Connections between Deviations from Lorentz Transformation and Relativistic Energy-Momentum Relation

Violations of Lorentz transformation for space-time scales render corrections to relativistic energy-momentum relations and vice versa. Some recent articles[1-31 on tests of the special theory of relativity (STR) were motivation to write down a result obtained several years ago: thft the transformation properties of intrinsic space and time scales depend on the dispersion relation (that is the energy-momentum relation) of a particular system; and vice versa: a deviation from the Lorentz transformation would result in nonrelativistic transformation of energy and mass, as well as nonrelativistic energy-momentum relations. Throughout this communication, Einstein's original operationalizations for the concepts [4] of simultaneity and two-way velocity of light are applied. This enables a better comparison between STR and alternate theories. A quantized two-state system { i l) , 12)) is considered, which can serve as a clock (present-day state of the art atomic clocks can be described very similarly). The system starts out in a state i l) , which is no eigenstate of the Hamiltonian H . It will, therefore, undergo oscillations between 11) and 12). Assume an arbitrary real number tA for the initial time, and an arbitrary real number tB > tA for a later time. Denoting the amplitudes of the states by c l ( t ) = (l /+(t)) and c2(t) = (2l+(t)); and assuming symmetric transition rules Hll = Hz2 and H12 = HZl, the Schrodinger equation reads (la) i ~1 = Hi1 ~1 + Hi2 ~2 , (1b) i e2 = Hz1 e1 + H2* e2 . Its solutions are oscillations in Hilbert space. With the above initial condition I c ~ ( ~ A ) I ' = 1 / c 2 ( t ~ ) 1 ~ = 1, a short calculation yields (2) d dt d dt Icl(tB)12 = cos2 [H12(tA tB)l 84 EUROPRYSICS LETTERS The time span tB tA has been defined arbitrarily; and (2) can be measured. Hence, the energy scale of H has to be calibrated, such that (2) is satisfied. This condition sets the energy scale in a particular frame of reference. Now consider two reference frames, a and b, both carrying (quantized) clocks with them. Suppose an observer in a watches the clock in 6, as b passes with a velocity U. The observer is comparing the times of synchronized clocks resting in a with the time of the clock resting in b. Since the quantum states of the clocks can be described similarly in both systems, state vectors and amplitudes can be identified: for instance 11,) = , lb) = 1) and c l ( t l d = cl(tLd. Since in both frames a Schrodinger equation similar to (1) can be written down, the dynamics is the same, and we obtain by identifying the amplitudes for state l ) , The indices a and b mean <<measured in a and b.. For infinitesimal time difference At = tB fA-+ dt, eq. (3) yields Next, some velocity c defined to be constant covention immediately (prefereably a sound velocity or the maximum signal velocity) is in all reference frames, in particular c, = c b = c. With eq. (4), this yields the dilatation laws for space scales parallel to v :