I want to explore the idea of symmetry as it pertains to physics. I generally learn better when I try to explain things to people, particularly in writing.
We have a physics enthusiast group, but things have been rather dead these days on the group. So I decided to post here.
Unlike my computer architecture thread, where I knew everything I was presenting, and started at the very most basic level, and lost people's interest, I will be exploring a topic I am just learning, and will jump pretty quickly into the challenging material.
So symmetry is a fairly simple concept right? You kinda know it when you see it (or many people think so anyway). Mathematically speaking, one can have various symmetries, including rotational symmetry or reflective symmetry.
In physics, a crucial set of symmetries lead to the conservation laws with which we are familiar. This phenomenon is due to Noether's First Theorem.
The informal statement of this Theorem is :
The hefty mathematical formulations are there in the wikipedia article. One of my goals is to be able to understand them, as well as general Gauge Theory and Variational Principles.Originally Posted by wikipedia
The summary of important results, however, is:
- The fact that the laws of physics are the same with respect to a translation of space coordinates leads to the conservation of linear momentum.
- The fact that the laws of physics are the same with respect to a rotation of space coordinates leads to the conservation of angular momentum.
- The fact that the laws of physics are the same with respect to a translation of the time coordinate leads to the conservation of energy.
- The fact that the laws of physics are the same with respect to what is known as the Gauge Transformation leads to the conservation of electrical charge.
Groups and Representations
Beyond Noether's Theorem, one very important mathematical concept is foundational, the notion of a mathematical Group.
A group is composed of a set, S, and a binary operator, *, such that:
- The operation is closed (i.e. if a is in S, and b is in S, then a*b is in S).
- There exist an element, e, in S, known as the "identity" such that e*a=a*e=a for all elements, a, of S.
- For every element, a, of S, there exists an element a^-1 of a, such that a*a^-1=a^-1*a=e.
- In all cases where a,b, and c are elements of S, a*(b*c)=(a*b)*c. This property is called "associativity."
Of course, our use of groups is to represent symmetries, by letting each element of a group be an operation under which something is the same, and letting composition of operations be the group operator.
Such symmetries are:
- ...closed--because after each operation the desired object remains the same. The composition of two such operations(applying the operation on after another) will obviously keep the operation the same, and so the composition itself is part of the group.
- ...has an identity--namely the operation of doing nothing.
- ...has an inverse--doing the reverse operation.
- ...is associative--composition is inherently associative.
Note something about the identity, that will give you a flavor of the type of thinking required to understand groups.
Suppose you have two elements, e and f, in S that satisfy the properties of an identity. Then we have e=e*f=f. So e = f. What this means in terms of symmetries is that the act of doing nothing EQUALS the act of doing something followed by the reverse operation, and all the other such things that are effectively equivalent. In the group of Rubik's Cube, turning a side 4 times 90 degrees, is the same as doing nothing. I think you get the idea.
Quite often we'd like to use a set of matrices with matrix multiplication as an operation to "represent" the symmetry group we are studying. This is especially useful in physics, since many of the equations of state can be turned into a linear system which in turn has matrix representations.
Representation theory is another thing I would like to understand more deeply.
There are some important classes of matrices to keep in mind.
I will denote the transpose of a matrix, A, with A^T. I will denote the inverse of a matrix, A, with A^-1. If A^-1=A^T (which means A*A^T=1), then A is called Orthogonal (because the column vectors of such a matrix are mutually orthogonal).
The group (you can verify it's a group) of all Orthogonal, nxn matrices is referred to as O(n). The group of all Orthogonal, nxn matrices with a determinant of 1 is referred to as SO(n).
Note that, generally speaking, orthogonal matrices have real value entries. But many things in physics use complex numbers. With this, the concept of a complex conjugate comes up.
I will denote the result of replacing all entries in A^T with their complex conjugate as A^(dagger). If A^-1=A^(dagger) (which means A*A^(dagger)=1), then A is called Unitary.
The group of all Unitary, nxn matrices is referred to as U(n). The group of all Unitary, nxn matrices, with determinant 1, is referred to as SU(n).
With this information, see if you can go from understanding this explanation of the standard model to this more advanced one.
Understanding the more advanced track of explanations at the official string theory web-site is another goal of mine.
I'll stop there, but there is a lot more to this subject. I'll summarize my learning goals (so far) here:
- Understand the derivations of Noether's Theorem, as well as general Gauge Theory and Variational Principles.
- Understand Group Representation Theory.
- Understand the more advanced track on the official string theory web-site.