Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 301 / 2 / ii

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- Solution ii

Solution

For a free real scalar field, the Lagrangian density contains its quadratic kinetic term and mass term:

L_{1} = \frac{1}{2} \partial_{μ} ϕ_{1} \partial^{μ} ϕ_{1} - \frac{1}{2} m_{1}^{2} ϕ_{1}^{2} = \frac{1}{2} \dot{ϕ}_{1}^{2} - \frac{1}{2} ∣\nabla ϕ_{1} ∣^{2} - \frac{1}{2} m_{1}^{2} ϕ_{1}^{2} .

The field's conjugate momentum is

π_{1} = \partial L_{1} / \partial \dot{ϕ}_{1} = \dot{ϕ}_{1}

. The positive sign of the time-derivative term uses the metric convention fixed in Question 1.

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