norm and are orthogonal to any physical states. Thus, as is well known, a physical state $|{\rm phys}_1\rangle$ can be regarded as equivalent to any state of the form $|{\rm phys}_1\rangle + a^{(0n)\dagger}_{\omega lm}|{\rm phys}_2\rangle$. We can take as the representative elements the states obtained by applying $a^{(\lambda n)\dagger}_{\omega lm}$, $\lambda = 1,2$, on $|0\rangle$.

Unphysical particles created by $a^{(3n)\dagger}_{\omega lm}$ will be in thermal equilibrium in the Hartle-Hawking vacuum [@HH] for a static black hole if we require the gauge-fixed two-point function be non-singular on the horizons as in the scalar case [@KW]. There will also be a flux of unphysical particles in the Unruh vacuum. Therefore, technically speaking, the Hartle-Hawking and Unruh vacua are unphysical if we impose the Gupta-Bleuler condition using the positive-frequency notion in the Boulware vacuum. Fortunately, the BRST quantization does not suffer from this problem since the physical state condition does not refer to any notion of positive frequency [@KO]. This issue, however, will not concern us in the following, since all our calculations will be at the tree level.

Mode expansions in spherical polar coordinates are not very widely used and are quite different from the ordinary method using Cartesian coordinates. For this 