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Quantum field theory course of the University of Oxford

Past exam of the quantum field theory course of the University of Oxford

2017

The two real <fields> have canonical normalization because
$$
\phi^*\phi=\frac{\phi_1^2+\phi_2^2}{2},\qquad
\partial_\mu\phi^*\partial^\mu\phi=\frac12\sum_{a=1}^2\partial_\mu\phi_a\partial^\mu\phi_a.
$$
Therefore
$$
\boxed{\mathcal L=\frac12\sum_{a=1}^2\left[(\partial\phi_a)^2-m_0^2\phi_a^2\right]
-\frac{\lambda_0}{16}\left(\phi_1^4+2\phi_1^2\phi_2^2+\phi_2^4\right).}
$$
The interaction is invariant under $O(2)$ rotations and reflections of $(\phi_1,\phi_2)$. The continuous rotations are the original complex-field $U(1)$ <symmetry>. The factor $1/16$ follows from both the original $1/4$ and the square of the $1/2$ in $\phi^*\phi$.

Solution

Past exam of the quantum field theory course of the University of Oxford / 2017 / 3 / b / Solution

Past exam of the quantum field theory course of the University of Oxford / 2017 / 3 / b / Solution

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Past exam of the quantum field theory course of the University of Oxford / 2017 / 3 / b / Solution

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