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Advanced quantum field theory for particle physics course of the University of Oxford

Past exam of the advanced quantum field theory for particle physics course of the University of Oxford

2025

{c}

Vary the <action>, taking the field variation to vanish at the boundary. <Integration by parts> gives
$$
\delta S_0=\int d^4x\left[\partial^\mu\phi\,\partial_\mu\delta\phi-m^2\phi\delta\phi-\frac\lambda6\phi^3\delta\phi\right]
=-\int d^4x\left(\Box\phi+m^2\phi+\frac\lambda6\phi^3\right)\delta\phi.
$$
Thus the <Euler-Lagrange field equation> is
$$
\boxed{\Box\phi+m^2\phi+\frac\lambda6\phi^3=0.}
$$
For the <field redefinition> in part <1/f>, define the <functional derivative> $E(\phi)=\delta S_0/\delta\phi=-(\Box\phi+m^2\phi+\lambda\phi^3/6)$. The sign does not affect the <equation of motion>, but it fixes the sign of an added <redundant operator>.

Solution

Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 1 / b / Solution

Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 1 / b / Solution

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Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 1 / b / Solution

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