APPENDIXTerms

For our purposes, a tiny object that flies in a straight line until something measures it. The "thing" emitted by the source. In real Bell experiments, particles are usually photons (particles of light) or electrons. The math is identical.

The device that emits two particles at once, in opposite directions, with the special property that they "remember each other" (are entangled). In real labs, this is usually a crystal that splits one photon into two, or an atom that decays into two photons.

The angle a scientist sets on the measurement box before the particle arrives. In real experiments, this is the orientation of a polariser or a Stern-Gerlach magnet. The point is that the experimenter has free choice over this angle.

The single bit of information that comes out of each measurement: ▲ or ▼. Real experiments call this "spin up" / "spin down" or "horizontal" / "vertical" depending on what kind of particle is involved.

The standard real-world apparatus that does what our "dial and light" does for spin-½ particles. It splits an incoming particle into one of two paths depending on the spin's projection along the magnet's orientation. The two paths correspond to the two flashes.

The property of having two particles whose individual states cannot be described separately — only the pair as a whole has a definite state. The "singlet" pair Bell uses is the cleanest example. Entanglement is what makes the EPR experiment interesting; without it, the two particles would be independent and there would be no correlation to puzzle over.

A specific entangled state of two particles in which the total spin is exactly zero. Whichever direction you measure spin in, the two particles give opposite answers. The most famous entangled state and the one Bell uses throughout.

The assumption that what happens at one place cannot instantaneously affect what happens at a distant place. More precisely: Alice's measurement result depends only on Alice's apparatus and the particle she has, not on what Bob has done. This is the assumption local hidden-variable explanations cannot keep.

The proposal that quantum mechanics is incomplete and that there are extra, unobserved properties of each particle that determine its measurement results. Einstein's preferred way to explain away the spookiness. Bell proves no such theory can match all of QM's predictions while also being local.

How often two events match up. In the experiment: across many trials, what fraction of the time do Alice's and Bob's flashes agree? Pure chance gives 50% (no correlation). 0% means perfect anti-correlation; 100% means perfect correlation. The Bell experiment is a story about a particular pattern of correlations across multiple dial settings.

A mathematical bound that any local hidden-variable theory must satisfy. The simplest form (chapter 3) says cross-dial match rate ≤ 2/3 for three dial settings 120° apart. The CHSH form says |S| ≤ 2 for a particular weighted sum. Quantum mechanics violates both.

Clauser-Horne-Shimony-Holt, the four physicists who in 1969 generalised Bell's 1964 inequality into a standard four-setting form for experimental tests. CHSH is what many modern Bell experiments test. The bound is |S| ≤ 2 for local hidden variables; ideal QM gives 2√2 ≈ 2.83.

A way the experimental data might still be explained by a hidden-variable theory — usually because of imperfect detectors, slow dial-switching, or correlations between source and detector. Modern Bell experiments are designed to close the main experimental loopholes simultaneously; they have all confirmed the QM prediction.

A theorem that says even though entanglement is "non-local," it cannot be used to send information faster than light. Bob looking only at his own data sees pure randomness; the correlation only appears once the two notebooks are compared. The universe enforces this strictly.

Shorthand for "+1 or −1." We use ±1 for the two possible flashes (▲ = +1, ▼ = −1) so we can multiply outcomes together to get a correlation. Alice ▲ × Bob ▼ = (+1) × (−1) = −1. Same flash → product = +1; opposite flash → product = −1.

The arithmetic average of many trials. If Alice and Bob's outcome product is +1 on 25% of trials and −1 on 75%, the average is 0.25 × (+1) + 0.75 × (−1) = −0.5. In quantum-mechanics-speak, this is called the "expectation value" of the joint measurement.

A specific smooth curve from trigonometry. cos(0°) = 1, cos(90°) = 0, cos(180°) = −1, with smooth transitions. The curve in chapter 1 is −cos θ. You don't need the formula — just the picture.

The pattern of how often each value of a hidden variable shows up across many trials. If λ can be any direction in space, ρ(λ) tells you how often the source emits a particle pair with each particular hidden direction. It's a "weighting." The integral / averaging over ρ(λ) is the formal way of saying "average across many trials."

A direction in space. We draw vectors as arrows. "Dial setting â" and "dial setting b̂" are vectors pointing along whatever direction the scientist has set. The little hat â means "this is a unit vector — its length is 1, only the direction matters."

For two unit vectors (length-1 arrows), the dot product equals cos θ where θ is the angle between them. So â · b̂ = 1 when they point the same way, 0 when perpendicular, −1 when opposite. Used everywhere in the paper as a compact way to write "cosine of the angle between two dial directions."

Northern Irish theoretical physicist working at CERN. Wrote the 1964 paper while on sabbatical at SLAC, Brandeis, and the University of Wisconsin. Died of a stroke in 1990, before the experimental confirmations of his theorem won the Nobel Prize.

Wrote the 1935 EPR paper (with Podolsky and Rosen) arguing quantum mechanics was incomplete. Spent the rest of his life convinced something deeper underlay it — convinced, in particular, that physics could be rescued by adding hidden variables without spooky distant influence. Bell's theorem is the (posthumous) proof that this hope cannot be realised without abandoning locality.

Einstein's two co-authors on the 1935 paper. The paper sets up the thought experiment Bell formalises 30 years later.

Reformulated EPR's 1935 argument for spin-½ particles in 1957 (with Yakir Aharonov), giving Bell the version of the experiment he uses in the paper. Bohm also constructed an explicit hidden-variable theory ("pilot wave"), which is non-local and so consistent with Bell's theorem.

The three experimentalists who, between 1972 and 2015, built increasingly tight tests of Bell's inequality. Clauser, with Stuart Freedman, ran the first one in 1972. Aspect closed the "switching loophole" in 1982 by switching dial settings during particle flight. Zeilinger pushed Bell tests to long distances and into quantum communication. All three shared the 2022 Nobel Prize in Physics.

Replied to the 1935 EPR paper a few months after it appeared, defending the standard quantum interpretation. Bohr's reply is famously hard to follow, and the resulting debate between Einstein and Bohr was inconclusive — which is part of why Bell's much later contribution was so important: he turned the philosophical disagreement into mathematics.