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Short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford

2018

For any one of the four anticommuting matrices $A$, choose another $B$. Since $B^{-1}=B$ and $BAB=-A$, cyclic invariance of the <trace> gives
$$
\operatorname{Tr}A=\operatorname{Tr}(B^{-1}AB)=\operatorname{Tr}(-A).
$$
Therefore
$$
\boxed{\operatorname{Tr}\alpha_i=0,\qquad\operatorname{Tr}\beta=0.}
$$
Equivalently, the paired positive and negative unit <eigenvalues> cancel. This argument does not depend on a particular matrix representation.

Solution

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S18 / 3 / a / iii / Solution

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S18 / 3 / a / iii / Solution

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S18 / 3 / a / iii / Solution

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