10.7. Casimir effect Consider two parallel horizontal planes, separated by a distance a along the axis z.

Suppose that a massless field theory is defined between the two planes, with boundary conditions that ensure a non-zero value of the expectation value of the stress-energy tensor Tμν, tμν(t, x) ≡ 0 | Tμν(t, x) | 0. The system is assumed to be time invariant.

Thanks to the symmetry of the problem, tμν can be written in terms of the metric tensor gμν and the tensors made of the unit vector ˆ zμ = (0, 0, 0,1).

a. Write the most general expression of tμν based on the considerations given above.

b. Show that the conservation law ∂μTμν(t, x) = 0 and the zero-trace condition of Tμν uniquely tμν up to a constant. Use the dimensional analysis to fix this constant (up to a numerical coefficient) in terms of the only dimensional parameter of the problem.

c. Use the final formof tμν to compute the force per unit area between the two planes.