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[Ki,Pj]=iℏc2 δij H[K_i,P_j]=\frac{i\hbar}{c^2}\,\delta_{ij}\,H

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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 [Ki,Pj]=iℏc2 δij H[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\,b\mathbf b⋅\cdotv\mathbf v/(ℏ\hbar c2c^2) that varies from state to state. A central extension, by definition, must give the same phase…

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[Ki,Pj]=iℏc2 δij H,[K_i,P_j]=\frac{i\hbar}{c^2}\,\delta_{ij}\,H,

Equation 51 · 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.

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 [Ki,Pj]=iℏc2 δij H[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\,b\mathbf b⋅\cdotv\mathbf v/(ℏ\hbar c2c^2) that varies from state to state. A central extension, by definition, must give the same phase…

Meanings in this article

  • HH: the total energy operator [ 2 ].
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