 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

2025

Paper 304

The product of generators satisfies
$$
T_aT_b=\frac1{2N}\delta_{ab}I
+\frac12(d_{ab}{}^c+if_{ab}{}^c)T_c,
$$
and therefore
$$
\operatorname{Tr}(T_aT_bT_c)=\frac14(d_{abc}+if_{abc}).
$$
Because $\phi^a\phi^b\phi^c$ is symmetric, its contraction with the antisymmetric <structure constant of a Lie algebra> $f_{abc}$ vanishes. Hence
$$
-\frac1{3!}\operatorname{Tr}(\phi^3)
=-\frac1{24}d_{abc}\phi^a\phi^b\phi^c.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 4 / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 4 / 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 / 2025 / iii / Paper 304 / 4 / b / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)