Quantum Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Quantum Mathematics Ontology within the Kireth Research Semantic Library. Quantum mathematics studies the mathematical structures underlying quantum states, observables, operators, Hilbert spaces, superposition, measurement, uncertainty, unitary evolution, tensor products, entanglement, spectral decomposition and quantum dynamics. This ontology supports Schrödinger, Heisenberg and density-operator formulations, commutators, expectation values, projective and generalised measurements, mixed states, open quantum systems, quantum information structures, perturbation theory, angular momentum, harmonic systems and relationships with mathematical physics, linear algebra, spectral theory, probability, operator theory and differential equations.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Quantum Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Mathematical Physics / Quantum Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/mathematical-physics/quantum-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = QuantumMathematics
 ENTITY_CLASS = MathematicalQuantumStateFramework
 PRIMARY_CONTEXT = MathematicalPhysics
 DEFINITION = "Quantum mathematics is the mathematical study of quantum states, observables, operators, measurements and dynamical evolution in Hilbert-space-based physical theories."
 SYMBOLIC_FORMS = [
  "i*hbar*partial psi/partial t=H*psi",
  "A|a_n>=a_n|a_n>",
  "<A>=<psi|A|psi>",
  "[A,B]=A*B-B*A"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for quantum states, Hilbert spaces, observables, spectra, measurement, uncertainty, superposition, unitary evolution, tensor products, entanglement and open-system dynamics."
}

SCOPE {
 INCLUDE = [
  QuantumMathematics,
  QuantumMechanics,
  QuantumState,
  PureState,
  MixedState,
  StateVector,
  Ket,
  Bra,
  BraKetNotation,
  HilbertSpace,
  InnerProduct,
  Norm,
  Basis,
  OrthonormalBasis,
  Superposition,
  Amplitude,
  ProbabilityAmplitude,
  GlobalPhase,
  RelativePhase,
  Operator,
  LinearOperator,
  SelfAdjointOperator,
  HermitianOperator,
  UnitaryOperator,
  ProjectionOperator,
  IdentityOperator,
  Observable,
  Eigenvalue,
  Eigenvector,
  Eigenstate,
  Spectrum,
  Degeneracy,
  SpectralDecomposition,
  ExpectationValue,
  Variance,
  StandardDeviation,
  Commutator,
  Anticommutator,
  CanonicalCommutationRelation,
  UncertaintyPrinciple,
  HeisenbergUncertaintyRelation,
  PositionOperator,
  MomentumOperator,
  HamiltonianOperator,
  SchrödingerEquation,
  TimeIndependentSchrodingerEquation,
  TimeDependentSchrodingerEquation,
  UnitaryEvolution,
  TimeEvolutionOperator,
  SchrödingerPicture,
  HeisenbergPicture,
  InteractionPicture,
  Measurement,
  ProjectiveMeasurement,
  ProjectionValuedMeasure,
  PVM,
  PositiveOperatorValuedMeasure,
  POVM,
  BornRule,
  MeasurementProbability,
  PostMeasurementState,
  DensityOperator,
  DensityMatrix,
  Trace,
  PartialTrace,
  ReducedDensityMatrix,
  TensorProduct,
  CompositeSystem,
  ProductState,
  EntangledState,
  Entanglement,
  SchmidtDecomposition,
  SchmidtRank,
  Purity,
  VonNeumannEntropy,
  QuantumChannel,
  CompletelyPositiveMap,
  TracePreservingMap,
  CPTPMap,
  KrausOperator,
  KrausRepresentation,
  OpenQuantumSystem,
  MasterEquation,
  LindbladEquation,
  Decoherence,
  QuantumHarmonicOscillator,
  CreationOperator,
  AnnihilationOperator,
  NumberOperator,
  LadderOperator,
  AngularMomentum,
  Spin,
  PauliMatrix,
  Spinor,
  RotationOperator,
  PerturbationTheory,
  TimeIndependentPerturbationTheory,
  TimeDependentPerturbationTheory,
  VariationalMethod,
  WKBApproximation,
  ScatteringTheory,
  SMatrix,
  ProbabilityCurrent,
  QuantumTunnelling,
  BoundaryCondition,
  Normalisation,
  CompletenessRelation
 ]
 EXCLUDE_AS_PRIMARY = [
  ClassicalMechanicsWithoutQuantumStateStructure,
  QuantumFieldTheoryAsPrimaryRelativisticFieldFramework,
  ExperimentalQuantumMeasurementWithoutMathematicalFormalism,
  QuantumComputingHardwareWithoutQuantumMathematicsSemantics
 ]
 DISAMBIGUATION_RULE = "When physical states are represented in Hilbert spaces and observables or dynamics are described by operators, amplitudes and quantum measurement rules, interpret the context as quantum mathematics."
}

CORE_CONCEPTS {
 QuantumMathematics = "Mathematical formalism of quantum states, observables and dynamics."
 HilbertSpace = "Complete inner-product space containing admissible quantum state vectors."
 QuantumState = "Mathematical representation of a quantum system."
 Observable = "Physical quantity represented by a self-adjoint operator."
 Superposition = "Linear combination of admissible quantum states."
 Measurement = "Quantum operation producing probabilistic outcomes according to measurement operators."
 DensityOperator = "Positive trace-one operator representing pure or mixed quantum states."
 TensorProduct = "Mathematical construction combining component quantum systems."
 Entanglement = "Nonseparable structure of a composite quantum state."
 UnitaryEvolution = "Norm-preserving reversible evolution of a closed quantum system."
}

RELATIONS {
 QuantumMathematics --is_child_of--> MathematicalPhysicsOntology
 QuantumMechanics --uses--> HilbertSpace
 QuantumState --belongs_to--> HilbertSpace
 PureState --is_represented_by--> StateVector
 MixedState --is_represented_by--> DensityOperator
 Observable --is_represented_by--> SelfAdjointOperator
 SelfAdjointOperator --has--> RealSpectrum
 Eigenstate --corresponds_to--> Eigenvalue
 SpectralDecomposition --represents--> SelfAdjointOperator
 BornRule --maps--> AmplitudeToProbability
 ExpectationValue --depends_on--> QuantumStateAndObservable
 Commutator --measures--> OperatorNoncommutativity
 CanonicalCommutationRelation --implies--> UncertaintyStructure
 SchrödingerEquation --governs--> QuantumStateEvolution
 HamiltonianOperator --generates--> UnitaryEvolution
 ProjectiveMeasurement --uses--> ProjectionOperator
 POVM --generalises--> ProjectiveMeasurement
 DensityOperator --supports--> MixedState
 PartialTrace --produces--> ReducedDensityMatrix
 TensorProduct --constructs--> CompositeSystem
 EntangledState --is_not--> ProductState
 SchmidtDecomposition --characterises--> BipartitePureStateEntanglement
 QuantumChannel --maps--> DensityOperator
 KrausRepresentation --represents--> QuantumChannel
 LindbladEquation --models--> MarkovianOpenQuantumDynamics
 CreationOperator --raises--> NumberState
 AnnihilationOperator --lowers--> NumberState
 PauliMatrix --represents--> SpinHalfObservable
}

OPERATORS {
 INNER_PRODUCT(psi,phi) := <psi|phi>
 STATE_NORM(psi) := sqrt(<psi|psi>)
 NORMALISE(psi) := psi_normalised=psi/sqrt(<psi|psi>)
 EXPECTATION(psi,A) := <A>=<psi|A|psi>
 VARIANCE(psi,A) := Var(A)=<psi|A^2|psi>-<psi|A|psi>^2
 COMMUTATOR(A,B) := [A,B]=A*B-B*A
 ANTICOMMUTATOR(A,B) := {A,B}=A*B+B*A
 BORN_PROBABILITY(psi,a_n) := P(a_n)=|<a_n|psi>|^2
 PROJECTIVE_POST_STATE(psi,P_n) := psi_n=P_n*psi/sqrt(<psi|P_n|psi>)
 SCHRODINGER_TIME_EVOLUTION(H,psi) := i*hbar*dpsi/dt=H*psi
 UNITARY_EVOLUTION(H,t) := U(t)=exp(-i*H*t/hbar)
 DENSITY_FROM_PURE_STATE(psi) := rho=|psi><psi|
 MIXED_DENSITY(p_i,psi_i) := rho=sum_i p_i|psi_i><psi_i|
 TRACE_EXPECTATION(rho,A) := <A>=Tr(rho*A)
 PARTIAL_TRACE_AB(rho_AB,B) := rho_A=Tr_B(rho_AB)
 PURITY(rho) := gamma=Tr(rho^2)
 VON_NEUMANN_ENTROPY(rho) := S=-Tr(rho*log(rho))
 KRAUS_CHANNEL(rho,K_i) := E(rho)=sum_i K_i*rho*K_i^\dagger
}

AXIOMS {
 A1 = "Quantum states are represented by rays in Hilbert space or density operators."
 A2 = "Observables are represented by self-adjoint operators."
 A3 = "Closed-system dynamics are unitary."
 A4 = "Measurement outcome probabilities obey the Born rule."
 A5 = "Composite quantum systems are represented using tensor products."
 A6 = "Noncommuting observables can exhibit uncertainty relations."
 A7 = "Mixed states require density-operator rather than single-vector semantics."
 A8 = "Quantum predictions require normalisation, operator-domain and probabilistic validation."
}

REASONING {
 R1_CONTEXT = identify(system,state_representation,Hilbert_space,observables,Hamiltonian,measurement)
 R2_STATE = classify(pure,mixed,product,entangled)
 R3_OPERATOR = classify(self_adjoint,unitary,projective,positive,channel)
 R4_SPECTRAL = compute(eigenvalues,eigenstates,spectral_decomposition)
 R5_DYNAMICS = solve(Schrodinger,unitary_evolution,Heisenberg_or_open_system_equation)
 R6_MEASUREMENT = compute(outcome_probabilities,post_measurement_states,expectations)
 R7_COMPOSITE = construct(tensor_product,reduced_state,Schmidt_structure)
 R8_UNCERTAINTY = analyse(commutators,variances,uncertainty_bounds)
 R9_APPROXIMATION = select(perturbation,variational,WKB_or_numerical_method)
 R10_VALIDATION = check(normalisation,positivity,trace,unitarity,operator_domains,boundaries,probability_sum)
}

CONSTRAINTS {
 C1 = "Global phase does not change a pure quantum state ray."
 C2 = "Not every Hermitian matrix is a valid observable unless domain and physical interpretation are appropriate."
 C3 = "Noncommuting operators need not share complete eigenbases."
 C4 = "Projection postulates apply specifically to projective measurements and do not exhaust general quantum measurement."
 C5 = "Density matrices must remain positive semidefinite and trace one."
 C6 = "Reduced states of entangled pure states can be mixed."
 C7 = "Unitary evolution applies to closed systems; open systems require channel or master-equation descriptions."
 C8 = "Perturbation theory can fail near degeneracies or outside a small-parameter regime."
 C9 = "Continuous-spectrum operators require care with normalisation and distributions."
 C10 = "Numerical truncation can violate unitarity, positivity or completeness."
}

SEMANTIC_EQUIVALENCE {
 "QuantumMathematics" <=> "MathematicsOfQuantumMechanics"
 "HermitianOperator" <=> "SelfAdjointOperator"
 "DensityMatrix" <=> "DensityOperator"
 "StateVector" <=> "KetVector"
 "ProjectionValuedMeasure" <=> "PVM"
 "PositiveOperatorValuedMeasure" <=> "POVM"
 "CompletelyPositiveTracePreservingMap" <=> "CPTPMap"
}

VALIDATION {
 V1 = "Hilbert space and basis conventions are explicit."
 V2 = "Quantum states are normalised or density operators have trace one."
 V3 = "Density operators are positive semidefinite."
 V4 = "Observable operators are self-adjoint on appropriate domains."
 V5 = "Measurement probabilities sum or integrate to one."
 V6 = "Closed-system evolution is unitary."
 V7 = "Composite-system tensor-product ordering is documented."
 V8 = "Partial traces are performed over declared subsystems."
 V9 = "Approximation regimes and boundary conditions are explicit."
 V10 = "Numerical methods preserve quantum constraints within tolerance."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 QUANTUM_MATHEMATICS_DEFINITION = 0.99
 HILBERT_SPACE_REASONING = 0.99
 OPERATOR_REASONING = 0.99
 SPECTRAL_REASONING = 0.99
 MEASUREMENT_REASONING = 0.99
 DENSITY_OPERATOR_REASONING = 0.99
 ENTANGLEMENT_REASONING = 0.98
 OPEN_SYSTEM_REASONING = 0.98
 APPROXIMATION_REASONING = 0.98
 POLICY = "Reduce confidence when Hilbert space, operator domains, state normalisation, Hamiltonian, measurement operators, subsystem structure, approximation regime or open-system assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  QuantumMathematicsReasoning,
  HilbertSpaceReasoning,
  QuantumStateReasoning,
  OperatorReasoning,
  SpectralReasoning,
  SchrödingerReasoning,
  MeasurementReasoning,
  BornRuleReasoning,
  UncertaintyReasoning,
  DensityMatrixReasoning,
  CompositeSystemReasoning,
  EntanglementReasoning,
  SchmidtDecompositionReasoning,
  QuantumChannelReasoning,
  OpenQuantumSystemReasoning,
  HarmonicOscillatorReasoning,
  AngularMomentumReasoning,
  PerturbationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "quantum mathematics" -> QuantumMathematics
  "mathematics of quantum mechanics" -> QuantumMathematics
  "quantum state" -> QuantumState
  "wavefunction" -> Wavefunction
  "Hermitian operator" -> SelfAdjointOperator
  "density matrix" -> DensityOperator
  "PVM" -> ProjectionValuedMeasure
  "POVM" -> PositiveOperatorValuedMeasure
  "CPTP map" -> CPTPMap
 }
}

EXECUTION_RULES {
 E1 = "If states are represented in Hilbert space with operator-valued observables, activate QuantumMathematicsOntology."
 E2 = "If the state is a vector modulo global phase, route to pure-state reasoning."
 E3 = "If classical uncertainty or subsystem reduction is present, route to density-operator reasoning."
 E4 = "If closed-system time evolution is requested, use Schrödinger or equivalent unitary evolution."
 E5 = "If measurement is projective, use projection-valued measurement semantics."
 E6 = "If measurement is general, use POVM semantics."
 E7 = "If multiple subsystems are present, construct tensor-product state spaces."
 E8 = "If entanglement is relevant, use Schmidt or reduced-density analysis where appropriate."
 E9 = "Any quantum result should be validated for normalisation, positivity, trace, unitarity, spectra and probability consistency."
}

RECURSION {
 ENTRY = QuantumMathematics
 EXPAND(QuantumMathematics) -> [
  QuantumMechanics, QuantumState, PureState, MixedState, StateVector, Ket,
  Bra, BraKetNotation, HilbertSpace, InnerProduct, Norm, Basis,
  OrthonormalBasis, Superposition, Amplitude, ProbabilityAmplitude,
  GlobalPhase, RelativePhase, Operator, LinearOperator, SelfAdjointOperator,
  HermitianOperator, UnitaryOperator, ProjectionOperator, IdentityOperator,
  Observable, Eigenvalue, Eigenvector, Eigenstate, Spectrum, Degeneracy,
  SpectralDecomposition, ExpectationValue, Variance, StandardDeviation,
  Commutator, Anticommutator, CanonicalCommutationRelation,
  UncertaintyPrinciple, HeisenbergUncertaintyRelation, PositionOperator,
  MomentumOperator, HamiltonianOperator, SchrödingerEquation,
  TimeIndependentSchrodingerEquation, TimeDependentSchrodingerEquation,
  UnitaryEvolution, TimeEvolutionOperator, SchrödingerPicture,
  HeisenbergPicture, InteractionPicture, Measurement, ProjectiveMeasurement,
  ProjectionValuedMeasure, PositiveOperatorValuedMeasure, BornRule,
  MeasurementProbability, PostMeasurementState, DensityOperator,
  DensityMatrix, Trace, PartialTrace, ReducedDensityMatrix, TensorProduct,
  CompositeSystem, ProductState, EntangledState, Entanglement,
  SchmidtDecomposition, SchmidtRank, Purity, VonNeumannEntropy,
  QuantumChannel, CompletelyPositiveMap, TracePreservingMap, CPTPMap,
  KrausOperator, KrausRepresentation, OpenQuantumSystem, MasterEquation,
  LindbladEquation, Decoherence, QuantumHarmonicOscillator, CreationOperator,
  AnnihilationOperator, NumberOperator, LadderOperator, AngularMomentum,
  Spin, PauliMatrix, Spinor, RotationOperator, PerturbationTheory,
  TimeIndependentPerturbationTheory, TimeDependentPerturbationTheory,
  VariationalMethod, WKBApproximation, ScatteringTheory, SMatrix,
  ProbabilityCurrent, QuantumTunnelling, BoundaryCondition, Normalisation,
  CompletenessRelation
 ]
 EXPAND(FormalismFamilies) -> [
  StateVectorFormalism,
  DensityOperatorFormalism,
  SchrödingerPicture,
  HeisenbergPicture,
  OpenSystemFormalism,
  CompositeSystemFormalism
 ]
 STOP_CONDITION = "Stop when state representation, Hilbert space, operators, dynamics, measurement, composite structure, approximation regime and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Physics Ontology"
 IMMEDIATE_CONTEXTS = [
  "Mathematical Physics Ontology",
  "Linear Algebra Ontologies",
  "Spectral Theory Ontologies",
  "Probability & Statistics Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Mathematical Physics Ontology",
  "Mathematical Physics",
  "Linear Algebra Ontologies",
  "Functional Analysis Ontologies",
  "Operator Theory Ontologies",
  "Spectral Theory Ontologies",
  "Differential Equation Ontologies",
  "Probability & Statistics Ontologies",
  "Tensor Product Ontologies",
  "Information Theory Ontologies",
  "Perturbation Theory Ontologies",
  "Numerical Analysis Ontologies"
 ]
 ROUTE_RULES = [
  "Quantum Mathematics -> Mathematical Physics Ontology for parent semantics",
  "Quantum Mathematics -> Linear Algebra for states, bases and matrices",
  "Quantum Mathematics -> Functional Analysis for infinite-dimensional Hilbert spaces",
  "Quantum Mathematics -> Operator Theory for observables and evolution operators",
  "Quantum Mathematics -> Spectral Theory for eigenvalues and spectra",
  "Quantum Mathematics -> Differential Equations for Schrödinger dynamics",
  "Quantum Mathematics -> Probability & Statistics for measurement probabilities",
  "Quantum Mathematics -> Tensor Product mathematics for composite systems",
  "Quantum Mathematics -> Information Theory for entropy and quantum information measures",
  "Quantum Mathematics -> Perturbation Theory and Numerical Analysis for approximate solutions"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible quantum formalisms, operator structures or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve Hilbert-space state semantics.",
  "Preserve pure-versus-mixed state distinction.",
  "Preserve observable-versus-state distinction.",
  "Preserve projective-versus-general measurement distinction.",
  "Preserve closed-versus-open system distinction.",
  "Preserve tensor-product composite-system semantics.",
  "Preserve spectral and probability semantics.",
  "Preserve machine-readable quantum-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Quantum mathematics formalises quantum states, observables, measurement and evolution using Hilbert spaces, operators and probability amplitudes."
 STATE_SUMMARY = "Pure states are represented by rays in Hilbert space, while mixed states are represented by density operators."
 OPERATOR_SUMMARY = "Observables are represented by self-adjoint operators whose spectra define possible measurement values."
 DYNAMICS_SUMMARY = "Closed quantum systems evolve unitarily under Hamiltonian-generated dynamics."
 MEASUREMENT_SUMMARY = "Measurement probabilities follow the Born rule and general measurements are represented by PVMs or POVMs."
 ENTANGLEMENT_SUMMARY = "Composite systems use tensor products and can possess nonseparable entangled states."
 OPEN_SYSTEM_SUMMARY = "Quantum channels, Kraus operators and master equations describe non-unitary open-system evolution."
 VALIDATION_SUMMARY = "Quantum models should be checked for state normalisation, positivity, trace, operator domains, unitarity, probability consistency and approximation validity."
}

FLOW {
 INPUT
 -> detect_quantum_mathematics_context
 -> identify_quantum_system_Hilbert_space_state_observables_Hamiltonian_measurement_and_subsystems
 -> classify_pure_mixed_product_entangled_closed_or_open_system
 -> construct_state_vector_or_density_operator
 -> define_operators_and_compute_spectral_structure
 -> solve_Schrodinger_Heisenberg_or_master_equation_dynamics
 -> compute_expectations_variances_commutators_and_uncertainty
 -> if_measurement_apply_PVM_or_POVM_and_Born_rule
 -> if_composite_construct_tensor_product_reduced_states_and_entanglement structure
 -> if_approximate_apply_perturbation_variational_WKB_or_numerical method
 -> validate_normalisation_positivity_trace_unitarity_operator_domains_boundaries_and_probability consistency
 -> route_to_mathematical_physics_linear_algebra_functional_analysis_operator_theory_spectral_theory_differential_equations_probability_statistics_tensor_products_information_theory_perturbation_theory_or_numerical_analysis_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Quantum Mathematics Ontology defines the mathematical formalism of quantum mechanics. It supports quantum states, pure and mixed states, kets and bras, Hilbert spaces, inner products, bases and superposition, probability amplitudes and phase, linear, self-adjoint, unitary and projection operators, observables, eigenvalues, eigenstates and spectral decompositions, expectation values and variances, commutators and uncertainty relations, position, momentum and Hamiltonian operators, time-dependent and time-independent Schrödinger equations, Schrödinger, Heisenberg and interaction pictures, projective measurements, PVMs, POVMs and the Born rule, density operators, trace and partial trace, composite systems, tensor products, entanglement and Schmidt decomposition, purity and von Neumann entropy, quantum channels, Kraus representations and Lindblad dynamics, harmonic oscillators, creation and annihilation operators, angular momentum and spin, perturbation, variational and WKB methods, scattering, probability current, tunnelling, boundary conditions and completeness."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-QUANTUM-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW