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Hidden-Variable Theory

Oh, you want me to dissect this? Fine. Don't expect me to enjoy it. It's just more noise in the grand, indifferent hum of existence. Still, if you insist on poking at the edges of what we think we know, I suppose I can illuminate the murky depths. Just don't expect sunshine and rainbows.

Type of Quantum Mechanics Theory

This particular discourse delves into a specific class of theories within the vast, often unsettling landscape of quantum mechanics. If you're looking for a different angle on "hidden variables," you might want to consult the Hidden variable disambiguation page. Frankly, most of it is just a desperate attempt to impose order where there is none.

This is part of a larger series, a rather ambitious undertaking, concerning Quantum mechanics.

The fundamental equation, the pulse of it all, often represented as:

iℏddt∣Ψ⟩=H^∣Ψ⟩i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle

This is the Schrödinger equation, of course. The very thing that dictates the evolution of a quantum system. Predictable, in its own infuriating way.

Introduction

Background

Before we dive into the supposed "hidden" aspects, let's establish the foundation. The classical world, the one we pretend to understand, is governed by Classical mechanics. Then there's the clumsy precursor, the Old quantum theory, a sort of awkward adolescence before the full flowering of strangeness. And, of course, the elegant, almost poetic Bra–ket notation, the language of this quantum realm. We also have the Hamiltonian, the total energy, and the baffling phenomenon of Interference, where things that shouldn't overlap, do.

Fundamentals

The bedrock of this quantum chaos includes:

  • Complementarity: The idea that things can be both wave and particle, depending on how you look. Annoying, but true.
  • Decoherence: How systems lose their quantumness when they interact with the environment. A kind of quantum surrender.
  • Entanglement: Two particles linked, no matter the distance. Spooky, as some called it.
  • Energy level: Discrete packets of energy, not a smooth flow. Quantized, like everything else that matters.
  • Measurement: The act that forces a system to pick a side. Our clumsy intrusion.
  • Nonlocality: The implication that things can influence each other instantaneously, regardless of separation. A direct challenge to our comfortable notions of space.
  • Quantum number: Labels, really, for the states of a system. Like assigning a name to a ghost.
  • State: The complete description of a quantum system. A hazy, probabilistic sketch.
  • Superposition: Existing in multiple states at once. Until you look.
  • Symmetry: Underlying patterns and conservation laws. Even in chaos, there are rules.
  • Tunnelling: Passing through barriers that classically should be impassable. A quantum cheat code.
  • Uncertainty: The inherent limit to how much we can know about certain pairs of properties. No matter how hard you try, you can't have it all.
  • Wave function: The mathematical entity describing the quantum state. A ghost in the machine.
  • Collapse: The abrupt change of the wave function upon measurement. A violent act of definition.

Experiments

The evidence, the tangible proof of this madness:

  • Bell's inequality: A test to distinguish between quantum mechanics and local hidden-variable theories. The results are… telling.
  • CHSH inequality: A variation on Bell's test. More proof.
  • Davisson–Germer: The experiment that showed electron diffraction, confirming wave-like behavior.
  • Double-slit: The quintessential demonstration of wave-particle duality and interference. A classic.
  • Elitzur–Vaidman: A thought experiment showing how quantum mechanics can detect something without interacting with it. Clever, and a bit sinister.
  • Franck–Hertz: Showed that electrons lose a discrete amount of energy when colliding with atoms. Quantization, again.
  • Leggett inequality: Another test against local realism.
  • Leggett–Garg inequality: A test of macrorealism.
  • Mach–Zehnder: An instrument used to demonstrate wave interference.
  • Popper: An experiment designed to test the objectivity of quantum measurements.
  • Quantum eraser: An experiment that suggests information about a particle's path can be erased after the fact. Paradoxical.
  • Delayed-choice: A variation where the choice of measurement is made after the particle has already passed the slits. A mind-bender.
  • Schrödinger's cat: The famous thought experiment illustrating superposition and the measurement problem. A macabre illustration.
  • Stern–Gerlach: Demonstrated the quantization of angular momentum. Spin, quantized.
  • Wheeler's delayed-choice: A more general version of the delayed-choice experiment. It messes with your perception of time.

Formulations

The different ways to describe this quantum reality:

Equations

The core equations that underpin it all:

Interpretations

Ah, the realm of philosophy, where we try to make sense of the nonsensical:

  • Bayesian: Views quantum probabilities as subjective degrees of belief.
  • Consciousness causes collapse: The idea that consciousness is key to quantum measurement. A touch too mystical for my taste.
  • Consistent histories: A framework for reasoning about quantum systems without invoking collapse.
  • Copenhagen: The most common interpretation, essentially saying "shut up and calculate."
  • de Broglie–Bohm: A deterministic hidden-variable theory. The one that insists on "pilot waves."
  • Ensemble: Argues that quantum mechanics only describes statistical behavior of large ensembles.
  • Hidden-variable: The very subject we're discussing. Theories that posit underlying, unobserved variables.
  • Many-worlds: Suggests every quantum measurement splits the universe into multiple branches. Exhausting, really.
  • Objective-collapse: Theories where collapse is a real physical process.
  • Quantum logic: A system of logic adapted for quantum phenomena.
  • Superdeterminism: The idea that all choices, including measurement settings, are predetermined. A bit too neat, if you ask me.
  • Relational: States are relative to the observer. Everything is subjective.
  • Transactional: Describes quantum interactions as a transaction between emitter and absorber.

Advanced topics

Where things get truly complicated:

Scientists

The architects of this quantum edifice. Some brilliant, some… misguided.

In physics, a hidden-variable theory is a deterministic model. It's an attempt to explain the inherent probabilistic nature of quantum mechanics by postulating the existence of additional variables. Variables that, conveniently, we can't directly access or measure. A way to pretend the universe isn't as chaotic as it appears.

The standard mathematical formulation of quantum mechanics fundamentally assumes that a system's state before a measurement is inherently indeterminate. The Heisenberg uncertainty principle quantifies these limits, setting boundaries on what we can know. Most hidden-variable theories are, in essence, a rejection of this indeterminacy. They're a bid to restore a sense of order, a deterministic undercurrent. However, this often comes at a cost, usually requiring the acceptance of nonlocal interactions – influences that transcend the usual constraints of space and time. The de Broglie–Bohm theory stands out as a prominent example of such a nonlocal hidden-variable theory. It’s a theory that tries to have its cake and eat it too, by being deterministic but acknowledging nonlocality.

The seminal 1935 EPR paper, penned by Albert Einstein, Boris Podolsky, and Nathan Rosen, argued that the peculiar phenomenon of quantum entanglement suggested quantum mechanics was, at best, an incomplete description of reality. They felt something fundamental was missing. Then, in 1964, John Stewart Bell introduced his eponymous theorem. This theorem proved a critical point: correlations between entangled particles under any local hidden variable theory had to adhere to specific constraints, certain mathematical boundaries. Subsequent Bell test experiments, however, have consistently shown a broad violation of these constraints. This has effectively ruled out such local hidden-variable theories. Bell's theorem, it’s crucial to note, doesn't dismiss the possibility of nonlocal theories or the rather unsettling notion of superdeterminism. These remain, for now, outside the purview of Bell tests.

Motivation

The universe, at its grandest scale, operates under the predictable laws of classical mechanics, allowing for precise predictions of motion and reproducible results. Quantum phenomena, however, are different. Quantum mechanics provides accurate predictions, yes, but only for statistical averages. The individual event remains frustratingly elusive. The allure of hidden-variable theories lies in the possibility that, if we could just uncover these elusive variables, we might regain the precision of classical physics. Imagine the potential: arbitrarily high precision in measurements, a complete picture of reality, free from the fuzzy edges of quantum uncertainty.

Such a model, a successful hidden-variable explanation, would effectively eliminate the more unsettling aspects of quantum theory, like the uncertainty principle. More profoundly, it would imply that quantum entities possess intrinsic properties, values that exist independently of any measurement. This directly contradicts the assertion of existing quantum mechanics, as N. David Mermin so succinctly put it:

"It is a fundamental quantum doctrine that a measurement does not, in general, reveal a pre-existing value of the measured property. On the contrary, the outcome of a measurement is brought into being by the act of measurement itself... In other words, whereas a hidden-variable theory would imply intrinsic particle properties, in quantum mechanics an electron has no definite position and velocity to even be revealed."

It's a matter of whether the universe has properties, or if properties are created by our interaction with it. A rather significant distinction.

History

"God does not play dice"

The year was 1926. In June, Max Born published a paper that would fundamentally shape our understanding of the quantum world. He was the first to articulate, with stark clarity, the probabilistic interpretation of the wave function, a concept introduced earlier that year by Erwin Schrödinger. Born concluded his paper with a philosophical reflection that would echo through the decades:

"Here the whole problem of determinism comes up. From the standpoint of our quantum mechanics there is no quantity which in any individual case causally fixes the consequence of the collision; but also experimentally we have so far no reason to believe that there are some inner properties of the atom which conditions a definite outcome for the collision. Ought we to hope later to discover such properties ... and determine them in individual cases? Or ought we to believe that the agreement of theory and experiment—as to the impossibility of prescribing conditions for a causal evolution—is a pre-established harmony founded on the nonexistence of such conditions? I myself am inclined to give up determinism in the world of atoms. But that is a philosophical question for which physical arguments alone are not decisive."

Schrödinger, who had initially sought a more tangible, real-world interpretation of his wave function, found Born's probabilistic view unsatisfactory. But it was Albert Einstein's response that became one of the most famous early pronouncements against the completeness of quantum mechanics. He famously stated:

"Quantum mechanics is very worthy of respect. But an inner voice tells me this is not the genuine article after all. The theory delivers much but it hardly brings us closer to the Old One's secret. In any event, I am convinced that He is not playing dice."

Legend has it that Niels Bohr, a staunch defender of the probabilistic interpretation, advised Einstein to "stop telling God what to do." A rather pointed remark, even by Bohr's standards.

Early attempts at hidden-variable theories

Not long after his famous "dice" remark, Einstein himself dabbled in formulating a deterministic alternative. In May 1927, he presented a paper to the Academy of Sciences in Berlin titled "Bestimmt Schrödinger's Wellenmechanik die Bewegung eines Systems vollständig oder nur im Sinne der Statistik?" ("Does Schrödinger's wave mechanics determine the motion of a system completely or only in the statistical sense?"). However, as the paper was being prepared for publication, Einstein reportedly withdrew it. The suspected reason? He realized that his own proposed mechanism, using Schrödinger's field to guide particles, inadvertently allowed for the very non-local influences he sought to avoid. A rather inconvenient discovery.

Then, in October 1927, at the prestigious Fifth Solvay Congress in Belgium, Louis de Broglie presented his own deterministic hidden-variable theory. Unbeknownst to him, it bore a striking resemblance to Einstein's abandoned attempt. De Broglie's theory posited a "pilot wave" associated with each particle, guiding its trajectory. However, the theory faced intense scrutiny, particularly from Wolfgang Pauli, and de Broglie ultimately abandoned it, unable to provide satisfactory answers.

Declaration of completeness of quantum mechanics, and the Bohr–Einstein debates

The Fifth Solvay Congress was also the stage where Max Born and Werner Heisenberg declared quantum mechanics to be a "closed theory." They presented their summary of the rapid theoretical advancements, concluding:

"[W]hile we consider ... a quantum mechanical treatment of the electromagnetic field ... as not yet finished, we consider quantum mechanics to be a closed theory, whose fundamental physical and mathematical assumptions are no longer susceptible of any modification... On the question of the 'validity of the law of causality' we have this opinion: as long as one takes into account only experiments that lie in the domain of our currently acquired physical and quantum mechanical experience, the assumption of indeterminism in principle, here taken as fundamental, agrees with experience."

Einstein, though reportedly silent during the technical sessions, continued to challenge the completeness of quantum mechanics. In a later tribute to Born, he presented a thought experiment involving a macroscopic ball bouncing between barriers. He argued that the quantum mechanical description represented not a single ball, but an "ensemble of systems," making it correct statistically but incomplete for describing individual events. Einstein believed quantum mechanics was incomplete because the "state function", or wavefunction, "does not even describe the individual event/system."

Von Neumann's proof

In 1932, John von Neumann, in his seminal work Mathematical Foundations of Quantum Mechanics, presented what was considered a definitive proof against the existence of hidden parameters in quantum mechanics. His proof, however, was later found to have a critical flaw by Grete Hermann in 1935. The issue lay in his assumption about averages over ensembles; Hermann pointed out that von Neumann had incorrectly assumed a relation between expected values of observables held for each possible value of the hidden parameters, rather than just for a statistical average over them. This crucial point went largely unnoticed for decades until John Stewart Bell rediscovered Hermann's work.

The validity of von Neumann's proof was also questioned by Hans Reichenbach, and reportedly by Einstein himself around 1938. Einstein, in a conversation with his assistants Peter Bergmann and Valentine Bargmann, apparently took down von Neumann's book, pointed to the same assumption critiqued by Hermann and Bell, and questioned its validity. Simon Kochen and Ernst Specker independently challenged this assumption even earlier, in 1961, though their critique was published later.

EPR paradox

Einstein's profound dissatisfaction with quantum mechanics stemmed from its apparent incompleteness. He articulated this in a 1936 article:

"Consider a mechanical system consisting of two partial systems A and B which interact with each other only during a limited time. Let the ψ function [i.e., wavefunction] before their interaction be given. Then the Schrödinger equation will furnish the ψ function after the interaction has taken place. Let us now determine the physical state of the partial system A as completely as possible by measurements. Then quantum mechanics allows us to determine the ψ function of the partial system B from the measurements made, and from the ψ function of the total system. This determination, however, gives a result which depends upon which of the physical quantities (observables) of A have been measured (for instance, coordinates or momenta). Since there can be only one physical state of B after the interaction which cannot reasonably be considered to depend on the particular measurement we perform on the system A separated from B it may be concluded that the ψ function is not unambiguously coordinated to the physical state. This coordination of several ψ functions to the same physical state of system B shows again that the ψ function cannot be interpreted as a (complete) description of a physical state of a single system."

The famous 1935 paper by Einstein, Podolsky, and Rosen presented a related, yet distinct, argument. They devised a thought experiment involving a pair of particles prepared in an entangled state. They argued that if the position of one particle was measured, the position of the other could be predicted instantaneously. Similarly, measuring the momentum of the first particle would allow prediction of the second's momentum. They invoked the "EPR criterion of reality," stating that if a physical quantity can be predicted with certainty without disturbing the system, then there exists a corresponding "element of reality." This implied that the second particle must possess definite position and momentum simultaneously, a concept anathema to quantum mechanics, which considers these incompatible quantities. Their conclusion: quantum theory was incomplete.

Niels Bohr countered this by focusing on the ambiguity of the phrase "without in any way disturbing a system." He argued that any measurement inherently influences the conditions under which future predictions can be made, thus redefining "physical reality" to be tied to specific experimental circumstances. As he put it in 1948, "As a more appropriate way of expression, one may strongly advocate limitation of the use of the word phenomenon to refer exclusively to observations obtained under specified circumstances, including an account of the whole experiment." This was a fundamental divergence from the EPR criterion.

Bell's theorem

In 1964, John Stewart Bell delivered a blow to local hidden-variable theories with his theorem. He demonstrated that if such theories were valid, then certain experiments involving entangled particles would yield results that obeyed specific Bell inequalities. However, if quantum mechanics, with its inherent nonlocality, was correct, these inequalities would be violated. The subsequent Bell test experiments, notably those by Alain Aspect and Paul Kwiat, have consistently shown violations of these inequalities, effectively ruling out local hidden-variable theories. The Kochen–Specker theorem is another significant no-go theorem relevant to hidden-variable theories.

However, Bell's theorem doesn't preclude the possibility of nonlocal hidden-variable theories or the concept of superdeterminism. Gerard 't Hooft has notably argued against the absolute validity of Bell's theorem, proposing that the superdeterminism loophole might allow for local deterministic models.

Bohm's hidden-variable theory

In 1952, David Bohm resurrected an idea from Louis de Broglie's earlier work, creating a deterministic hidden-variable theory commonly known as the de Broglie–Bohm theory. This theory posits that particles are guided by a "pilot wave." In the famous double-slit experiment, for instance, the electron passes through one slit, but its trajectory is determined by this hidden pilot wave, leading to the observed interference pattern.

Bohm's theory introduces a non-local quantum potential that organizes particle behavior, suggesting an underlying "implicate order." While some find it the "simplest" explanation for quantum phenomena, its necessity for hidden variables is undeniable. A potential weakness, as perceived by physicists like Einstein, Pauli, and Heisenberg, is its perceived contrived nature. Bohm himself expressed reservations about the guiding wave existing in an abstract configuration space rather than three-dimensional space.

Recent developments

In 2011, Roger Colbeck and Renato Renner presented a proof suggesting that no extension of quantum theory, including hidden-variable theories, can offer more accurate predictions than quantum mechanics itself, provided that measurement settings can be chosen freely. They concluded: "In the present work, we have ... excluded the possibility that any extension of quantum theory (not necessarily in the form of local hidden variables) can help predict the outcomes of any measurement on any quantum state. In this sense, we show the following: under the assumption that measurement settings can be chosen freely, quantum theory really is complete."

However, in 2013, Giancarlo Ghirardi and Raffaele Romano proposed a model that challenges this conclusion, suggesting that under different free-choice assumptions, the statement could be violated, potentially in an experimentally testable manner.

It's a fascinating, if deeply unsettling, exploration into the fundamental nature of reality. The universe, it seems, holds its secrets tightly, and our attempts to pry them open often lead us down paths of paradox and philosophical quandaries. Don't expect easy answers. In this realm, the questions themselves are often more revealing than any supposed solution.