Please post further comments or answers to   The probability that an initial quantum state $|\psi_i\rangle$ evolves to become the final quantum state $|\psi_f\rangle$ is given by  \begin{eqnarray} P_{i \rightarrow f} &amp;=&amp; |\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle|^2 \tag{1}\\ &amp;=&amp; \langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle^*\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|U^\dagger_{i \rightarrow f}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|U_{f \rightarrow i}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \end{eqnarray} where $U_{i \rightarrow f}$ is the forward-time evolution operator and $U_{f \rightarrow i}=U^\dagger_{i \rightarrow f}$ is the corresponding backward-time evolution operator.  Equation (1) seems to show that the probability $P_{i\rightarrow j}$ can be interpreted as the system first evolving forwards in time and then evolving backwards in time.  Perhaps this is an example of Murray Gell-Mann's  that "Everything not forbidden is compulsory"? At the quantum level, below observable probabilities, there is nothing to stop time flowing both forwards and backwards.  Does this reasoning help to explain the Born rule?
The probability that an initial quantum state $|\psi_i\rangle$ becomes the final quantum state $|\psi_f\rangle$ is given by  \begin{eqnarray} P(i \rightarrow f) &amp;=&amp; |\langle\psi_f|\psi_i\rangle|^2 \tag{1}\\ &amp;=&amp; \langle\psi_f|\psi_i\rangle^*\langle\psi_f|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|\psi_f\rangle\langle\psi_f|\psi_i\rangle. \end{eqnarray}  Equation (1) seems to show that the probability for the transition ($i\rightarrow f$) can be interpreted as the system both moving forward in time ($i\rightarrow f$) with amplitude $\langle\psi_f|\psi_i\rangle$ and backward in time ($f\rightarrow i$) with amplitude $\langle\psi_i|\psi_f\rangle$ simultaneously.  Does this reasoning help to explain the Born rule? (Is it like the Transactional Interpretation of QM?)  I guess we must experience the macroscopic direction of time ($i\rightarrow f$) in accord with increasing entropy in an expanding universe whereas microscopically QM works both forwards and backwards in time.  Addition  This is an improved version of the argument including time-evolution operators.  The probability that an initial quantum state $|\psi_i\rangle$ evolves to become the final quantum state $|\psi_f\rangle$ is given by  \begin{eqnarray} P_{i \rightarrow f} &amp;=&amp; |\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle|^2 \tag{2}\\ &amp;=&amp; \langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle^*\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|U^\dagger_{i \rightarrow f}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|U_{f \rightarrow i}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \end{eqnarray} where $U_{i \rightarrow f}$ is the forward-time evolution operator and $U_{f \rightarrow i}=U^\dagger_{i \rightarrow f}$ is the corresponding backward-time evolution operator.  Equation (2) seems to show that the probability $P_{i\rightarrow j}$ can be interpreted as the system first evolving forwards in time and then evolving backwards in time.  Perhaps this is an example of Murray Gell-Mann's  that "Everything not forbidden is compulsory"? At the quantum level, below observable probabilities, there is nothing to stop time flowing both forwards and backwards.
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