$$\rho_F(g)=\operatorname{Tr}_{R\bar F} \!\left[\mathcal U\bigl(\rho_R(g)\otimes\rho_E\bigr)\right]$$

Equation 27 · Evolutionary Physics

A Reference Frame Becomes Classical by Publishing Its Orientation

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Let the rotor interact with an environment E=$E_1$$\otimes$$\cdots$$\otimes$ $E_N$ through a channel $\mathcal U$ . For a candidate fragment F , the conditional fragment state is $\rho_F(g)=\operatorname{Tr}_{R\bar F} \!\left[\mathcal U\bigl(\rho_R(g)\otimes\rho_E\bigr)\right]$. This equation defines what the observer receives. $\bar F$ is everything except fragment F , including the remaining environment; $\mathcal U(\rho)$=U$\rho$ $U^\dagger$ in a closed unitary model. The trace is not an assertion that the discarded systems cease to exist. It says the fragment observer has no access to them.

Meanings in this article

• $\rho_F$: the same for every g , no measurement on F can reveal orientation.
• $\bar F$: everything except fragment F , including the remaining environment.
• $\rho_R$: the chosen fiducial state.

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