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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2018

Paper 1

Anti-Hermiticity makes $\bar z_jA_{jk}z_k$ purely imaginary, so $I_A$ is real. Its complex derivatives are
$$
\frac{\partial I_A}{\partial z_l}
=\frac1{2i}\bar z_jA_{jl},
\qquad
\frac{\partial I_A}{\partial\bar z_l}
=\frac1{2i}A_{lk}z_k.
$$
The complex-coordinate bracket from part (c) therefore gives
$$
\begin{aligned}
\{I_A,I_B\}
&=\frac{i}{2}\bar z_j(A_{jl}B_{lk}-B_{jl}A_{lk})z_k\\
&=\frac{i}{2}\bar z[A,B]z\\
&=-\frac1{2i}\bar z[A,B]z.
\end{aligned}
$$
Hence the <anti-Hermitian quadratic Hamiltonian> identity is
$$
\boxed{\ \{I_A,I_B\}=-I_{[A,B]}.\ }
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 32A / d / Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 32A / d / Solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 32A / d / Solution

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