Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 2 / b

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- Solution b

Solution

The electric components of the electromagnetic field tensor are

F_{0 i} = \dot{A}_{i}^{V} - \partial_{i} Φ_{A},

while

F_{ij}

depends only on the transverse vector. The scalar-vector cross term integrates to zero because

\partial_{i} A_{i}^{V} = 0

, so the scalar action is

S_{scalar} = \frac{1}{2} \int d t d^{3} x a (\partial_{i} Φ_{A})^{2} .

Varying the nondynamical scalar gives the constraint equation in field theory

\nabla^{2} Φ_{A} = 0 .

For every nonzero Fourier momentum this fixes

Φ_{A} = 0

, confirming that a source-free massless vector has no propagating scalar polarization.

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