Tim Maudlin

(1958-)

Tim Maudlin's book *Quantum Non-Locality and Relativity* is a critical analysis of Bell's Theorem.

Maudlin says that the "interaction among distantly separated particles presents profound interpretive difficulties." (p.20). He cites three features of this "quantum connection" between particles as surprising, even "weird." (pp.22-23)

- The quantum connection is unattenuated.
It appears to be unaffected by distance. Quantum theory predicts that exactly the same correlations will continue unchanged no matter how far apart the two wings of the experiment are

- The quantum connection is discriminating.
It is a private arrangement between two particles. When one is measured, its twin is affected, but no other particle need be. Only particles which have interacted with each other in the past seem to retain this power of private communication.

- The quantum connection is faster than light (Instantaneous).
[N]o relativistic theory can permit instantaneous effects or causal processes. We must therefore regard with grave suspicion anything thought to outpace light.

Maudlin does not discuss the possibility that there is a "common cause" for these distant but perfect correlations, coming along with the particles from the past light cone, so not violating relativity. Nor does he mention David Bohm's claim that no such common cause is possible.

Bohm suggested that "local hidden variables" traveling with the particles might explain the perfect correlations (which John Bell said seemed "pre-determined"), but it could not explain the apparent randomness (indeterminism) of the sequence of measurement outcomes of each particle (critically needed for the random bit strings of quantum cryptography).

Maudlin's method is a logical analysis of the "questions" and "answers" in a game that reproduces the results of a sequence of Bell-test experiments, similar to that in David Mermin's 1985 "contraption."

Maudlin writes...

Over a long run of this game you are aiming to reproduce the behavior of the photons in similar circumstances, That is, after a long series of plays, you want to ensure that
- Fact 1: When you and your friend happen to be asked the same question you always give the same answer.
- Fact 2: When your questions differ by 30, that is, when one is asked "0?" and the other "30?" or one is asked "30?" and the other "60?", you and your friend agree 3/4 of the time
- Fact 3: When your questions differ by 60, that is, when one of you is asked "0?" and the other "60?", your answers agree 1/4 of the time.

After all, this is what the photons manage to do. (*Non-Locality*, p.14)

*Quantum Non-Locality and Relativity*, p.14

The "questions" in Maudlin's *logical* game correspond to the *physical angle* settings of the particle detectors at positions A and B. The "answers" correspond to the spin directions ("up" or "down") found as outcomes of the measurements. When A and B measure by pre-agreement at the same angle (ask the same "questions"), their spins are always perfectly correlated in opposite directions. This is Maudlin's fact 1.

When their "questions" (the measurement angles) differ by angle θ, their correlations are diminished by the square of the angle's cosine - cos^{2}θ, as Maudlin explains.

The mathematics of quantum theory predicts precisely the observed experimental results.

The Dirac/Schrödinger "superposition" equation for Schrödinger's two-particle wave function is

| *ψ*_{12} > = (1/√2) | *+* *-* > - (1/√2) | *-* *+* >

The coefficients 1/√2, when squared, tell us that there is a 50/50 chance that the particles will be found in the state *+* *-* or in the state *-* *+*.

When measurements by A and B (the "questions") are made at angles differing by angle 30°, since the cosine of 30° is 1/2, the "answers" agree cos^{2}30° = 1/4 of the time. This is Maudlin's fact 2.

When measurements by A and B (the "questions") are made at angles differing by angle 60°, since the cosine of 60° is √3/2, the "answers" agree cos^{2}60° = 3/4 of the time. This is Maudlin's fact 3.

Now when measurements differ by 30°, the "answers" *disagree* sin^{2}30° = 1/4 of the time.

Instead of the state *+* *-* or the state *-* *+*, the outcomes that disagree are found randomly in states *+* *+* or *-* *-*. Either outcome *appears to violate* the conservation of total spin angular momentum zero.

We call the conserved total spin zero a "hidden constant of the motion". The initial entangled state is a "singlet" state that is spherically symmetric. The rotational symmetry means it has spin angular momentum zero in any and all directions. Measurements will find the spins opposite as long as the measurements are made in perfectly parallel or perfectly opposite directions.

Conservation ensures that this *shared property* of the two particles is true at all times up to the moment of (simultaneous) measurements. If measurements are not made symmetrically, the measurement apparatus imparts additional spin angular momentum to the particles, and its loss of that spin balances the particles' gain, so the particles plus the apparatus continue to conserve the total spin angular momentum zero.

The conservation law is the implicit reason why David Bohm, John Bell, and many others say that when one particle is measured spin-up, we instantly know the other must be spin-down.

Exactly how the bit strings of data at A and at B are indeterministically random, even as the combined A and B results appear to be deterministically correlated, Maudlin does not discuss.

But the Dirac/Schrödinger "superposition" equation also explains this perfectly. The fact there is a 50/50 chance that the particles will be found in the state *+* *-* or in the state *-* *+*, just as observed, means that the bit strings at A and B can be used as quantum keys that have been distributed to A and B in a way that cannot be intercepted by an eavesdropper.

Quantum key distribution (QKD) does not require impossible faster-than-light instantaneous actions at a distance. No "hidden variables" are needed. The "hidden constant" will suffice.

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