Is my understanding of Gauge Symmetries correct?

Question

I'm currently working on a project about Symmetry Breaking for my physics bachelor.

Right now I'm trying to understand Gauge Symmetries (although I guess it's not much of a symmetry). And I've been thinking about it and want to know if my understanding of the concept is correct.

Now for a global symmetry:

Suppose I am in empty space and there is a chair facing me. In my understanding a symmetry is e.g. if we translate and then rotate the chair a bit so that it is still facing me. For me this operation is entirely symmetric.

Now to think of a gauge symmetry I visualized an axis system with an origin at e.g. my location. Would a gauge symmetry be moving the origin at the location of the chair? Because it doesn't change anything to the system and it is not measurable.

In other words, I came to think of it like this:

Normal Symmetry, with some unitary matrix U (e.g.that translates then rotates the chair) \begin{align} \hat U|a⟩=|b⟩ \end{align}

Gauge Symmetry \begin{align} |a⟩=|a⟩ \end{align} So basically you've changed nothing that can be measured.

Is this correct? Thanks

Edit:

Are there similar examples one could think of for a local gauge symmetry and a local symmetry?

Answer 1

I would say what you're describing in your question has more to do with the difference between "active" and "passive" transformations, than it does global vs local/gauge symmetries. In particular, an active transformation is one where we perform some operation on the actual physical state, wheres a passive transformation is a change of our description keeping the physical state fixed. So for example, an active translation of a chair is, actually moving the chair. A passive translation of a chair is us moving our origin of coordinates. In the rest of the answer I'll use the language appropriate to active transformations, as they make more physical sense to me.

Global vs local symmetries are a different beast entirely. Exactly how to describe them depends a bit on your level of sophistication.

The precise answer, which is possibly useless depending on your level of knowledge, is that a generic symmetry transformation (assuming a continuous symmetry) has the form $\phi(x) \rightarrow \phi'(\phi(x),\theta)$. In other words, we take all our fields $\phi$, and transform them all to new fields $\phi'$ in a way that depends on some parameters $\theta$. What makes these transformations symmetries is that after doing this we find that the physics looks the same in our transformed point of view. The relationship between the transformed fields $\phi'$ and the original fields $\phi$ can be linear or non-linear. Then the key question is whether the parameter of the transformation, $\theta$, depends on space or not. If it does not, we call it a global symmetry: this is a genuine physical symmetry of the system. If $\theta$ does depend on space, we call it a local symmetry: this isn't a genuine symmetry, it really represents a redundancy of our description of the system.

To get an intuitive picture of what's going on let's imagine a field (ie a set of numbers of each point in space) which is made as simply the time at every point on the surface of the earth. Imagine we've covered the surface of the earth with lots of people each wearing wristwatches, the value of a field at point x on the earth is the reading on the wristwatch of the person standing there.

An example of a global symmetry would be time translations. Imagine moving forward in time by one second. Everyone's watch has been shifted forward in time by the same amount, one second. This is a symmetry in the sense that the laws of physics are still the same.

A local symmetry, on the other hand, would be if everyone suddenly and independently changed the settings on their wristwatches. This is a local transformation in the sense that each person can adjust their watch by different amounts. Note that in this example, the actual time has not changed, all that's happened is that our description of time in terms of people's watches has changed.

One thing that's not present in this example is the notion of curvature. It is possible for everyone to synchronize their watches with each other, and for them to remain synchronized? Where gauge theory really gets interesting is when you consider situations where that is not possible.

One way you can go with the wristwatch analogy I'm making is to start thinking about curvature of spacetime: black holes, clocks run at different speeds in a gravitational fields. Observers at different radial distances from the center of a black hole cannot synchronize their clocks.

Another way to describe the situation, from a less gravitational perspective, is given by Maldacena in this amusing essay: http://arxiv.org/abs/1410.6753.

Answer 2

You will get a better answer than this but, assuming all you want is a rough analogy compared to your chair idea, here goes.

For a local gauge symmetry analogy, forget the chair.

Imagine you are building a house on rough surface, with the ground sloping all over the place. Obviously you want the floor to be level, no matter how uneven the ground is.

So you have to take into account how to get a level line between two points, say the corners of the building.

A local symmetry has to allow for changes in the ground to keep the line level, so its going to be a function/transformation that does this. We need independent transformations at each point to keep the line level, this is the connection field.

With this connection field come the forces, such as the photons associated with the electromagnetic field, for example.

Compare this to a global symmetry, which as you say, means you've changed nothing that can be measured.

A global symmetry is basically an overall phase factor, it is a trivial symmetry, whereas a local symmetry has to preserve properties across a wide range of "uneven ground".