Our usual host, plf515, was last seen intrepidly staggering into the vortex of a wormhole, but I can confirm for you that he did survive the journey since he just sent me back this message:
Obama's acceptance speech in Denver was awesome, but take up a collection for bail... The Kossacks there were one wild group! Next week's Math Mania was our best ever, too!
Fortunately for all of us, it appears that plf515 will be back next week, but his message got me thinking about two things: Cosmology and Groups.
UNfortunately for all of us, I'm not an expert on either topic, but I do love learning about both, so I figured that while plf515 prepares for us to catch up to him on the other side of the wormhole, I'd take over today's Math Mania and try to explain in a very basic way how the physics of the universe, called Cosmology, is connected to one of the most beautiful topics in modern mathematics, Group Theory.
Of course, the ususal Math Mania disclaimer applies:
This series is for anyone. There will be no advanced math used. Nothing beyond high school, usually not beyond grade school. But it'll go places you didn't go in elementary school or high school.
If you "hate math" please read on.
If you love math, please read on.
I welcome thoughts, ideas, or what-have-you. If anyone would like to write a diary in this series, that's cool too. Just ask me. Or if you want to co-write with me, that's fine.
The rules: Any math that is required beyond arithmetic and very elementary algebra will be explained. Anything much beyond that will be VERY CAREFULLY EXPLAINED.
Anyone can feel free to help me explain, but NO TALKING DOWN TO PEOPLE. I'll hide rate anything insulting, but I promise to be generous with the mojo otherwise.
First, a pretty video that unites Cosmology and Group Theory:
Well... MAYBE it unites Cosmology and Group Theory. There's still a raging controversy about that among scientists who have taken to exasperated blogging and childish name-calling which is a sure sign that you're on the cutting edge of serious theoretical physics! So let's take them both apart and see how they are built.
If you're reading this diary, you probably already know that there are two major branches of Cosmology: General Relativity Theory and The Standard Model of Quantum Mechanics. Relativity works on the scale of stars and galaxies, and The Standard Model works on the scale of quarks and electrons.
You also probably already know that each theory is brilliantly accurate at making predictions on its own scale, but they are completely incompatible with each other at the other scale.
But since both are theories of the same universe, the holy grail of cosmology would be a single scientific theory that combines the two in a way that has the accuracy of each theory but without the compatability problem. Cosmologists call it a Grand Unified Theory (GUT) or Theory of Everything (ToE).
Many theoretical physicists have tried and failed to find this ToE (including Einstein himself!), but one physicist/surfer dude by the name of Garrett Lisi may, repeat MAY, have done it.
Lisi's proposed Theory of Everything (and DO look at that link before going on!) relies one of the most beautifully symmetric structures in all of Mathematics: E8
If you took my advice and followed that link, you probably feel like an idiot right about now. Don't worry about that. When I look at that link, I feel like an idiot, too, and I'm not even going to TRY to explain it all in detail.
But I do want to help you understand what this diagram is at least talking about, and in order to do that, I need to explain what a Mathematical Group is.
So, what is a group?
The idea is simple, really, and you already know a few groups:
For example, when you use addition on the set of Integers {...,-4,-3,-2,-1, 0, 1, 2, 3, 4,...}, you are working with an infinite cyclic group.
And when you add 4 hours to 10 o'clock and get 2 o'clock (Does 10 plus 4 really equal 2?), you are working with a finite cyclic group.
And when you try(!) to solve a Rubik's cube, you are working with a permutation group.
The formal definition of a group has several key points to it:
First, you need a well-defined set of elements:
The integers, the hours on a clock, or the positions of a Rubik's cube
Second, you need a well-defined operation:
Addition, modular (clock) addition, or a rotation of the cube's face
Finally, you need the set and the operation to obey four rules:
- Closure, 2. Associativity, 3. Identity, 4. Inverses
Closure: If you operate on two elements, then the result itself is an element in the set.
For example, 2+(-5) is itself an integer. [-3]
Associativity: If you perform two operations in succession, the result is the same if you perform the two operation in the other order.
For example, (2 hours + 5 hours) + 6 hours is the same as 2 hours +(5 hours + 6 hours). [1 hours]
Identity: There is an element in the set that leaves all other elements alone when you operate with it.
For example, 0 is the identity in integer addition, and adding 12 hours is the identity in modular (clock) addition, and a 360-degree rotation of any Rubik's cube face is the identity permutation
Inverses: Every element in the set is related to some other element in the set that turns it into the set's identity.
For example, the additive inverse of 3 is (-3) because 3 plus (-3) results in 0, and 3 hours plus 9 hours results in 12 hours, and 90-degree rotation followed by 270-degree rotation results in 360-degree rotation.
What is a subgroup?
Once you know you have a group, it's possible sometimes to find another smaller group embedded within it.
For example, we know from above that the integers with addition form a group. But the set of EVEN integers with addition also form a group, and since the even integers are contained within the (regular) integers, we say that the Evens are a subgroup of the Integers. (You should explore the Evens for Closure, Associativity, Identity, and Inverses to be sure. Are the Odds a subgroup of the Integers?)
Well, it turns out that Group Theorists working in the mid-1900's were able to classify and identify ALL possible finite groups at a fundamental level. This is a stunning achievement that makes working with group theory as a tool for cosmology much, much easier. It is akin to identifying all possible chess moves and thus classifying all possible chess games, only it's far more complicated than that.
One family of groups are called Exceptional Groups which are nothing exceptional except that they don't fall into an otherwise nice clean pattern of groups classifications. E8 is one of those Exceptional Groups.
The interactions of the elements of E8 is what the video above is all about. The subgroups of E8 form the backbone of the motion.
What's the connection between Groups and Cosmology?
Physicists working on The Standard Model long ago found that particles and forces can be organized completely by seeing them as mathematical groups, and the incredible success of The Standard Model as a predictive tool serves as a proud testament to the correctness of this approach.
Any Theory of Everything MUST incorporate this group theoretic approach on the microscopic scale of the universe or it will be instantly dismissed, but so far, Relativity has defied a similar description.
Lisi's proposal is that all the particles and forces of The Standard Model, as well as all the particles and force involved in General Relativity, can be merged into the single theory that their related groups are all subgroups of E8.
In other words, if you look at the top of Lisi's diagram, the shape of the Universe is E8, and it's subgroups below represent all the forces and particles of... well... everything.
Is Lisi correct?
Maybe, maybe not. As for me, I'm paying attention to some of the predictions made by the theory, and I am excited that the Large Hadron Collider which is about to begin its work may be able to prove it (in Lisi's words) "spectacularly right or spectacularly wrong".
In the meantime, I hope you enjoy this Mathematical/Musical Interlude: