
- Chapter 29 is essentially about the measurement paradox and the various interpretations of quantum mechanics - succinctly but beautifully written.
- The next chapters are essentially about the current unification attempts to reconcile QFT with GR. Penrose's FELIX experiment proposal (essentially testing the hypothesis of automatic quantum state-reduction as an objective gravitational effect) is utterly fascinating, but I feel that I should not bet any money on this daring hypothesis.
- I must say though that I was disappointed by the second part of chapter 33 (twistor theory): in section 33.8 Penrose uses several verbose sentences to express mathematical relationships, rather than writing down the actual underlying equations - result: something quite close to incomprehensible. I also fail to get the underlying physical intuition. Pity, as the other 33 chapters and a half are mostly beautifully written, and generally very rewarding.
- The last chapter (34) is beautifully written, and it deals primarily with aspects of philosophy of science and philosophy of mathematics.

To summarize: this is an immensely rewarding, even exhilarating book - a fantastic reading and learning experience. And it has opened my eyes on many aspects and connections that I was not aware of, and new beautiful vistas in modern mathematics (for example: tensor calculus, and exterior calculus in general) that I will now pursue in more depth.
After reading this great book, I can tell you that I now feel that the majority of popular science books I have read are dull and superficial by comparison.
It is not a perfect book by any means, but overall it is a great book, unique in its approach and contents. Not exhaustive, but with a huge and ambitious scope, virtually unrivaled in its category. It is much more than a "standard" popular science book. Very highly recommended, to be bought, studied, enjoyed and kept for future reference.
4.5 stars (rounded up to 5).

UPDATE: with link to a beautiful series of online lectures about Tensor Calculus and the Calculus of Moving Surfaces: (highly recommended to anybody interested in this subject).