Local Hidden-Variable Theory

Ah, you want to dissect the messy, contradictory heart of quantum mechanics. Fascinating. It's like trying to understand a shadow that insists it's the source of light. Fine. Let's pull this apart, shall we? But don't expect me to hold your hand. This is a minefield, and you're the one walking into it.

Interpretation of Quantum Mechanics

This whole section is a desperate attempt to make sense of a reality that seems to have a penchant for the absurd. It's part of a larger series about Quantum mechanics, as if that's supposed to make it any less bewildering.

The core of it all, the Schrödinger equation, is presented with all the elegance of a cryptic incantation: $i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle$ . It’s supposed to describe how the quantum state of a physical system evolves over time. But "evolves" is a rather polite word for the sheer, unadulterated chaos it unleashes.

This is all built on a foundation that’s less solid rock and more shifting sand. We have the Introduction, a Glossary for those who like their paradoxes neatly defined, and a tangled History that reads like a series of arguments that never quite resolved.

Background

Before we dive into the abyss, a little context. We're told about Classical mechanics, the predictable, Newtonian world that quantum mechanics gleefully ripped apart. Then there's the Old quantum theory, a clumsy precursor, like a child's drawing of a masterpiece. And of course, Bra–ket notation, a shorthand for the incomprehensible. We also have the Hamiltonian, the energy of the system, and Interference, the stubborn refusal of particles to behave like simple little billiard balls.

Fundamentals

This is where it gets truly interesting, or perhaps, truly unsettling.

Complementarity: Niels Bohr's notion that certain properties of a quantum system can only be observed in mutually exclusive ways. You can know its position, or its momentum, but never both with perfect clarity. It’s like trying to see a ghost and a solid object simultaneously.
Decoherence: The process by which a quantum system loses its quantum properties and starts behaving classically, usually due to interaction with its environment. The universe, it seems, has a way of tidying up loose quantum ends.
Entanglement: The spooky connection between particles, where they remain linked no matter the distance. Measure one, and the other instantly "knows." Einstein hated it. I can see why. It’s a cosmic prank.
Energy level: Quantum systems can only possess discrete amounts of energy. Like steps on a ladder, not a ramp.
Measurement: The act of observation. It’s here that the quantum world gets particularly awkward. The act of looking seems to fundamentally alter what you're looking at. A truly invasive spectator sport.
Nonlocality: The idea that events can influence each other instantaneously across vast distances, defying the speed of light. A direct consequence of entanglement. Spooky, indeed.
Quantum number: These are the labels that describe the allowed energy levels and other properties of a quantum system. Discrete, precise, and utterly arbitrary.
State: The complete description of a quantum system, often represented by the wave function. It's a superposition of all possibilities until measured.
Superposition: The ability of a quantum system to be in multiple states at once. A particle can be here and there, until you look.
Symmetry: Underlying principles that govern the behavior of quantum systems. Nature, it seems, has a preference for balance, even in its most chaotic manifestations.
Tunnelling: The bizarre phenomenon where a particle can pass through a potential energy barrier, even if it doesn't have enough energy to do so classically. It's like walking through a wall. Because why not?
Uncertainty: Heisenberg's famous principle, stating that there are fundamental limits to how precisely certain pairs of physical properties of a particle, like position and momentum, can be known simultaneously. The more you know about one, the less you know about the other. A cosmic trade-off.
Wave function: The mathematical description of a quantum state. It contains all the information about the system, but it's inherently probabilistic. A map of possibilities, not certainties.
Collapse: The presumed abrupt change of the wave function into a single definite state upon measurement. The moment the possibilities snap into one reality, often with a violent shudder.

Experiments

The proofs, or rather, the persistent confirmations of quantum weirdness.

Bell's inequality and CHSH inequality: These are tests designed to see if quantum mechanics is compatible with local hidden-variable theories. The results have consistently shown that it is not. The universe, it turns out, is not as sensible as we'd like.
Davisson–Germer, Double-slit, Franck–Hertz, Stern–Gerlach: These are foundational experiments that demonstrated the wave-particle duality of matter, the quantization of energy, and the quantization of angular momentum. They are the bread and butter of quantum weirdness.
Elitzur–Vaidman and Popper: Thought experiments, and some real ones, exploring the implications of quantum measurement, like detecting a bomb without detonating it. The quantum world is full of clever, albeit terrifying, tricks.
Quantum eraser and Delayed-choice, Wheeler's delayed-choice: These experiments suggest that the choice of measurement after a particle has already passed through an apparatus can influence its past behavior. The past, it seems, is not as fixed as one might hope.
Schrödinger's cat: The famous thought experiment illustrating the paradox of superposition and measurement. A cat, both alive and dead, until the box is opened. A rather morbid metaphor for quantum uncertainty.
Leggett inequality and Leggett–Garg inequality: These are more recent inequalities designed to test the limits of classical intuition in quantum systems, particularly concerning macrorealism and the assumption that macroscopic variables have definite values independent of measurement. They push the boundaries of what we consider "real."
Mach–Zehnder: Used to demonstrate quantum interference and superposition. It’s a classic setup for observing wave-like behavior.

Formulations

Different ways of wrangling the equations, each with its own quirks.

Overview: The general approach to describing quantum mechanics mathematically. A necessary evil, I suppose.
Heisenberg, Interaction, Schrödinger: Different perspectives on how quantum states and operators evolve. It's like looking at the same distorted landscape from slightly different angles.
Matrix: An early formulation of quantum mechanics using matrices. Less intuitive, more algebraic.
Phase-space: A way of representing quantum mechanics in a phase space, often associated with the Wigner quasi-probability distribution. It tries to bridge the gap between classical and quantum descriptions, with limited success.
Sum-over-histories (path integral): Feynman's approach, where a particle takes all possible paths between two points, and the probability is determined by summing the contributions of each path. A rather extravagant way to travel.

Equations

The specific incantations that govern the quantum realm.

Dirac, Klein–Gordon, Pauli, Schrödinger: These are the fundamental equations describing the behavior of particles and fields in quantum mechanics. They are the lawgivers.
Rydberg: An empirical formula for predicting the wavelengths of photons emitted by an electron in an atom when it changes energy levels. It predates full quantum mechanics but hinted at the quantized nature of energy.

Interpretations

This is where the real fun begins. The attempts to graft meaning onto the mathematical framework.

Bayesian: Views quantum probabilities as subjective degrees of belief, rather than objective properties of nature. A philosophical sidestep.
Consciousness causes collapse: The idea that conscious observation is what causes the wave function to collapse. A rather self-important notion for consciousness.
Consistent histories: A formalism that allows reasoning about sequences of events in quantum mechanics without invoking measurement collapse. It tries to impose a narrative, but the universe resists.
Copenhagen: The "standard" interpretation, championed by Bohr and Heisenberg. It emphasizes the role of measurement and complementarity, and frankly, tells you to stop asking awkward questions.
de Broglie–Bohm: A deterministic, pilot-wave theory that postulates hidden variables guiding the particles. It's deterministic, but it's also non-local. A compromise that satisfies no one completely.
Ensemble: Suggests that quantum mechanics only makes probabilistic statements about large collections of systems, not individual ones. A statistical shrug.
Hidden-variable: The general idea that quantum mechanics is incomplete and that there are underlying variables that determine the outcome of measurements. Bell’s theorem, however, has made this very difficult for local hidden-variable theories.
Many-worlds: Proposed by Everett, this suggests that every quantum measurement causes the universe to split into multiple branches, with each outcome occurring in a separate universe. A rather extravagant explanation, but it avoids wave function collapse. More universes than we can possibly comprehend.
Objective-collapse: Theories that propose a physical mechanism for wave function collapse, independent of observation. A more physical, less philosophical approach.
Quantum logic: The idea that the logic governing quantum systems is different from classical logic. The rules of reality change at this scale.
Superdeterminism: The notion that all events, including the choices made by experimenters, are predetermined. It’s a way to preserve locality, but at the cost of free will. A bleak outlook.
Relational: States that the quantum state of a system is not absolute but relative to the observer. Reality is defined by interactions.
Transactional: A rather peculiar interpretation involving "offer" and "confirmation" waves traveling forward and backward in time. It’s elegant, in a time-traveling sort of way.

Advanced topics

The rabbit hole goes deeper.

Relativistic quantum mechanics and Quantum field theory: Merging quantum mechanics with special relativity. The foundation of modern physics, describing particles as excitations of fields.
Quantum information science, Quantum computing, Quantum machine learning: The practical (and often overhyped) applications of quantum phenomena.
Quantum chaos: The study of quantum systems that exhibit chaotic behavior. Even randomness can have patterns, it seems.
EPR paradox: The famous thought experiment that highlighted the strangeness of entanglement and led to Bell's theorem. Einstein's attempt to show quantum mechanics was incomplete. It backfired spectacularly.
Density matrix: A way to describe quantum states, especially mixed states (probabilistic mixtures of pure states). Useful when you don't know the exact state.
Scattering theory: Describes how quantum particles interact and change direction. The physics of collisions.
Quantum statistical mechanics: Applying quantum mechanics to systems with many particles.

Scientists

The names associated with this glorious mess. Bohr, Einstein, Heisenberg, Schrödinger, Dirac, Feynman, Bell, Everett... a constellation of brilliant minds wrestling with a universe that refuses to play by their rules.

Now, about this "local hidden-variable theory." It's a concept born from a desire for order, a hope that the quantum world isn't quite as fundamentally random as it appears. The idea is simple: if we just had access to more information – these "hidden variables" – we could predict outcomes with certainty, just like in the good old days of Classical mechanics. And crucially, these variables must obey the principle of locality, meaning what happens here can't instantly affect what happens light-years away. It’s a comforting thought, isn't it? A universe where cause and effect are strictly local.

But then came John Stewart Bell. In 1964, he took this comforting idea and systematically dismantled it. His theorem proved that any theory that relies on local hidden variables cannot possibly reproduce the correlations predicted by quantum mechanics, especially when it comes to quantum entanglement. It’s a brutal, elegant refutation. The universe, it turns out, is either non-local, or it's not deterministic in the way we'd like, or both. And the subsequent Bell test experiments have, with unnerving consistency, sided with quantum mechanics. The "spooky action at a distance" is, apparently, very real. [1]

Models

Even within this flawed framework, there are attempts to build models.

Single qubit

Bell himself, bless his contrarian heart, managed to construct a local hidden-variable model for a single qubit, or a spin-1/2 particle. It’s a concession, a narrow escape. N. David Mermin later simplified it, and Simon B. Kochen and Ernst Specker contributed their own related models. [2] [3] [4] [5] This works because Gleason's theorem, which generally forbids such models for quantum mechanics, doesn't quite apply to the simplest case of a single qubit. It’s like finding a loophole in a law that’s otherwise ironclad.

Bipartite quantum states

When you have two particles, things get more complicated, and more interesting. Initially, discussions of quantum entanglement focused on perfect correlations or anti-correlations – cases where measuring one particle instantly told you everything about the other. These, too, could be explained by local hidden variables. [2] [7] [8]

But the real surprise is that even some entangled states—states that demonstrably violate Bell inequalities—can sometimes be described by hidden-variable models, provided you allow for von Neumann measurements. [9] These states are entangled, yes, but they don't exhibit the full "spookiness" that Bell's theorem targets.

Take the Werner states. They're a family of states, essentially noisy versions of entangled states. Reinhard F. Werner showed that for certain levels of "noise" (specifically, when $p \leq 1/2$ ), a hidden-variable model is possible, even though the state is still entangled if $p > 1/3$ . [10] The bound for these models has been pushed to $p=2/3$ . And these models can even accommodate positive operator-valued measurements (POVM), not just the more restrictive von Neumann measurements. [11] This has been extended to noisy maximally entangled states and even arbitrary pure states mixed with white noise. [12] Even in multipartite systems, models have been constructed, like for a three-qubit state. [13] It seems the universe is determined to find ways to be both entangled and, in some limited sense, explainable.

Time-dependent variables

There’s been some rather contentious speculation about the role of time. K. Hess and W. Philipp proposed that time-dependent hidden variables might offer a way out. However, this idea has been met with considerable skepticism from figures like Richard D. Gill, Gregor Weihs, Anton Zeilinger, and Marek Żukowski, as well as D. M. Appleby. [14] [15] [16] The core criticism seems to be that even with time-dependent variables, the fundamental non-locality persists. It’s a desperate attempt to pin down a phantom.