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Section 5.7 Representations of the Lorentz group

So far we studied representations of \(SU(N)\text{,}\) and ordinary and spin representations for \(SO(3)\text{.}\) But the symmetry group of spacetime in special relativity is the Lorentz group, which is composed of Lorentz transformations. Thus in quantum field theory, which is the unique formalism that naturally combines quantum mechanics and special relativity, we want to think of our fields in terms of how they transform under the Lorentz group. That is, we want to think of them in terms of representations of the Lorentz group. This is what we study in this section.

Subsection 5.7.1 Representations of \(SO(4)\)

Before we look at the Lorentz group it is instructive to look at the group of rotations in four dimensions, \(SO(4)\text{.}\) This is because the Lorentz group \(SO(3,1)\) is also a group of rotations in four dimensions, but with respect to the Minkowski metric instead of the Euclidean metric. As a result, many of the properties of representations of \(SO(4)\) carry through to the Lorentz group \(SO(3,1)\text{.}\)

Remark 5.7.1.

More precisely, in this section we focus on the component of \(SO(3,1)\) that is connected to the identity, which is often denoted by \(SO(3,1)^+\text{.}\) This is the group that is physically represented in terms of Lorentz transformations of spacetime. To avoid cluttering notation in the following we will denote this component by \(SO(3,1)\text{,}\) but keep in mind that we always only consider its component connected to the identity.

Subsubsection 5.7.1.1 The spin group \(Spin(4)\)

The double cover of \(SO(4)\) is the spin group \(Spin(4)\text{.}\) We know that the representations of \(Spin(4)\) all descend to either ordinary or spin representations of \(SO(4)\text{,}\) depending on whether they assign the same matrix to the elements of the kernel of the map \(Spin(4) \to SO(4)\) (if they do, they descend to ordinary representations; if they do not, they descend to spin representations). Thus, to understand the representations of \(SO(4)\text{,}\) we want to study representations of \(Spin(4)\text{.}\)

In the case of \(SO(3)\text{,}\) we found that \(Spin(3) \cong SU(2)\text{,}\) which was nice and easy. It turns out that \(Spin(4)\) is also isomorphic to nice and easy group: \(Spin(4) \cong SU(2) \times SU(2)\text{.}\)

Remark 5.7.2.

Note that the spin groups \(Spin(N)\) in general are not necessarily isomorphic to such nice and easy groups. But there are such isomorphisms for small enough \(N\text{.}\) For instance, \(Spin(6) \cong SU(4)\text{.}\)

How can we see that \(Spin(4) \cong SU(2) \times SU(2)\text{?}\) Intuitively, one can think of it as follows. The group \(SU(2)\) is in fact the unit three-sphere \(S^3\) as a manifold, as we saw in Example 4.1.10. In other words, an element of \(SU(2)\) corresponds to a point on a \(S^3\text{,}\) that is, a point in \(\mathbb{R}^4\) that satisfies the equation

\begin{equation*} x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1. \end{equation*}

Thus, specifying an element of the product group \(SU(2) \times SU(2)\) is equivalent to specifying two elements of \(SU(2)\text{,}\) or, in other words, two points on the three-sphere \(S^3\text{.}\) But the group of symmetries of a three-sphere is the group of rotations \(SO(4)\) in four dimensions. Thus, these two points will be related by a rotation. In summary, to an element of \(SU(2) \times SU(2)\text{,}\) we can assign an element of \(SO(4)\text{,}\) corresponding to the rotation that brings the first point on \(S^3\) to the second. Finally, one sees that this mapping is not one-to-one, since the two points \(U,V \in S^3\) and the opposite points \(-U, -V \in S^3\) will be mapped to the same rotation in \(SO(4)\text{.}\)

From the point of view of Lie algebras, the statement that \(SU(2) \times SU(2)\) is a double cover of \(SO(4)\) implies that they must share the same Lie algebra \(\mathfrak{so}(4) \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)\) (the Lie algebra of a product group is the direct sum of the Lie algebras of the individual factors). Let us see this explicitly.

Following along the same lines as we did for \(\mathfrak{so}(3)\) and \(\mathfrak{su}(2)\text{,}\) we can find an explicit description of the Lie algebra \(\mathfrak{so}(4)\text{.}\) It is six-dimensional, and there is a natural choice of basis, which we denote by \(J_i, K_i\) with \(i=1,2,3\text{,}\) with the bracket:

\begin{align} [J_i, J_j] =\amp i \sum_{k=1}^3 \epsilon_{ijk} J_k,\tag{5.7.1}\\ [J_i, K_j] =\amp i \sum_{k=1}^3 \epsilon_{ijk} K_k,\tag{5.7.2}\\ [K_i, K_j] =\amp i \sum_{k=1}^3 \epsilon_{ijk} J_k.\label{equation-lie-so4}\tag{5.7.3} \end{align}

To see the isomorphism with the Lie algebra \(\mathfrak{su}(2) \oplus \mathfrak{su}(2)\text{,}\) we do a change of basis. We define a new basis

\begin{equation*} J_{\pm, i} = \frac{1}{2}(J_i \pm K_i), \qquad i=1,2,3. \end{equation*}

It is straightforward to check that the bracket becomes, in this new basis,

\begin{align*} [J_{+,i}, J_{-,j}] =\amp 0, \qquad \text{for all } i,j=1,2,3,\\ [J_{+,i}, J_{+,j} =\amp i \sum_{k=1}^3 \epsilon_{ijk} J_{+, k},\\ [J_{-,i}, J_{-,j} =\amp i \sum_{k=1}^3 \epsilon_{ijk} J_{-, k}. \end{align*}

We recognize that the \(J_{+,i}\) and the \(J_{-,i}\) generate two copies of the \(\mathfrak{su}(2)\) Lie algebra. Moreover, those commute, and hence the Lie algebra is a direct sum \(\mathfrak{su}(2) \oplus \mathfrak{su}(2)\text{.}\) We conclude that \(\mathfrak{so}(4) \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2).\)

Subsubsection 5.7.1.2 Representations of \(SO(4)\)

Now that we have established that \(Spin(4) \cong SU(2) \times SU(2)\text{,}\) we can find the irreducible representations of \(Spin(4)\text{.}\) The irreducible representations of \(SU(2) \times SU(2)\) are given by tensor products \(T \otimes S\) of irreducible representations \(S\) and \(T\) of \(SU(2)\text{.}\) But from Section 5.3, we already know the irreducible representations of \(SU(2)\text{.}\) There is an infinite tower of finite-dimensional irreducible representations, indexed by a non-negative half-integer \(j\text{,}\) of dimension \(2j+1\text{.}\) Therefore, the irreducible representations of \(SU(2) \times SU(2)\) are indexed by two non-negative half-integers \(j_1\) and \(j_2\text{,}\) and have dimensions \((2j_1 + 1)(2j_2 + 1)\text{.}\)

We know that all these representations descend to either ordinary irreducible representations of \(SO(4)\text{,}\) or spin representations. We also know that all irreducible ordinary and spin representations of \(SO(4)\) arise from irreducible representations of \(Spin(4)\text{.}\) So to conclude our study of representation theory of \(SO(4)\text{,}\) all we need to do is determine which of the representations above descend to ordinary representations of \(SO(4)\text{,}\) and which descend to spin representations.

We will simply give a rough argument here. To determine whether a representation of \(SU(2) \times SU(2)\) descends to ordinary representation of \(SO(4)\text{,}\) we look at the kernel of the double covering \(SU(2) \times SU(2) \to SO(4)\text{.}\) If the given representation assigns the same matrix to all elements of the kernel, then it descends to an ordinary representation of \(SO(4)\text{.}\) If it doesn't, it descends to a spin representation of \(SO(4)\text{.}\)

The kernel of the double covering \(SU(2) \times SU(2) \to SO(4)\) is \(\mathbb{Z}_2\text{.}\) Its non-trivial element is given by the tensor product of the two non-trivial elements in the kernel of the mapping \(SU(2) \to SO(3)\) that we studied before. In terms of the fundamental representation of \(SU(2)\) given by \(2 \times 2\) special unitary matrices, the kernel of \(SU(2) \to SO(3)\) was given by the two matrices \(\{I, -I \}\text{,}\) where \(I\) is the \(2 \times 2\) identity matrix. For the mapping \(SU(2) \times SU(2) \to SO(4)\text{,}\) the kernel is then given by the tensor products \(\{U \otimes I, (-I) \otimes (-I) \}\text{.}\)

Let us continue with this particular representation. It corresponds to \(j_1 = j_2 = 1/2\text{.}\) It is a four-dimensional representation of \(SU(2) \times SU(2)\text{.}\) Does it descend to an ordinary representation of \(SO(4)\text{?}\) To the identity element it assigns the matrix \(I \otimes I\text{,}\) which is of course the \(4 \times 4\) identity matrix. To the non-trivial element in the kernel, it assigns the \(4 \times 4\) matrix given by the tensor product

\begin{equation*} (-I) \otimes (-I) = I \otimes I, \end{equation*}

which is again the \(4 \times 4\) identity matrix. Therefore, it descends to an ordinary representation of \(SO(4)\text{:}\) it is in fact the fundamental representation of \(SO(4)\) (which is interestingly built by taking the tensor product of two copies of the spin-\(1/2\) representation of \(SU(2)\)).

In general, the non-trivial element of the kernel is given by the matrix obtained as the tensor product \(e^{2 \pi i T_3^{(1)}} \otimes e^{2 \pi i T_3^{(2)}}\text{,}\) where we use the notation of Section 5.4. Recall that for a given irreducible representation of \(SU(2)\) indexed by a half-integer \(j\text{,}\) the matrix \(T_3\) is a diagonal matrix with entries that are half-integers (but not integers). Thus \(e^{2 \pi i T_3}\) becomes minus the identity matrix. If \(j\) is an integer, then \(T_3\) is a diagonal matrix with integer entries, and hence \(e^{2 \pi i T_3}\) is the identity matrix. Therefore, since minus signs cancel in the tensor product, we conclude that a representation of \(SU(2) \times SU(2)\) descends to an ordinary representation if and only if \(j_1 + j_2\) is an integer. It is spin otherwise.

In the case of \(SO(3)\text{,}\) we showed in Lemma 5.5.2 that the \(2\)-dimensional spin representation of \(SO(3)\text{,}\) which descends from the fundamental representation of \(Spin(3) \cong SU(2)\text{,}\) is pseudo-real. For \(SO(4)\text{,}\) we have found two lowest-dimensional spin representations, the \((1/2,0)\) and \((0,1/2)\text{,}\) which also have dimension \(2\text{.}\) One can ask: are those complex conjugate of each other, or are they both independently real or pseudo-real?

We will not prove this here, but it turns out that for \(SO(4)\text{,}\) the two \(2\)-dimensional spin representations are pseudo-real. In fact, there is a nice classification result about spin representations of \(SO(N)\) in general. One needs to distinguish between \(SO(2N)\) and \(SO(2N+1)\text{.}\) For \(SO(2N)\text{,}\) there are always two lowest-dimensional spin representations, and they have dimension \(2^{N-1}\text{.}\) For \(SO(2N+1)\text{,}\) there is only one lowest-dimensional spin representation, and it has dimension \(2^{N-1}\text{.}\) The type of these spin representations (complex, real or pseudo-real) is summarized in the table below. It turns out that the pattern is periodic, with periodicity eight. So, if we write \(SO(8 k +m)\) for some integers \(k \in \mathbb{Z}\) and \(0 \leq m \leq 7\text{,}\) then the type of spin representations depends only on \(m\text{.}\)

\(m\) \(0\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\)
Type Real Real Complex Pseudo-real Pseudo-real Pseudo-real Complex Real
Table 5.7.5. Spin representations for \(SO(8 k + m)\)

To end this section, let us check that the table is consistent with what we have found so far. \(SO(3)\) corresponds to \(k=0\) and \(m=3\text{,}\) which says that the one \(2\)-dimensional spin reprepresentation is pseudo-real, which is what we have found. For \(SO(4)\text{,}\) we have \(k=0\) and \(m=4\text{,}\) and hence from the table we conclude that the two \(2\)-dimensional spin representations are pseudo-real, which is indeed what we stated above. Great!

Subsection 5.7.2 Representation theory of the Lorentz group

This is all very nice, but what does it have to do with the Lorentz group \(SO(3,1)\text{?}\) The Lorentz group is certainly more complicated. For instance, it is non-compact, which implies that it has no non-trivial finite-dimensional unitary representations. Fortunately, it is however still semi-simple, so all representations are equivalent to direct sums of irreducible representations. So we only need to care about irreducible representations. In any case, at the level of Lie algebras, the irreducible representations of \(SO(3,1)\)are indexed in a way very similar to those of \(SO(4)\text{.}\) Let us see why.

The Lie algebra \(\mathfrak{so}(4)\) was presented in (5.7.3). It turns out that the Lie algebra \(\mathfrak{so}(3,1)\) of the Lorentz group is very similar. It is also six-dimensional, and can be written in terms of generators \(J_i, K_i\) with \(i=1,2,3\) as

\begin{align} [J_i, J_j] =\amp i \sum_{k=1}^3 \epsilon_{ijk} J_k,\tag{5.7.4}\\ [J_i, K_j] =\amp i \sum_{k=1}^3 \epsilon_{ijk} K_k,\tag{5.7.5}\\ [K_i, K_j] =\amp - i \sum_{k=1}^3 \epsilon_{ijk} J_k.\label{equation-lie-so31}\tag{5.7.6} \end{align}

The only difference with (5.7.3) is the sign in the bracket \([K_i, K_j]\text{.}\) This sign is however quite important.

To determine the representations of \(SO(3,1)\text{,}\) we want to construct its double cover, as we did for \(SO(4)\text{.}\) Let us first do it at the level of Lie algebras. As in (5.7.3), we want to find a change of basis to construct an isomorphism of Lie algebras.

The key is to realize that if we do a change of basis \(K_j \mapsto i K_j\text{,}\) the brackets in (5.7.6) become exactly equal to those in (5.7.3). But we are only allowed to do such a change of basis if we consider the “complexified Lie algebra”, that is, we consider the complex vector space with basis given by the \(J_i, K_i\text{.}\) We denote this complex Lie algebra as \(\mathfrak{so}(3,1)_{\mathbb{C}}\text{.}\) Thus what we have found is that the complexified Lie algebra \(\mathfrak{so}(3,1)_{\mathbb{C}}\) and \(\mathfrak{so}(4)_{\mathbb{C}}\) are isomorphic. In particular, doing the change of basis as for (5.7.3), we conclude that

\begin{equation*} \mathfrak{so}(3,1)_{\mathbb{C}} \cong \mathfrak{su}(2)_{\mathbb{C}} \oplus \mathfrak{su}(2)_{\mathbb{C}}. \end{equation*}

But in the end we want to work with the real Lie algebra \(\mathfrak{so}(3,1)\text{.}\) So we want to turn the right-hand-side into the complexification of a single Lie algebra (instead of direct sum of complex Lie algebras).

To do so we proceed through a sequence of isomorphisms of Lie algebra. The key is the isomorphism \(\mathfrak{su}(2)_{\mathbb{C}} \cong \mathfrak{sl}_2(\mathbb{C})\text{,}\) which can be checked from the commutation relations of those. Then we get:

\begin{align} \mathfrak{so}(3,1)_{\mathbb{C}} \cong\amp \mathfrak{su}(2)_{\mathbb{C}} \oplus \mathfrak{su}(2)_{\mathbb{C}}\tag{5.7.7}\\ \cong \amp \mathfrak{sl}_2(\mathbb{C}) \oplus \mathfrak{sl}_2(\mathbb{C})\tag{5.7.8}\\ \cong \amp \mathfrak{sl}_2(\mathbb{C}) \oplus i \mathfrak{sl}_2(\mathbb{C})\tag{5.7.9}\\ \cong \amp \mathfrak{sl}_2(\mathbb{C})_{\mathbb{C}},\label{equation-iso-lorentz}\tag{5.7.10} \end{align}

where in the last line we mean the complexification of the real Lie algebra \(\mathfrak{sl}_2(\mathbb{C})\) associated to the Lie group \(SL(2,\mathbb{C})\text{.}\)

As a result, we get \(\mathfrak{so}(3,1)_{\mathbb{C}} \cong \mathfrak{sl}_2(\mathbb{C})_{\mathbb{C}}\text{,}\) which is an isomorphism between the complexifications of two Lie algebras. Upon restriction to the real Lie algebras this becomes the isomorphism

\begin{equation*} \mathfrak{so}(3,1) \cong \mathfrak{sl}_2(\mathbb{C}). \end{equation*}

Thus the Lie groups \(SO(3,1)\) and \(SL(2,\mathbb{C})\) share the same Lie algebra.

It turns out that \(SL(2,\mathbb{C})\) is simply connected, and is in fact the universal cover of \(SO(3,1)\text{.}\) Moreover, the map \(SL(2,\mathbb{C}) \to SO(3,1)\) is a double covering, and hence \(Spin(3,1) \cong SL(2,\mathbb{C})\text{.}\) We have found the spin group \(Spin(3,1)\text{!}\) Thus all finite-dimensional irreducible projective representations of \(SO(3,1)\) descend from irreducible finite-dimensional representations of \(SL(2,\mathbb{C}).\)

What are the irreducible representations of \(SL(2,\mathbb{C})\text{?}\) Well, it is easier to work at the level of Lie algebras. I will be rather sketchy here for brevity. The isomorphisms (5.7.10) are very useful. First, it turns out that the complex representations of the complexification \(\mathfrak{sl}_2(\mathbb{C})_{\mathbb{C}}\) are in one-to-one correspondence with the real representations of \(\mathfrak{sl}_2(\mathbb{C})\text{.}\) Thus, the latter are in one-to-one correspondence with the complex representations of \(\mathfrak{su}(2)_\mathbb{C} \oplus \mathfrak{su}(2)_\mathbb{C}\text{.}\) But those are equivalent to the representations we constructed earlier when we studied \(SO(4)\text{.}\) They are indexed by two half-integers \(j_1\) and \(j_2\) and have dimensions \((2j_1 + 1)(2j_2+1)\text{.}\) Therefore, the same is true of the real irreducible representations of \(SL(2,\mathbb{C})\text{.}\)

It turns out that as for \(SO(4)\text{,}\) the representations with \(j_1 + j_2 \in \mathbb{Z}\) descend to ordinary irreducible representations of \(SO(3,1)\text{,}\) while the other ones are spin. The \((1/2,0)\) and \((0,1/2)\) are the two \(2\)-dimensional spin representations; the \((1/2,1/2)\) is the four-dimensional representation (which corresponds to how a four-vector in spacetime transforms).

As for \(SO(4)\text{,}\) we can ask whether the spin representations are real, pseudo-real, or complex. It turns out that for \(SO(3,1)\text{,}\) the two \(2\)-dimensional spin representations are in fact complex conjugate of each other. This is one key difference with \(SO(4)\text{,}\) in which case they were pseudo-real.

In fact, the table Table 5.7.5 can be generalized for arbitrary special orthogonal groups \(SO(p,q)\text{.}\) The type of spin representations is again periodic with periodic eight, but it now depends on \(p-q\) mod \(8\text{.}\) We get the following generalized table, which reduces to Table 5.7.5 for \(q=0\text{.}\)

\(p-q\) mod \(8\) \(0\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\)
Type Real Real Complex Pseudo-real Pseudo-real Pseudo-real Complex Real
Table 5.7.6. Spin representations for \(SO(p,q)\)
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