Math 2.0

Quantum theory Definition

Hidden-variable theory

Builds on Probability measureHilbert spaceQuantum stateBorn rule

Definition

Let ℋ\mathcal{H} be a finite-dimensional complex Hilbert space. A hidden-variable theory, or ontological model, of quantum mechanics on ℋ\mathcal{H} consists of the following data:

  1. a measurable space (Λ,ℱ)(\Lambda, \mathcal{F}), whose points λ\lambda are the hidden variables, or ontic states;
  2. for each pure quantum state ψ\psi, a probability measure μψ\mu_\psi on Λ\Lambda;
  3. for each orthonormal basis M=(m1,…,md)M = (m_1, \dots, m_d) of ℋ\mathcal{H}, measurable functions ξM(k∣⋅) ⁣:Λ→[0,1]\xi_M(k \mid \cdot) \colon \Lambda \to [0, 1] with ∑k=1dξM(k∣λ)=1\sum_{k=1}^d \xi_M(k \mid \lambda) = 1 for all λ\lambda.

It must reproduce the Born rule: ∫ΛξM(k∣λ) dμψ(λ)=∣⟨mk,ψ⟩∣2\int_\Lambda \xi_M(k \mid \lambda)\, d\mu_\psi(\lambda) = \abs{\langle m_k, \psi \rangle}^2 for all ψ\psi, MM and kk. The theory is deterministic if all values ξM(k∣λ)\xi_M(k \mid \lambda) are 00 or 11.

Remarks

Example: the Beltrametti–Bugajski model takes Λ\Lambda to be the set of unit vectors up to phase, μψ\mu_\psi the point mass at ψ\psi, and ξM(k∣λ)=∣⟨mk,λ⟩∣2\xi_M(k \mid \lambda) = \abs{\langle m_k, \lambda \rangle}^2. The Born rule holds by construction, so every quantum system has this trivial model. Non-example: taking ξM(k∣λ)=1/d\xi_M(k \mid \lambda) = 1/d for all kk satisfies clause 3, but it gives probability 1/d1/d instead of 11 when ψ=m1\psi = m_1 and k=1k = 1. A deterministic theory is noncontextual if the value of ξM(k∣λ)\xi_M(k \mid \lambda) depends only on the ray of mkm_k and not on the rest of the basis MM. For a bipartite system, the theory is local if the response functions for a product measurement factorize into a function of the local setting on each side.

Reading path

To understand this page from the ground up, read these 26 pages in order. The foundations come first.

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Adjoint of an unbounded operator, concept, no evidence levelAdjoint of an unbound…conceptBorel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptContinuous functional calculus, concept, no evidence levelContinuous functional…conceptDensely defined operator, concept, no evidence levelDensely defined opera…conceptDensity matrix, concept, no evidence levelDensity matrixconceptExpectation value, concept, no evidence levelExpectation valueconceptHilbert space, concept, no evidence levelHilbert spaceconceptHilbert space adjoint, concept, no evidence levelHilbert space adjointconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionObservable, concept, no evidence levelObservableconceptProbability measure, concept, no evidence levelProbability measureconceptProjection-valued measure, concept, no evidence levelProjection-valued mea…conceptQuantum state, concept, no evidence levelQuantum stateconceptSpectrum of an operator, concept, no evidence levelSpectrum of an operat…conceptTrace class operator, concept, no evidence levelTrace class operatorconceptUnbounded operator, concept, no evidence levelUnbounded operatorconceptTranspose, definition, no evidence levelTransposedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionNormal matrix, definition, no evidence levelNormal matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionSpectral theorem for normal matrices, statement, no evidence levelSpectral theorem for …statementSpectral theorem for bounded normal operators, statement, no evidence levelSpectral theorem for …statementSpectral theorem for unbounded self-adjoint operators, statement, no evidence levelSpectral theorem for …statementBorn rule, statement, no evidence levelBorn rulestatementHidden-variable theory, definition, no evidence levelHidden-variable theorydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Adjoint of an unbounded operator concept
  2. Borel sigma-algebra concept
  3. Continuous functional calculus concept
  4. Densely defined operator concept
  5. Density matrix concept
  6. Expectation value concept
  7. Hilbert space concept
  8. Hilbert space adjoint concept
  9. Identity matrix object
  10. Matrix definition
  11. Observable concept
  12. Probability measure concept
  13. Projection-valued measure concept
  14. Quantum state concept
  15. Spectrum of an operator concept
  16. Trace class operator concept
  17. Unbounded operator concept
  18. Transpose definition
  19. Conjugate transpose definition
  20. Normal matrix definition
  21. Unitary matrix definition
  22. Spectral theorem for normal matrices statement
  23. Spectral theorem for bounded normal operators statement
  24. Spectral theorem for unbounded self-adjoint operators statement
  25. Born rule statement
  26. Hidden-variable theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hidden-variable_theory
About this page
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Kinddefinition
WritingW0 stub
MSC81P05
DefinesHidden-variable theory
UsesProbability measure, Hilbert space, Quantum state, Born rule
Changed3 November 2026

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