What does this equation mean?

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 group throughout. This is the content of Bargmann’s own cohomology theorem: semisimple factors such as the Lorentz group admit no continuous central charge, so relativistic quantum mechanics carries no analogue of the mass superselection rule that follows from the Galilei case [ 1 , 4 ] . Mass in special relativity shows up instead as the ordinary Casimir invariant $P^\mu$ $P_\mu$=$m^2$$c^2$ : a quantum number you diagonalize, not a phase a loop leaves behind.