The Born rule states that if an observable of a particle in state $| ψ ⟩$ with eigenvalues ${λ_{1}, λ_{2}, \dots}$ is measured then the probability of yielding a particular eigenvalue $λ_{i}$ equals $| ⟨ λ | ψ ⟩ |^{2}$ .

In quantum mechanics, time-reversal is performed by complex conjugation $ψ \to ψ^{*}$ . If we apply time-reversal to the Born rule then all the math stays the same except $| ψ ⟩$ becomes $| ψ^{*} ⟩$ .

The mathematics for time-reversing the Born rule is straightforward complex algebra. I use the variable $V$ to indicate an arbitrary vector of complex values ${v_{1}, v_{2}, \dots}$ .

$\begin{matrix} ⟨ V | V ⟩ & = & \sum i | v_{i} |^{2} = & \sum i | v_{i}^{*} |^{2} = & ⟨ V^{*} | V^{*} ⟩ \end{matrix}$

Consider the particular case $ψ = V$ .

$\begin{matrix} ⟨ ψ | ψ ⟩ & = & ⟨ ψ^{*} | ψ^{*} ⟩ (⟨ ψ | ψ ⟩)^{*} & = & (⟨ ψ^{*} | ψ^{*} ⟩)^{*} | ψ^{*} ⟩ ⟨ ψ^{*} | & = & | ψ ⟩ ⟨ ψ | ⟨ λ | ψ^{*} ⟩ ⟨ ψ^{*} | λ ⟩ & = & ⟨ λ | ψ ⟩ ⟨ ψ | λ ⟩ | ⟨ λ | ψ^{*} ⟩ |^{2} & = & | ⟨ λ | ψ ⟩ |^{2} \end{matrix}$

The Born rule is invariant under time reversal because $| ⟨ λ | ψ ⟩ |^{2} = | ⟨ λ | ψ^{*} ⟩ |^{2}$ . The Born rule is time-symmetric because $| ⟨ λ | ψ ⟩ |^{2}$ is invariant under time reversal. But what, intuitively, does it mean?

Alternate Histories

We traditionally think of the multiverse as several possible futures branching off of the present.

But we have just shown that the arrow can be reversed. All possible futures are also possible pasts.

Distinguishing past from future is a tricky problem when the set of possible futures equals the set of possible pasts.

The physics establishment sidesteps this quandary by defining time to progress in the direction of increasing entropy.

Entropy

An ontology $O$ is an associative operator that buckets a set of microstates $m = {m_{i}}$ into a smaller set of macrostates $M = {M_{i}}$ .

$\begin{matrix} | m | > | M | O : m \to M \end{matrix}$

The entropy of a state equals the natural log of its macrostatic degeneracy.

The entropy $S$ of a macrostate $M_{i}$ equals the natural log of the number of microstates in its bucket.

$S (M_{i}) = ln | O^{- 1} M_{i} |$

The entropy $S$ of a microstate $m_{i}$ equals the natural log of the number of microstates in its bucket.

$S (m_{i}) = ln | O^{- 1} O m_{i} |$

The Random Walk of Time

The only thing you directly experience is this instant right now. The past and future are inferences. They are not empirical observations.

You are a classical being. Therefore your experience of this instant is a degenerate macrostate. A degenerate macrostate is a macrostate wi

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Quantum Gravity