# Difference between $SU(2)$ and $SU(2)$ gauge transformations?

I hear this jargon all the time, so what is the difference?

(Of course this is nothing special to $SU(2)$, but rather I just took it as an example)

• What is a gauge transformation ? Jan 23 '14 at 13:24
• Huh, what? I can't make head or tail of your question.
– Siva
Jan 23 '14 at 13:34
• The word gauge here relates to the redundant degrees of freedom in a Lagrangian (specifically). The transformations between all these possible gauges are called gauge transformations and the will form a Lie group—referred to as the symmetry group or the gauge group of the theory. Is this what you're getting at? Jan 23 '14 at 13:55
• See article "Gauge symmetry is not a symmetry" physics.stackexchange.com/q/13870 Jan 23 '14 at 19:40

$SU(2)$ is a finite-dimensional Lie group. The elements of this group are $2 \times 2$ matrices $A$ which are unitary ($A^\dagger = A^{-1}$) and have determinant $1$. More generally, $G$ may be a reductive Lie group. Physicists usually prefer to talk about products of $U(1)$ and semisimple groups.

A gauge transformation is a function from spacetime $\Sigma$ into the group $G$. This is a map $g$ which assigns to every $x \in \Sigma$ a matrix $g(x) \in SU(2)$. ('$SU(2)$ gauge transformation' denotes the special case $G=SU(2)$.) The set $\mathcal{G} = Map(\Sigma,G)$ of all maps $g: \Sigma \to G$ is a group because we can take two maps $g$ and $h$ and define a new map $gh$ via $gh(x) = g(x)h(x)$. The identity is the constant map given by $e(x) = 1$, where $1$ is the identity in $G$.

When being precise, physicists tend to reserve the term 'gauge transformation' for gauge transformations which belong $\mathcal{G}_0$, the connected component of the identity in $\mathcal{G}$; these are points in $\mathcal{G}$ which are connected to the identity by a continuous path. (It is considered impolite on physics boards to pay too much attention to what the topology is on $\mathcal{G}$ and exactly how well-behaved the maps ought to be. However, for lattice gauge theory, you really want to ensure that you can approximate any gauge transformation well by a map from the lattice to $G$.)

The reason for doing this is that physicists don't usually want to allow constant functions (or, at least, constant and valued in the identity component of $G$) to count as gauge symmetries. Gauge symmetries are supposed to leave the physical system invariant. But in electromagnetism, for example, we want the global group $U(1)$ to be a Noether group, whose conserved charge is electric charge. Likewise, in QCD, the statement that only color singlets are observable at large distances is meant to have physical content. Confinement is not a triviality.

Sometimes people use the terminology 'global gauge transformation' to refer to such constant functions to $G$. This is perverse terminology, but good luck getting a newspaper to publish a letter to the editor complaining about this problem.

• @Hunter I'm pretty bad at coming up with references. However, you should be able to see that this is the case just by thinking about classical QED. The global transformations are the group that gives rise to the conservation of charge. They can't act trivially! Jan 23 '14 at 20:02
• I understand that global transformaions cannot act trivially. But why are local transformations that do not tend to the identity as $x \to \infty$ considered to be a global symmetry? Jan 23 '14 at 20:06
• Ah, I get you. If I think of a nice answer beyond what David Bar Moshe has given, I'll give it as an answer to your linked question. Jan 23 '14 at 21:05

I am not an expert but lets give this a shot. I really recommend reading up on classical theory of gauge fields by Rubakov, first; Its a life saver. All I have below is just mostly words, but the book I recommend is great.

There are a few terms in the business :

Gauge group: Some group of principal bundles on which a field is connected. eg $SU(N)$

Gauge transformation: $A \rightarrow A- \nabla \psi$

and $\phi \rightarrow \phi + a$

Gauge Invariance: Your New Lagrangian $L \rightarrow L'$ does not change the theory

Also consult comments @Hunter's comment below

Some other things:

You will surely come across two motivated responses to your questions, A mathematical one, and a physical one, I recommend spending time on both

I have to again say, I am not any sort of expert, but I hope this helps. I am open to any suggestions on how to edit my response.

• The way I learnt it (in the framework of Yang-Mills at least), is that a Lagrangian has some global symmetry. Then we impose that this symmetry must also hold locally which is only then considered a gauge symmetry. (And as a result of imposing this gauge symmetry, we need to introduce covariant derivatives.) Jan 23 '14 at 19:12
• You are most likely more experienced than I am. Can you comment in some mathematical details. I am really bad at learning anything without seeing some equations.
– user37343
Jan 23 '14 at 19:38
• You could have a look at the answer I gave here (although this answer is not directly related classical/quantum field theory, the principle stays the same; also the gauge group discussed there is $U(1)$): physics.stackexchange.com/questions/94699/… Jan 23 '14 at 19:44
• I think you are correct. I think you should edit my response and also in-link your previous post.
– user37343
Jan 23 '14 at 20:03

SU(2) is a type of group, that is a collection of objects that have certain properties.

An SU(2) gauge transformation refers to the observation that certain objects in certain equations remain invariant if you multiply them by the elements of the SU(2) group.

How this works is easiest to see with rotations on vector dot products. Consider an equation that only has terms like $\vec{v} \cdot \vec{v}$. Now let us rotate our coordinate system by some matrix $R$ (in this case $R$ belongs to the group SO(3)). Then,

$\vec{v} \cdot \vec{v} = v^T v -> v^T R^T R v = v^T v$

since $R^T R = 1$ is one of the defining properties of the group SO(3).

To recap, SU(2) in your example refers to a group of objects and an SU(2) gauge transformation refers to the act of multiplying objects in an equation by the objects in the group SU(2) and noting that the equation doesn't change under such multiplications (just like the length of a vector doesn't change under rotations).

• Downvote: your answer is not correct. Gauge transformation corresponds to a local transformation (in you example, $R(x)$), not to the fact of doing a global transformation. In condensed matter, people sometimes consider gauge transformation and global transformation to be equivalent, but that's a mistake.