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Section 5.4 Projective and spin representations

Now that we understand the irreducible representations of the Lie algebra \(\mathfrak{su}(2) \cong \mathfrak{so}(3)\text{,}\) let us now go back to the representations of the Lie group \(SO(3)\text{.}\) We know that all irreducible representations of \(\mathfrak{su}(2)\) correspond to irreducible representations of the universal cover \(SU(2)\) by exponentiation. But what do they correspond to in terms of the Lie group \(SO(3)\text{?}\)

To answer this question we need to define projective and spin representations, to which we now turn to.

Subsection 5.4.1 Projective representations

One motivation for the definition of projective representations comes from quantum mechanics. In quantum mechanics, the state of a system is specified by a vector in some vector space (or Hilbert space) of possible states. This state is called the “wave-function” of the system.

But this is not quite precise. The physical probability of finding the system in a state specified by a wave-function \(\psi\) is equal to the norm square \(|\psi|^2 = \psi^* \psi\text{.}\) In particular, two wave-functions that only differ by a phase give the same physical probability. So the wave-function is only really defined up to a phase factor: mathematically, the statement is that the state of a system is a vector in a projective vector space (or projective Hilbert space), where two vectors that differ by overall rescaling are identified.

Now suppose that the system has a group of symmetries. As we have already seen, the group of symmetries act on the space of states of the system. In other words, we get a representation of the group as acting on the space of states. But in fact, since states are only defined up to overall rescaling, we do not really need an honest representation of the group; it is sufficient for the symmetry operations to transform the wavefunctions only up to rescaling. This is the idea behind projective representations.

Definition 5.4.1. Projective representations.

A projective representation of a group \(G\) is a collection of matrices \(T(g)\text{,}\) \(g \in G\text{,}\) such that it preserves the group structure up to a constant:

\begin{equation*} T(g) T(h) = c(g,h) T(g h) \end{equation*}

for some constants \(c(g,h)\text{.}\)

More mathematically, a projective representation of a group \(G\) on a vector space \(V\) over a field \(F\) (such as \(\mathbb{C}\) or \(\mathbb{R}\)) is a group homomorphism from \(G\) to the projectivization of \(GL(V)\text{:}\)

\begin{equation*} T: G \to GL(V)/F^*, \end{equation*}

where elements of \(GL(V)/F^*\) are equivalence classes of invertible linear transformations of \(V\) that differ by overall rescaling.

This is precisely what we want in quantum mechanics, since overall rescaling of the wave-function by a phase factor does not change the physics. In other words, given a group of symmetry for a quantum mechanical system, we know that it acts on the space of states of the system as a projective representation. So our goal to understand the system is to classify projective representations of the group of symmetries.

But how do we construct projective representations of a group \(G\text{?}\) We have already seen that the irreducible representations of a Lie algebra are in one-to-one correspondence with the irreducible ordinary representations of the associated simply-connected Lie group. As for projective representations, the key result is Bargmann's theorem, which tells us that for a large class of groups (which includes \(SO(3)\text{,}\) the Lorentz group and the Poincare group), every projective unitary representation of \(G\) comes from an ordinary representation of the universal cover. Great! In fact, for finite-dimensional representation this is always true regardless of the group. But in quantum mechanics the Hilbert space is often infinite-dimensional, so one has to be careful, but Bargmann's theorem holds for the groups of interest such as rotations, translations and Lorentz transformations.

In the context of \(SO(3)\text{,}\) we know that its universal cover is \(SU(2)\text{.}\) So what this means is that all irreducible projective representations of \(SO(3)\) come from ordinary irreducible representations of \(SU(2)\text{,}\) which in turn correspond to irreducible representations of the Lie algebra \(\mathfrak{su}(2) \cong \mathfrak{so}(3)\text{,}\) which we have already constructed. So we already know all projective representations of \(SO(3)\text{!}\)

Before we look at those representations more closely, let us define a particular type of projective representations that is very important in physics.

Subsection 5.4.2 Spin representations

There is a particular type of projective representations that is fundamental in physics. Those are called spin representations. They are defined for the special orthogonal groups \(SO(n)\) (and more generally for \(SO(p,q)\) - the discussion below can be easily adapted to this more general setup).

Roughly speaking, spin representations are representations of the elements of \(SO(n)\) (which you can think of as rotations) that pick a sign when you rotate by an angle of \(2 \pi\) about an axis. Those are particular types of projective representations, since they preserve the group law only up to a sign. We call spinors the objects that transform according to spin representations of \(SO(n)\text{,}\) which should be contrasted with tensors, which are objects that transform according to ordinary representations of \(SO(n)\text{.}\)

The more precise definition of spin representations is in terms of the double cover of \(SO(n)\text{.}\) We have already seen that \(SO(3)\) has a double cover, which is given by the group \(SU(2)\text{.}\) It turns out that in general, the group \(SO(n)\) has a unique connected double cover. We call this double cover the spin group, and denote it by \(Spin(n)\text{.}\) Then there is a group homomorphism \(Spin(n) \to SO(n)\) whose kernel is \(\mathbb{Z}_2\text{.}\) Spin representations of \(SO(n)\) are defined as being the ordinary representations of the spin group \(Spin(n)\) that do not come from ordinary representations \(SO(n)\text{.}\)

In the case of \(SO(3)\text{,}\) its unique connected double cover is \(SU(2)\text{,}\) and hence \(Spin(3) \cong SU(2)\text{.}\) Therefore, we know that all ordinary representations of \(SU(2)\) that do not come from ordinary representations of \(SO(3)\) are spin representations.

Subsection 5.4.3 Spin representations for \(SO(3)\)

With this under our belt, we can now go back to representation theory of \(SO(3)\text{.}\) In the previous section we constructed all irreducible representations of the Lie algebra \(\mathfrak{su}(2) \cong \mathfrak{so}(3)\) using the highest weight construction. We obtained an infinite tower of irreducible representations, indexed by a non-negative half-integer \(j\text{,}\) with dimensions \(2j+1\text{.}\)

We know that these representations are in one-to-one correspondence with the irreducible representations of the simply connected Lie group \(SU(2)\text{.}\) But what we are often interested in in physics is the group of rotations \(SO(3)\text{,}\) not \(SU(2)\text{.}\) What do the representations of the Lie algebra \(\mathfrak{su}(2) \cong \mathfrak{so}(3)\) correspond to in terms of the non-simply connected Lie group \(SO(3)\text{?}\)

From the previous discussion, from Bargmann's theorem we know that exponentiating the representations of \(\mathfrak{su}(2)\) will generally give projective representations of \(SO(3)\text{.}\) Moreover, since \(SU(2)\) is the unique connected double cover of \(SO(3)\text{,}\) we know that they will all be either ordinary or spin representations of \(SO(3)\text{.}\) How do we distinguish between the two?

Let us first work out an example. Consider the spin-\(1/2\) representation of \(\mathfrak{su}(2)\text{,}\) with \(j=1/2\text{.}\) The representation is given by the Pauli matrices (4.4.1), reproduced here for convenience:

\begin{equation*} T_1 = \frac{1}{2} \begin{pmatrix} 0 \amp 1 \\ 1 \amp 0 \end{pmatrix},\qquad T_2 = \frac{1}{2} \begin{pmatrix} 0 \amp -i \\ i \amp 0 \end{pmatrix},\qquad T_3 = \frac{1}{2} \begin{pmatrix} 1 \amp 0 \\ 0 \amp -1 \end{pmatrix}. \end{equation*}

Now consider two elements of the Lie algebra given by \(\theta_1 T_3\) and \(\theta_2 T_3\) for some \(\theta_1, \theta_2 \in \mathbb{R}\text{.}\) By exponentiation, these give rise to matrices

\begin{equation*} R(\theta_1) = e^{i \theta_1 T_3}, \qquad R(\theta_2) = e^{i \theta_2 T_3}. \end{equation*}

Since exponentiating a diagonal matrix means exponentiating the diagonal components, we get:

\begin{equation*} R(\theta_1) = \begin{pmatrix} e^{i \theta_1 /2} \amp 0 \\ 0 \amp e^{-i\theta_1 /2} \end{pmatrix}, \qquad R(\theta_2) = \begin{pmatrix} e^{i \theta_2 /2} \amp 0 \\ 0 \amp e^{-i\theta_2 /2} \end{pmatrix}. \end{equation*}

Now suppose that \(\theta_1 + \theta_2 = 2 \pi\text{.}\) Then we get:

\begin{align*} R(\theta_1)R(\theta_2) =\amp \begin{pmatrix} e^{i (\theta_1 + \theta_2)/2} \amp 0 \\ 0 \amp e^{-i (\theta_1 + \theta_2)/2} \end{pmatrix} \\ =\amp - \begin{pmatrix} 1 \amp 0 \\ 0 \amp 1 \end{pmatrix}\\ =\amp - I, \end{align*}

where \(I\) is the \(2 \times 2\) identity matrix. But in \(SO(3)\text{,}\) rotating about an axis by \(2 \pi\) should be the identity transformation, not minus the identity transformation! Thus we conclude that this representation does not quite preserve the group structure: it preserves it only up to sign. That is, it is a spin representation of \(SO(3)\text{!}\)

This was somehow expected, because when we constructed representations of \(SO(3)\) in terms of how tensors transform, we only obtained \((2j+1)\)-dimensional representations with \(j\) a non-negative integer: we never saw the representations with \(j\) a half-integer.

But what about the other representations of \(SO(3)\) with half-integer \(j\text{?}\) Are they ordinary or spin representations?

Consider the highest weight construction in the previous section. In this construction, \(\Gamma_3\) was always a diagonal matrix, whose diagonal entries corresponded to the values of \(m\) between \(-j\) and \(j\text{.}\) If \(j\) is a half-integer, then those diagonal entries are all half-integers. Then if we construct matrices \(R(\theta)\) by exponentiating the matrices \(\theta \Gamma_3\text{,}\) the same argument above will go through, and we will always end up with the statement that a rotation by \(2 \pi\) gives minus the identity matrix. (Note that this does not happen if the diagonal values of \(\Gamma_3\) are integers, which is the case when \(j\) is an integer.)

Therefore we obtain the very important result that finite-dimensional irreducible projective representations of \(SO(3)\) come in two classes:

  • An infinite class of ordinary representations, indexed by a non-negative integer \(j\text{,}\) with dimensions \(2j +1\text{.}\) Those are tensor representations, and we call objects that transform according to these representations tensors (for \(j=0\text{,}\) we call them scalars, and for \(j=1\text{,}\) we call them vectors).
  • An infinite class of spin representations, indexed by a positive half-integer \(j\) (which is not an integer), with dimensions \(2j+1\text{.}\) Those are spin representations, and we call objects that transform according to these representations spinors.

Isn't that cool? Now you see how spinors come about in physics. As you may already know, in physics we call particles that transform according to ordinary representations bosons (they have integer spin), and particles that transform according to spin representations fermions (they have half-integer spins). Those have very different physical properties, due to the difference between Bose-Einstein and Fermi-Dirac statistics (the Pauli exclusion principle). For instance, particles that exchange forces, such as photons, gluons, the W and Z bosons, and the Higgs, are bosons. On the other hand, particles that make up matter, such as electrons, protons, muons, quarks, etc., are fermions. Cool, hey?

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