What is the basis of gauge theory?

Question

I’m learning about gauge concepts. I’ve always had the idea that by looking at a phenomenon from different viewpoints, that symmetries could be derived – in fact, that was what an equal sign signified. In other words, the underlying phenomenon remained constant, only the viewpoints changed, and therefore could be equated. But it seems that gauge theory is the opposite – identical observable quantities are seen, although the configurations of the underlying fields change. Is this right, or am I completely off-track?

Answer 1

PhotonicBoom has already provided a nice overview of the basic idea behind gauge theories, let me lay it on a bit thicker with the abstraction:

A gauge theory is a theory that has a local gauge symmetry induced by a gauge group $G$, which is required to be a Lie group. Now what do we mean by that?

Let $\Sigma$ be our spacetime (of arbitrary dimension and signature). Suppose we know that there should be some field $A_\mu : \Sigma \rightarrow \mathfrak{g}$ on $\Sigma$ ($\mathfrak{g}$ is the Lie algebra of $G$, you can think of the (vector) potential of classical electrodynamics (henceforth referred to as ED) for this. But, whenever we look, we can look only locally at this field, so we have some open sets $U_\alpha$ with $\alpha$ some index covering $\Sigma$, and on each of these $U_\alpha$, we have some $(A_\alpha)_\mu$ (e.g. as solution to the Maxwell equations). We get a global definition for the field if we require

$$ A_\alpha = g_{\alpha\beta}A_\beta g_{\alpha\beta}^{-1} + \mathrm{i}g_{\alpha\beta}\mathrm{d}g_{\alpha\beta}^{-1} \; \text{with} \; g_{\alpha\beta} : U_\alpha \cap U_\beta \rightarrow G \text{ a smooth function}$$

for all pairs of sets $U_\alpha,U_\beta$ that have non-zero intersection. This may look very strange, but consider the case of ED, where $G = \mathrm{U}(1)$: There, for $f : \mathbb{R}^4 \rightarrow \mathbb{R}$, we can write such functions as $g(x) = \mathrm{e}^{\mathrm{i}f(x)}$. Everything commutes, and the above formula reduces to

$$A_\alpha = A_\beta - \mathrm{d}f$$

which is precisely the gauge freedom we have in classical ED! So, in this sense, the ugly looking equation above is the generalization of the familiar case to general, non-abelian symmetry groups $G$. In fact, the above data (the sets $U_\alpha$ and the transition functions $g_{\alpha\beta}$) define what is called a $G$-principal bundle. Now, on this $G$-principal bundle, let's call it $P$, you can define the notion of a gauge transformation $g : P \rightarrow G$, and the transformation of $A$ under this will again be $A \mapsto gAg^{-1} + g\mathrm{d}g^{-1}$ (ignoring annoying factors of $\mathrm{i}$). What does this have to do with our intuitive idea of gauge? Well, $P$ locally looks like $U_\alpha \times G$, where letting a group element $h \in G$ act on a point $(x,k)$ just means $(x,k)h := (x,kh)$, and doing the gauge trafo is just $(x,k) \mapsto (x,kg(x,k))$. So all the trafo does is switching the group elements above a point $x \in \Sigma$ around, or, in other words, chooses a new point in what looks like $\{x\} \times G$ to be the group identity. This is (in a vague sense) the generalization of the freedom of setting the "zero" for some potential that PhotonicBoom talked about.

Now, having this weird symmetry going on is alright, but how do we get things that do not change under gauge transformation? Right now, $A$ is changing, the points in $P$ are switching around, isn't something supposed to be gauge invariant here?

Define the covariant derivative w.r.t. to the connection $A$ as

$$ \mathrm{d}_A f := \mathrm{d}f + A \wedge f$$

(every covariant derivative of a field transforming in a representation of $G$ will also transform in that representation, but I've already written a wall of text here, so I'll not go into matter fields, the buzzword is associated vector bundles) and define the curvature or field strength

$$ F := \mathrm{d}_A A = \mathrm{d}A + A \wedge A$$

It can now, by direct computation, be shown that $F$ transforms as $F \mapsto gFg^{-1}$. In ED, everything commutes, and you have already a gauge invariant quantity (since $gFg^{-1} = F$ there), which is good, since the field strength should, as a physical quantity , not change under gauge transformation! For general $G$, which can in all cases of interest be written as matrix groups, just take the trace. $\mathrm{tr}(gFg^{-1}) = \mathrm{tr}(F)$ is invariant, since the trace is invariant under cyclic permutations.

And, we're done! The action for this pure Yang-Mills theory is

$$ S[A] := \frac{1}{4e^2}\int_\Sigma \mathrm{Tr}(F \wedge \star F)$$

with $e$ some coupling constant. I'm not quite certain if that's refining your intuition as you may have hoped, but that's how it is (provided I've not made some major technical blunder somewhere, pointers of course appreciated).

Answer 2

You are basically right. A gauge theory is a field theory that leaves the equations of motion invariant under local (important distinction pointed out by @joshphysics) transformations of the coordinates. It gives physicists the ability to introduce arbitrary degrees of freedom to play with and simplify problems, as long as the physical quantities remain the same.

For example in Electrodynamics you can redefine the potential as long as the gradient stays the same. The electric field $E(r)$ (our physical quantity) is given by: $$E = -\nabla\phi$$

But $\phi$ can be transformed by adding a constant term $k$ which will give: $$\phi' \rightarrow \phi + k$$

Substituting this into the previous equation we get: $$E = -\nabla(\phi + k) = -\nabla \phi$$ which corresonds to the same physical quantity as above (in this case the electric field).

This is just an example. Another example, which probably makes the usefulness of this theory clearer, is the gravitational potential $V = mgh$. We can choose the origin to be anywhere since we are only interested in the potential energy difference, and this simplifies the calculations a lot (you don't have to care about the origin, only the distance between the points you are investigating).

In Quantum Field Theory, gauge transformations lead to conserved quantities via Noether's Theorem which says that for every continuous symmetry there exists a conserved quantity. A group of independent gauge transformations gives rise to gauge fields. Each generator of the gauge group corresponds to a gauge field that describes gauge bosons.

Answer 3

Another take on gauge theories, to add to ACuriousMind's answer: as well as adding degrees of freedom which allow greater wriggle room to bring a broader class of solution techniques to bear, a gauge theory is a way for a theorist to encode experimentally observed symmetries into the a candidate theory. You might, for example, know from the experimental literature that a certain kind of interaction conserves some experimentally measured continuous quantities, let's call them "blooblehood", "twangleness" and "thargledom" to emphasise the idea's generalness (I fancy these were studied by the Thargoids under their leader theoretical physicist Gort). One way of making a candidate theory conserve blooblehood, twangleness and thargledom in its descriptions of the interaction is to make it a theory described by a Lagrangian and then to set that Lagrangian up so that it is invariant with respect to a Lie group $\mathfrak{G}$ of transformations on its co-ordinates. Noether's theorem then tells you that there will be one conserved quantity for each basis member of the Lie group's Lie algebra $\mathfrak{g}$. This Lie group is then the structure (gauge) group for the fibre bundle formed with $\Sigma$ as the base space (in ACuriousMind's notation and the "gauged" fields are the fibres, as in ACuriousMind's answer. So we postulate a Lagrangian that has a symmetry group of dimension 3 for $\mathfrak{G}$.

Other answers wherein I talk about similar things are here and here. The latter answer has what I hold to be excellent references for nonspecialists, and were how I began to feel I understand such things (albeit still fairly dimly).

Of course, this is not the only way things can be conserved, so conserved quantites don't prove the description has to be a gauge theory. It's just a suck and see kind of approach that makes an analogy with the first gauge theory (Maxwell's electrodynamics) and other kinds physical theories: you hope you can make some falsifiable foretelling with your Lagrangian so an experimentalist can see whether you're on the right track. What's good about a gauge theory is that conservation holds throughout smooth transformations that pass through a smooth path in spacetime, so we're not talking about jumping discontinuously from one point to one displaced a nonzero distance away and our theory stays "local": Richard Feynman talks about this nonuniqueness of a conservative theory when he derives Poynting's theorem in his second volume: see the beginning of Chapter 27 of Vol II

As an aside, a most interesting and highly unusual use for gauge theories is in the field of anholonomic control theory of dynamical systems. A most excellent framework for thinking about the way a falling cat flips over whilst conserving angular momentum is the following: The cat can be described by a manifold $\Sigma$ (to use ACuriousMind's notation) called the "space of cat shapes" and the cat can deform his or her shape to move smoothly between points of the manifold. These shapes are described in a co-ordinate frame that is fixed with respect to axes mounted on the tumbling (i.e. falling and rotating) cat. In the language of fibre bundles, the space of shapes is the base space $\Sigma$, the fibre is the space $SO(3)$ (or $SO(2)\cong U(1)$) of the cat's orientations in space. The bundle's topology is defined by the "parallel transport" notion or connexion one gets by computing the shift in cat orientation that arises, owing to the conservation of angular momentum, from the cat's following of a piecewise $C^1$ path through the space of shapes. The structure (gauge) group is some connected Lie subgroup of $SO(3)$ acting on the fibre $SO(3)$ itself.

I say more about the falling cat in my article:

"Of Cats and their Most Wonderful Righting Reflex" at my website Wet Savanna Animals.

Richard Montgomery's seminal work is:

Richard Montgomery, "Gauge Theory of the Falling Cat", in M.J. Enos, "Dynamics and Control of Mechanical Systems", American Mathematical Society, pp. 193–218, 1993

I wrote my article in the course of reading and understanding Montgomery's ideas. So you MAY find my article gentler, but different kinds of technical exposition work better for different mindsets.