Past exam of the quantum field theory course of the University of Oxford / 2024 / 1 / a

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Solution

Write

\int_{p} = \int d^{3} p / (2 π)^{3}

. The canonical commutation relations and the product rule for a commutator give

[a_{p}, a_{q}^{†} a_{q}] = (2 π)^{3} δ^{3} (p - q) a_{q}, [b_{p}^{†}, b_{q}^{†} b_{q}] = - (2 π)^{3} δ^{3} (p - q) b_{q}^{†} .

The other-species terms vanish. Integrating the Dirac delta distributions in the Hamiltonian therefore gives

[a_{p}, H] = E_{p} a_{p}, [b_{- p}^{†}, H] = - E_{p} b_{- p}^{†},

where

E_{- p} = E_{p}

. Consequently the complex scalar field satisfies

[ϕ (t, x), H] = \int_{p} \frac{E _{p}}{2 E _{p}} (- b_{- p}^{†} e^{i E_{p} t} + a_{p} e^{- i E_{p} t}) e^{i p \cdot x} .

Direct differentiation of the time factors gives the same expression, proving the Heisenberg equation of motion

i \partial_{t} ϕ (t, x) = [ϕ (t, x), H] .

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