Notes on Basic Concepts of Group Theory
August 30, 2020
Background
These notes are made based on the purpose of quick exposure to basic concepts of group theory, a pervasive tool used in the research of condensed-matter QFT. The following text will function as a lookup table or a prelude for my possible future application of group theory in QFT problems.
Transformation in the Special Relativity
The allowed transformation in the special relativity has to preserve the following quadratic form of infinitesimal Minkowski spacetime,
\[(ds)^2=(cdt)^2-(dx)^2-(dy)^2-(dz)^2=dx_{\mu}dx_{\nu}\eta^{\mu\nu}\tag{1},\]where \(\eta^{\mu\nu}\) is the Minkowski space metric tensor. If we define a linear transformation so that
\[dx_{\mu}^{\prime}=\Lambda_{\mu}^{\sigma}dx_{\sigma}\tag{2},\]then the invariant of \((ds)^2\) leads to
\[dx_{\mu}dx_{\nu}\eta^{\mu\nu}=dx_{\mu}^{\prime}dx_{\nu}^{\prime}\eta^{\mu\nu}=dx_{\mu}dx_{\nu}\Lambda_{\lambda}^{\mu}\eta^{\lambda\sigma}\Lambda_{\nu}^{\sigma}\tag{3},\]where the second equivalence in (3) is valid as \(\eta^{\mu\nu}=\eta^{\nu\mu}\). Eq.(3) also indicates that
\[\Lambda^T\eta\Lambda=\eta\tag{4}.\]The transformations that satisfy (4) are Lorentz transformations, belonging to Poincaré group. For an arbitrary function \(F(a,b,c,\cdots)\) if the function value remains unchanged after the arguments are transformed into \(a^{\prime},b^{\prime}\), i.e.,
\[F(a,b,c,\cdots)=F^{\prime}(a^{\prime},b^{\prime},c^{\prime},\cdots),\]We say the function is invariant under the transformation. Another useful concept that relates to invariance is covariance, and we illustrate the concept by considering the following polynomial,
\[E=ax^2+bxy+cz^4.\]After we transform \(x,y,\) and \(z\) into \(x^{\prime},y^{\prime},\) and \(z^{\prime}\), if the form of the function remains unchanged, instead of preserving function value,then we call such transformation as covariant. In special relativity, all the physical laws has to be covariant under Lorentz transformation.
Practical definition of Group
A group \(G\) is a set that contains symmetric transformations related by a binary operator \(\circ\) (e.g. matrix multiplication). A group has the following properties:
- Closure: if \(g_1,g_2\in G\), then \(g_1\circ g_2\in G\);
- Must contain an identity element \(e\);
- Must contain inverse elements,i.e. \(g^{-1}\in G\) and \(g\circ g^{-1}=e\);
- Associativity: \(g_1\circ(g_2\circ g_3)=(g_1\circ g_2)\circ g_3\)
Some Commonly Used Groups
Rotation in 2D space
- \(O(2)\): group of all orthogonal \(2\times2\) matrices that perform sysmmetric rotation on 2D space, with the binary operator being matrix multiplication and \(O^TO=I\).
- \(SO(2)\): similar to \(O(2)\) with one more constraint of \(det[O]=1\), which remains coordinate system orientation.
- \(U(1)\): group of unit compex number \(z\) with the binary operator being the number multiplication.
We can show that the group \(SO(2)\) relates to \(U(1)\) by representing “\(i\)” using a \(2\times2\) matrix,
\[i\rightarrow\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right),\]and the elements in \(U(1)\) becomes
\[z=e^{i\theta}=\left(\begin{array}{cc} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right),\]the rotation matrix in 2D space.
Rotation in 3D space
The rotation in 3D space is accomplished using 4-dimensional complex number, quaternions,
\[q=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}\tag{5},\]with
\[\mathbf{i^2=j^2=k^2=1},|q|^2=1,\mathbf{ij=k}.\]If we represent \(i,j,k\) as three different \(2\times2\) matrices,
\[\mathbf{i}=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right),\quad\mathbf{j}=\left(\begin{array}{ll} 0 & i \\ i & 0 \end{array}\right) \quad, \quad \mathbf{k}=\left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array}\right)\]and demand that \(det(q)=1\), we obtain the group of \(SU(2)\). For any arbitrary vector \(\vec{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\), the 3D rotation is applied to \(\vec{v}\) as
\[\vec{v}^{\prime}=q\vec{v}q^{-1}.\]Lie Theory for Continuous Symmetry
For any finite transformation \(h(\theta)\) with a parameter \(\theta\) (e.g. rotate an object by \(50^{\circ}\)), we can define the generator \(\mathbf{X}\) for \(h(\theta)\) as
\[h(\theta)=\lim_{N\rightarrow\infty}(I+\frac{\theta}{N}\mathbf{X})^N=e^{\theta \mathbf{X}}\tag{6}.\]Based on the concept of generator we can define Lie algebra \(\mathfrak{g}\) as a collection of generator that gives the elements of group \(G\) when exponentiated and a Lie bracket \([,]\) between generators. With proper math symbols, the relationship between two generator \(\mathbf{X},\mathbf{Y}\in\mathfrak{g}\) can be casted in the Baker-Campbell-Hansdorff formula,
\[g\circ h=e^{\mathbf{X}}\circ e^{\mathbf{Y}}=exp\{\mathbf{X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[X,[X,Y]]\cdots}\}\tag{7}.\]For the group \(SU(2)\) the Lie bracket is simply the commutator bracket. Usually, another possible candidate for the Lie bracket is Poisson bracket. In general, for a Lie algebra on a vector space \(\mathfrak{g}\), the Lie bracket \([,]:\mathfrak{g\times g}\rightarrow\mathfrak{g}\) satisfies:
- \[[a\mathbf{X}+b\mathbf{Y},\mathbf{Z}]=a[\mathbf{X,Z}]+b[\mathbf{Y,Z}]\]
- \[[\mathbf{X,Y}]=-[\mathbf{Y,X}]\]
- Jacobi identity: \([\mathbf{X,[Y,Z]}]+\mathbf{[Y,[Z,X]]+[Z,[X,Y]]}=0\)
Example: get elements in the group \(SO(3)\)
\(SO(3)\) is typically represented by a group of \(3\times3\) matrices that rotate objects in 3D space. Here we provide a general process of getting elements in \(SO(3)\) by first calculating the generators. Let \(O\) and \(J\) be the group element and its generator, i.e. \(O=e^{\theta J}\). For \(SO(3)\), the operators satisfy the following conditions:
- \(O^TO=I\),
- \(\text{ and } det(O)=1\).
The two conditions above indicate, separately:
\[J^T+J=0\tag{8}\] \[det(O)=det(e^{\theta J})=e^{\theta Tr(J)}=1\text{ and }Tr(J)=0\tag{9}\]The \(3\times3\) matrices that obey (8) and (9) are
\[J_{1}=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array}\right), J_{2}=\left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{array}\right), J_{3}=\left(\begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right).\]Toe get a mateix operator from \(J_1\), we first focus on the submatrix at the bottom right corner,
\[j_{1}=\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right).\]For any integer \(n\), we have
\[(j_1)^{2n}=(-1)^nI\quad (j_1)^{2n+1}=(-1)^nj_1.\]Thus,
\[\begin{aligned}\mathrm{e}^{\theta j_{1}}&=\sum_{n=0}^{\infty} \frac{\theta^{n} j_{1}^{n}}{n !}\\ &=\sum_{n=0}^{\infty} \frac{\theta^{2 n}}{(2 n) !} \underbrace{\left(j_{1}\right)^{2 n}}_{-1^nI}+\sum_{n=0}^{\infty} \frac{\theta^{2 n+1}}{(2 n+1) !} \underbrace{\left(j_{1}\right)^{2 n+1}}_{(-1)^{n}j_{1}}\\ &=\underbrace{\left(\sum_{n=0}^{\infty} \frac{\theta^{2 n}}{(2 n) !}(-1)^{n}\right)}_{=\cos (\theta)} I+\underbrace{\left(\sum_{n=0}^{\infty} \frac{\theta^{2 n+1}}{(2 n+1) !}(-1)^{n}\right)}_{=\sin (\theta)} j_{1}\\ &=\cos (\theta)\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)+\sin (\theta)\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)=\left(\begin{array}{cc} \cos (\theta) & -\sin (\theta) \\ \sin (\theta) & \cos (\theta) \end{array}\right). \end{aligned}\]Since \(e^{0J_1}=1\), we get the full operator \(O_1\) as
\[O_{1}=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos (\theta) & -\sin (\theta) \\ 0 & \sin (\theta) & \cos (\theta) \end{array}\right)\tag{10}.\]Following the similar steps, the other two elements of \(SO(3)\) group are
\[O_{2}=\left(\begin{array}{ccc} \cos (\theta) & 0 & -\sin (\theta)\\ 0 & 1 & 0\\ \sin (\theta) & 0 & \cos (\theta) \end{array}\right),O_{3}=\left(\begin{array}{ccc} \cos (\theta) & -\sin (\theta) & 0\\ \sin (\theta) & \cos (\theta) & 0\\ 0 & 0 & 1 \end{array}\right).\]Conventionally, we introduce “\(i\)” in the generator \(J_i\) to make them Hermitian, i.e. \(J^{\dagger}=J\). The Lie algebra is then
\[[J_i,J_j]=i\epsilon_{ijk}J_k\tag{11},\]with
\[J_{1}=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{array}\right) \quad J_{2}=\left(\begin{array}{ccc} 0 & 0 & i \\ 0 & 0 & 0 \\ -i & 0 & 0 \end{array}\right) \quad J_{3}=\left(\begin{array}{ccc} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{array}\right).\]Proof: \(SU(2)\) and \(SO(3)\) have the same Lie algebra
To derive Eq.(11) for \(SU(2)\) we notice that the elements in \(SU(2)\) (i.e. unit quaternions) follow:
\[\mathcal{U^{\dagger}U}=1\quad det(\mathcal{U})=1.\]Let \(J_i\) be the generator for the element \(\mathcal{U}_i\), and we use Baker-Campbell-Hansdorff formula and the conditions above to build the relationship between \(J_i\) and its Hermitian conjugate \(J^{\dagger}_i\) as
\[\begin{aligned} e^{-iJ^{\dagger}_i}e^{iJ_i}&=1\\ e^{-iJ^{\dagger}_i+iJ_i+\frac{1}{2}[J^{\dagger}_i,J_i]+\cdots}&=e^0, \end{aligned}\]and thus, \(J_i^{\dagger}=J_i\). Because \(\mathcal{U}\) has unit determinant, we have
\[det(e^{iJ_i})=e^{iTr(J_i)}=1\quad Tr(J_i)=0.\]We conclude that the generators of SU(2) must be Hermitian traceless matrices. A basis for Hermitian traceless 2 × 2 matrices is given by the following 3 Pauli matrices:
\[\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)\tag{12}.\]This means every Hermitian traceless \(2\times2\) matrix can be written as a linear combination of these matrices. It is straightforward to show that
\[\left[\sigma_{i}, \sigma_{j}\right]=2 i \epsilon_{i j k} \sigma_{k}\]To get rid of “2” in the equation above, we define the generator of \(SU(2)\) as \(J_i=\frac{1}{2}\sigma_{i}\), and we recover the Lie algebra of (11).
The Abstract Definition of a Lie Group
Furthermore, the group operation \(\circ\) must induce a differentiable map of the manifold into itself. This is a compatibility requirement that ensures that the group property is compatible with the manifold property. Concretely this means that every group element, say \(A\) induces a map that takes any element of the group \(B\) to another element of the group \(C = AB\) and this map must be differentiable. Using coordinates this means that the coordinates of \(AB\) must be differentiable functions of the coordinates of \(B\).
A manifold is a set of points, for example a sphere that looks locally like flat Euclidean space \(R^n\). Another way of thinking about a n-dimensional manifold is that it’s a set which can be given \(n\) independent coordinates in some neighborhood of any point.
Each group corresponds to a simply-connected geometric object. Take \(SU(2)\) as an example, the unit quaternion in the group satisfies:
\[a^2+b^2+c^2+d^2=1\]which is the condition for a sphere in 4D space (i.e. \(S^3\) sphere). Because \(S^3\) sphere is a simply-connected object, the \(SU(2)\) is considered as a fundamental group. Generally, we represent the symmetry in nature only using fundamental groups, and there is precisely one simply-connected Lie group corresponding to each Lie algebra.
Representation Theory
Elements in any group can simply act on objects that are different in nature, which could be a vector, or geometric shapes. This idea motivates the definition of a representation of a group: a representation \(R\) for a group element is a map between the group elements, \(g\), and linear transformation \(R(g)\) of some vector space \(V\) in such a way that the group properties are preserved.
Using representation theory, we are able to investigate systematically how a given group acts on very different vector spaces.
One of the most important examples in physics is \(SU(2)\). For example, we can investigate how \(SU(2)\) acts on \(\mathbb{C}^{2}\). The objects living in \(\mathbb{C}^{2}\) are 2D complex and therefore the elements in \(SU(2)\) acts on them as \(2\times2\) matrices. The matrices (=linear transformations) acting on \(\mathbb{C}^{2}\) are just the “usual” \(\mathbb{SU}(2)\) (this is where the “2” in \(SU(2)\) comes from!) matrices that we already know. In addition, we can examine how \(SU(2)\) acts on \(\mathbf{C}^{3}\) or even higher dimensional vector spaces. Since the objects living in \(\mathbb{C}^3\) are 3D imaginary vectors, the elements in \(SU(2)\) are then \(3\times3\) matrices. A basis for the generator of \(SU(2)\) elements is given by
\[J_{1}=\frac{1}{\sqrt{2}}\left(\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right), J_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{array}\right), J_{3}=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{array}\right)\tag{13}.\]To avoid confusion, researchers sometimes refer \(SU(2)\) group as \(S^3\) sphere.
Useful notions
- Similarity transformation
If \(R(G)\) is a representation of \(G\), then \(S^{-1}R(G)S\) is also representation.
This means if we have a representation, we can transform it using any non-singular matrix to get nice-looking matrix representations.
- Invariant subspace
When we have a representation \(R\) of a group \(G\) on a vector space \(V,\) we call \(V^{\prime}\subseteq V\) an invariant subspace if for \(v\in V^{\prime}\) we have \(R(g)v\in V^{\prime}\) for all \(g\in G\).
- Irreducible representation (the fundamental representation)
An irreducible representation is a representation of a group \(G\) on a vector space \(V\) that has no invariant subspaces besides the zero space \(\{0\}\) and \(V\) itself.
Conclusion
This blog is dry in a sense of the “information density” as I only expect this blog serves as a reminder for a layman like me who first encoutered group theory in a condensed matter QFT book. For various applications of group theory you might find Jakob’s book Physics from Symmetry worth reading. If you’re a math-driven head, Prof. John Stillwell’s Naive Lie Theory is so far the best choice we have to get used to the mathematical language behind the theory.