Particle Physics and Representations of Groups

Question

This question is asked from a point of complete ignorance of physics and the standard model.

Every so often I hear that particles correspond to representations of certain Lie groups. For a person completely ignorant of anything physics, this seems very odd! How did this come about? Is there a "reason" for thinking this would be the case? Or have observations in particle physics just miraculously corresponded to representation theory? Or has representation theory of Lie groups grown out of observations in particle physics?

In short: what is the chronology of the development of representation theory and particle physics (with relation to one another), and how can one make sense of this relation in any other way than a freakish coincidence?

Answer 1

The "chronology" isn't clear to me, and having looked through the literature it seems much more convoluted than it should be. Although it seems like this is basically how things were done since the beginning of quantum mechanics (at least, by the big-names) in some form or another, and was 'partly' formalized in the '30s-'40s with the beginnings of QED, but not really completely carefully formalized until the '60s-'70s with the development of the standard model, and not really mathematically formalized until the more careful development of things in terms of bundles in the '70s-'80s. (These dates are guesses--someone who was a practicing physicist during those periods is more than welcome to correct my timeline!)

Generally speaking, from a 'physics' point of view, the reason particles are labeled according to representations is not too different than how, in normal quantum mechanics, states are labeled by eigenvalues (the wiki article linked to mentions this, but it's not as clear as it could be).

In normal QM, we can have a Hilbert space ('space of states') $\mathcal{H}$, which contains our 'physical states' (by definition). To a physicist, 'states' are really more vaguely defined as 'the things that we get the stuff that we measure from,' and the Hilbert space exists because we want to talk about measurements. The measurements correspond to eigenvalues of operators (why things are 'obviously' like this is a longer historical story...).

So we have a generic state $| \psi \rangle \in \mathcal{H}$, and an operator that corresponds to an observable $\mathcal{O}$. The measured values are

$\mathcal{O} |\psi\rangle = o_i | \psi \rangle$.

Because the $o_i$ are observable quantities, it's useful to label systems in terms of them.

We can have a list of observables, $\mathcal{O}_j$, (which we usually take to be commuting so we can simultaneously diagonalize), and then we have states $|\psi\rangle$,

$\mathcal{O}_j | \psi \rangle = {o_i}_j | \psi \rangle$.

So, what we say, is that we can uniquely define our normal QM states by a set of eigenvalues $o_{ij}$.

In other words, the $o_{ij}$ define states, from the physics point of view. Really, this defines a basis where our operators are diagonal. We can--and do!--get states that do not have observables which can be simultaneously diagonalized, this happens in things like neutrino oscillation, and is why they can turn into different types of neutrinos! The emitted neutrinos are emitted in states with eigenvalues which are not diagonal in the operator that's equivalent to the 'particle species' operator. (Note, we could just as well define the 'species' to be what's emitted, and then neutrinos would not oscillate in this basis, but would in others!)

This has to do with representations, because when we talk about particles with spin, for example, we're talking about operators which correspond to 'angular momentum.' We have an operator:

$L_z = i \frac{\partial}{\partial\phi}$

and label eigenvalues by half-integer states which physically correspond to spin. Group theoretically, $L_z$ comes from the lie algebra of the rotation group, because we're talking about angular momentum (or spin) which has associated rotational symmetries.

Upgrading from here to quantum field theory (and specializing that to the standard model) is technically complicated, but is basically the same as what's going on here. The big difference is, we want to talk there about 'quantum fields' instead of states, and have to worry about crazy things like apparently infinite values and infinite dimensional integrals, that confuse the moral of the story.

But the idea is simply, we want to identify things by observables, which correspond to eigenvalues, which correspond to operators, which correspond to lie algebra elements, which have an associated lie group.

So we define states corresponding to things which transform under physically convenient groups as 'particles.'

If you want a more mathematically careful description, that's still got some physical intuition in it, you can check out Gockler and Schuker's "Differential Geometry, Gauge theory, and Gravity," which does things from the bundle point of view, which is slightly different than I described (because it describes classical field theories) but the moral is similar. At first it might seems surprising that the classical structure here is the same, when it seemed to rely on operators and states in Hilbert spaces, but it only technically relied on it, but morally, what's important is actions under symmetry groups. And that is in the classical theory as well. But it's not as physically clear from the beginning from that point of view.

Answer 2

Let me add a little bit to what has already been written above.

The current framework for quantum mechanics motivates the study of projective (anti)unitary representations of the symmetry groups of the physical system. In the context of four-dimensional relativistic quantum field theory (with some mild assumptions), it follows from a celebrated theorem of Coleman and Mandula that the symmetry group is a direct product of (the universal cover of) the Poincaré group and a compact Lie group. (One can get around this theorem by considering not Lie groups but Lie supergroups, but that's another story.) Let me focus on the Poincaré group. The determination of the (physically relevant) unitary irreducible representations of the Poincaré group is due to Wigner, generalising (though perhaps not consciously) Frobenius's method of induced representations. It was Mackey who extended Wigner's method and placed it firmly in the "right" mathematical context.

The point I would like to make is that approaching the representation theory of the Poincaré group (however one motivates this study) in this fashion naturally makes contact with particle physics.

Induced representations

Let us start with finite groups. Let $G$ be a finite group and $H$ be a subgroup and let $\delta: H \to \mathrm{U}(W)$ be a unitary representation on a finite-dimensional hermitian vector space $W$. Consider the vector space $V$ of functions $f:G \to W$ subject to the equivariance condition $f(gh) = \delta(h^{-1}) f(g)$ for all $g \in G$ and $h \in H$. The homomorphism $\rho:G \to \mathrm{GL}(V)$ defined by $$(\rho(g) f)(g') = f(g^{-1}g')$$ makes $V$ into a representation of $G$. (One has to check that $\rho(g) f \in V$ again.)

Moreover, it is possible to define on $V$ a hermitian structure relative to which $\rho$ is a unitary representation. This is best seen by viewing $V$ in a different light. Let $X= G/H$ be space of left $H$-cosets in $G$. Then $V$ is isomorphic to the vector space of functions $\psi: X \to W$, but not canonically. The isomorphism depends a choice of coset representative $\sigma: X \to G$, a section through the surjection $\pi: G \to X$. Then given $f \in V$ we define $\psi(x)= f(\sigma(x))$. Conversely, given $\psi:X \to G$ we define $f \in V$ by writing $g = \sigma(\pi(g))h(g)$ for some $h(g) \in H$ and declaring $f(g) = \delta(h(g)^{-1}) \psi(\pi(g))$. Then we define the inner product of $f_i \in V$ to be $$ \langle f_1,f_2\rangle_V = \sum_{x\in X} \langle f_1(\sigma(x)), f_2(\sigma(x)) \rangle_W.$$ One can show that this is independent of the coset representative precisely because $W$ is a unitary representation of $H$.

The representation $V$ of $G$ is said to be induced from the representation $W$ of $H$.

Wigner's method

Wigner's method is formally very similar: $G$ is the Poincaré group; that is, the semidirect product $L \ltimes T$, where $L = \mathrm{Spin}(3,1)$ is the spin cover of the Lorentz group and $T$ is the translation ideal. Wigner starts by choosing a character $p$ of $T$, which physically is interpreted as a momentum. A version of Schur's Lemma says that on an irreducible representation of the $G$, all the characters of $T$ which appear share the same minkowskian norm $p^2$. Physically relevant representations have $p^2 = - m^2$, where $m\geq 0$ is the mass.

Let $H < G$ denote the stabilizer of $p$. Wigner induces a unitary representation of the Poincaré group from a finite-dimensional unitary representation of $H$. Now $H$ is non-compact, so such representations are necessarily not faithful. They factor through faithful representations of a group known as Wigner's little group. It is isomorphic to $\mathrm{Spin}(3)$ for $m>0$ and $\mathrm{Spin}(2)$ for $m=0$. Irreducible finite-dimensional representations of the little groups are labelled by integers: the spin (a non-negative integer) for $m>0$ the helicity for $m=0$. So Wigner tells us that to a unitary irreducible representation of the Poincaré group one can associate a mass and a spin/helicity, which are the basic data specifying relativistic particles. But there's more.

The space $G/H$ is the hyperboloid $p^2 = - m^2$ (for a fixed mass $m\geq 0$) in the dual to the Lie algebra of $T$. The vector space carrying the induced representation of $G$ consists of (square-integrable) sections of homogeneous vector bundles over $G/H$ associated to the representation of $H$ from which we induce. Thus this gives naturally a representations on geometric objects defined in the space $G/H$ of momenta. Fourier transforming to Minkowski spacetime we arrive at sections of homogeneous bundles over Minkowski spacetime satisfying (linear) partial differential equations coming from Fourier transforming the condition $p^2 = -m^2$ and the other irreducibility conditions. And the nice surprise is that these partial differential equations are precisely the linearised free field equations for the corresponding particles: the Klein-Gordon, Dirac, Weyl, Maxwell,... equations!

Answer 3

My understanding is that this started with the Dirac wave equation. This was a relativistic equation for an electron. However it happened to also introduce the idea that a point particle could have an internal state space. This was a successful theory and was taken up and imitated when it came to probing the structure of the nucleus.

For a description of the standard model a good place to start is: http://arxiv.org/abs/0904.1556

Answer 4

There is an excellent introduction by John Baez and John Huerta to the Standard Model and Lie groups theory in the Bulletin of the AMS, vol 47, no. 3, July 2010. In particular, it gives plenty of references and historical notes.

I just noticed that this is the same article as the one suggested by Bruce, sorry!

Answer 5

As a physicist, I may be able to give a different perspective on this question. In particular, many of the responses so far have been about quantum mechanics and quantum field theory (which involve Lie groups), but if the question is, "Why is the particle content of physics theories derived from Lie groups?" then the answer is not specifically about the theories' quantumness. It's about their geometry, which can be discussed separately from quantum effects.

In 1926, Kaluza and Klein attempted to unify electromagnetism with gravity by proposing a theory of General Relativity with 5 dimensions (4 spatial). Since we don't macroscopically experience this extra degree of freedom, they proposed that it is topologically like a cylinder with a small radius, so small that the extra degree of freedom can't be probed as a direction. This degree of freedom does, however, allow us to encode classical electromagnetism as part of the geometry of space-time. We'll see in a moment that while this formulation isn't exactly right, it does show how the differential geometry concepts of General Relativity can be used in particle physics theories, leading to a unification of all four forces at a classical level. (It's the quantization of gravity that's the hard part.)

The Lagrangian of quantum electrodynamics (late 1940's) is just the Lagrangian of the Dirac equation with an additional requirement: that the spinor field $\psi$ has a local $\mathcal{U}(1)$ symmetry. I'll use the same notation as the Wikipedia article, except that I'll use $c = \hbar = 1$. The Dirac Lagrangian

$\mathcal{L_D} = m\bar{\psi}\psi - \frac{i}{2}\left(\bar{\psi} \gamma^\mu (\partial_\mu\psi) - (\partial_\mu\bar{\psi}) \gamma^\mu \psi \right)$ (1)

has a global $\mathcal{U}(1)$ symmetry in that the complex phases of components of $\psi$ cancel in the $\bar{\psi}\psi$ terms: multiplying all instances of $\psi$ by $e^{i\alpha}$ for some constant $\alpha$ would not change the value of $\mathcal{L}$. The Dirac equation does not have a local $\mathcal{U}(1)$ symmetry, that is, invariance under

$\psi(x,t) \to e^{i\theta(x,t)} \psi(x,t)$ where everything is a function of 4-D space-time points $(x,t)$ (2).

If we want to create a new Lagrangian which does have a local $\mathcal{U}(1)$ symmetry, we find that we would need to replace the derivative operators $\partial_\mu$ with

$D_\mu = \partial_\mu - iqA_\mu$ (3)

where $A$ is a new field with the transformation property

$A_\mu(x,t) \to A_\mu(x,t) + \frac{1}{q}\partial_\mu \theta(x,t)$ (4).

The new theory has a Lagrangian

$\mathcal{L_{QED}} = m\bar{\psi}\psi - \frac{i}{2}\left(\bar{\psi} \gamma^\mu (D_\mu\psi) - (D_\mu\bar{\psi}) \gamma^\mu \psi \right) + \frac{1}{4}\left((\partial_\mu A_\nu - \partial_\nu A_\mu)(\partial^\mu A^\nu - \partial^\nu A^\mu)\right)$ (5)

where the last term is required to preserve symmetry under Lorentz boosts (conservation of energy in the new $A$ field). Just following the consequences of a local $\mathcal{U}(1)$ symmetry, we have turned freely-streaming Dirac Lagrangian into the interacting electromagnetic Lagrangian, where we can interpret $\psi$ as charged particle (e.g. electron) waves and $A$ as the vector potential of electromagnetism, which is to say, the photon waves. The transformation of Eqns (2) and (4) is the gauge transformation of electromagnetism: we've learned that the electromagnetic gauge symmetry is fundamentally a local $\mathcal{U}(1)$ symmetry.

Getting back to Kaluza and Klein's theory, a 5th compactified dimension is a little like having a $\mathcal{U}(1)$ invariance at every point in 4-D space, since it's hard to see where we are in the loop of the 5th dimension. It's not exactly the same thing: with an extra dimension, we should in principle be able to perform rotations in which spatial dimensions and the extra dimension mix, while that would not be possible in a 4-D space plus $\mathcal{U}(1)$ fiber bundle. (This difference is perhaps related to the reason Kaluza and Klein's original theory didn't work...?) If we generalize our notion of space-time to include the $\mathcal{U}(1)$ fibers, we can think about electromagnetism and General Relativity in the same terms. For instance, the photon field $A$ plays the same role in the $\mathcal{U}(1)$ symmetry as the connection/covariant derivative in the local Lorentz symmetry of the space-time metric. That is, the classical photon field is the "curvature" of the fiber bundle in the same sense that gravitation is the curvature of space-time.

Moreover, this picture unifying the geometry of electromagnetism with the geometry of gravity also works for all the other known forces. In 1954, Yang and Mills generalized the "local $\mathcal{U}(1)$-to-electromagnetism" idea to work for any Lie group, including non-Abelian ones. The Yang-Mills idea wasn't popular at first because it didn't seem to describe the nuclear strong force (but that was based on a wrong assumption that the nuclear force is a Yukawa interaction). By the late 1960's, Weinberg derived a unified electro-weak theory from local $\mathcal{SU}(2)\times\mathcal{U}(1)$, and Han and Nambu derived a theory of nuclear strong force from $\mathcal{SU}(3)$. (I'm skipping over many important contributions for brevity.) By the mid-1970's or early 1980's, depending on who I ask, this became known as the Standard Model of particle physics because of its experimental success.

We can think about the Standard Model geometrically as an $\mathcal{SU}(3)\times\mathcal{SU}(2)\times\mathcal{U}(1)$ at every point in 4-D space-time, with the gluon, W and Z bosons, and photon being connections through groups at neighboring points of space-time, constantly arranging themselves to hide information about the components of matter fields in all of these "internal" degrees of freedom. The structures of the groups are directly responsible for the charges and interactions of the matter fields (quarks and leptons), but the matter fields themselves are not derived from the groups (supersymmetry might change that part of the picture). There is a direct analogy between these group connections (the gluon, W, Z, and photon) and the space-time connection in General Relativity (which we could call a graviton field, if you wish). I have said nothing at this point about the quantization of all of these fields, which further complicates the picture, especially in the case of gravity!

By the way, I would love to know more about the curvature of fiber bundles, in order to understand the above at a deeper mathematical level. If you have any suggested reading, I'm interested. Thanks!