Past exam of the quantum field theory course of the University of Oxford / 2021 / 2 / b

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- Solution b

Solution

The canonical anticommutation relations are

{a_{p}^{s}, a_{q}^{r †}} = {b_{p}^{s}, b_{q}^{r †}} = (2 π)^{3} δ^{sr} δ^{(3)} (p - q),

with the mixed and equal-type anticommutators zero. Insert the mode expansion of a Dirac field. The phases cancel at equal time, and a gives

{ψ_{α} (t, y), ψ_{β}^{†} (t, x)} = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (y - x)} \frac{\sum _{s} [ u _{α}^{s} ( p ) u _{β}^{s *} ( p ) + v _{α}^{s} ( - p ) v _{β}^{s *} ( - p )]}{2 E} = δ_{α β} δ^{(3)} (x - y) .

Since

\overset{ˉ}{ψ} γ^{0} = ψ^{†}

, the requested componentwise answer is

{(i \overset{ˉ}{ψ} γ^{0})_{α} (t, x), ψ_{β} (t, y)} = i δ_{α β} δ^{(3)} (x - y) .

This is the Equal-time canonical anticommutator of a Dirac field: the canonical momentum conjugate to

ψ_{α}

is

π_{α} = i ψ_{α}^{†}

. It is the fermionic counterpart of the canonical field canonical commutation relation, ensuring the correct local canonical quantization. For this first-order Lagrangian density, the conjugate variables are constrained; the displayed relation is the standard result after handling that constraint. It does not assert a nonzero anticommutator between the fields at separated spatial points.

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