45.5 The non-Abelian case

45.5 The non-Abelian case

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Pauli matrices · Lie bracket

We saw in section 38.2.2 that quantum field theories with a $U(m)$ group acting on the fields can be constructed by taking m-component complex fields and a Hamiltonian that is the sum of the single complex field Hamiltonians for each component. The constructions of this chapter generalize from the $U(1)$ to $U(m)$ case, getting a gauge group $\mathcal{G}$ of maps from ${\mathbf{R}}^4$ to $U(m)$ , as well as a generalized notion of connection and curvature. In this section we’ll outline how this works, without going into full detail. The non-Abelian $U(m)$ case is a relatively straightforward generalization of the $U(1)$ case, except for the definition of the curvature, where new terms with diferent behavior arise, due to the non-commutative nature of the group.

The diagonal $U(1)\subset U(m)$ subgroup is treated using exactly the same formalism for the vector potential, covariant derivative, electric and magnetic fields as above. It is only the $SU(m)\subset U(m)$ subgroup which requires a separate treatment. We will do this just for the case $m=2$ , which is known as the Yang-Mills case, since it was first investigated by the physicists Yang and Mills in 1954. We can think of the $i\phi (x)$ in the $U(1)$ case as a function valued in the Lie algebra $\mathfrak{u}(1)$ , and replace it by a matrix-valued function, taking values in the Lie algebra $\mathfrak{su}(2)$ for each x. This can be written in terms of three functions ${\phi }_a$ as

$$i\phi (x)=i\sum _{a=1}^3{\phi }_a(x)\frac{{\sigma }_a}{2}$$

using the Pauli matrices. The gauge group becomes the group ${\mathcal{G}}_{YM}$ of maps from space-time to $SU(2)$ , with Lie algebra the maps $i\phi (x)$ from space-time to $\mathfrak{su}(2)$ . Unlike the $U(1)$ case, this Lie algebra has a non-trivial Lie bracket, given by the point-wise su $.(2)$ Lie bracket (the commutator of matrices).

In the $SU(2)$ case, the analog of the real-valued function $A_{\mu }$ will now be matrix-valued and one can write

$$A_{\mu }(x)=\sum _{a=1}^3{A}_{\mu }^a(x)\frac{{\sigma }_a}{2}$$

Instead of one vector potential function for each space-time direction $\mu$ we now have three (the $A_{\mu }^a(x))$ , and we will refer to these functions as the connection or “gauge field”. The complex fields $\psi$ are now two-component fields, and the covariant derivative is

$$D_{\mu }^A\left(\genfrac{}{}{0ex}{}{{\psi }_1}{{\psi }_2}\right)=\left(\frac{\partial }{\partial x_{\mu }}-ie\sum _{a=1}^3{A}_{\mu }^a(x)\frac{{\sigma }_a}{2}\right)\left(\genfrac{}{}{0ex}{}{{\psi }_1}{{\psi }_2}\right)$$

For the theories of complex fields with $U(m)$ symmetry discussed in chapters 38 and 44, this is the $m=2$ case. Replacing derivatives by covariant derivatives yields non-relativistic and relativistic theories of matter particles coupled to background gauge fields.

In the Yang-Mills case, the curvature or field strengths can still be defined as a commutator of covariant derivatives, but now this is a commutator of matrix-valued diferential operators. The result will as in the $U(1)$ case be a multiplication operator, but it will be matrix-valued. The curvature can be defined as

$$\sum _{a=1}^3{F}_{\mu \nu }^a\frac{{\sigma }_a}{2}=\frac{i}{e}[D_{\mu }^A,D_{\nu }^A]$$

which can be calculated much as in the Abelian case, except now the term involving the commutator of $A_{\mu }$ and $A_{\nu }$ no longer cancels. Distinguishing electric and magnetic field strength components as in the $U(1)$ case, the equations for matrix-valued electric and magnetic fields are:

$$E_j(x)=\sum _{a=1}^3{E}_j^a(x)\frac{{\sigma }_a}{2}=-\frac{\partial A_j}{\partial t}+\frac{\partial A_0}{\partial x_j}-ie[A_j,A_0]\text{\hspace{1em}(45.3)}$$

and

$$B_j(x)=\sum _{a=1}^3{B}_j^a(x)\frac{{\sigma }_a}{2}={ϵ}_{jkl}\left(\frac{\partial A_l}{\partial x_k}-\frac{\partial A_k}{\partial x_l}-ie[A_k,A_l]\right)\text{\hspace{1em}(45.4)}$$

The Yang-Mills theory thus comes with electric and magnetic fields that now are valued in $\mathfrak{su}(2)$ and can be written as 2 by 2 matrices, or in terms of the Pauli matrix basis, as fields ${\mathbf{E}}^a(x)$ and ${\mathbf{B}}^a(x)$ indexed by $a=1,2,3$ . These fields are no longer linear in the $A_{\mu }$ fields, but have extra quadratic terms. These non-quadratic terms will introduce non-linearities into the equations of motion for Yang-Mills theory, making its study much more dificult than the $U(1)$ case.

Previous section: 45.4 The geometric significance of the connection · Next section: 45.6 For further reading · 中文版本

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 484、485、486、487、488、489、490、491、492、493

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：d3d285500af2024670a515c59ee5503cd8bd26667c0e1e839c22ee489820b440

OCR 产物 SHA-256：d3d285500af2024670a515c59ee5503cd8bd26667c0e1e839c22ee489820b440