## Archive for **September 2010**

## Proof of the Jacobi Identity in Classical Mechanics

I finally thought I should write it down…for once in my life.

## A lemma on a rather frequently encountered set of Hermitian matrices

I’m documenting a small ‘lemma’ that I think is worth mentioning. Suppose are 4 Hermitian matrices () satisfying

where denotes the identity matrix. Then these matrices have eigenvalues , are traceless and are necessarily of even dimension.

For , the anticommutator above gives . So, for any eigenvector and eigenvalue , we have

or equivalently .

Traceless-ness has a neat proof:

Suppose . Then

where the last equality follows from . But , so

.

Finally, suppose the numbers of +1 eigenvalues and -1 eigenvalues are and respectively. The dimension of the matrix is then . Since the trace equals the sum of eigenvalues, we have

So, the dimension is , which is clearly always an even number.

## Group Theory II – SU(2)

In a previous post, I cursorily touched upon the standard orthogonal and unitary groups. It turns out that in quantum mechanics, the operators of principal interest are either Hermitian or unitary. So it is only natural that we should be interested in transformations from the Special Unitary Group . In particular, we dwell on the case.

First of all, has 3 independent parameters, something that is obvious if one demands that a matrix ,

be unitary, that is

and have a determinant equal to . It follows that any such matrix can be written as

in which there are three independent parameters. If is expanded about the identity, i.e.

then unitarity of demands that be Hermitian, and also **traceless**. For a finite transformation, this generalizes to

That is more generally, implies that for . So, is a traceless matrix.

We can write

From nonrelativistic quantum mechanics, we know that the effect of a rotation through an angle on a spin-1/2 particle about an axis is given by the unitary matrix,

where is the Pauli spin matrix “vector” given by,

Clearly, in this development, we can identify for as the **generators** of the group. The commutation relation for the generators is

This is called the * fundamental representation of *. An -dimensional representation of consists of unitary matrices satisfying such commutation relations.

## Group Theory I – O(n), SO(n), U(n), SU(n)

I have been trying to learn group theory for a long time. Invariably, the books I come across are either too formal or too cursory. But if I ignore this, there are of course a number of very well written introductions to group theory available on the internet. In this post, I will try not to bore you with what a group is (chances are, you already know, if you’re reading this), but will present a somewhat different perspective of how it fits into our ‘daily’ physics.

**Orthogonal and Special Orthogonal Groups**

The set of all orthogonal matrices, i.e. matrices satisfying

forms a group denoted by . Now, if is a real orthogonal matrix of dimension , it has independent parameters. This is easily seen by taking . Let me try a more general proof here..

Clearly,

There are of these equations with on the RHS, and there are such equations with a on the RHS. But of them, only half are unique, because yields the same equation. So, the number of independent conditions is only .

Why is important? It is the rotation group in n-dimensional Euclidean space. The only restriction we need to impose on it is for the matrices to have a determinant of though, which makes them **special**, and part of the group . It is well known that rotation matrices in 3 dimensions are orthogonal and have a unit determinant. Note that orthogonality guarantees a determinant of , but **proper **rotations can only be part of .

There is a caveat to this. Elements of both and are parametrized in general by **continuous** variables. For a rotation in -dimensional space, one angle is sufficient. For -dimensions, we have the three Euler angles. So each rotation matrix is a function of angles, which are continuous. Groups that depend on continuously varying parameters are called **Lie Groups**. Also, since angles vary over closed, finite intervals, these groups are also said to be **compact**.

**Unitary and Special Unitary Groups**

The set of unitary matrices, i.e. matrices satisfying

forms a group denoted by . Let’s use the above idea to find the number of independent parameters of a unitary matrix of dimension .

The unitarity condition translates to

In this case, the (diagonal) entries contribute to the only nonvanishing RHS, and the upper and lower triangular equations are all distinct because of additional conditions imposed by the elements being complex. So, in all there are independent elements of matrix in .

Additionally, if the determinant of the matrix is , the matrix is said to be **special**, and part of the special unitary group of order n, i.e. . By the way, I haven’t shown that any of these groups are indeed groups. For that, you have to show closure under group multiplication, associativity, existence of a unique unit element and of an inverse for every element. Why is important? Well, that requires a motivation through , the ‘spin group’ in nonrelativistic quantum mechanics. I will address this in a subsequent post.

Much of group theory involves a lot of jargon, which can be a bit tricky to connect to the “real” world, if you think like me. But it helps to know the jargon, as in any field you want to grapple with.

A subset of a group which is closed under multiplication, is called a **subgroup **of G.

Let and . If for every and , then is called an **invariant subgroup** of G.

**Representations**

So far, we have looked at matrix representations of the standard orthogonal and unitary groups. These are useful because they let us employ familiar rules of matrix algebra to study the properties of the group in question. Matrix representations are closely associated with symmetries. A good example is the time independent Schrodinger equation,

which is an eigenvalue problem for a stationary state . If there exists a group of transformations under which the Hamiltonian stays invariant, then

for some represents the group action. In particular, this implies that every element of commutes with the Hamiltonian (and hence, by a result from linear algebra, it is possible to diagonalize and simultaneously for every — quite a strong result!), i.e. .

Now,

but

So, all the transformed states are degenerate and constitute a **multiplet**.

Suppose denotes the vector space of all transformed solutions, and has a finite dimension . Then, it has a basis, which we can denote by . Since belongs to the multiplet, it can be expanded in terms of this basis, as

This means that there is a matrix associated with every group element .

Group representations are of two kinds:

1. **Irreducible**, which means by rotating any element of with **all **elements of , we can recover **all** other elements of , or

2. **Reducible**, in which case, the vector space splits into a direct sum of vector subspaces each of which is mapped into itself (but not into another) under the action of , i.e. .