 Solution

ID: past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2022/5/a/solution

\frac{δ ϕ _{a} ( x )}{δ ϕ _{b} ( y )} = δ_{ab} δ^{(3)} (x - y), \frac{δ ϕ _{a} ( x )}{δ π _{b} ( y )} = 0.

Putting

F = Q

G = ϕ_{a} (x)

{Q, ϕ_{a} (x)}_{PB} = - \int d^{3} y \frac{δ Q}{δ π _{b} ( y )} δ_{ba} δ^{(3)} (x - y), {Q, ϕ_{a} (x)}_{PB} = - \frac{δ Q}{δ π _{a} ( x )} .

The superscript on the species label

a

in the printed denominator has no spacetime-metric significance; with the ordinary species Kronecker metric it is the same expression.

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