Today:

• Spin
• The Bell tolls for local realism.

Uncertainty relations for Spin

In QM, many physical systems have complementary pairs of observables which cannot be measured at the same time. Position (x) and momentum (p_x) are one such pair. The product of the uncertainties must exceed .

Another physical quantity, spin, will be important in arguments to follow. Think of a spinning ball. Its spin angular momentum points along the axis of rotation and has a length equal to the rate of rotation times the moment of inertia. It is a vector, , and all three components can be specified.

That is the classical picture. In QM, pairs of spin components satisfy uncertainty relations . This means that one can only know, or measure, one component of spin at a time. (One can know the magnitude as well.)

When one measures s_x, one always obtains a multiple of .

When one measures s², one always obtains . s is a multiple of 1/2. Results of spin measurements are quantized. A new feature: The angular momentum of states in an atom can be zero. This runs counter to the classical picture of orbits. In fact, the lowest energy state of the hydrogen atom has L=0.

Experimental implications

If you separate a beam of neutrons into s_x= +1/2 and s_x =-1/2 beams (by running through magnet pole-faces), you can discard the (-1/2) part and make a beam of pure s_x =1/2 neutrons.

Now try measuring s_y (just using a magnet turned 90°): you find that the measurements still give ±1/2, with a random pattern of + and - results.

If you take either the s_y= 1/2 or the s_y= 1/2 beam, and again try measuring s_x, you also find random results. The neutrons don't seem to be able to remember both values at once. (the uncertainty relation)

But if you recombine the s_y= 1/2 and the s_y= 1/2 beams without measuring, i.e. without letting them interact with some sort of detector, the resulting beam is still all s_x= +1/2.

Each s_x= +1/2 was BOTH s_y= +1/2 and s_y= -1/2, and follows BOTH pathways . Only a "measurement" makes it choose one or the other. Apparently s_y is not specified by a hidden variable, because each s_x= +1/2 neutron seems to have both values of s_y.

A Science Fiction Story

Say that you want to find out why people like pepperoni, mushrooms, and olives on pizza.

You ask many people, and they give Yes/No answers when asked about each. (Say 50% yes for each.) But you don't notice any distinguishing properties of the people who say Y.

Does that mean you can conclude that the answer is random, some sort of momentary glitch that occurs in a person's response?

Of course not- you may just have missed the "hidden variables" needed to understand taste. There MAY be something in each person's mind ahead of time that determines their answer, or maybe not.

Try asking couples. You find that each couple gives the same answer to the one question you ask them, so you can be sure that there was already something in their heads that determined the answer. Otherwise how could they all get their stories straight? So there WAS some hidden variable determining the outcome.

(There's a slight complication- all these folks seem to get confused after one question. If you ask each member of the couples a second question, the perfect correlation between them is lost. So you're only allowed to ask one of each person.)

Now you try asking one person about pepperoni, and the other about mushrooms. You find that 85% of the time, they give the same answer (YY or NN). So you conclude that whatever makes people like mushrooms usually makes them like pepperoni.

You try the same thing asking about mushrooms and olives. Again 85% of the time, they agree. (YY or NN)

Now you ask about pepperoni and olives. What extent of agreement do you expect? The 15% who had different opinions on M-P and the 15% with different opinions on M-O might be the same people. Then there would be 100% agreement on P-O. Or maybe they're all different people. Then there would be 100%-15%-15% = 70% agreement on P-O. Or it could come out anywhere in between.

You now ask the couples do you like pepperoni/olives?

And you find they give the same answer 50% of the time.

What gives?

Maybe there was a statistical fluke. You try the same thing again with 100,000 couples- and get the same result.

Maybe the couples were secretly signaling each other, getting their stories coordinated (on some questions) AFTER the questions were asked.

You ask the questions in sealed boxes. Same result.

You ask the questions SIMULTANEOUSLY (in Earth frame). Same result.

Maybe the couples were getting their stories straight only on the questions which they knew they would be asked.

You draw the questions out of hats, in the sealed boxes, simultaneously. And get the same result.

Conclusion: this story is obviously fiction. The identical results on any ONE question tell you that (whether or not you believe that people in general are spontaneous and random) on these particular issues there had to be some prior content in their heads determining the answers. But there is NO WAY of assigning answers to the 3 questions (M,O,P) which gives these results for the disagreements:

Violation of triangle inequality!

Now we tell exactly the same story about particles, where "couples" are pairs emitted together in a decay process.

"Like pepperoni?" becomes "spin up along x?"

"Like olives?" becomes "spin up along y?"

"Like mushrooms?" becomes "spin up along direction halfway between x and y?"

Experimentally the results are as we have described. (Most of the real experiments use a slightly different set-up with photon polarization, with the same logical structure.)

Obviously when we concluded that this story is science fiction we assumed something that's false.

What did we assume about reality?

Note that we assumed nothing whatever about QM- we didn't even mention it. It happens that QM precisely predicts the results, but we are now finding how the world differs from our assumptions, not how quantum theory differs from our assumptions.

WE assumed

Realism. Not necessarily that the world is deterministic, but only that those aspects of it which can be predicted perfectly are determined by some element of reality. You can predict the result on any particle by making the corresponding measurement on its partner.

"If, without in any way disturbing a system, we can predict with certainty (i.e., probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." (Einstein, Podolsky, Rosen, 1935)

Local causality. The value of a property possessed by an isolated system cannot be affected by any operations carried out at a sufficient (i.e., spacelike) separation from it.

Induction. The properties of an ensemble are defined completely by the preparation conditions. In particular, the distribution of "possessed values" of a variable for the subensemble which we actually measure is identical to the distribution for the complete ensemble. (random sampling)

Which one are you willing to give up?

There's a conceivable loophole: the initial passage of the particles through the angular momentum selectors are space-like separated. However, in the current generation of experiments, the conversion of those micro-events to some large-scale device setting occurs slowly enough for it to be conceivable that a time-like signal could propagate between the detectors before "measurement" is complete. This again is a real long-shot, but one that can be addressed experimentally just by speeding up the detectors and setting them farther apart.

This loophole was recently narrowed substantially in some Austrian experiments, in which the distance between the detectors was several hundred meters, and in which the analyzer settings were set by local random number generators. Experiments are planned in which the detectors will be on satellites separated by several hundred km. (These expts are funded by agencies interested in quantum cryptography.)

Does the problem lie with probabilities?

Our discussions have concentrated on probabilities and correlations. This may give the incorrect impression that the fundamental issue is the probabilities. It is possible to devise a 3-particle spin measurement in which the distinction between QM and LR can be seen in every single measurement.

Prepare a 3-particle (each spin 1/2) state with the z-components of the spins described by ( - ¯¯¯)/2^1/2. Measure the x-components of the spins.

• QM predicts s_1xs_2xs_3x = -1, always,
• LR predicts s_1xs_2xs_3x = +1, always.

A single measurement of the 3 spins could do the job. (See article by N. D. Mermin in Physics Today , June 1990.)

Here's some old notes, left in for historical interest:

"Although it is conceivable that this hypothetical experiment, if it is done, will agree with LR rather than QM, don't bet on it. It would be very strange for QM to do a perfect job of predicting the violations of LR so far, then break down and have LR work in the non-probabilistic context. There's no theory that predicts that, and there was only a vague hunch that somehow the application of probabilistic reasoning might have something to do with the failure of LR."

The new parable:

You can ask one question each of Anne,(A) Belinda (B), or Cary (C). The questions can be "olives?" (X) or "Onions?" (Y) . The answers can be "yes" (+1) or "no" (-1).

You always find that if you ask one "olives?" and two "onions?" the product of the answers is +1. So it sounds as if each answer is determined by an "element of reality" since it is predictable by asking two OTHER questions.

X₁Y₂Y₃ = Y₁X₂Y₃= Y₁Y₂X₃=1

So if those are just numbers we're talking about:

X₁Y₂Y₃Y₁X₂Y₃Y₁Y₂X₃ = X₁X₂X₃ Y₁² Y₂² Y₃² = X₁X₂X₃ = 1 ALWAYS

BUT QM says something strange: X₁X₂X₃ = -1 ALWAYS

Essentially this experiment (using photons rather than spin=1/2) has just been done, and the result supports QM.

More than 85% of the time, X₁X₂X₃ = -1, was found which is impossible classically. The remaining 15% or so is attributed to imperfections in the analyzers, etc.

Now we can return to the "measurement" question, but without any false hope of finding a mundane solution in terms of local hidden variables.

Which have we lost:

1. realism?

2. Local causality?

3. Induction?

Microscopic realism

The CI explicitly abandons microscopic realism. However, the structure of our experiment said nothing about assigning "reality" to the wave function or any other microscopic construct. We argued only about macroscopic outcomes: which detectors said 'up' or 'down'. There was a slight loophole- the time it would take to send a signal between detectors allows only partial conversion to a full-scale macro-outcome. However, recent experiments with the detectors farther apart have kept pushing up the size of events to which we cannot apply ordinary local realism. On a time scale of microseconds, very many logic operations in a computer can be performed, etc.

Extrapolation of the abandonment of realism to a slightly larger size scale would imply that it is not true that counterfactual statements about fully macroscopic (brain-size) events have definite truth value. The postulate of counterfactual definiteness would be false. As Asher Peres said, "Unperformed experiments have no results."

It becomes impossible to say that IF the horizontal spin had been measured then the result would have been (+ or -) *

This claim does not deny the definiteness of results of experiments which are actually performed ("would have" ≠ "did").

* unless you change the rules of logic, so that you cannot "distribute" "or" i.e.
(Up &(left or right)) doesn't imply ((up & left) or (up & right))

If Induction is lost, keeping local realism:

The principle of induction says that the properties of the subensemble on which some measurements are made are the same as those of the entire ensemble of prepared systems. This seems natural if we are selecting the objects "at random" from the ensemble. Is it possible that we are wrong about this?

Breakdown of induction implies that either

1. the emitter already knows what is going to be measured and produces the appropriate subensemble. The particles get their stories straight on just those questions which they shall be asked, and not others. Or

2. the measurer is clued as to just what questions are ok to ask, i.e. what questions the emitter will be ready to answer.

But in recent expts, those questions are set by some sort of "random" chance (e.g. another quantum event) in a space-like separated (non-causal) way, off near the detectors.

That separation requires either communication from the future to the past (detectors signaling emitter) or that somehow the "random" events picking what will be measured at the detectors have some prior cause which also sets which subensemble the emitter will produce, i.e. some very intricate causal web of self-consistent connections between remote events, all of which look random in local measurements.

So dropping induction while keeping local realist causation really is the same as postulating some entirely different theory of intricate local causes producing distant connections which manage to look completely random locally. It is impossible in principle to rule out all such conspiracies.

Why can we not rule out conspiracies?

The Bell argument only "rules out" LOCAL hidden variables. But any HV which permeates all parts of the experimental apparatus looks GLOBAL. For any finite apparatus, there are always past points which could have sent conspiratorial signals to all parts of the finite apparatus, by purely causal channels, giving pseudo-global variables.

Formally, Bell assumes that the result at one point is independent of the apparatus setting at a remote point, which is not strictly required by locality, as argued above.

However, any particular conspiracy theory should have some testable additional implications.

If Local causality is lost:

Remember what we needed in Special Relativity:

1. Nothing with mass can travel faster than c.

2. No conserved quantity can travel faster than c.

3. No information can travel faster than c.

We found it tempting to generalize this to the idea that "ALL connections between events are a causal network flowing forward in time at speeds not greater than c" but that generalization was not logically necessary.

What happens if we say "The correlation between the spins of the two particles exists, but not at any particular locus in space-time." Someone trying to describe the measurement by a picture completely imbedded in space-time will say "One particle measured 'up", and that info was transmitted faster than c to the other, making it spin 'down'". An observer in another frame would be forced to say that the measurements happened in reverse order, and would describe the flow of information as going the other way.

Does that mean signals could be sent backward in time?

SRI said to the Navy: You have to threaten retaliation if there's a nuclear attack. You don't want your missiles blown up, and you certainly don't want to accidentally fire when there's a false warning. SRI requested $$$$ to develop a way to wait until there was a confirmed attack, then signal backwards in time and shoot first! Fortunately, the Navy didn't bite.

What was nuts about the SRI idea?

The correlated events which one might interpret as having backwards causation are random quantum events. They are not predictable on the basis of anything else in the universe (if QM is correct) and thus cannot be used to send information. The output of any receiver is a random string of bits, regardless of whether some other receiver is getting a correlated random string of bits. You cannot send information via random bit streams if neither you nor anything else in the universe can do anything to affect what's in those bit streams

Notice that everything is ok so long as the apparently random QM variables are really and truly completely random, i.e. uninfluenced by any prior cause.

Things are also ok (if obscure and terribly complicated) if these variables all have some hidden deterministic cause.

A mixed case would seem to lead to contradictions, in that signals could be sent backward in time.

Akin to the uncertainty relations: consistent as a whole, but inconsistent if you removed just one.

Local causality lost? (cont)

In other words, this option requires that certain facts about the world not have any "place" in it.

The direction of "causation" (if you wish to use that concept) for this class of events ceases to be Lorentz invariant.

The world has become much stranger than we had imagined, but not, perhaps, stranger than we can imagine- until we return to the current descriptions of the measurement process.