E=mc^2 and E=hf. In Special Relativity, how can y=h/p be derived from E=hf?
E2(p) = m2 c4 + p2 c2 [energy in the reference frame with momentum p]
E(p)=h/T(p) [T(p) is the time periodicity in the reference frame p ]
m c2 = E(0) [the mass is the energy in the rest frame p=0]
m c2 = h/T(0) [T(0) is the time periodicity in the reference frame p=0]
by putting all things together you find:
1/T2(p) = 1/T2(0) + c2/y2(p) [from the relativistic dispersion relation]
where y(p)= h / p [is the induced spatial periodicity in the reference frame with momentum p].
See http://arxiv.org/abs/0903.3680" [Broken] "Compact time and determinism: foundations"
Then if you impose the above periodicities as constraints to a string (field in compact space-time, similarly to the harmonic frequency spectrum of a vibrating string with fixed ends) you obtain the following energy quantization
E2n(p) = n2 E2(p) = n2( M2 c4 + p2 c2)
which is actually the energy quantization coming from the usual field theory with second quantization, after normal ordering. In arXiv:0903.3680 it is shown that this procedure provides an exact matching with ordinary quantum field theory, including Path integral and the commutation relations.
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