Can one explain the relativistic energy transformation formula:
$$E = \gamma\ E',$$
where the primed frame has a velocity $v$ relative to the unprimed frame, in terms of relativistic time dilation and the quantum relation $E=h\nu$?
I imagine a pair of observers, A and B, initially at rest, each with an identical quantum system with oscillation period $T$.
Now A stays at rest whereas B is boosted to velocity $v$.
Just as in the "twin paradox" the two observers are no longer identical: B has experienced a boost whereas A has not. Both observers should agree on the fact that B has more energy than A.
From A's perspective B has extra kinetic energy by virtue of his velocity $v$. Relativistically A should use the energy transformation formula above.
But we should also be able to argue that B has more energy from B's perspective as well.
From B's perspective he is stationary and A has velocity $-v$. Therefore, due to relativistic time dilation, B sees A's oscillation period $T$ increased to $\gamma\ T$.
Thus B finds that his quantum oscillator will perform a factor of $\gamma\ T/T=\gamma$ more oscillations in the same period as A's quantum system.
Thus B sees that the frequency of his quantum system has increased by a factor of $\gamma$ over the frequency of A's system.
As we have the quantum relation, $E=h\nu$, this implies that B observes that the energy of his quantum system is a factor of $\gamma$ larger than the energy of A's stationary system.
Thus observer B too, using his frame of reference, can confirm that his system has more energy than observer A's system.
Is this reasoning correct?