### by Shlomo Sternberg

In this book Professor Sternberg of Harvard University takes a fairly detailed look at the deep links between mathematical group theory and the underpinnings of much of modern physics. It’s heavy going in many parts, requiring at least an undergraduate degree in mathematics (one containing lots of group theory) to even vaguely follow it. If you can at least follow along (and I was very much reduced to skimming the deeper technical material), then you’ll find beauty, and deep truth. Eugene Wigner talked about the “unreasonable effectiveness” of mathematics in explaining the physical world, and that’s what you will find herein.

In Chapter 1, Prof. Sternberg introduces key ideas like the action of a group on a set, and uses them to classify the finite subgroups of O(3), the orthogonal group of order 3 (i.e. the group of ways in which you can transform things in three dimensions such that distances between things are preserved). There is an immediate physical application in crystallography, and Prof. Sternberg takes us through this, introducing me at least to some marvelously-named crystals and compounds (iodosuccinimide, diaboleite, or wulfenite, anyone?)

Chapter 2 introduces representation theory for finite groups, which is a way of representing abstract objects as matrices, with the operations on those objects as equivalent to things like matrix multiplication. Using these techniques, you can produce character tables, which are a simple two-dimensional table that can capture much of the deep structure of a particular group. This material is set up for use in the following chapters.

Chapter 3 starts to build on this, first with a simple physical model of molecules as point masses connected by springs, and then adding in some quantum mechanics. This allows us to look at the ways in which such model molecules would “resonate” in vibration, which in turn leads to direct physical applications in spectrography. (Appendix F contains an interesting, detailed, and entirely non-technical review of how spectography evolved through the 19th century – one of the highlights of the book, for me.) This leads to a discussion of the symmetries of space-time, and out of this emerges a very natural representation of the fundamental properties of subatomic particles. Finally we come to Wigner’s truly remarkable 1951 paper in which he identifies these subatomic particles “as” the irreducible representations of the group. By doing this, Wigner was able to lay out a model that has directed particle physics for at least the last 60 years – he earned the Nobel Prize for Physics for this in 1963, and subsequent modifications of his work by Gell-Mann and others have lead to further Nobel Prize level discoveries.

It’s worth pausing on this point: here we have a use of the most abstract parts of mathematics, out of which directly flows deep and powerful results about the universe in which we live. It’s not that we can (say) simply model a molecule as a bunch of weights joined by springs: the types and properties of the most fundamental things we know about in the world spring directly from this abstract mathematics. It’s truly a remarkable, and very beautiful, thing.

Chapter 4 continues the journey, moving on to compact groups. From this we get models which apply directly to the hydrogen spectrum, models of atomic nuclei, and the group underlying the periodic table. Again, we see the powerful linkages back to Wigner’s work on the Poincare group.

Chapter 5 moves on to quarks, and the predictions of and subsequent discoveries of the elements of quantum chromodynamics. Because this book was written in the early 1990s, it doesn’t include any discussion of grand unified models or sypersymmetric models like string theory or quantum gravity, which is a shame, as once again we see the mathematical and group theoretical underpinnings.

There is much to like about this book, and I enjoyed the journey. However, I spotted a number of annoying typographical errors, and I very much fear that if I’d been able to follow some of the denser passages, my journey would have been made harder by further typos. The index is also weak, and I was often reduced to paging back and fro to find what a particular notation or definition was. I also felt that since this book set out to link group theory and physics, I was often not at all certain what the point of a particular mathematical section was – a brief up-front discussion of the motivation would have been very helpful to me. The fact that it ends in the mid-1990s also means it doesn’t cover a lot of recent, equally interesting, work in theoretical physics.

Finally, to be honest, a lot of the book was somewhat beyond me: the author says “a course in multivariate calculus and linear algebra, together with an elementary physics course, should suffice”. It’s been a few years since I took my courses in these subjects, but I have very good retention of them, and they in no way adequately prepared me for some of the long technical passages!

Three stars for me – quite possibly more for you, if you’re further along the maths curve than I started.