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The speed of a photon

In the electromagnetism, we refer to the electromagnetic wave as a light.  On the other side, in quantum mechanics, we refer to the same electromagnetic wave as a photon.  Inn relativistic theory, the light runs through the vacuum space by the speed c irrespective of the speed of an observer.  In quantum mechanics, a photon having the wave-particle dual nature takes only the wave nature while it is travelling through the space according to the Copenhagen School interpretation.  That wave nature must be quantum mechanically described by some equation like,

F(x, y, z, t) = 0              (1)

Logically speaking, since the light and the photon are identical, the photon ought to run through the vacuum space by the same speed c irrespective of the speed of the observer.  Then I wonder what will the form of the quantum mechanical equation (1) for a photon be like.  I would repeat that all of the indivisual photon need to run through the space by the universal speed c whichever inertial frame one is standing on to measure the speed of the photon.  Is the speed of the propagation of the wave function of the photon relativistically invariant, isn't it?  Or does the above question involve something wrong?

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