Ok. As per the title this will be (at least at times) technical. This is for temur, especially, who asked a question about this a month ago or so, in another higgs thread.

I am at a summer school these next two weeks in Princeton (which means I don't have a lot of spare time to write long threads about gauge invariance), so I will try to make posts in stages.

Also, some ground rules. This is not a place to talk about alternatives to the higgs. If you want to learn some of the maths behind the higgs mechanism, I you can ask. Please don't interrupt this thread with alternative theories to the higgs mechanism, unless you can do it with as much mathematical detail as I am doing here---in other words, unless you can point out a flaw in my (and many Nobel Lauriate's) maths, start another thread.

I'll try to put excercizes here and there, that you can try and post solutions to if you're interested. For me, at least, I never learn anything unless I actually CALCULATE something, so it may help you to think about some of these questions (which I have struggled with in the past, too).

Finally, if I am wrong, point it out. If you read some of my responses on this Forum, you'll know that I often screw things up (even things I know well, like what holds nucleons together, or how a bubble chamber works). I will try to keep these posts mistake free, but I can only try!

Ok. The higgs.

It may help to know about representations of the algebra $$SU(N)$$, so that is what I will do before anything else. Next we'll move to Young's tableaux, and then I'll show you how to make mass terms in the standard model lagrangian. Then I will be able to state the problem, in full detail. After that, I'll show you how the higgs mechanism works, and why it generates mass. Hopefully this will convince you that the higgs mechanism is very elegant, and at least a very clever solution to a very hard problem. So clever, in fact, that no other acceptable solution has been found, in the forty years that very smart people have been thinking about it.

Consider some set of $$n \times n$$ matrices which obey:

$$

\det U = 1,\\

U^{\dagger} U = 1,

$$ (1a)

where the dagger means take the complex conjugate of the transpose, specifically:

$$

U^{\dagger} = (U^T)^*

$$ (1b)

The matrices are

$$

\left|\eta\right\rangle \rightarrow U\left|\eta\right\rangle,\\

\left\langle\eta\right| \rightarrow \left\langle\eta\right| U^{\dagger},

$$(2)

with a similar expression for $$\left|\xi\right\rangle$$. The inner product of those two states then transforms as

$$

\left\langle\xi\left|\eta\right\rangle \rightarrow \left\langle\xi\right|U^{\dagger}U\left|\eta\right\rangle = \left\langle\xi\left|\eta\right\rangle.

$$(3)

[Aside]

Here is where I will pause and appologize to the mathematicians---if you are unfamiliar with the Dirac notaion, I will direct you to the Wikipedia page http://en.wikipedia.org/wiki/Bra-ket_notation. If you are just trying to follow along, the |>'s are called kets and the <|'s are called bras. They are vectors in some abstract space described by (in this case) $$SU(N)$$. What I did above amounts to taking a dot product, or scalar product, if you like. The main thing I wanted to show (and if you believe me then good) is that the unitary matrices we're talking about leave the inner products alone. This is crucial to understand, otherwise we wouldn't be assured of things like a good interpretation of probability.

[/Aside]

Now I will appeal to your geometrical intuition. Suppose you wanted to describe your location, somewhere on the face of the Earth, to someone else. The easiest way to do this is to use lattitude and longitude---for example, the US GPS system can tack things down to within a meter or so, I think. But either way, you only need two numbers to describe your position. Another example is locating points in a plane---you need an x coordinate and a y coordinate, or r and theta. But either way, you've expressed things in terms of basis vectors. There is a vector of unit length that points in the x direction, often called $$\hat{i}$$ and a vector of unit length that points in the y direction, called $$\hat{j}$$ which obey:

$$

\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = 1,\\

\hat{i} \cdot \hat{j} = 0.

$$(4)

By taking linear combinations of the basis vectors, you can generate any other vector in the space, which is described by the group $$O(2)$$.

What does this have to do with $$SU(N)$$ representations? Well, it turns out that you can also find a basis of the space described by $$SU(N)$$---the basis vectors are called generators. And for physics, we're interested in generating unitary matrices, so the basis vectors we want to use are going to be matrices.

The first trick is to see that we can always write a unitary matrix as

$$

U = e^{i H},

$$(5)

where $$H$$ is an $$n \times n$$ matrix.

The matrices $$H$$ must be traceless and hermitian, which you should work out and convince yourself of. To a physicist, hermitian means

$$

H^{\dagger} = H.

$$ (6)

Now we need some knowledge of the dimension of the space we're dealing with. Rather than derive this, I'll just tell you. But first, keep in mind that the meaning of the word ``dimension'' hasn't really changed. It just means the number of basis vectors we need to cover the whole space---just as above, we needed $$\hat{i}$$ and $$\hat{j}$$ to describe the x-y plane, we'll need enough basis vectors to get all of the $$SU(N)$$ matrices we can possibly think of.

The dimension of $$SU(N)$$ is $$N^2 - 1$$---this means that we need this many basis vectors. The canonical exaple is $$SU(2)$$, and I'm not one to break with tradition. Plus, $$SU(2)$$ is one of the most important groups to particle physics, and studying it will help us down the road.

The dimension of SU(2) is three, so let's try and find three traceless, hermitian matrices. To further simplify things, let's look for 2 x 2 matrices, although this is by no means necessary. We could do the same thing with 50 x 50 matrices, but not quite as easily.

First, traceless. This means that the main diagonal entries have to be either all zeros, or they have to be equal and opposite.

The matrices must also be hermitian, as per Eq. (6). After some thinking, you should be able to convince yourself that a good basis (called the Pauli basis) of SU(2) is

$$\sigma_1 = \left(\begin{array}{cc}0&1\\1&0\end{array}\right), \sigma_2 = \left(\begin{array}{cc}0&-i\\i&0\end{array}\right), \sigma_3 = \left(\begin{array}{cc}1&0\\0&-1\end{array}\right). $$ (7)

Now we can make any 2x2 unitary matrix, with determinant 1, by simply doing this:

$$

U = e^{i \vec{\alpha} \cdot \vec{\sigma}}.

$$

This is shorthand notation for

$$

U = e^{i \left(\alpha_1\sigma_1 + \alpha_2\sigma_2 + \alpha_3\sigma_3\right)}.

$$

What we have done is to find the fundamental representation of the algebra SU(2). In this context, ``fundamental representation'' just means that the $$\sigma$$ matrices we found have the same size as the number in the parenthesis of the group. SU(2) ---> 2x2 matrices.

Next, I will talk about Young's Tableaux, and an easy way to figure out what happens when we deal with MANY representaitons of differing sizes.

I am at a summer school these next two weeks in Princeton (which means I don't have a lot of spare time to write long threads about gauge invariance), so I will try to make posts in stages.

Also, some ground rules. This is not a place to talk about alternatives to the higgs. If you want to learn some of the maths behind the higgs mechanism, I you can ask. Please don't interrupt this thread with alternative theories to the higgs mechanism, unless you can do it with as much mathematical detail as I am doing here---in other words, unless you can point out a flaw in my (and many Nobel Lauriate's) maths, start another thread.

I'll try to put excercizes here and there, that you can try and post solutions to if you're interested. For me, at least, I never learn anything unless I actually CALCULATE something, so it may help you to think about some of these questions (which I have struggled with in the past, too).

Finally, if I am wrong, point it out. If you read some of my responses on this Forum, you'll know that I often screw things up (even things I know well, like what holds nucleons together, or how a bubble chamber works). I will try to keep these posts mistake free, but I can only try!

Ok. The higgs.

It may help to know about representations of the algebra $$SU(N)$$, so that is what I will do before anything else. Next we'll move to Young's tableaux, and then I'll show you how to make mass terms in the standard model lagrangian. Then I will be able to state the problem, in full detail. After that, I'll show you how the higgs mechanism works, and why it generates mass. Hopefully this will convince you that the higgs mechanism is very elegant, and at least a very clever solution to a very hard problem. So clever, in fact, that no other acceptable solution has been found, in the forty years that very smart people have been thinking about it.

*SU(N)*and some representations.Consider some set of $$n \times n$$ matrices which obey:

$$

\det U = 1,\\

U^{\dagger} U = 1,

$$ (1a)

where the dagger means take the complex conjugate of the transpose, specifically:

$$

U^{\dagger} = (U^T)^*

$$ (1b)

The matrices are

*unitary*, in that they preseve the length of inner products. For example, consider two vectors, $$\left|\eta\right\rangle$$ and $$\left|\xi\right\rangle$$, which transform under $$SU(N)$$ as:$$

\left|\eta\right\rangle \rightarrow U\left|\eta\right\rangle,\\

\left\langle\eta\right| \rightarrow \left\langle\eta\right| U^{\dagger},

$$(2)

with a similar expression for $$\left|\xi\right\rangle$$. The inner product of those two states then transforms as

$$

\left\langle\xi\left|\eta\right\rangle \rightarrow \left\langle\xi\right|U^{\dagger}U\left|\eta\right\rangle = \left\langle\xi\left|\eta\right\rangle.

$$(3)

[Aside]

Here is where I will pause and appologize to the mathematicians---if you are unfamiliar with the Dirac notaion, I will direct you to the Wikipedia page http://en.wikipedia.org/wiki/Bra-ket_notation. If you are just trying to follow along, the |>'s are called kets and the <|'s are called bras. They are vectors in some abstract space described by (in this case) $$SU(N)$$. What I did above amounts to taking a dot product, or scalar product, if you like. The main thing I wanted to show (and if you believe me then good) is that the unitary matrices we're talking about leave the inner products alone. This is crucial to understand, otherwise we wouldn't be assured of things like a good interpretation of probability.

[/Aside]

Now I will appeal to your geometrical intuition. Suppose you wanted to describe your location, somewhere on the face of the Earth, to someone else. The easiest way to do this is to use lattitude and longitude---for example, the US GPS system can tack things down to within a meter or so, I think. But either way, you only need two numbers to describe your position. Another example is locating points in a plane---you need an x coordinate and a y coordinate, or r and theta. But either way, you've expressed things in terms of basis vectors. There is a vector of unit length that points in the x direction, often called $$\hat{i}$$ and a vector of unit length that points in the y direction, called $$\hat{j}$$ which obey:

$$

\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = 1,\\

\hat{i} \cdot \hat{j} = 0.

$$(4)

By taking linear combinations of the basis vectors, you can generate any other vector in the space, which is described by the group $$O(2)$$.

What does this have to do with $$SU(N)$$ representations? Well, it turns out that you can also find a basis of the space described by $$SU(N)$$---the basis vectors are called generators. And for physics, we're interested in generating unitary matrices, so the basis vectors we want to use are going to be matrices.

The first trick is to see that we can always write a unitary matrix as

$$

U = e^{i H},

$$(5)

where $$H$$ is an $$n \times n$$ matrix.

__Excercize 1__: What does Equation (1) mean for $$H$$?The matrices $$H$$ must be traceless and hermitian, which you should work out and convince yourself of. To a physicist, hermitian means

$$

H^{\dagger} = H.

$$ (6)

Now we need some knowledge of the dimension of the space we're dealing with. Rather than derive this, I'll just tell you. But first, keep in mind that the meaning of the word ``dimension'' hasn't really changed. It just means the number of basis vectors we need to cover the whole space---just as above, we needed $$\hat{i}$$ and $$\hat{j}$$ to describe the x-y plane, we'll need enough basis vectors to get all of the $$SU(N)$$ matrices we can possibly think of.

__Excercize 2__: What is the dimension of $$SU(N)$$? Hint: How many degrees of freedom does a traceless, hermitian matrix have?The dimension of $$SU(N)$$ is $$N^2 - 1$$---this means that we need this many basis vectors. The canonical exaple is $$SU(2)$$, and I'm not one to break with tradition. Plus, $$SU(2)$$ is one of the most important groups to particle physics, and studying it will help us down the road.

The dimension of SU(2) is three, so let's try and find three traceless, hermitian matrices. To further simplify things, let's look for 2 x 2 matrices, although this is by no means necessary. We could do the same thing with 50 x 50 matrices, but not quite as easily.

First, traceless. This means that the main diagonal entries have to be either all zeros, or they have to be equal and opposite.

__Exercize 3:__Why can't there be any*i*'s on the main diagonal?The matrices must also be hermitian, as per Eq. (6). After some thinking, you should be able to convince yourself that a good basis (called the Pauli basis) of SU(2) is

$$\sigma_1 = \left(\begin{array}{cc}0&1\\1&0\end{array}\right), \sigma_2 = \left(\begin{array}{cc}0&-i\\i&0\end{array}\right), \sigma_3 = \left(\begin{array}{cc}1&0\\0&-1\end{array}\right). $$ (7)

__Exercize 4:__Verify that these matrices are traceless and hermitian.Now we can make any 2x2 unitary matrix, with determinant 1, by simply doing this:

$$

U = e^{i \vec{\alpha} \cdot \vec{\sigma}}.

$$

This is shorthand notation for

$$

U = e^{i \left(\alpha_1\sigma_1 + \alpha_2\sigma_2 + \alpha_3\sigma_3\right)}.

$$

What we have done is to find the fundamental representation of the algebra SU(2). In this context, ``fundamental representation'' just means that the $$\sigma$$ matrices we found have the same size as the number in the parenthesis of the group. SU(2) ---> 2x2 matrices.

Next, I will talk about Young's Tableaux, and an easy way to figure out what happens when we deal with MANY representaitons of differing sizes.

Last edited: