Good morning RTU followers. Today I want give you a very brief introduction to an integral part of Theoretical Physics – Group Theory. When studying for my MSc I knew I had to get my head around this subject which to me initially seemed very dry and written in very purely mathematical language (i.e. not easy to read!) However once I’d dedicated some time the beauty of group theory began to fall into place – it’s a slow burner. The purpose of group theory is to classify and understand symmetries in the natural world. So without further ado let me lay down the principles of group theory as clearly as I can.
Firstly a group, denoted G, is a collection of elements call them g(1), g(2), g(3)…. and there exists a group operation, denote it *, which determines how the elements act on each other. Now the elements of the group must obey the 4 axioms of group theory. I’ll lay them out first and everything may seem rather abstract to begin with – but bear with me, all will become clear after an example.
Axioms of Group Theory
In each group their must exist an element called the identity, it is denoted e. When the identity element acts on any of the other group elements it essentially does nothing, the element remains the same. In group theory language this is written:
e * g = g or g * e = g
The element is unchanged when acted on by the identity.
This principle states that the product of any two group elements will produce an element that is also part of the same group.
For example if g(1) and g(2) belong to G, then g(1)*g(2) must also belong to G.
This principle states that the order of operation between elements can be fluid. If g(1) acts on the product of g(2) and g(3), this is the same as the product of g(1) and g(2) acting on g(3). In group theory language this is written:
g(1) * (g(2)*g(3)) = (g(1)*g(2)) * g(3)
Finally there must exist an element which is the inverse for each pre-existing element. The inverse is denoted with a superscript -1 after the element but to save me from introducing math-type I will denote it with a strikethrough.
So the inverse of element g(1) is
g(1) When each element is acted on by its inverse it gives… the identity!
In group theory language this is written:
g(1) = e or more generally g * g = e
Ok this must be seeming extremely abstract without an example so let’s introduce the square – one of the most simple examples we can work with. Group theory is all about respecting and classifying symmetries in nature so the question we want to ask is what transformations exist that preserve the symmetry of the square?
A square has four sides, forming four right angles. What action can we perform on the square that will preserve its shape/symmetry? If we rotate the square by 90 degrees, we will take point a to point b, point b to point c, point c to point d and point d to point a – but the square will still look exactly the same. In fact if we rotate the square by 180 degrees we’ll still get a square as well except point a will go to c, point b will go to d, point c will go to a and point d will go to b! Ok very nice. I think we now see if we rotate by 270 degrees a will go to d, b will go to c, c will go to b and d will go to a. And finally if we rotate by 360 everything goes back to its original place and nothing changes!
These transformations/rotations form the elements of the group G – in this case the cyclic group of a square.
Let g(1) = clockwise rotation by 90 degrees
g(2) = clockwise rotation by 180 degrees
g(3) = clockwise rotation by 270 degrees
and what’s rotation by 360 degrees? Of course! It’s e – the identity element.
Let’s now test the axioms to make sure these elements fit our definition for a group.
1. Identity – Already checked, we can see that a rotation by 360 degrees leaves all sides as they were to begin with. The e element exists.
2. Closure – Is for example g(2)*g(3) a member of the original group? This would be a rotation of 180 degrees followed by 270 degrees, so in total 450 degrees. Perform this on the square, it’s rotating through by 360 then adding an extra 90. So yes it gives back the same operation as you would if you just performed g(1).
g(2)*g(3) = g(1) – Check
You can try this with any combination of elements and check it works!
3. Associativity – Does g(1) * (g(2)*g(3)) = (g(1)*g(2)) * g(3) ?
Let’s try it: The left hand side is just 90 degrees then 450 degrees (total 540 degrees). The right hand side is just 270 degrees then 270 degrees (total 540 degrees). All rotations are taken to be clockwise in this case and so it does not matter in which order you perform them, they will produce the same outcome.
4. Inverse – This one is very straight forward. What are the inverse of clockwise rotations? Anti-clockwise rotations.
So for g(1),
g(1) is an anticlockwise rotation by 90 degrees. If we perform g(1) then g(1) we undo our first rotation and get back what we started with a.k.a e
g(1) = e g(2), g(3) are anticlockwise rotations of 180 and 270 degrees respectively.
So there we have it the lay-out of the group G – in this case the cyclic group of a square.
Many other, far more beautiful, groups exist in nature and this was by far the simplest explanation I could give whilst still having enough elements to explain the axioms clearly. At some point in the future I’ll write again on more complex groups and may even dare to venture into the famous Lie Group. If you’re interested in reading more now wikipedia does a decent job or I would recommend ‘Physics from Symmetry’ for an extremely clear approach, which I must say i’ve found hard to come by in this subject. For now, I hope you have found some closure on the subject.