48.1 Non-Abelian gauge fields

48.1 Non-Abelian gauge fields

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · U(1)

The Standard Model includes gauge fields for a $U(1)\times SU(2)\times SU(3)$ gauge group, with Hamiltonian given by the sum 46.10

$$h_{YM}=\sum _{j=1}^3\frac{1}{2}{\int }_{{\mathbf{R}}^3}t{r}_j(|{\mathbf{E}}_j{|}^2+|{\mathbf{B}}_j{|}^2)d^3\mathbf{x}\text{\hspace{1em}(48.1)}$$

The ${\mathbf{E}}_j$ and ${\mathbf{B}}_j$ take values in the Lie algebras $u(1)$, su(2), su(3) for $j=1,2,3$ respectively. $t{r}_j$ indicates a choice of an adjoint-invariant inner product for each of these Lie algebras, which could be defined in terms of the trace in some representation. Such an invariant inner product is unique up to a choice of normalization, and this introduces three parameters into the theory, which we will call $g_1,g_2,g_3$

Some method must be found to deal appropriately with the gauge-invariance problems associated with quantization of gauge fields discussed in section 46.6. Interacting non-Abelian gauge field theory remains incompletely understood outside of perturbation theory. The theory of interacting quantum fields shows that, to get a well-defined theory, one should think of the parameters $g_j$ as being dependent on the distance scale at which the physics is being probed, and one can calculate the form of this scale-dependence. The $SU(2)$ and $SU(3)$ gauge field dynamics is “asymptotically free”, meaning that g₂, $g_3$ can be defined so as to go to zero at short-distance scales, with behavior of the theory approaching that of a free field theory. This indicates that one should be able to consistently remove short-distance cutofs necessary to define the theory, at least for those two terms in the Hamiltonian.

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