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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)

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    $\begingroup$ What is a gauge transformation ? $\endgroup$
    – Trimok
    Jan 23 '14 at 13:24
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    $\begingroup$ Huh, what? I can't make head or tail of your question. $\endgroup$
    – Siva
    Jan 23 '14 at 13:34
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    $\begingroup$ 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? $\endgroup$
    – Autolatry
    Jan 23 '14 at 13:55
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    $\begingroup$ See article "Gauge symmetry is not a symmetry" physics.stackexchange.com/q/13870 $\endgroup$ Jan 23 '14 at 19:40
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$SU(2)$ 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.

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  • $\begingroup$ Interesting, in the second paragraph you are referring to the difference between large and small gauge transformations, right? I was wondering if you could provide me with a reference where you have read about this? A while ago, I asked a question about this on this forum but I still don't feel confident about this (physics.stackexchange.com/questions/72815/…). $\endgroup$
    – Hunter
    Jan 23 '14 at 19:58
  • $\begingroup$ @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! $\endgroup$
    – user1504
    Jan 23 '14 at 20:02
  • $\begingroup$ 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? $\endgroup$
    – Hunter
    Jan 23 '14 at 20:06
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    $\begingroup$ 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. $\endgroup$
    – user1504
    Jan 23 '14 at 21:05
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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.

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  • $\begingroup$ 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.) $\endgroup$
    – Hunter
    Jan 23 '14 at 19:12
  • $\begingroup$ 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. $\endgroup$
    – user37343
    Jan 23 '14 at 19:38
  • $\begingroup$ 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/… $\endgroup$
    – Hunter
    Jan 23 '14 at 19:44
  • $\begingroup$ I think you are correct. I think you should edit my response and also in-link your previous post. $\endgroup$
    – user37343
    Jan 23 '14 at 20:03
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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).

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    $\begingroup$ 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. $\endgroup$
    – Adam
    Jan 23 '14 at 14:27
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    $\begingroup$ Gauge transformations can be global or local. I didn't include that because it just adds additional detail that is not immediately relevant to the OP's question. Regardless, nothing changes about my answer for local gauge transformations. $\endgroup$
    – mcFreid
    Jan 23 '14 at 14:29
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    $\begingroup$ You are mistaken. I think that's exactly what is relevant to the OP's question. (S)he wants to know what is a "SU(2) gauge transformation" compare to "SU(2)". And you don't say anything about what is really a gauge transformation. And you use a global transformation, calling that a gauge transformation. I think that your answer is really confusing. $\endgroup$
    – Adam
    Jan 23 '14 at 14:46
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    $\begingroup$ I have also down-voted as I agree with Adam. $\endgroup$
    – Hunter
    Jan 23 '14 at 15:34
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    $\begingroup$ @Siva: a local gauge transformation transforms between different gauge fixings on overlapping coordinate charts; consistently re-fixing the gauges on all charts of a coordinate atlas - a global gauge transformation - is equivalent to an automorphism of the underlying principal bundle; gauge transformations in the latter sense form a group, which shouldn't be confused with the 'gauge group' (the structure group of the principal bundle - stuff like the gauge potential and curvature 2-form take its values in the corresponding Lie-algebra) $\endgroup$
    – Christoph
    Jan 23 '14 at 16:49

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