Numerator: langle Hrangle

The passage around this formula

The two pictures are not independent theories that happen to share a symbol. The Galilei phase is the c$\to$$\infty$ residue of the Poincaré non-extension, and the residue can be taken explicitly rather than asserted. Write the energy operator for a system at rest as H=M$c^2$+$H_{\rm NR}$ , separating the rest energy from whatever nonrelativistic energy — kinetic, internal — sits on top of it. The relativistic loop phase becomes $\frac{\langle H\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v=\frac{M}{\hbar}\,\mathbf b\cdot\mathbf v+\frac{\langle H_{\rm NR}\rangle}{\hbar c^2}\,\mathbf b\cdot\mathbf v$. Holding $\mathbf b$ , $\mathbf v$ , M , and $\langle$ $H_{\rm NR}$$\rangle$ fixed and sending c$\to$$\infty$ , the second term vanishes and the first survives unchanged, converting a state-dependent relativistic phase into a fixed, universal constant multiplying every…

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