Born’s Rule

C. Born’s Rule

Superdeterministic theories are hidden variables theories. This means they explain the seeming randomness of quantum mechanics as due to our lack of information about variables λ which do not appear in standard quantum physics. To recover a probabilistic theory, therefore, we must now incorporate the hidden variables.

For this, we will assume that the probability of a prepared state to go from an initial state |Ψs⟩ := |Ψ(ts)⟩ into an end state |Ψe⟩ := |Ψ(te)⟩ is determined by random variables X(|Ψs⟩, |Ψe⟩, λ), which are, to a good approximation, independent for each possible end-state. The end state into which the initial state evolves is that with the smallest X.

We define the random variables so that they have rate (inverse mean value)

r(|Ψs⟩, |Ψe⟩) := e −2A(|Ψs⟩,|Ψe⟩), (31)

where

… (32)

and so that the probability distribution has maximum entropy, which means it is given by

… (33)

This distribution depends on the end state |Ψe⟩ via r and hence on the measurement settings and thereby violates measurement independence. To understand this expression, we note that for the rotation in section II C, we have C(t) = cos(θ(t) − θs) , so that

… (34)

and hence r = |α2| 2 .

In the case of the enforced product state of section II B we have instead

… (35)

Its absolute value generically gives a correction of order one. That is, by order of magnitude we get

A ∼ τm|Φ12| , (36)

where τ is the time that the superposition remains in existence. To move on, we define ∆t = te − ts and |Ψ′

as the final state under the Schrodinger evolution.

We have previously seen that if the final state is to good accuracy a product state of metric and matter it can serve as a measurement eigenstate. We will denote these states as |ΨI ⟩,

that is, the values have to fulfill the condition λ1 < λ2.

We can then calculate the probability of this happening by integrating over the probability distributions

.. (38)

where r1 and r2 are the rates associated with end state 1 and 2, respectively. The result of this integral is

… (39)

We see that if the end state |Ψe1⟩ is that of the usual Schrodinger evolution but not a product state, then the prob- ¨ ability is exponentially suppressed. We hence never observe these outcomes. The best possible path is one with a sudden branch rotation. In this case P = |α1| 2 for the state to go to end state |Ψ1⟩, which is what Born’s rule requires. Do the probabilities come out correctly for arbitrary numbers of |ΨI ⟩, I ∈ {1, 2, . . . D}? Yes, they do. One can confirm this the hard way by actually calculating the integral for D probability distributions. But we can make it simpler by denoting PI := P(|Ψe⟩ = |ΨI ⟩). It follows from the above that

|αI | 2 |αI | 2 + |αJ | 2 = PI PI + PJ ∀ I ̸= J . (40)

Since this equation is trivially also true for I = J, we get

|αI | 2 |αJ | 2 = PI PJ ∀ I, J . (41)

And because both the PI and the |αI |

sum up to 1, we have

1 = X D I=1 PI = PJ |αJ | 2 X D I=1 |αI | 2 = PJ |αJ | 2 . (42)

So, PJ = |αJ | 2 ∀ J.

We also see that since the integral in A is additive and in the exponent, the probabilities of consecutive local rotations will factorise, as they should.

The probability distribution of the hidden variables (33) appears like a bunny out of a magic hat rather than following from anything in particular. This is entirely true, but absent an underlying theory really the best I can do is to provide an example for the probability distribution that reproduces quantum mechanics3

For what I am concerned, it doesn’t pop any more out of the hat than Born’s rule pops in the standard formulation of quantum mechanics. Of course one hopes that eventually one would be able to derive this (or some other) expression from some deeper theory, but we will leave this to future work. . . A second question one might have is why I introduced both S and A and not just used A for both purposes. The reason is not only that this doesn’t make physical sense, it doesn’t work.

S is the functional whose variation determines the possible paths that the system can take. A is a statistical measure that counts paths in the presumed to be underlying theory. They should not be the same. Worse, if we set them to be the same, we would run into the problem that δA = A = 0 on any path of the form e

iφ(t) |Ψ′ ⟩, φ(t) ∈ R.

That is, we would lose the usual Schrodinger evolution as a unique solution in the case ¨ when gravity can be neglected.

Another question one might have here is how this can possibly reproduce Born’s rule if there are other time evolutions from the initial to the final state that we could realise with local interactions even disregarding gravity. To be concrete, consider a Mach-Zehnder interferometer. Under the usual Schrodinger evolution, the state goes in a superposition over ¨ both paths, and then recombines. But we could alternatively use a local rotation at the first beam splitter to either one of the paths, and then another local rotation on the second beam splitter into the wrong port. Why does this not happen? The reason this cannot happen is that such a path would not be a stationary solution of S. As long as the outcome of

the usual Schrodinger evolution is a detector eigenstate, the ¨ product state constraint does nothing and the usual solution of the Schrodinger equation is the optimal path. It is only ¨ when the forward evolution of the Schrodinger solution is not ¨ a product state between matter and geometry that the product states become the stationary solutions.

In case you noticed that (35) could be equal to zero for α1/2 = α ′ 1/2 , |α1| = |α2| = p 1/2, and a suitably chosen phase, do not worry. This path is not stationary since any nudge towards smaller |α1| or |α2| will decrease the residual, and there always must be a local path for this, since otherwise the state that we actually measure could not come about locally.

That is, it was somewhat unnecessary to include this case here. However, it will make it easier to interpret the mathematics, which we turn to next.

Born’s Rule

C. Born’s Rule

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