Lie algebras

2.2 Lie algebras

We next introduce the concepts of Lie groups and Lie algebras, which are highly useful in understanding the structure and representations of continuous groups.

Definition 2.2.1. A Lie group is a group that is also a differentiable manifold, or “smooth”. Virtually all continuous groups we consider in physics are Lie groups. What this means is that we can think of the operation of any arbitrary group element as equivalent to $N$ successive infinitesimal operations of the form

g (𝜀_{A}) = 𝟙 + i 𝜀_{A} T_{A},

(2.2.1)

where $𝜀_{A}$ are infinitesimal and indexing the continuous group parameters, e.g. rotation angles for $SO (n)$ , $T_{A}$ are called the generators of the group, and we are using Einstein notation, implicitly summing over the index $A$ . Thus, for a general element $g (𝜃_{A})$ , where $𝜃_{A} = N 𝜀_{A}$ as defined above, we have

g (𝜃_{A}) = {(𝟙 + i \frac{𝜃_{A}}{N} T_{A})}^{N} \to_{}^{N \to \infty} e^{i 𝜃_{A} T_{A}} .

(2.2.2)

This is somewhat analogous to Taylor expansion in calculus, except for Lie groups only the first order / derivative term is necessary to capture the group behavior.²

Definition 2.2.2. The Lie algebra, $𝔤$ , of a group is defined by the set of commutation relations between its generators:³

[T_{A}, T_{B}] = i f_{ABC} T_{C},

(2.2.3)

where $[T_{A}, T_{B}] = T_{A} T_{B} - T_{B} T_{A}$ is the commutator of $T_{A}$ and $T_{B}$ , and $f_{ABC}$ are called the structure constants of $𝔤$ . As $[T_{A}, T_{B}] = - [T_{B}, T_{A}]$ , the structure constants must be totally antisymmetric in the swapping of their indices.

Example 2.2.1. For the $U (1)$ group, we can see directly from Eq. 2.1.5 that the sole generator of the group is $T = 1$ . This has the rather uninteresting Lie algebra $𝔲 (1)$ of $[1, 1] = 0$ , stemming from the fact that the group is abelian. Next, we look at the more interesting $SO (3)$ and $SU (2)$ groups, where the power of Lie algebras shines.

Fundamental and adjoint representations of the $𝔰𝔬 (3)$ and $𝔰𝔲 (2)$ algebras

We now introduce two important representations of Lie algebras, using the $SO (3)$ and $SU (2)$ groups as examples — both because of their importance in physics, and as their derivation introduces a number of useful concepts for the following sections. $SO (3)$ and $SU (2)$ are very closely related: $SU (2)$ is a double cover of $SO (3)$ , which means that every rotation in $SO (3)$ can be mapped to two elements of $SU (2)$ . Importantly, however, they are locally isomorphic near the identity, meaning they have the same Lie algebra.

We can derive the generators of $SO (3)$ by using the properties of the special orthogonal group ( $R^{T} R = 𝟙$ ). From Eq. 2.2.1, we have

\begin{matrix} \begin{aligned} R (𝜀) & ≊ 𝟙 + i𝜀T \\ R^{T} R & = 𝟙 + i𝜀 (T^{T} + T) + O (𝜀^{2}) \overset{!}{=} 𝟙 \\ \Rightarrow T^{T} & = - T . \end{aligned} \end{matrix}

(2.2.4)

Thus $T$ are antisymmetric matrices, of which for $N = 3$ dimensions there are three linearly independent ones:

J_{x} = i (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{matrix}), J_{y} = i (\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ - 1 & 0 & 0 \end{matrix}), J_{z} = i (\begin{matrix} 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}),

(2.2.5)

labeled as $x$ , $y$ , $z$ as they represent rotations around the respective axes. The factor of $i$ ensures the reality of the infinitesimal rotations in Eq. 2.2.4 and also that the generators are Hermitian.⁴ These provide us with the fundamental representation of $𝔰𝔬 (3)$ , and should be familiar as the angular momentum operators in quantum mechanics (QM). By exponentiating these, as in Eq. 2.2.2, we obtain the fundamental representation of the $SO (3)$ group: $R (𝜃) = e^{i 𝜃_{i} J_{i}}$ .

To find the fundamental representation of $𝔰𝔲 (2)$ , we can follow the same procedure as above, using the unitarity constraint $R^{†} R = 𝟙$ for $N = 2$ dimensional complex matrices, which yields:

\begin{matrix} \begin{aligned} T_{1} = \frac{1}{2} σ_{x} = \frac{1}{2} (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), & T_{2} = \frac{1}{2} σ_{y} = \frac{1}{2} (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}), \\ T_{3} = \frac{1}{2} σ_{z} = & \frac{1}{2} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}), \end{aligned} \end{matrix}

(2.2.6)

where $σ_{i}$ are the Pauli matrices — the angular momentum operators for the spin of spin- $1 ∕ 2$ particles in QM. Either set of generators yield the following Lie algebra of both groups:

[T_{A}, T_{B}] = i 𝜖_{ABC} T_{C},

(2.2.7)

where the structure constants $f_{ABC}$ of the algebra are simply $𝜖_{ABC}$ , the totally antisymmetric Levi-Civita tensor. Structure constants themselves furnish the following representation of the corresponding Lie algebra:

{[T_{A}]}_{BC} = - i f_{ABC} .

(2.2.8)

This can be confirmed by plugging this representation into the commutator in Eq. 2.2.3 and using the Jacobi identity [85]. As $B, C$ index the number of generators, we see that this representation has a dimension equal to the number of generators of the Lie algebra, and it is called its adjoint representation. It is analogous to the regular representation (Definition 2.1.5) for a Lie algebra, with the underlying vector space spanned by the generators $V = span {| T_{A} ⟩}$ and the requirement that $D (T_{A}) | T_{B} ⟩ = i f_{ABC} | T_{C} ⟩$ .

Definition 2.2.3. The dimension of a Lie group is defined as the number of generators of the group. Thus, it is the same as the dimension of the adjoint representation.

As it turns out, for $𝔰𝔬 (3)$ and $𝔰𝔲 (2)$ , the adjoint representation ${[T_{A}]}_{BC} = - i 𝜀_{ABC}$ is simply the fundamental representation of $𝔰𝔬 (3)$ . More generally, the dimensions of the fundamental and adjoint representations of $SO (n)$ and $SU (n)$ are given in Table 2.1. The significance of these representations, as we will see, is that the force carriers (i.e., gauge bosons) of the SM live in the adjoint representation of their associated gauge group, while the matter particles live in either their fundamental or trivial representations.

-------------------------------------------------------

Group dim (Fundamental ) dim (Adjoint )
-------------------------------------------------------

SO (n ) n n(n − 1)∕2

SU (n ) n n2 − 1
------------------------------------------------------- — Table 2.1. Dimensions of the fundamental and adjoint representations of the $SO (n)$ and $SU (n)$ groups.

General representations

So far we have discussed two representations of the $𝔰𝔬 (3)$ and $𝔰𝔲 (2)$ algebras. The general representations can be derived in much the same way as finding the eigenstates of the angular momentum operator in QM. We first choose a basis in which one of the generators, conventionally $J_{z}$ , is diagonal, and label eigenvectors of $J_{z}$ as $| m ⟩$ with eigenvalue $m$ :

J_{z} | m ⟩ = m | m ⟩ .

(2.2.9)

These eigenvectors, by definition, form a basis for the representations of the generators, so counting them tells us the dimensions of allowed representation. To do so, we define the “raising” and “lowering” operators $J_{\pm} = J_{x} \pm i J_{y}$ , with commutation relations

[J_{z}, J_{\pm}] = \pm J_{\pm}, [J_{+}, J_{-}] = 2 J_{z} .

(2.2.10)

These are named so because

J_{z} J_{\pm} | m ⟩ = [J_{\pm} Jz \pm J_{\pm}] | m ⟩ = (m \pm 1) J_{\pm} | m ⟩,

(2.2.11)

i.e., $J_{\pm} | m ⟩$ are eigenvectors of $J_{z}$ with eigenvalues $m \pm 1$ , implying

J_{\pm} | m ⟩ = c_{m \pm 1}^{\pm} | m \pm 1 ⟩,

(2.2.12)

where $c_{m \pm 1}$ are normalization constants. Now if we assume that the representation is finite-dimensional and label the highest-weight state $| j ⟩$ — such that $J_{+} | j ⟩ = 0 \Leftrightarrow c_{j + 1}^{+} = 0$ — we can iteratively lower the state and solve for the normalization constants until we reach the lowest-weight state. By doing so we find that $c_{- j - 1}^{-} = 0 \Rightarrow$ the lowest weight state is in fact $| - j ⟩$ .⁵

Thus, we conclude the algebra allows $2 j + 1$ -dimensional representations spanned by ${| - j ⟩, | - j + 1 ⟩, \dots, | j - 1 ⟩, | j ⟩}$ , with $j \in ℤ^{\geq 0} ∕ 2$ (non-negative integers and half-integers only). Each possible $j$ indexes a different representation of the group, and any eigenstate can thus be labeled by $| j, m ⟩$ . We have already seen the $j = 1 ∕ 2$ and $j = 1$ representations explicitly in Eqs. 2.2.6 and 2.2.5, respectively, while the $j = 0$ is simply the trivial representation of the group ( $D (g) = 1 \forall g \in G$ ).

Definition 2.2.4. More generally, irreducible representations of a group are labeled by eigenvalues of the Casimir invariants, or Casimirs, of the group. Casimirs are operators that commute with all generators of the group. For $𝔰𝔬 (3)$ and $𝔰𝔲 (2)$ , there is only one Casimir,

J^{2} = J_{x}^{2} + J_{y}^{2} + J_{z}^{2} .

(2.2.13)

This is the total angular momentum operator, which we know from QM commutes with all the $J_{i}$ s and for any eigenstate $| j, m ⟩$ has eigenvalue $j (j + 1)$ :

J^{2} | j, m ⟩ = j (j + 1) | j, m ⟩ .

(2.2.14)

As expected, since the Casimir commutes with all the generators, its eigenvalues depend only on the irrep $j$ . We have also seen that individual states can be further labeled using the eigenvalues of a set of maximally commuting operators, in this case ${J^{2}, J_{z}}$ .

These representations can directly be used to derive those corresponding to the $SU (2)$ and $SO (3)$ group except that, surprisingly, the latter does not admit the half-integer irreps; essentially, $SU (2)$ has double the irreps because it is the double cover of $SO (3)$ . Overall, the irreps of $𝔰𝔲 (2)$ and $𝔰𝔲 (3)$ are quite significant in physics, with direct applications to classical and quantum mechanics, and, moreover, they will also serve as the building blocks for the representations of the Lorentz and Poincaré groups in the next section.

²This is because, based on the Campbell-Baker-Hausdorff [84] formula, higher order terms in the expansion of exponential form of $g$ in Eq. 2.2.2 involve only commutators of the generators.

³An algebra $(V, ∙)$ is a vector space $V$ with a bilinear operation $∙ : V \times V \to V$ . Examples include the cross product of vectors and matrix multiplication of square matrices. The Lie algebra is the special case where $∙$ is the commutator.

⁴Note that the conventions around this factor are inconsistent in the literature and likely, despite our best efforts, will be inconsistent in this chapter as well.

⁵See, for example, Chapter IV.2 in Zee [79] for a more detailed derivation.

2.2 Lie algebras

Fundamental and adjoint representations of the 𝔰𝔬(3) and 𝔰𝔲(2) algebras

General representations

Fundamental and adjoint representations of the $𝔰𝔬 (3)$ and $𝔰𝔲 (2)$ algebras