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The passage around this formula

The hole’s evaporation is modeled, as it is throughout this line of work, as an isometry V from the interior-plus-infalling-system Hilbert space to a radiation-plus-remaining-interior Hilbert space. What makes the conservation law bite is that V must commute with the group action on both sides: $V\,U_{\mathrm{in}}(g) \;=\; \big[U_{\mathrm{rad}}(g)\otimes U_{\mathrm{rem}}(g)\big]\,V$. Every symbol here is a unitary representation of the same rotation on a different Hilbert space — dimensionless linear operators, none of them observables in their own right. The equation says the hole’s internal dynamics cannot tell which way “up” was chosen before the compass fell in: rotate the input, and the output rotates the same way, split between radiation and remainder.…

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