content_image/334199.jpg content_image/334198.jpg content_image/334197.jpg content_image/334196.jpg content_image/334195.jpg
longer diagonalize the polarization operator already because the vectors ( ˜ Fk )$_{µ}$ and ( Fk )$_{µ}$ stop being mutually orthogonal, since their scalar product − k ˜ F Fk = G k 2 is now nonzero. 

When G ̸ = 0, the first eigenvector is expressed in terms of the fields by the same formula as in (7): 

<img src='content_image/334199.jpg'>

and the first eigenvalue is given by the formula 

<img src='content_image/334198.jpg'>

where the scalar function of the fields and momentum Λ$_{1}$ here, as well as other Λ’s below, is a linear superposition of the polarization tensor components Π$_{µν}$. The other two eigenvectors are the linear combinations 

(where the square root is understood algebraically: √$_{Z}$ 2 = Z , and not | Z | ) of two orthonor- malized vectors : 

<img src='content_image/334197.jpg'>

<img src='content_image/334196.jpg'>

thereby of the former basic vectors ( ˜ Fk )$_{µ}$ and ( Fk )$_{µ}$ , too. The corresponding two eigenvalues are 

<img src='content_image/334195.jpg'>

The scalar coefficients in the linear combination (10) cannot be expressed in a universal way in terms of the field and momentum, but are irrational functions of the polarization tensor components. The reason is that the polarization operator is a linear combination of four independent matrices with four scalar coefficients, whereas there may be only three eigenvalues in accordance with three polarization degrees of freedom of a vector field. (When 

