"I don't demand that a theory correspond to reality because I don't know what it is. Reality is not a quality you can test with litmus paper. All I'm concerned with is that the theory should predict the results of measurements."—Stephen Hawking

I'm watching my old dog sleep in his chair. The tip of his tail curls under his snout. I look away and look back. Nothing has changed. I have every reason to believe that my observation of him has no effect on him whatsoever. What do sleeping dogs do when we look away? Exactly what they do when we look back.

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But what do microscopic particles do when we're not watching them? That's another question entirely. We'll try, with trepidation, to apply our common sense. The problem is that common sense does not always work.

Suppose that your observation of particles merely reveals properties that the particles already had prior to measurement. No surprise there. Suppose, further, that the measurement of one particle has no effect on a distant particle. Also unsurprising — why would it? Suppose, too, that we have free will to choose how and when to measure the particles. Suppose, finally, that a particle deflected upward is not, in an unreachable parallel universe, deflected downward.

What happens when we combine all these reasonable everyday assumptions? We find that our everyday assumptions impose constraints on measurable quantities. This means that we can do experiments to test whether nature obeys the constraints mandated by our common sense.

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I'd like to give you an example of a constraint imposed by common sense. We require nothing more than simple logic, arithmetic, and a few minutes of concentration.

(We're about to do a simple but rigorous proof. Proofs were studiously excluded from early books about the philosophical implications of quantum mechanics. Fritjof Capra wrote, "I have suppressed all the mathematics," and Gary Zukav wrote that the proof is "indecipherable to a mathematician." In the decades since those words were written, physicists have developed many accessible—even enjoyable—proofs. Some of these are included in my recent book, "Quantum Entanglement," published by MIT Press.)

Consider Alice and Bob, the indefatigable physicists frequently employed in these thought experiments. A pair of particles is emitted such that one particle flies into Alice's measuring instrument, and the other flies into Bob's measuring instrument. Alice's measuring instrument has two settings: A and A'. Regardless of the setting, only two measurement outcomes are possible: for example, the particle is deflected either upward or downward. If the particle is deflected upward, Alice records +1. If the particle is deflected downward, Alice records -1. Similarly, Bob's measuring instrument has two settings, B and B', and the result of his measurement is either +1 or -1.

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Suppose Alice sets her instrument to A, and Bob sets his instrument to B. They each measure a particle and record the result, +1 or -1. Then they multiply Alice's result by Bob's result, and the product of the multiplication must also be +1 or -1. Let's call this product AB, where A and B represent the individual results, +1 or -1. Suppose Alice and Bob repeat this process for many pairs of particles, and they record AB for each pair. Then they compute the average value of AB.

Next, Alice keeps her instrument set to A, while Bob switches his to B'. They perform many measurements and compute the average value of AB'. Similarly, Alice and Bob determine the average value of A'B, and the average value of A'B'.

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So, Alice and Bob have obtained four measured values: the averages of AB, AB', A'B, and A'B'. Now I'm going to write one equation, and I promise it's the only equation in this article. I'm going to define a quantity called S, as follows:

S = AB - AB' + A'B + A'B'.

S doesn't have any obvious physical meaning; it's just something we can calculate. Since A, A', B, and B' can only be +1 or -1, the only possible values of S are +2 or -2. If you don't believe me, check out the math for yourself..

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Now, the average value of S is the average value of AB, minus the average value of AB', plus the average value of A'B, plus the average value of A'B'. Using the four experimental results obtained by Alice and Bob, we can compute the average value of S.

Since the only possible values of S are +2 and -2, surely the average value of S is between -2 and +2. This is a constraint mandated by common sense.

And yet nature disobeys this constraint! Experimentally, we find that the average value of S exceeds 2!

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Where did we go wrong? All we used was basic logic and arithmetic. Did we just invalidate arithmetic? Do we need to inform elementary math teachers, so they stop spreading their lies? No, arithmetic isn't the problem here. But some false assumption, whatever it may be, leads to a constraint flagrantly disregarded by nature.

Let's look again at our equation: S = AB - AB' + A'B + A'B'. If A, A', B, and B' are physical properties that exist all along, independent of measurement, and each of these properties is represented by +1 or -1, then S must indeed be either +2 or -2, and the average of S must indeed be between -2 and +2. Since this constraint is disobeyed, we can only conclude that the particles' properties (A, A', B, and B') do not exist all along, independent of measurement.

What can this mean? Although Alice can measure either A or A', perhaps these two values never exist at the same time for a single particle. This could mean that the measured value was not merely unknown prior to measurement; it didn't even exist. Alternatively, perhaps Alice's decision to measure A (for example) was predetermined, so that A existed all along, whereas A' never existed. Yet another possibility is that A and A' depend on whether Bob measures B or B'. In this case, A is not a single value to plug into the equation for S, and we no longer find that S must be +2 or -2. At least one pillar of common sense must collapse to accommodate the experimental result that S exceeds 2.

We find in the laboratory, again and again, that nature disobeys our common sense. While disobeying our common sense, nature simultaneously adheres to a different set of formulas, those established by quantum mechanics.

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This puts physicists in a pickle. Quantum mechanics accurately predicts the outcomes of measurements, but we don't know what to say about particles when we're not looking at them; all we know for sure is that our common sense gets it wrong. So, the interpretation of quantum mechanics remains a topic of speculation, controversy, equivocation, or indifference.

Some physicists argue that unobserved particles are simply none of our business; the business of physics is predicting observations. Let philosophers handle the unobserved particles. Physicists thus divest themselves of awkward questions and focus on what they do best. This viewpoint is admirably humble in its acknowledgment of the limitations of physics. Or is it just lazy?

Some physicists argue that every possible measurement outcome occurs simultaneously in parallel universes. Impassioned arguments are made for and against this "many worlds interpretation" . In its favor, the many worlds interpretation avoids the distinction between abrupt measurements and smooth evolution of probabilities. The laws of quantum mechanics provide probabilities of different outcomes. What, ultimately, determines which outcomes occur, and which don't? The many worlds interpretation circumvents this question entirely because all outcomes occur. The arguments against the many worlds interpretation are plentiful, and include the fact that we simply have no evidence of parallel universes.

Perhaps, instead, the error in our common sense is the belief in free will. Perhaps we are preprogrammed automatons, or the particles under observation diabolically influence our decisions. This possibility, though unpalatable and strange, is duly considered by physicists and philosophers.

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Or, perhaps our error is the assumption that the measurement of one particle can't affect a distant particle. Quantum entanglement is a subtle connection undiminished by distance. Can we assert that the connection isn't so subtle, but that the measurement of one particle physically alters the distant particle? I believe this claim can be neither proven nor disproven. If we say that the measurement of one particle affects the other, we really mean that the first measurement, of either particle, affects both; subsequent results are determined by this first measurement.

We'd like to look at Alice's particle before and after Bob measures his particle, to see if Bob's measurement affects Alice's particle. But then the initial measurement of Alice's particle becomes the first measurement that determines subsequent results! Since it's impossible to do a measurement before the first measurement, it's impossible to observe whether the measurement of one particle physically alters the other.

As we continue to expand the frontiers of knowledge, the rigorous mystery of quantum entanglement remains stubbornly unsolved. I wonder if this is a salutary check on human hubris, a reminder of our place in a world we never made. Physics achieves the goal of medieval alchemy and astrology, to illuminate the invisible forces that govern the destinies of all things. But something always scurries away from the light, and the thirst for complete understanding is forever unquenched.

I struggle against the temptation to close with a joke about parallel universes. Perhaps in another universe, I succeed.