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Quantum approach (super photons)

If we look at the expressions (2.15)  and (2.16) in derivation of planar hyperspace waves, and we try to calculate the energy of wave, we will soon realize, that there is something odd with this approach. The reason is that  the energy is going to be infinite in both cases (STL and FTL frames) i.e. the overall energy is infinite for regular planar electromagnetic wave according to equation (2.15) , as well as for  planar hyperspace waves according to equation  (2.16).  Explanation of this apparent problem is though quite obvious  - the waves span infinitely in time and space end therefore they posses infinite energy. But in real world we don't have infinite energies and waves with infinite span. So how to deal with that problem? The answer is, that we should use quantum approach instead, and thus calculate the energy of wave carrying particles.

Quantum theory suggests, that electromagnetic waves are radiated in wave packets or quanta called photons. Quantum theory introduces wave-particle duality and also a probability function Ψ , which must satisfy so called Schrődinger equation. Let's start with cutoff  Gaussian envelope solution of Ψ for photon particle [31] (7.9)  (1.1) :





The above expression considers only one spatial dimension x, but it's possible to re-write this expression in 3D. Let's introduce wave vector and space vector (1.2):


 
Now the 3D solution (still timeless)  for photon looks like (1.3):




Now introduce time t in equation and assume that regular photon propagates as a non-dispersive wave packet at the speed c (1.4):
 


(Note: the variation of wave packet in  direction normal to direction of propagation was neglected, as it's not important for the derivation)



CASE 1 - STL (v<c)

Lorentz transform for STL (slower then light) case looks like (2.1) :
                

Now let's substitute this transformation to argument of equation 1.4, which describes photon wave packet (2.2):





From that, after argument comparison, we will get red shift formula for  photons (2.3):




The above formula for  ω'  is formula of the relativistic Doppler effect as described in reference [43]. Please note that the value of ω' depends also on κ ratio defined as (2.4):


Ratio κ express direction of the propagation of the photon with the respect to the direction of STL frame propagation. For κ=0 the formula
2.3  becomes the formula of  transversal Doppler effect, while for κ=1 it is formula of the  regular Doppler effect.


CASE 2 - FTL (v>c)

Now let's analyze the FTL (faster then light) moving frame case. For simplicity, but without loss of generality, let's assume, that the dashed frame is moving at speed v>c along to x-x' direction.
We can rewrite 2.1 for v>c as Lorentz transformation for FTL frames (3.1):

                                                                                                                                                    


Please note, that x and t variables have now purely imaginary (!) values. Now let's substitute this transformation to argument of equation 1.4  of photon wave packet (3.2):




From that, after argument comparison, considering equality [44] (3.3):





 we will get finally Doppler effect formula for  photons in FTL frames (3.4):




Therefore we can rewrite Doppler effect formula for FTL frames  as (3.5):




From equation 3.5 it's obvious, that we have now in general 4 solutions depending on selection of s1=±1 and s2=±1.  It is assumed, that quantities s1 and s2, which select the final solution are quantum parameters of given photon (seen from FTL moving frame), similar  e.g. to spin of a particle. Like in STL case ω' and kx'  depend also on κ ratio defined in 2.4. But please note, that both  ω' and k have now also purely imaginary (!) values. This is the most important conclusion from whole derivation. 

Photon formula now gets different form, form of  dumped exponential. Moreover from the transformed formula it is obvious, that particles are now located in "infinity" and have 2 new quantum parameters
s1 and s2. Therefore lets introduce new name for photons seen from FTL frames. We will call them  "super photons".
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