In general, how are representations used in physics?

Question

I want is a basic overview, if there is one, of the meaning (and purpose) of the word representation in general terms. I have looked up sources such as Particle Physics and Representation Theory, but I can't see the general purpose of using representations, beyond its particular applications in particle physics.

In general, what is the point of representation theory?

Answer 1

G. Smith already said everything relevant, I try to dive a bit more into the physical side, which will be only special cases but they cover more than just one field of physics:

Let's assume we are given a field theory described by the Lagrange Density $\mathcal{L}$. Now, I assume you are familiar with Noethers Theorem (which also holds for discrete systems described by a Lagrange Function $L$). Now, given a symmetry of the system the theorem tells us that there have to be conserved quantities. In order to understand how representations are relevant in this context, its enough to understand what exactly a symmetry is. We can assume that there is a Matrix Group $G$ which is acting on the state space (or just the space in which our field takes it's values), and acting in the natural way, e.g. $g \psi$ is the matrix multiplication pointwise: $g \psi (x) = g \cdot \psi(x)$ for $g \in G$. Now let us generalize this! First of all, $g$ might depend on $x$, secondly we can take any group $G$ (not necesarily a matrix group!) and definie a representation, call it $\rho$, on $GL(V)$, where we assumed $\psi: M \to V$, and $M$ is just the space on which $\psi$ lives, so $M= \mathbb{R}^{1,3}$ in the case of minkowskispace (which we assum in what follows, this does not really matter for the whole picture).

That is (by definition), a group homomorphism $$\rho: G \to GL(V). $$

Now, being a symmetry group of a system means (not exactly, but for simplicity let's assume the following, it really does not matter for the whole picture) that $\mathcal{L}$ be invariant under the action of the given group, that is: $$\mathcal{L}(x,\rho(g)\psi, \partial_{\mu}\rho(g) \psi) = \mathcal{L}(x,\psi, \partial_{\mu} \psi) \ \forall g \in G$$ where we allow $g$ to depend on $x$. One can then show, that $\rho(G) \subset GL(V)$ is a Lie Group, so one typically assumes $G$ itself being a Lie Group and $\rho$ be continous.

Thus, in the conetxt of symmetries of a field theory (which is pretty general I'd say!), representations are needed to understand how the symmetry Group affects the system, which in turn is needed to study the conserved quantities corresponding to this symmetry group!

In short we say, that each symmetry gives rise to conserved quantities, by which really the above is meant, and the way we call it is more or less a consequence of being lazy about technical details. Again, this holds for any field theory (classical or quantum) as well as for discrete systems, so for almost any theory we have!

Answer 2

You're asking about "representations" in the group-theoretic sense. You can think of a representation simply as a way to make an abstract group concrete, by expressing each group element as a numerical matrix.

This may seem odd because a lot of groups in physics are defined as matrix groups... for example, SU(3) is defined as the group of unitary 3x3 matrices with determinant 1. But it turns out that there are faithful irreducible representations of SU(3) as matrices of an infinite number of other sizes: 6x6, 8x8, 10x10, 15x15, ... (but not 2x2, 4x4, 5x5, 7x7, ...).

So, for example, in an 8-dimensional representation of SU(3), each group element is represented by an 8x8 matrix rather than a 3x3 matrix.

("Faithful" means that the mapping is one-to-one. "Irreducible" means that a representation can't be broken down into smaller representations.)

In particle physics, the color force that binds quarks into protons and neutrons has SU(3) as its gauge symmetry. The 3x3 matrices tell you how the three colors of, say, an up quark mix with each other under a gauge transformation. The 8x8 matrices tell you how the eight gluons mix with each other.

Some groups have non-matrix definitions and can seem extremely abstract. For example, G₂ can be defined as the symmetry group of the octonions. But each representation of G₂ is just a set of nxn matrices. The basic idea from a mathematical standpoint is that one can represent any abstract group as a set of linear transformations of an n-dimensional vector space, but the specific values of n for which this is possible depend on the group.

Answer 3

Let me give a slightly different point of view.

A group can be thought of as a set of symmetries. Different possible symmetries correspond to different groups, and since symmetry is an underlying principle in most of physics, groups find their way into most aspects of physics. For instance, reflection in a plane corresponds to the group $\mathbb{Z}_2$.

For a given symmetry group, a group action is a way in which that symmetry can act on an object. This is clearly of relevance in physics, since we care about how different physical objects behave when we perform the symmetry operations in question. Under a reflection in space, for instance, we know that the momentum of a particle flips sign, whilst its energy stays the same.

Indeed, the way that objects transform under symmetry operations can be a useful way of classifying those objects. At school one learns about the difference between a vector and a scalar. We can define a scalar as something which is unchanged by a rotation, whereas a vector is something which is, well, rotated. A tensor is yet another class of object, and this also changes when we perform a rotation, but it changes in a different manner to an ordinary vector.

A representation is merely an action of a group on a vector space. The prevalence of representations in physics is then a consequence of the prevalence of vector spaces. In particular, nearly all physical quantities are vectors – at least in the abstract sense. Recall that both $\mathbb{R}$ and $\mathbb{C}$ are vector spaces. So energies are vectors; position in space is a vector; the inertia tensor is a vector; quantum states are elements of a Hilbert space – so they are vectors; the electric field at a point is a vector; real- and complex-valued fields are vectors (in some function space); and so on. Remember that for an object to be a vector, it is only required that we can add such objects together and multiply them by numbers.

This is why representations crop up in so many areas of physics.