 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 304

Requiring $D_\mu\phi$ to transform as $U^\dagger(D_\mu\phi)U$ and using the gauge-field transformation law gives
$$
D_\mu\phi=\partial_\mu\phi-ig[A_\mu,\phi].
$$
Indeed, differentiating $U^\dagger\phi U$ produces two inhomogeneous derivative terms, and those cancel against the inhomogeneous part of the transformed connection.

For $U=1+i\alpha^aT_a+O(\alpha^2)$,
$$
\phi'=U^\dagger\phi U
=\phi+i[\phi,\alpha^aT_a]+O(\alpha^2).
$$
If $[T_a,T_b]=if_{ab}{}^cT_c$, then
$$
\delta\phi^c=f_{ab}{}^c\alpha^a\phi^b.
$$
This is the infinitesimal <Adjoint representation of a Lie algebra>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 3 / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 3 / b / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 3 / b / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)