# A simple explanation of the Born rule?

The probability that an initial quantum state $$|\psi_i\rangle$$ becomes the final quantum state $$|\psi_f\rangle$$ is given by

$$\begin{eqnarray} P(i \rightarrow f) &=& |\langle\psi_f|\psi_i\rangle|^2 \tag{1}\\ &=& \langle\psi_f|\psi_i\rangle^*\langle\psi_f|\psi_i\rangle \\ &=& \langle\psi_i|\psi_f\rangle\langle\psi_f|\psi_i\rangle. \end{eqnarray}$$

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.

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

$$\begin{eqnarray} P_{i \rightarrow f} &=& |\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle|^2 \tag{2}\\ &=& \langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle^*\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &=& \langle\psi_i|U^\dagger_{i \rightarrow f}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &=& \langle\psi_i|U_{f \rightarrow i}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \end{eqnarray}$$ 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 Totalitarian Principle that "Everything not forbidden is compulsory"? At the quantum level, below observable probabilities, there is nothing to stop time flowing both forwards and backwards.

• I think all you're seeing here is that the Born rule has to be consistent with the time-reversal symmetry possessed by quantum mechanics in general. Your way of stating the Born rule also seems wrong to me. The Born rule only makes sense as a way of talking about measurements. It's not a rule for time evolution. Time evolution occurs via the Schrodinger equation. – Ben Crowell Sep 25 at 14:18
• @BenCrowell "The Born rule only makes sense as a way of talking about measurements. It's not a rule for time evolution." – this statement is what proponents of different interpretations disagree about. – Prof. Legolasov Sep 27 at 11:42
• You should merge the material from your new question into this one; that one was closed as a duplicate of this one (as per general Stack Exchange policy), so it's no longer possible to post answers to it. – PM 2Ring Sep 27 at 11:57

The Born rule and the continuity equation are just the Noether charge conservation law. The Nöther charge of the Schrödinger equation is simply $$e|\psi|^2$$. The probability density is proportional to the charge density by $$e$$.