subscript

The passage around this formula

The obvious next question is why ordinary relativistic quantum mechanics gets along without any of this. Run the identical calculation on the Poincaré group. The relativistic analogue of the boost–momentum commutator is $[K_i,P_j]=\frac{i\hbar}{c^2}\,\delta_{ij}\,H$. 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…

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