Quantum field theory 2, lecture 09

Generators for SU\((N)\)

We consider now an SU\((N)\)-symmetry with fermions in the fundamental \(n\)-component representation. The generators of the Lie algebra associated with \(\text{SU}(N)\) are a complete basis for the vector space (over the real numbers) of hermitian, traceless \(N \times N\) matrices, \[\begin{split} T_z^\dagger=T_z,\quad\quad\quad \text{tr}\{T_z\}=0. \end{split}\] This can be easily checked for infinitesimal transformations, and implies also \(U^\dagger = \exp\left(-i \alpha^z(x)T_z\right)\).

Generators for SU\((2)\) and SU\((3)\)

For SU\((2)\), one has \(z=1,\ldots,3\), and the generators can be written in terms of the three Pauli matrices, \[\begin{split} T_z = \frac{1}{2}\sigma_z, \quad\quad \sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}, \quad\quad \sigma_2=\begin{pmatrix} 0 & -i \\ i & 0\end{pmatrix}, \quad\quad \sigma_3=\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}. \end{split}\] For SU\((3)\), there are eight generators, \(z=1,\ldots,8\), \[\begin{split} T_z = \frac{1}{2}\lambda_z, \end{split}\] and \(\lambda_z\) are the eight “Gell-Mann matrices”, to be given explicitly later. The normalization is \[\begin{split} \text{tr}\{ T_z T_y \} = \frac{1}{2}\delta_{zy}. \end{split}\] This normalization can be chosen for the fundamental representation of general \(\text{SU}(N)\).

Composing Lie group elements

To combine two transformations, one needs the Baker-Campbell-Hausdorff formula \[\begin{split} e^u e^v = e^{w(x,y)}, \end{split}\] with \[w(u,v) = u+v+\frac{1}{2}[u,v]+ \frac{1}{12}[u,[u,v]]-\frac{1}{12}[v,[u,v]]+ \cdots\] This shows that it is enough to know how to calculate commutators between the Lie algebra generators \(T_z\). To the order given here, the Baker-Campbell-Haussdorff formula can easily be derived from series expansions of exponentials and logarithms.

Commutator and structure constants

The commutator between generators for a given Lie group is of the form \[\label{eq:comRel} \begin{split} [T_y,T_z] = i f_{yz}^{~~w}T_w, \end{split}\] where the structure constants \(f_{yz}^{~~w}\) characterize the Lie algebra and therefore indirectly the Lie group. Obviously, the structure constants are anti-symmetric in the first two indices, \(f_{yz}^{~~w}=-f_{zy}^{~~w}\). When the generators are hermitian (which is the case for compact Lie groups), the structure constants are real, \(f_{yz}^{~~w}\in \mathbb{R}\).

As an example, the generators \(T_j=\sigma_j/2\) of \(\text{SU}(2)\) fulfill \[ [T_j, T_k] = i \epsilon_{jkl} T_l,\] so that the structure constants for \(\text{SU}(2)\) are given by the Levi-Civita symbol, \(f_{jk}^{~~l} = \epsilon_{jkl}\).

Representations of the Lie algebra

The generators \(T_z\) can be realized through different representations. For example, this might be matrices such that the above commutation relation is fulfilled. But representations can also be constructed in different ways, for example as differential operators acting on fields.

For example, rotations in three dimensions are defined for many different objects, e. g. fields, particles or solid bodys, corresponding to many different representations of the Lie algebra of \(\text{SU}(2)\) (which is equal to the Lie algebra of \(\text{SO}(3)\) in the sense that the structure constants agree).

Jacobi identity and adjoint representation

The generators also satisfy the Jacobi identity \[\begin{split} [T_x,[T_y,T_z]] + [T_y,[T_z,T_x]] + [T_z,[T_x,T_y]] = 0. \end{split}\] For matrix Lie algebras this idendity is easily checked by writing it out explicitly. More generally it is part of the requirements for a Lie algebra. For the structure constants, this implies \[\begin{split} f_{xu}^{~~v}f_{yz}^{~~u}+f_{yu}^{~~v}f_{zx}^{~~u}+f_{zu}^{~~v}f_{xy}^{~~u}=0. \end{split}\] From the Jacobi identity, one can see that the structure constants can be used to construct another generic representation, the adjoint representation. Here one sets the generator matrices to \[\begin{split} (T_z^{(A)})^{v}_{~u} &= i f_{zu}^{~~v}. \end{split}\] Indeed, one has now \[\begin{split} [T_x^{(A)},T_y^{(A)}] = if_{xy}^{~~w} T_w^{(A)}, \end{split}\] as a consequence of the Jacobi idendity.

The adjoint representation for the Lie algebra of \(\text{SU}(2)\) is given by the three \(3\times 3\)-matrices \[(T^{(A)}_j)^k_{~l} = -i \epsilon_{jkl}.\] These are also the generators for the Lie algebra of \(\text{SO}(3)\) in the fundamental representation. This shows again that these two Lie algebras agree, \(\text{SU}(2)\cong \text{SO}(3)\). In contrast, for \(\text{SU}(3)\) the adjoint representation is given by \(8\times 8\) matrices.

The fundamental and the adjoint representation are the most important representations of Lie algebras needed in the following. However, there are many more and they all induce corresponding representations of the Lie group through the exponential map.

Gauge fields and covariant derivatives

Partial derivatives do not transform homogeneously under local gauge transformations, \[\partial_\mu\psi(x) \to \partial_\mu \left[U(x) \psi(x) \right]= U(x) \left[\partial_\mu \psi(x)\right] + \left[\partial_\mu U(x) \right] \psi(x).\] For constant \(U(x)\) (a global transformation) one would only have the first term on the right hand side, but the second term arises in addition for local transformations. One would like to avoid such an additional term, in order to construct invariant actions etc.

Similarly to the local \(\text{U}(1)\) symmetry of electromagnetism, one introduces gauge fields and defines a covariant derivative. For the SU\((2)\) gauge symmetry of the weak interaction, these additional gauge fields give three \(W/Z\)-bosons, and for quantum chromodynamics (QCD), there are eight gluons. We denote the gauge fields by \(A^z_\mu(x)\). There is one field for each generator \(T_z\) (e. g. \(z=1,\ldots,3\) for SU\((2)\) or \(z=1,\ldots,8\) for SU\((3)\)).

The covariant derivative is defined as \[\begin{split} D_\mu \psi(x) = \left[\partial_\mu - i g A^z_\mu(x) T_z \right]\psi(x), \end{split}\] with \(g\) the gauge coupling.

Transformation law for the gauge field

We want a transformation of the gauge fields such that the covariant derivative transforms homogeneously, or, in other words, just like the field \(\psi(x)\) itself, \[D_\mu\psi(x) = \left[ \partial_\mu - i g A_\mu^z(x) T_z \right] \psi(x) \to U(x) D_\mu \psi(x).\] This requires \[A_\mu^z(x) T_z \to U(x) \left[ A_\mu^z(x) T_z \right] U^{-1}(x) - \frac{i}{g} [\partial_\mu U(x)] U^{-1}(x).\] With this, a covariant derivative transforms as \[D_\mu \to U(x) D_\mu U^{-1}(x).\] For an infinitesimal transformation of the form \[U(x) = \mathbb{1} + i \alpha^y(x) T_y, \quad\quad\quad U^{-1}(x) = \mathbb{1} - i \alpha^y(x) T_y,\] one finds the required transformation law for the gauge field \[\begin{split} A^z_\mu(x) T_z \to & A_\mu^z(x) T_z + \frac{1}{g}\partial_\mu \alpha^z(x) T_z + i \left\{ \alpha^y(x) T_y A_\mu^z(x) T_z - A_\mu^z(x) T_z \alpha^y(x) T_y \right\} \\ & = A_\mu^z(x) T_z + \frac{1}{g}\partial_\mu \alpha^z(x) T_z + i [\alpha^y(x) T_y, A^z_\mu(x) T_z] \\ & = A_\mu^z(x) T_z + \frac{1}{g}\partial_\mu \alpha^z(x) T_z + i \alpha^y(x) A^w_\mu(x) [T_y, T_w]. \end{split}\]

Adjoint representation

The commutator of two generators in the last line may be expressed itself as a linear combination of commutators using the structure constants, \[ [T_y, T_w] = i f_{yw}^{~~z} T_z,\] and we find the transformation law for the gauge field components \[A^z_\mu(x) \to A_\mu^z(x) + \frac{1}{g}\partial_\mu \alpha^z(x) - \alpha^y(x) f_{yw}^{~~z} A^w_\mu(x) .\] Interestingly, the last term is formally of the same form as the change in a matter field \[\delta \psi(x) = i \alpha^y(x) T_y \psi(x),\] except that the generator is here in the adjoint representation, \[i \alpha^y(x) (T_y^{(A)})^v_{~w} A^w_\mu(x) = - \alpha^y(x) f_{yw}^{~~v} A^w_\mu(x).\] In this sense, a non-Abelian gauge field is actually itself charged, and transforms in the adjoint representation of the gauge group.

Another useful way to write the transformation law for the gauge field is \[A_\mu^z(x) \to A_\mu^z(x) + \frac{1}{g} (D_\mu[A])^z_{~y} \alpha^y(x),\] where \[\begin{split} (D_\mu[A])^z_{~y} = & \partial_\mu \delta^z_{~y} - i g A^w_\mu(x) (T_w^{(A)})^z_{~y}\\ = & \partial_\mu \delta^z_{~y} + g A^w_\mu(x) f_{wy}^{~~z}\\ = & \partial_\mu \delta^z_{~y} - g A^w_\mu(x) f_{yw}^{~~z} \end{split}\] is the covariant derivative in the adjoint representation.xx

Different representations of covariant derivatives

We have seen that the generators of the Lie algebra exist in different representations and so do the covariant derivatives that can be constructed out of them, \[D_\mu = \partial_\mu - i g A_\mu^z(x) T_z.\] In fact, the appropriate generator for a covariant derivative depends on what object the derivative is acting on. For a field in some representation \(R\), one must use \[D_\mu^{(R)} \psi^{(R)}(x) = \left[\partial_\mu - i g A_\mu^z(x) T_z^{(R)} \right] \psi^{(R)}(x).\] For example, if the field is in the fundamental representation as for quarks, we ned to use \(T_z^{(F)}=T_z\) and for neutral fields, one has the trivial representation, \(T_z^{(0)}=0\), so that a covariant derivative becomes an ordinary derivative.

Covariant derivatives fulfills the Leibniz rule \[D_\mu[AB] = [D_\mu A] B+ A [D_\mu B],\] even though \(A\) and \(B\) may be in different representations. This is in particular useful for partial integration.