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“What is especially striking and remarkable is that in fundamental physics a beautiful or elegant theory is more likely to be right than a theory that is inelegant. A theory appears beautiful or elegant (or simple, if you prefer) when it can be expressed concisely in terms of mathematics we already have.”
– Murray GellMann 
The Symmetry of Reality
Any reader of popular science books in the last few years will have come across a reference to Special Unitary Groups ^{[1]}, their more quotidian Unitary Group cousins and the central role that both play in the Standard Model of Particle Physics. The Standard Model is a way of cataloguing the menagerie of subatomic particles that we have discovered over the last few decades and explaining how they respond to / mediate the three nongravitational fundamental forces of Nature:
 the electromagnetic force – the one we experience most daytoday and which is responsible for a range of phenomena from you not falling through the seat you are sitting on, to most of chemistry, to the power running the device you are using to read this article (and providing its memory), to light in all its forms
 the strong nuclear force – which ensures that groups of positively charged protons [generally] cohabit the nucleus of atoms contrary to what the electromagnetic force would typically dictate and also binds quarks into hadrons (such as protons, neutrons and their more esoteric relatives)
 the weak nuclear force – which governs things like radioactive decay
While the Standard Model neither encompasses gravity, nor explains why a range of physical constants (e.g. the strength of the fundamental forces, the mass of certain particles and so on) hold the values that we observe ^{[2]}, it is rightly viewed as the most wideranging, accurate and successful scientific theory we have yet developed. This theory helps us understand many, many aspects of the natural world in exquisite detail and with incredible precision. Perhaps of least import, it has also powered our current information age where our mastery of physics has enabled creativity in engineering and technology that our forefathers and mothers would never have dreamt of.
So the Standard Model is officially a PBD (Pretty Big Deal). Given its importance, the fact that the Standard Model is written in the language of Group Theory and dependent on Unitary Groups in general, suggests that these mathematical constructs are of great importance as well ^{[3]}.
Few popular science books attempt to delve too far into the intricacies of Group Theory. Marcus Du Sautoy is massively more qualified to do this than I am, but even he abjures this approach in his latest volume (though given the extremely broad scope of the book perhaps this is more than understandable). Some authors may go as far as quoting the seminal formulation which underpins this model.
SU(3) × SU(2) × U(1)
However I have not come across anyone who then goes on to explain this in much detail. It is this task to which I will now address myself.
To be clear, I’m not going to try to explain the Physics, I don’t have the background to do that. Instead I’m going to look to give at least some flavour of the Mathematics, a task I am hopefully better qualified to attempt. In order to do this I am inevitably going to have to introduce some Mathematical concepts which may not be familiar to a general audience. My goal is to convey these while losing the minimum amount of readers on the journey. I am well aware that this is a fairly Herculean labour to undertake, but I’ll give it my best shot.
Let’s start with SU(3) × SU(2) × U(1). This snippet of letters and numbers is couched in the notation of Mathematics and may be read as
“the direct product of the Special Unitary Group of degree 3 with the Special Unitary Group of degree 2 and the [not so special] Unitary Group of degree 1”
I’m sure every element of the text in quotation marks is crystal clear to all readers, but on the infinitesimal chance that some aspects are a little more recherché, I’m going to try to explain at least the essence of what these concepts mean.
On the Physics side, popular science books may go so far as to suggest that Groups are constructs relating to symmetry and that (as referenced in the Foreword), in the expression SU(3) × SU(2) × U(1), SU(3) governs symmetries to do with Quantum Chromodynamics (QCD) and SU(2) × U(1) plays the same role with respect to the Electroweak force ^{[4]}.
As previously mentioned, I’m going to focus on the Mathematics, not its physical interpretation. As I reference above, in order to do this, I’m going to have to do what most popular science books aver, cover a number of aspects of Mathematical theory. If at this point you feel yourself recoiling in horror, then I understand. Mathematics tends to be one of those things that elicits visceral reactions in people, frequently on the negative side. As someone with a background in the area, my reactions are of course somewhat different. However, the process of reacquainting myself with a once familiar subject in order to write these articles left me confused and disorientated at points, so I have a degree of empathy with those who view Mathematics as a closed book.
What I will try to do is to introduce new Mathematics as and when it is needed in telling the story I want to tell. This may lead to what might at first sight appear to be some meandering and also some doubling back to finish a thought which was introduced earlier. While this could end up being confusing, I hope that this is a price worth paying in order to ensure what I am covering is relevant rather than straying into a general primer on Group Theory. My aim will always be to provide a rudimentary understanding of Unitary Groups, Special or otherwise and I’ll try not to wander too far from the path to this destination.
With my objectives and intended approach laid out, let’s jump in to the deep end and explore what actually constitutes this Group thingamabob which I keep talking about. This is the subject of Chapter 2 and Chapter 3.


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Introduction – Notes
^{ [1]}  Where I have provided links to subjects in previous chapters, I won’t repeat these later in the text; so here you can refer back to the links in the Foreword. 
^{ [2]}  At the time of writing, just to cover myself! 
^{ [3]}  Of course, as a Pure Mathematician, I’d argue that any group is also of intrinsic interest, but even I would agree that this deep connection to the fundamentals of how our world works adds a further dimension. 
^{ [4]}  The Electroweak force is an amalgam of the Electromagnetic and Weak Nuclear Forces developed by Sheldon Glashow, Abdus Salam and Steven Weinberg. The search for such unifications has been a driving force for new developments in Physics in recent decades. 
Text: © Peter James Thomas 201617. 