Tim Maudlin
Here is a "lively discussion" with Professor Tim Maudlin who teaches the Philosophy of Science and who has a long-standing interest in the controversies around what to make of Bell's Theorem.
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Here is my answer.
What percentage of practicing professional physicists understand this?
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Tim Maudlin's answer: Bell’s Inequality is a mathematical constraint (in the form of an inequality) on the observed statistics of experimental outcomes that every *local* physical theory must respect. Suppose you have two (or three) laboratories that are spatially highly separated, and in each l.....
Barry Kort In "The Defeat of Reason" in Boston Review (June 01, 2018), you begin the section on Bell's Inequality thusly:
«John Stewart Bell queried whether Einstein’s dreaded spooky action at a distance could be avoided.»
In John Bell's 1964 paper, the derivation of his inequalities is mathematically valid if one of these two crucial assumptions is met:
1) The presumptive hidden variables are static (not time-varying), or
2) The presumptive hidden variables are time-varying and time ticks at the same rate everywhere and everywhen in the cosmos.
Several decades later, in experiments pioneered by Alain Aspect, Bell's inequalities were discovered not to hold in our cosmos. Why not? Where did Bell's calculus go awry? The resolution of this mystery can be found by reconsidering those two crucial assumptions in Bell's 1964 paper.
As it happens neither of those two essential assumptions in Bell's model is valid in our cosmos. General Relativity, in particular, reveals that there is no universal cosmic clock. The rate at which time ticks varies from one location to the next and (among other things) depends on the strength of the local gravitational field.
Instead of the presumptive hidden variable vanishing in the middle of the derivation of Bell's Inequalities, there necessarily arises a non-vanishing "beat frequency" term that survives to the bottom line.
The presumptive hidden variable turns out to be time, itself (and any parameter that is time-varying).
The fact that Bell's Inequalities do not hold in our cosmos is confirmation that any hidden variables are necessarily time-varying and Einstein's GR is right about time-keeping being a function of the local gravitational field strength.
What Einstein derided as "spooky action at a distance" turns out to be not-so-spooky time-keeping at a distance.
Barry Kort When it comes to not-so-hidden variables, here are some examples:
To a good approximation, photons obey Maxwell's Equations, which characterize photons as a traveling wavefront composed of sinusoidal oscillations of electromagnetic fields.
String Theory posits that quarks can be modeled with mathematics of the sort associated with vibrating strings.
Quantum Spin is modeled in a way similar to a toy top that is precessing around its mean spin axis.
Let's take a look at one way to model Quantum Spin by considering how a mathematical model of a toy top provides a suitable framework.
Almost everyone has played with a toy top.
You observe a toy top spinning on the tabletop and you can say a few things about it.
1. It's spinning clockwise or counterclockwise.
2. It's spinning around its own axis at a rate that encodes as angular momentum.
3. It's also precessing around the upright direction with some angle from the vertical and with some period.
In quantum spin, we can imagine precession, but we can't directly observe it. But we can reasonably infer that the precession angle must be 45° from the mean spin axis and the period of precession must be very fast; there is no way to measure the phase of the precession at any instant.
Entanglement means two particles are not only spinning with their mean spin axes aligned, but their precession phases are also perfectly synched, like two ballet dancers pirouetting in sync.
But what happens when you move them apart? Since time-keeping is local, their instantaneous precession phases drift out of sync. That's called "decoherence" and it's a feature of General Relativity (because time-keeping depends on the strength of the local gravitational field). That simple observation explains why Bell's Inequalities are violated in our cosmos.
Once you have that model in mind, all the paradoxes vanish.
To summarize, the key observation is that Bell's Inequalities do not apply to our cosmos because they only would hold for an imaginary cosmos where timekeeping is uniform everywhere and everywhen in the cosmos. That simplifying assumption was demonstrably overthrown with General Relativity.
Discard the simplifying assumption about universal timekeeping, and Bell's derivation becomes a homework problem in the perils of oversimplifying crucial aspects of the model. Or as the saying goes, the devil is in the details.
Barry Kort Now, given that timekeeping is demonstrably local, here's a follow-up homework problem. In the following narrative (from the same article in Boston Review), which of these conclusions is confirmed? And which of these conclusions is falsified?
«Bell proved that the nonlocality is unavoidable. No local theory—the type Einstein had sought—could recover the predictions of quantum mechanics. The predictions of all possible local theories must satisfy the condition called Bell’s inequality. Quantum theory predicts that Bell’s inequality can be violated. All that was left was to ask nature herself. In a series of sophisticated experiments, the answer has been established: Bell’s inequality is violated. The world is not local. No future innovation in physics can make it local again. The spookiness that Einstein spent decades deriding is here to stay.»
Tim Maudlin No, you are confused about the content of the theorem.
Any distribution of additional variables at the initial time yields a distribution over results, no matter what the dynamics of the additional variable (and indeed the wavefunction) is. Bell can prove his result at a level of abstraction that simply does not care about those details. (In his 1964 paper, he does take the result of the EPR argument for granted, and so can focus on deterministic dynamics, so there is a map from initial conditions and detector settings to outcomes.)
There is no step in Bell's derivation where he gets anything to cancel with anything. You seem to be commenting on a proof that bears no relation at all to what he did. I suggest you study his paper more carefully.
Barry Kort Any hidden variable, λ(x,t) that includes a sinusoidal time-varying component has to take into account that the age of the particles at +x and -x drift apart by some phase difference associated with Δt, the difference in timekeeping at locations separated in space. Sinusoidal time-varying components of λ(x,t) vary with their local time by definition. Photons traversing a gravitational field gradient undergo a corresponding "frequency modulation" otherwise known as the gravitational red shift.
There is a key step in Bell's derivation where he manages to get λ(+x,t) to cancel with λ(-x,t) because he is assuming that the value of t (the age of the particles) is exactly the same. He is assuming perfect phase-locked synchrony, which in our cosmos is an untenable simplifying assumption. If Bell had considered λ(+x,t+Δt) versus λ(-x,t-Δt), his hidden variables would not have vanished. Rather they would have combined to yield a non-zero beat frequency term that survives to the bottom line.
Tim Maudlin Barry Kort No, you just paid zero attention to what I wrote.
Any distribution of additional variables at the initial time yields a distribution over results, no matter what the dynamics of the additional variable (and indeed the wavefunction) is. Bell can prove his result at a level of abstraction that simply does not care about those details. (In his 1964 paper, he does take the result of the EPR argument for granted, and so can focus on deterministic dynamics, so there is a map from initial conditions and detector settings to outcomes.)
There is no step in Bell's derivation where he gets anything to cancel with anything. You seem to be commenting on a proof that bears no relation at all to what he did. I suggest you study his paper more carefully.
Barry Kort I paid attention to what you wrote, but you are saying that Bell doesn't care about a crucial detail that he overlooked. When I pay attention to that overlooked detail, I can no longer carry out the derivation.
I can see how λ(x,t) vanishes if you assume λ(x,t) = λ(-x,t) for all x and t, but not if you drop that simplifying assumption. Equivalently, the math would work if you assume that λ is and odd function, such that λ(x,t) = λ(x,-t).
If λ(x,t) is a sinusoid (e.g. the E-field of a photon under Maxwell's model), then it follows that a sinusoid integrates to zero over any integer number of complete wavelengths. But that doesn't mean the sinusoid itself is exactly zero.
Similarly if λ(+x,t) ≠ λ(-x,t), then you get a beat frequency term that's still a sinusoid, and like any sinusoid it still integrates to zero over an integer number of "beats" in the beat frequency. But you cannot say the beat frequency itself is identically zero.
But more to the point, how can anyone infer there can be no hidden variable at all? Feynman simplified Maxwell's model down to a rotating vector for the photon, where the speed of rotation corresponds to the frequency (color) of the photon.
Let's say Alain Aspect aligns his apparatus along an east-west line and runs it at dawn on the day of the New Moon. Then the eastbound photon is descending the gravitational gradient associated with the monthly Spring Tide. It undergoes a blue shift (meaning the speed of Feynman's rotating vector is increasing). Conversely the westbound photon is ascending a gravitational gradient and undergoes a red shift. The two rotating vectors drift out of phase. That model suffices to yield a prediction that when the two photons reach their detectors at +x and -x respectively, they will be at different phases in their underlying sinusoidal model. That random phase difference suffices to explain the outcome of Aspect's experiment.
Bell's Inequality does not hold in our cosmos because his model fails to include the crucial detail that time-keeping is local. How else would you explain that Bell's model generates a prediction that does not agree with experimental results?
Feynman says it best:
«Now I am going to discuss how we would look for a new law. In general we look for a new law by the following process. First we guess it. Then we compute the consequences of the guess to see what would be implied if this law that we guessed is right. Then we compare the result of the computation to nature, with experiment or experience, compare it directly with observation, to see if it works. If it disagrees with experiment it is wrong. In that simple statement is the key to science. It does not make any difference how beautiful your guess is. It does not make any difference how smart you are, who made the guess, or what his name is — if it disagrees with experiment it is wrong. That is all there is to it.»
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Barry Kort «The additional variable lambda that Bell integrates over is a distribution of whatever additional physical variables there happen to be *at the initial time of the experiment*.»
Ok, let's say that λ represents an analytic function such that λ(0) encodes everything you need to know to predict the state of the photons when they arrive at the detectors at +x and -x, respectively. Then λ(0) has to encode not only the initial phase and color of the twin photons, it also has to encode the strength of the gravitational field at every point along the future path of the two photons as they speed apart. In other words the analytic function, λ, has to have countable derivatives at λ(0) sufficient to encode the future, as it pertains to the journeys of the two photons.
No such omniscient analytic function could possibly exist, and any model that posits such an analytic function would be relying on an absurd assumption. We know that the photons will in fact experience (largely unknown) gravitational field gradients along their journeys, and hence they will ineluctably drift out of phase (decohere), but no such λ(0) can encode that amount of knowledge in advance. At best we can anticipate aspects of the quantum state that are, to a good approximation, nominally time invariant. The instantaneous phase of any rapidly time-varying component of the quantum state is elusively beyond our reach.
Tim Maudlin Barry Kort No. No. Lambda is a complete speficiation of the state at the initial time, save for the settings of the later experiments. The proof has nothing to do do with what we know or can predict: it is a purely mathematical result. Whether the functions are analytic is neither here nor there.
Again, you seem to have no grasp at all of what Bell proved. It is not even a matter of correcting a misconception: you are basically just making things up, as you did about Bell postulating anything canceling anything else.
Barry Kort Bell "proved" a result that disagreed with experiment. The question to be answered is simple:
What is wrong with his model? What did he assume that is at odds with observable reality?
He assumed there exists a continuous and differentiable function λ, that characterized the state of twin particles that are arbitrarily separated in space, even if they obeyed the same model, λ(•) at time zero. But a single λ cannot specify all the relevant details of the state of two particles, once they are separated in space. They each need their own independent λ function, because λ(x) ≠ λ(-x) for any x > 0.
Bell's "purely mathematical result" is based on an unrealistic assumption that a single function, λ, suffices to capture all the relevant characteristics of twin particles, once they become separated in space. In particular the instantaneous phase of any time-varying element in λ(•) necessarily decoheres as they go their separate ways, and λ(0) cannot capture that ineluctable dechorence.
Tim Maudlin "He assumed there exists a continuous and differentiable function λ, that characterized the state of twin particles that are arbitrarily separated in space,"
No he didn't. He nowhere differentiates anything.
Once again, all you are proving is that you have no idea what you are talking about.
Barry Kort In Equations 2 and 19, he has dλ inside the integral.
How can he have dλ unless he assumes λ is a differentiable function?
Barry Kort Right after Equation 2, Bell writes:
«Some might prefer a formulation in which the hidden variables fall into two sets, with A dependent on one and B on the other; this possibility is contained in the above, since λ stands for any number of variables and the dependences thereon of A and B are unrestricted.»
Note that I suggested above that using the same function λ(•) for the two sides is problematic if λ(•) includes a time-varying component.
In the same paragraph, Bell continues:
«In a complete physical theory of the type envisaged by Einstein, the hidden variables would have dynamical significance and laws of motion; our λ can then be thought of as initial values of these variables at some suitable instant.»
And this is where you assert that only λ(0) matters, and not the concern that the particles evolve independently in time and space after time zero.
But the initial values then omit the crucial information, namely that a dynamic model that includes a time-varying component must somehow account for the fact that time-keeping is local, which is why the two particles age independently.
The fact that Bell's derived inequality does not hold in our cosmos (for reasons explained in GR), impels us to conclude that the information contained in the initial value λ(0) omits the crucial information that would have been taken into account had Bell expressly included the disparate local time-keeping of the particles as they sped away.
Tim Maudlin I have pointed out over and over that you have made plainly false claims about the structure and presuppositions of Bell's theorem. The passage you cite does nothing to respond to the facts I pointed out. There is nothing about anything canceling anything, there is nothing about the exact nature of the time evolution, there are no derivatives taken, there is no assumption of analyticity. Every claim you have made about a tacit presupposition of the theorem is false. If there would be any point of citing part of the paper, it would be to show these presuppositions are there. They aren't.
Once more: you have no idea what you are talking about. You seem to have no interest in understanding what Bell did. Please stop posting this nonsense.
Mate Jagnjic Barry, one can show Bell's inequality by considering spin measurement along different 3 axes (with 120° angle between any of them) preformed on the singlet state pair of particles, with respecting only one prediction of QM formalism: probability of opposite results of the measurement ("up" spin for one particle and "down" for another) is 0.5(1+cos(angle)).
E.g. you can show that any set of pre-existing random variables (whatever you can think of, be it number or a function or something else) cannot reproduce predictions of QM.
Tim Maudlin Mate Jagnjic Well, one also needs the locality constraint, of course! That’s why the result is not just a theorem of the probability calculus.
Barry Kort In integrating dλ, one is obliged to consider the independent variables in λ(•).
if λ is a function of x and t, then dλ(•) = ∂λ/∂x + ∂λ/∂t, where (under GR) timekeeping, t, is a function of x, so that one also needs to include ∂t/∂x (which, to a first approximation is a constant if one models the gravitational gradient as linear along Alain Aspect's east-west axis on the morning of the New Moon.
Einstein had been dead for nine years when Bell's 1964 paper came out. I have no doubt that if Einstein had been alive to review Bell's 1964 paper, he would have raised similar objections.
Barry Kort Mate Jagnjic ~ The mean spin axis can be treated as a fixed direction in space, but doing so leaves out the instantaneous precession of the spin axis around its mean direction in space.
We can infer from the actual experiments (as well as from modern understanding of Larmor Precession in Magnetic Resonance Imaging) that the Precession Angle for Spin ½ must be 45° if the model is to agree with observations. The rapid precession frequency means the twin particles rapidly decohere (drift out of phase with each other) due to differential timekeeping at points separated in space.
Once you take this overlooked detail into account, Bell's inequalities can no longer be derived, and instead one arrives at a model that makes predictions consistent with actual experimental measurements. At the instant of measurement, the unknown (i.e. random) phase of the precession around the mean spin axis would explain the results of the Bell Test Experiments on Spin ½ particles.
Tim Maudlin Barry Kort Once again, all you are doing is displaying a shocking lack of understanding. |x| is not differentiable at 0, but that creates zero problems integrating it there.
Please stop.
Barry Kort Let's go back to Bell's crucial simplification, as cited above:
«In a complete physical theory of the type envisaged by Einstein, the hidden variables would have dynamical significance and laws of motion; our λ can then be thought of as initial values of these variables at some suitable instant.»
In that simplifying assumption, Bell cavalierly discards GR.
Why is that crucial simplifying assumption so bloody untenable?
Because GR tells us that timekeeping is local, being dependent on the strength of the local gravitational field!
In discarding GR, Bell tacitly assumes that clocks keep the same time everywhere and everywhen in the cosmos. In discarding GR, the phase and frequency of the photons at time zero would have sufficed to predict the phase and frequency of the photons forever and ever. But lo and behold, that model (and its consequential prediction) do not agree with experimental measurements.
Therefore, that simplifying assumption must be eliminated and replaced by a more appropriate model that embraces GR and admits that timekeeping is local, and not governed by the clock located at x = 0.
And if one does that — if one adopts a working model consistent with Einstein's model — one would derive a different result than Bell's result where he ignored the differential timekeeping as revealed in GR.
F.A. Muller 3rd paragraph (locality as outcome-setting independence: violated) in tension with the closing paragraph (outcome-setting independence: proved). 3rd paragraph should have been: outcome-outcome independence, right?