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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 354

For a scalar operator of dimension $\Delta=2$ in a boundary theory of dimension $d=3$, the <AdS scalar mass--dimension relation> gives
$$
m^2=\Delta(\Delta-d)=-2
$$
in unit-radius $\operatorname{AdS}_4$. In Poincare coordinates,
$$
ds^2=\frac1{z^2}(\eta_{\mu\nu}dx^\mu dx^\nu+dz^2),
$$
so the free <Klein-Gordon equation> is
$$
\boxed{\left[z^2(\partial_z^2+\Box_{2+1})
-2z\partial_z+2\right]\phi=0}.
$$
This mass is exactly that of a <conformally coupled scalar field> in four-dimensional anti-de Sitter spacetime. The <Weyl transformation>
$$
\boxed{\phi(x,z)=z\,\psi(x,z)}
$$
therefore maps the equation to the flat half-space wave equation
$$
\boxed{(\partial_z^2+\Box_{2+1})\psi=0,
\qquad z>0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 / 1 / b / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 / 1 / b / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 / 1 / b / i / Solution

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