Nick Herbert



Abstract: In a truly non-causal world, Bell's Theorem cannot be formulated because in such a world elemental events are not stable enough for Bell-type non-locality to even be defined.

The justly celebrated theorem of John Stewart Bell purports to prove from the existence of certain quantum facts that any reality that underlies these facts must be non-local--happenings at distant regions of space influence one another by instantaneous (faster-than-light) connections. I examine here a loophole in Bell's Theorem which renders it invalid in some worlds.

Bell's Theorem is formulated for two entangled particles A & B (usually photons) whose properties (usually polarizations) are measured by two detectors (call them AA & BB) whose angular settings can be changed. In the photon case these settings are the Angle of Polarization (z). For photons in what I call the "twin state" whenever the detectors AA and BB are set at the same angle z the photons are observed to always have the same polarization (either UP or DOWN, symbolized as 1 or 0). Each individual photon pair event occurs at random so that whether it registers 0 or 1 is unpredictable but whatever the outcome of an individual detector (say AA) the other detector BB will always register an identical outcome. Total agreement at x = zero degrees (where x is the relative angle between detectors).

On the other hand when the detectors are misaligned by 90 degrees, each detector registers the opposite outcome. For each photon pair, if AA registers 0, then BB will inevitably register 1. Total disagreement at x = ninety degrees.

When the detectors are misaligned by some angle x between zero and ninety degrees, then the agreement is described as the cosine squared of x--a function that smoothly varies between 1 (total agreement) and 0 (total disagreement) as the angle x varies between zero and ninety degrees. This cosine squared dependence is what quantum mechanics predicts (quantum theory) and what is actually measured (quantum fact).

Bell showed in 1964 that this cosine-squared dependence of polarization correlations is incompatible with a local reality. Therefore any reality (quantum reality) that lies behind these facts must be non-local. It is important to realize that Bell's Theorem is based solely on the facts not on the details of quantum theory so that if someday quantum theory is falsified or replaced by a better theory, Bell's Theorem will still be valid. Even if quantum theory is wrong, reality must be non-local.

Although Bell's Theorem is based on facts, it goes beyond facts to make confident assertions about the reality behind those facts. Because it deals with reality, not facts or theory, Bell's Theorem is metaphysical, not physical, and hence is vulnerable to certain metaphysical assumptions about the nature of the world that are essential to Bell's proof. If we live in a world where these metaphysical assumptions are not justified, then Bell's theorem cannot even be formulated. Here I examine a class of worlds in which Bell's Theorem does not apply, and in which non-locality, in the way it's defined by Bell, does not even make sense.

The proof of Bell's theorem begins with both detectors aligned. Then at each detector we observe a particular binary sequence of photon polarization states (say 1101001100) that is identical for both detectors. Perfect match at zero degrees.

Now we imagine changing the orientation of the BB detector to an angle (30 degrees) where the the sequences are mismatched by 25%. And we apply the famous Locality Assumption: that moving the BB detector can only change BB events, and cannot change events at distant detector AA. 25% sequence mismatch at 30 degrees.

Proceeding in this fashion we can show that the Locality Assumption leads to a mathematical contradiction with the measured facts (the cosine-squared match function) so we conclude that the Locality Assumption is mistaken and that Reality must be non-local. In a non-local reality the results at detector AA must depend not only on the setting of detector AA but also on the setting of the distant detector BB--a dependence which for very distant detectors (and rapid changes in detector settings) would necessitate faster-than-light signaling.

But this non-local signaling takes place "only" on the level of reality not on the level of fact. Nothing that humans can observe about this two-photon experiment ever changes faster-than-light (no observed non-local facts). And even though quantum theory is profoundly non-local in its structure, the predictions of this non-local theory are always local (no predicted non-local facts). So what Bell's Theorem seems to be telling us is that we don't really have to trouble ourselves about these mathematically necessary non-localities in Nature, because these Bell non-localities are not factual but "merely real".


One might be justifiably suspicious about the claim that "reality is non-local" when this claim seems to have no direct experimental consequences. What other assumptions enter into Bell's Theorem besides the quantum facts and the assumption that detector AA's result is independent of the setting of detector BB?

The most crucial move in the Bell Theorem proof is this. Each quantum event N (the paired results at AA and BB for a particular setting of the two detectors) occurs only once but Bell's proof considers the results of not only the one event N that actually happened but three hypothetical versions of the event N that could have happened but did not.

Only one of these configurations can actually happen for a given photon pair N (say N = 3, 947) yet all four configurations (for the same N) figure into the proof.

1. Both detectors aligned.
2. AA misaligned by 30 degrees.
3. BB misaligned by -30 degrees.
4. AA misaligned by 30 degrees and BB misaligned by -30 degrees (total misalignment = 60 degrees)

So to make Bell's Theorem work we must consider the hypothetical results of three experiments (on the single photon pair N) which we could have done but which in fact we did not. This is where metaphysics enters. Certain kinds of worlds will allow the consideration of the results of such hypothetical experiments and certain kinds of worlds will not.

In the world of classical physics, for instance, it makes sense to talk about the results of experiments that you could have performed but did not. Since the classical world

How can we reason correctly about causeless events?

is governed by deterministic laws, one can not only talk about such hypothetical experiments but specify their outcome. Therefore, if underneath the statistical predictions of quantum theory lies a classical world (a so-called hidden-variable theory), which specifies the outcome of each event then the results of hypothetical experiments (on the same system N) make sense. Bell's Theorem then proves that such a hidden variable model MUST BE NON-LOCAL.

But what if the world is not undergirded by a hidden variable theory and is irreducibly random and a-causal? Can we still prove Bell's Theorem in such a world? I will argue that we cannot.

The essence of classical determinism is that the same initial state leads to the same outcome. The essence of a-causality is that the same initial state can lead to many different outcomes. What could it mean to live in a truly a-causal world?

In an a-causal world each event is truly "uncaused", fresh and without precedent, governed only by statistical laws. The only control one has over such a world is to change the possible outcomes. Each event is brand-new but if more outcomes favor B, then B will happen more often than A. This is how dice work: there are more elementary dice events that add up to seven so seven comes up more often than two. We believe that dice are governed by classical physics but it is easy to imagine dice-like systems that are utterly causeless and whose outcomes are determined solely by the number of open possibilities and nothing else.

Besides the issue of determinism, quantum mechanics (without hidden variables) differs from classical mechanics in another essential way--the matter of Identity. In classical mechanics (the solar system, for example) all manner of orbits and masses are possible; no two classical systems are exactly alike. In quantum mechanics, it is the very essence of quantization that all electrons are considered to be alike, all protons are alike and (in analogy with the solar system) all hydrogen atoms are alike.

For example, we can imagine an atom excited to a particular state. It emits a photon at an unpredictable time in an unpredictable direction. The same atom, excited again emits a photon at an unpredictable time and direction. Quantum mechanics asserts that there is no intrinsic difference between these two excited atoms. They are EXACTLY the same. Yet they behave differently. I claim that this peculiar behavior--the same thing acting differently--is the very essence of the New Physics.

This quantum identity principle clarifies and intensifies the a-causal nature of quantum events. The essence of a-causality is that the same initial situation can lead to many different outcomes. One can always wiggle out of this stricture by claiming that seemingly identical situations were in fact slightly different--hence the different outcomes. But the quantum identity principle closes that loophole. Quantum identity says that these two systems are "truly and essentially" equal; quantum a-causality says that these truly identical systems behave (for no reason at all!) differently. Same system (quantum identity); unsame outcome (quantum a-causality).

Now let's reconsider photon pair N (where N = 3, 947) in the situation where both detectors are aligned and both detectors clicked "1".

Now we imagine doing this experiment again (same N) changing only the orientation of the BB detector. We assume (locality) that the result of the AA detector will still be "1" but this assumption seems to smack of an unexamined faith in determinism. In a truly a-causal world, all bets are off, and both AA and BB events are utterly uncertain (constrained only by statistical rules) so AA could either be "0" or "1", not changed by non-local influences but by the fundamental a-causal nature of reality.

In a truly a-causal world, even in the aligned case (where N = 3,947) where both counters clicked "1", a hypothetical repeat of the "same experiment" would be equally likely to give a result in which both counters clicked "0". In classical physics, a hypothetical repetition of the same experiment will give the same result. In what I am calling "a truly a-causal reality", repeating the "same experiment" is likely to give different results.

Thus in an a-causal world the locality assumption--that the result at AA remains unchanged when we change the BB detector--is intrinsically meaningless. The result at AA is not a well-defined concept. In an a-causal world the result at AA is likely to be different even when everything else in the world (itself included) is unchanged.


If quantum reality consists of hidden variables that give a unique and hypothetically repeatable result to each experiment then Bell's Theorem constrains those hidden variables to be non-local. In what I have called "a truly a-causal reality", Bell's Theorem cannot even be formulated because such a reality is not stable enough for us to be able to define the concept of non-locality that is essential to the proof. Thus for certain kinds of realities, Bell's Theorem is refuted. It is not a question for physicists but for metaphysicians to decide whether, for better or worse, we live in a world where Bell's Theorem makes sense, or if we live in a truly non-causal reality where classically-contaminated thinking leads to false conclusions. Perhaps we have yet to come to grips with a world whose elemental events are truly without cause and whose only reliable "realities" are probabilities and correlations (another type of probability for two or more).

Nick Herbert
May 2007

1. See http://quantumtantra.com/bell2.html for a simple proof of Bell's Theorem.

2. Quantum reality aficionados may recognize this refutation as an explicit version of the CFD (contrafactual definiteness) loophole first publicized (to my knowledge) by Berkeley physicist Henry Stapp. The contrafactual definiteness assumption requires that if I had repeated experiment N with a different setting I would have got a definite result. However, in an a-causal world each instance of a system is completely "new" (as if you had pushed a "reset button") and for each new system the outcome is likely to be different. Neither the notion of "repeating an experiment" nor the notion of "definite outcome" is well defined in an a-causal world. The "reset button" severs all connections with the previous experiment (so what are you repeating?) and the a-causality produces not one outcome but a different outcome for each reset.

3. This refutation applies only to those Bell's Theorem proofs (such as that found in note #1) that define "locality" in terms of individual events ("locality" means that the individual events at A are not changed by BB's choice of measurement at B). Proofs that define "locality" in terms of probabilty ("locality" means that the probabilty of events happening at A are not changed by BB's choice of measurement at B) may still be valid in an a-causal world.