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{{Short description|Deterministic models extending quantum mechanics via additional hidden parameters}}

{{Quantum book backlink|Conceptual and interpretations}}
'''Hidden-variable theory''' refers to a class of theoretical models in [[Physics:Quantum mechanics|quantum mechanics]] that attempt to explain its probabilistic nature by introducing additional, unobserved parameters—called ''hidden variables''—that determine the outcomes of measurements.<ref name="Bell1966">{{cite journal |last=Bell |first=J. S. |title=On the problem of hidden variables in quantum mechanics |journal=Reviews of Modern Physics |volume=38 |issue=3 |pages=447–452 |year=1966 |doi=10.1103/RevModPhys.38.447}}</ref>

These theories are typically motivated by a desire to restore [[Determinism|determinism]] and provide a more complete description of physical reality than standard quantum mechanics.
[[File:Hidden_variable_theory_concept.jpg|thumb|400px|Conceptual illustration of hidden-variable theories: underlying variables determine outcomes that appear probabilistic in standard quantum mechanics.]]

== Concept ==

In standard quantum mechanics:

* Physical systems are described by a [[Physics:Wave function|wave function]]
* Measurement outcomes are inherently probabilistic
* Properties do not have definite values prior to measurement

Hidden-variable theories instead assume:

→ Physical systems possess definite properties at all times
→ Probabilities arise from ignorance of underlying variables

This contrasts with the orthodox view, where measurement plays a fundamental role in defining outcomes.<ref name="Mermin1993">{{cite journal |last=Mermin |first=N. David |title=Hidden variables and the two theorems of John Bell |journal=Reviews of Modern Physics |volume=65 |issue=3 |pages=803–815 |year=1993 |doi=10.1103/RevModPhys.65.803}}</ref>

== Motivation ==

Hidden-variable theories aim to resolve conceptual issues in quantum mechanics, including:

* [[Physics:Quantum indeterminacy|Quantum indeterminacy]]
* The [[Physics:Measurement problem|measurement problem]]
* The absence of definite physical properties prior to measurement

They attempt to provide a framework closer to classical physics, where:

* Systems have well-defined states
* Evolution is deterministic
* Measurement reveals pre-existing values

== Historical background ==

The idea dates back to the early development of quantum theory.

=== Early debates ===

In 1926, [[Biography:Max Born|Max Born]] introduced the probabilistic interpretation of the wave function. This was challenged by [[Biography:Albert Einstein|Albert Einstein]], who argued that quantum mechanics must be incomplete.<ref>{{cite book |title=The Born–Einstein Letters |year=1971 |publisher=Macmillan}}</ref>

Einstein’s famous remark:

→ “God does not play dice”

expressed his belief that a deeper deterministic theory should exist.

=== EPR argument ===

In 1935, Einstein, Podolsky, and Rosen proposed the [[Physics:EPR paradox|EPR paradox]], arguing that quantum mechanics does not provide a complete description of reality.<ref name="EPR1935">{{cite journal |last1=Einstein |first1=A. |last2=Podolsky |first2=B. |last3=Rosen |first3=N. |title=Can quantum-mechanical description of physical reality be considered complete? |journal=Physical Review |volume=47 |issue=10 |pages=777–780 |year=1935 |doi=10.1103/PhysRev.47.777}}</ref>

They suggested that:

* Additional hidden variables might exist
* These would restore determinism and locality

=== Bell’s theorem ===

In 1964, [[Biography:John Stewart Bell|John Bell]] showed that:

→ No ''local'' hidden-variable theory can reproduce all predictions of quantum mechanics<ref name="Bell1964">{{cite journal |last=Bell |first=J. S. |title=On the Einstein Podolsky Rosen paradox |journal=Physics Physique Физика |volume=1 |issue=3 |pages=195–200 |year=1964 |doi=10.1103/PhysicsPhysiqueFizika.1.195}}</ref>

This result fundamentally constrained hidden-variable approaches.

== Local vs nonlocal theories ==

Hidden-variable theories are divided into two main classes:

=== Local hidden variables ===

* Respect [[Physics:Principle of locality|locality]]
* No faster-than-light influence
* Ruled out by Bell test experiments

Experiments consistently show violations of Bell inequalities, excluding this class.<ref name="Hensen2015">{{cite journal |last=Hensen |first=B. |title=Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres |journal=Nature |volume=526 |pages=682–686 |year=2015 |doi=10.1038/nature15759}}</ref>

=== Nonlocal hidden variables ===

* Allow instantaneous correlations
* Compatible with quantum predictions
* Example: [[Physics:De Broglie–Bohm theory|de Broglie–Bohm theory]]

These models preserve determinism but require nonlocality.

== de Broglie–Bohm theory ==

The most well-known hidden-variable theory is the '''de Broglie–Bohm theory'''.

Key features:

* Particles have definite trajectories
* A guiding wave (pilot wave) determines motion
* Evolution is deterministic

In this framework:

* The wave function evolves via the Schrödinger equation
* Particle positions evolve via a guiding equation

This theory reproduces all predictions of standard quantum mechanics while remaining deterministic.<ref>{{cite book |last=Bohm |first=D. |title=The Undivided Universe |publisher=Routledge |year=1993}}</ref>

However, it is explicitly nonlocal.

== Modern developments ==

Recent theoretical work has placed further constraints on hidden-variable theories.

A notable result is:

→ No extension of quantum theory can improve its predictive power (under reasonable assumptions)<ref name="ColbeckRenner">{{cite journal |last1=Colbeck |first1=R. |last2=Renner |first2=R. |title=No extension of quantum theory can have improved predictive power |journal=Nature Communications |volume=2 |pages=411 |year=2011 |doi=10.1038/ncomms1416}}</ref>

This suggests that:

* Even with hidden variables, predictions cannot surpass quantum mechanics

== Conceptual implications ==

Hidden-variable theories highlight fundamental questions:

* Is reality deterministic or intrinsically probabilistic?
* Do physical properties exist prior to measurement?
* Is nonlocality a fundamental feature of nature?

Bell’s theorem shows that at least one classical assumption must be abandoned:

→ locality, realism, or measurement independence

== Physical significance ==

Although local hidden-variable theories are ruled out, the concept remains important because it:

* Clarifies the foundations of quantum mechanics
* Motivates experimental tests (Bell tests)
* Informs interpretations of quantum theory

It also plays a central role in:

* [[Physics:Quantum information theory|quantum information]]
* [[Physics:Quantum nonlocality|quantum nonlocality]]
* [[Physics:Foundations of quantum mechanics|foundations of physics]]

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Hidden variable theory|1}}

Physics:Quantum Hidden variable theory - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template