It is certainly obeyed by the free massive particle: k$_{0}$ = ( k 2 + m $^{2}$)1 / 2 ≤ m + | k | , where m = ( k$_{0}$ ) | k = 0 . The gauge invariance property (21) implies via equation (14) that for each mode there always exists a dispersion curve with m 2 = f$_{a}$ (0) = 0, which passes through the origin in the ( k 2 0 − k 2 $_{‖}$, k 2 $_{⊥}$)-plane. It is such branch that is to be called a photon , since it is massless in the sense that the energy k$_{0}$ turns to zero for the excitation at rest, k$_{3}$ = k$_{⊥}$ = 0 (although, generally, k 2 = 0 where k = 0 ). Other branches for each polarization mode a may also appear provided that a dynamical model includes a massive excitation of the vacuum with quantum numbers of a photon, for instance the positronium atom [5, 17–19] or a massive (pseudo)scalar particle (axion) in a gauge-invariant interaction with the electromagnetic field [36]. Note that while the number of polarization modes of a vector particle is three – in correspondence with the dimension of the space and, hence, with the number of degrees of freedom, – dispersion curve for each of the three modes may have any number of branches, e.g. an infinite number of excited positronium branches. The energy on a dispersion curve should be real, since the dispersion equation (14) supplies poles to the photon propagator [4, 5] D$_{µν}$ ( k ) = 3 ∑ a =1 ♭ ( a ) µ ♭ ( a ) ν ( ♭ ( a $^{)}$)2 1 k 2 − κ $_{a}$( k ) , (28) (defined up to arbitrary longitudinal part ∼ k$_{µ}$k$_{ν}$ ), and these should not get into complex plane. If the state is unstable and should therefore decay, its energy must have an imaginary part, indeed, but in this case the pole is located on an nonphysical sheet of the complex plane, whose presence must be provided by branching points in the polarization operator to be introduced within an approximation where the state is expected to be unstable. An example of such situation is given by the cyclotron resonance [14] approximation of the polarization operator in a magnetic field. The corresponding dispersion equations are cubic with respect to energy squared. Out of its three solutions, one corresponds to a stable state and, therefore, is real, whereas the other two mutually complex conjugated branches responsible for the photon decay/capture to electron-positron pairs belong to nonphysical sheets of the complex energy plane. Neither of these solutions can be disregarded. As applied to unstable branches our appeal to the group velocity may be reasonable only provided that the absorption is small, i.e. the imaginary part is much less than the real one. On the other hand, the number of massless modes is, as a matter of fact, not three, 15