More late night thinking...

(Probably mad, but worth saving for posterity)


Today's post follows on from this previous post in which I argued that the entire universe is in a superposition of states. I showed how this can lead to the impression that only the system under study is in a superposition, whilst the rest of the world is in a single - classical - state.  To summarize that argument:

  • Reality is a superposition of states for the entire universe, each of which can be thought of as a complete classical block universe.
  • We don't directly experience the whole universe, so let's arbitrarily place a closed surface around us to demarcate what we "directly" experience.  Remember that this is a surface in 4 dimensional spacetime and so is itself 3 dimensional.  Also remember that it is completely arbitrary: if you like you can let it contain your entire body for your entire life; or it could just be your brain for some duration.  You could even let the surface enclose everything except some system you happen to be thinking about.  It's just a construct.
  • We don't feel like we are in a superposition of states.  So consider just a single classical state for "our" side of the surface, but let the other side range over all matching extensions of this classical state. In this way we have constructed a subset of all universal states.  If our side of the closed surface contains a representation of what's on the other side, it has to represent it as a superposition or be wrong for most members of the subset.
  • Our experience doesn't seem completely random.  We observe patterns.  For this reason we assert that each subset so constructed has a "weight". We postulate that this weight is given by the amplitude of our side of the closed surface multiplied by the sum of the amplitudes making up the other side.
  • Let's name the sum of the amplitudes making up the other side the "amplitude of the surface".  If the surface has zero amplitude, the subset will have zero weight - i.e. the subset represents a situation that is never experienced.
  • If we wish to calculate the amplitude of a surface we can make a clever shortcut.  If there's an embedded surface (think of it like the patterned wall of a nucleus inside a cell) which itself has zero amplitude, then states that intersect with that embedded surface can be ignored.  This means that although our side has to represent the other side as a superposition, it doesn't have to include every possibility!
In this post I want to give an example of how this approach resolves an apparent paradox known as the EPR paradox, after its discoverers Einsten, Podolsky, and Rosen.  In the version I am using, a pair of electrons are placed in the entangled singlet state, moved apart, and then the vertical component of spin, $\sigma_z$ is measured on each.  Critically, the two measurements are performed a space-like interval apart so that no causal connection between the two is possible without breaking locality.

The singlet state is normally described as
$$
\frac{1}{\sqrt{2}}\left( \left| \uparrow\downarrow\right> - \left| \downarrow\uparrow \right> \right)
$$
However, it is only described in this way because we arbitrarily chose $\{\left| \uparrow\uparrow\right>,\left| \uparrow\downarrow\right>,\left| \downarrow\uparrow\right>,\left| \downarrow\downarrow\right>\}$ as our basis.  We could easily have chosen any other two diametrically opposite directions. A less arbitrary way of thinking about the singlet state is as an evenly distributed superposition of every state in which the two spins are pointing in opposite directions.

If the experiment described above is performed the result is always either that the spin is measured "up" on the LHS and "down" on the RHS, or "down" on the LHS and "up" on the RHS.  But you never get "up"/"up"!

First, let's try to interpret this result using the universe-is-classical-at-some-scale interpretation and see why this result is problematic.

Universe-is-classical-at-some-scale interpretation

I'm using this tag to collect together all the various non-many-worlds interpretations. These all require that "at some scale" the universe is not in a superposition, but allow it to be a superposition at some smaller scale. The details are always a bit vague about where the crossover occurs (because there's no evidence for it) but whatever the scale at which it happens some sort of collapse postulate is required such that when superposed "quantum" systems interact with "large scale" ones a single state is chosen at random, with the probabilities derived from the wavefunction.

Let's suppose that the collapse occurs prior to the spins reaching the "up"/"down" spin-o-meters. At this point the singlet state breaks down from a superposition of pairs of opposite spins to a specific pair of opposite spins. If this were to happen then occasionally you'd get a "left"/"right" pair, and then you'd get "up" or "down" with a 50/50 probability when $\sigma_z$ is measured.  But the spins would not be entangled at the point the measurements are taken, and so the results would not be correlated. Therefore you'd at least occasionally get "up"/"up" readings.  Since this never happens we can reject the idea that the singlet state collapses into two independent spins prior to measurement.

Okay, so let's suppose it collapses at or shortly after measurement. In this case the space time region in which we consider superposition to exist is shown in yellow in the diagram below

In this case we are saying that there's nothing "up" or "down" about either spin until they're measured, yet when they're measured you always get a matching pair. The two measurement events are correlated without a common cause, and this breaks locality.

Spacetime-is-a-superposition-of-block-universes interpretation

The supposed paradox dissolves with the interpretation I set out at the start of this post.  All we need to do is imagine a closed surface on which "up"/"up" is measured by the spin-o-meters and demonstrate that this surface has an amplitude of zero.  It will then follow that the outcome in the diagram above is one with weight zero and is therefore never observed.

Now, the amplitude of a closed surface is just the sum of the amplitudes for each possible extension into the interior. Let's look at just one of these extensions where the interior consists of two opposite spins inclined at $\theta$ to the vertical.  The state representing the left hand spin is

$$ cos \left(\frac{\theta}{2}\right) \left|\uparrow\right> + sin\left(\frac{\theta}{2}\right) \left| \downarrow\right> $$

and for the right hand spin it's
$$ -sin\left(\frac{\theta}{2}\right) \left|\uparrow\right> + cos\left(\frac{\theta}{2}\right) \left| \downarrow\right> $$
I haven't proved the above but note that the expectation value for the vertical component of spin comes out as $cos(\theta)$ for the left hand spin, and $-cos(\theta)$ for the right hand spin (taking "up" to be +1 and "down" to be -1) and this is what is expected.

So the amplitude for getting "up" on the LHS is $cos\left(\frac{\theta}{2}\right)$ and the amplitude for getting "up" on the RHS is $-sin\left(\frac{\theta}{2}\right)$ meaning the amplitude for both occurring is
$$ -cos\left(\frac{\theta}{2}\right) sin\left(\frac{\theta}{2}\right) =\ -\frac{1}{2}sin(\theta) $$
But this is the probability amplitude for just one extension into the interior. It clearly cancels the amplitude you get when the spins are inclined at $-\theta$.  It follows that the boundary of the yellow region, which contains two "up" measurements, has amplitude zero, and so cannot be one that is ever experienced.

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