$$P^\mu P_\mu=m^2c^2$$

Equation 56 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

where H is the total energy operator [ 2 ] . The right-hand side is not proportional to the identity. It is proportional to H , an operator with a spectrum, different on every energy eigenstate. Repeating the loop calculation of the previous section on a state with sharp energy $\langle$ H$\rangle$ gives a phase $\langle$ H$\rangle$\,$\mathbf b$$\cdot$$\mathbf v$/($\hbar$ $c^2$) that varies from state to state. A central extension, by definition, must give the same phase to every vector in the representation; a state-dependent phase is not a central extension, it is an ordinary consequence of ordinary dynamics, and it can be removed by working with true, non-projective unitary representations of the Poincaré…

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