Mathematical Physics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Mathematical Physics Ontology within the Kireth Research Semantic Library. It formalises the core mathematical structures used to represent physical systems, including state spaces, dynamical equations, operators, variational principles, symmetries, conservation laws, fields, tensors, spectral structure, boundary conditions, approximation methods and numerical solution.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = MathematicalPhysicsOntology
 ENTITY_CLASS = FormalPhysicalMathematicsOntology
 PRIMARY_CONTEXT = MathematicalPhysics
 DEFINITION = "The Mathematical Physics Ontology is a formal semantic framework for mathematical representations of physical systems, laws, states, fields, operators, symmetries and evolution."
 SYMBOLIC_FORMS = [
  "F=ma",
  "d/dt(partial L/partial qdot)-partial L/partial q=0",
  "i*hbar*partial psi/partial t=H psi",
  "L=D-A or physical_operator_equivalent"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for core physical mathematics across mechanics, fields, quantum theory, relativity, statistical physics, continuum mechanics, spectral analysis and computational physics."
}

SCOPE {
 INCLUDE = [
  MathematicalPhysicsOntology,
  MathematicalPhysics,
  PhysicalSystem,
  PhysicalState,
  StateSpace,
  ConfigurationSpace,
  PhaseSpace,
  PhysicalQuantity,
  Observable,
  Parameter,
  Constant,
  Unit,
  Dimension,
  PhysicalLaw,
  DynamicalLaw,
  Symmetry,
  Invariance,
  ConservationLaw,
  NoetherPrinciple,
  ClassicalMechanics,
  NewtonianMechanics,
  LagrangianMechanics,
  HamiltonianMechanics,
  GeneralisedCoordinate,
  GeneralisedMomentum,
  Lagrangian,
  Hamiltonian,
  Action,
  VariationalPrinciple,
  EulerLagrangeEquation,
  CanonicalEquation,
  QuantumMechanics,
  QuantumState,
  Wavefunction,
  HilbertSpace,
  Operator,
  SelfAdjointOperator,
  Eigenvalue,
  Eigenstate,
  Spectrum,
  Commutator,
  UncertaintyRelation,
  Electromagnetism,
  ElectricField,
  MagneticField,
  ElectromagneticField,
  MaxwellEquations,
  ScalarPotential,
  VectorPotential,
  GaugeTransformation,
  Relativity,
  SpecialRelativity,
  GeneralRelativity,
  Spacetime,
  MetricTensor,
  ChristoffelSymbol,
  CovariantDerivative,
  CurvatureTensor,
  RicciTensor,
  EinsteinTensor,
  Geodesic,
  StatisticalMechanics,
  Thermodynamics,
  Microstate,
  Macrostate,
  Ensemble,
  PartitionFunction,
  Entropy,
  Temperature,
  FreeEnergy,
  FieldTheory,
  ClassicalFieldTheory,
  QuantumFieldTheory,
  GaugeTheory,
  Field,
  LagrangianDensity,
  ActionFunctional,
  EulerLagrangeFieldEquation,
  WaveEquation,
  HeatEquation,
  DiffusionEquation,
  LaplaceEquation,
  PoissonEquation,
  HelmholtzEquation,
  ContinuumMechanics,
  Elasticity,
  FluidMechanics,
  StressTensor,
  StrainTensor,
  TensorField,
  BoundaryCondition,
  InitialCondition,
  GreenFunction,
  FourierAnalysis,
  FourierTransform,
  LaplaceTransform,
  SpectralTheory,
  Eigenfunction,
  EigenfunctionExpansion,
  SturmLiouvilleProblem,
  PerturbationTheory,
  AsymptoticAnalysis,
  DimensionalAnalysis,
  ScalingLaw,
  NonlinearDynamics,
  Stability,
  Bifurcation,
  Chaos,
  Soliton,
  IntegrableSystem,
  NumericalSimulation,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  SpectralMethod,
  MonteCarloMethod,
  NumericalStability,
  Convergence
 ]
 EXCLUDE_AS_PRIMARY = [
  ExperimentalMeasurementWithoutMathematicalModel,
  PureMathematicsWithoutPhysicalInterpretation,
  EngineeringDesignWithoutPhysicalTheorySemantics,
  DescriptivePhysicsWithoutFormalMathematicalRepresentation
 ]
 DISAMBIGUATION_RULE = "When physical entities, states, fields or laws are represented through formal mathematical structures and the mathematical structure is central to reasoning, interpret the context as the Mathematical Physics Ontology."
}

CORE_CONCEPTS {
 MathematicalPhysicsOntology = "Formal ontology of the mathematical structures used in physical theories."
 PhysicalSystem = "Defined physical entity or collection of entities subject to physical laws."
 PhysicalState = "Mathematical representation of a physical system at a specified condition or time."
 StateSpace = "Space containing admissible mathematical states of a physical system."
 PhysicalLaw = "Mathematical relationship governing physical quantities or evolution."
 Symmetry = "Transformation preserving specified physical structure."
 ConservationLaw = "Quantity remaining invariant under physical evolution."
 Operator = "Mathematical mapping acting on states, fields or observables."
 VariationalPrinciple = "Physical principle expressed through extremisation of a functional."
 Field = "Physical quantity assigned throughout space, spacetime or another domain."
 Spectrum = "Set of eigenvalues or spectral values associated with an operator."
}

RELATIONS {
 MathematicalPhysicsOntology --is_child_of--> MathematicalPhysics
 MathematicalPhysicsOntology --formalises--> PhysicalTheory
 PhysicalSystem --has--> PhysicalState
 PhysicalState --belongs_to--> StateSpace
 PhysicalLaw --governs--> PhysicalSystem
 Symmetry --supports--> ConservationLaw
 NoetherPrinciple --relates--> ContinuousSymmetryAndConservation
 LagrangianMechanics --uses--> VariationalPrinciple
 Lagrangian --defines--> Action
 EulerLagrangeEquation --follows_from--> ActionStationarity
 HamiltonianMechanics --uses--> PhaseSpace
 Hamiltonian --generates--> CanonicalEvolution
 QuantumMechanics --uses--> HilbertSpace
 Observable --is_represented_by--> SelfAdjointOperator
 Operator --has--> Spectrum
 Electromagnetism --uses--> MaxwellEquations
 GaugeTransformation --preserves--> GaugeEquivalentPhysics
 GeneralRelativity --uses--> MetricTensor
 MetricTensor --defines--> SpacetimeGeometry
 CurvatureTensor --describes--> SpacetimeCurvature
 StatisticalMechanics --uses--> Ensemble
 PartitionFunction --generates--> ThermodynamicQuantities
 FieldTheory --uses--> LagrangianDensity
 ContinuumMechanics --uses--> TensorField
 GreenFunction --represents--> LinearOperatorInverseResponse
 SpectralTheory --analyses--> Operator
 PerturbationTheory --approximates--> PhysicalSystem
 NumericalSimulation --approximates--> MathematicalPhysicsModel
}

OPERATORS {
 NEWTON_SECOND_LAW(m,x,t) := m*d2x/dt2=F(x,t)
 LAGRANGIAN(T,V) := L=T-V
 ACTION(L,t0,t1) := S=integral_t0^t1 L(q,qdot,t)dt
 EULER_LAGRANGE(L,q) := d/dt(partial L/partial qdot)-partial L/partial q=0
 HAMILTON_EQUATION_Q(H,p) := dq/dt=partial H/partial p
 HAMILTON_EQUATION_P(H,q) := dp/dt=-partial H/partial q
 SCHRODINGER(psi,H) := i*hbar*partial psi/partial t=H*psi
 COMMUTATOR(A,B) := [A,B]=A*B-B*A
 EXPECTATION_VALUE(psi,A) := <A>=<psi|A|psi>
 MAXWELL_GAUSS_E(E,rho) := div(E)=rho/epsilon_0
 MAXWELL_GAUSS_B(B) := div(B)=0
 MAXWELL_FARADAY(E,B) := curl(E)=-partial B/partial t
 MAXWELL_AMPERE(B,E,J) := curl(B)=mu_0*J+mu_0*epsilon_0*partial E/partial t
 GEODESIC(metric,x) := d2x^mu/dtau2+Gamma^mu_ab*dx^a/dtau*dx^b/dtau=0
 EINSTEIN_FIELD_EQUATION(Gmunu,Tmunu) := G_munu=8*pi*G*T_munu/c^4
 WAVE_EQUATION(u,c) := partial2u/partial t2=c^2*nabla^2u
 HEAT_EQUATION(u,alpha) := partial u/partial t=alpha*nabla^2u
 POISSON_EQUATION(phi,rho) := nabla^2 phi=-rho
 FOURIER_TRANSFORM(f,k) := F(k)=integral f(x)*exp(-i*k*x)dx
}

AXIOMS {
 A1 = "Mathematical-physics entities require explicit physical interpretation."
 A2 = "Physical equations must preserve dimensional consistency."
 A3 = "Initial and boundary conditions are part of a complete dynamical problem."
 A4 = "Symmetries constrain physically admissible laws and solutions."
 A5 = "State spaces and operator domains affect admissible physical states."
 A6 = "Approximation methods require explicit validity regimes."
 A7 = "Continuous, discrete, classical, quantum and relativistic models must not be conflated."
 A8 = "Mathematical-physics results require analytical, numerical and physical validation."
}

REASONING {
 R1_CONTEXT = identify(system,state_variables,parameters,units,geometry,interactions)
 R2_THEORY_CLASS = classify(classical,quantum,electromagnetic,relativistic,statistical,field,continuum)
 R3_STATE_SPACE = define(configuration_space,phase_space,Hilbert_space_or_field_space)
 R4_FORMULATION = construct(equations,operators,functionals,initial_conditions,boundary_conditions)
 R5_SYMMETRY = analyse(transformations,invariants,conservation_laws)
 R6_SOLUTION = solve_or_characterise(trajectories,fields,waves,eigenstates,equilibria)
 R7_APPROXIMATION = choose(perturbative,asymptotic,variational,spectral_or_numerical_method)
 R8_STABILITY = analyse(stability,resonance,bifurcation,chaos_or_dispersion)
 R9_SPECTRAL = analyse(operator_spectrum,eigenfunctions,modes)
 R10_VALIDATION = check(units,boundaries,symmetries,conservation,limits,numerical_convergence,physical_admissibility)
}

CONSTRAINTS {
 C1 = "A mathematically valid solution may be physically inadmissible."
 C2 = "Boundary conditions can alter spectra and physical modes."
 C3 = "Classical approximations can fail in quantum or relativistic regimes."
 C4 = "Linearisation can hide nonlinear branches or instabilities."
 C5 = "Perturbation theory can fail outside its expansion regime."
 C6 = "Coordinate singularities are not necessarily physical singularities."
 C7 = "Gauge-dependent quantities must not be treated as gauge-invariant observables."
 C8 = "Operator domains can change spectral conclusions."
 C9 = "Numerical discretisation can violate physical invariants."
 C10 = "Dimensional consistency alone does not prove physical correctness."
}

SEMANTIC_EQUIVALENCE {
 "MathematicalPhysicsOntology" <=> "CoreMathematicalPhysicsOntology"
 "ActionFunctional" <=> "Action"
 "Wavefunction" <=> "QuantumStateAmplitude"
 "SelfAdjointOperator" <=> "HermitianOperator"
 "GeneralisedMomentum" <=> "CanonicalMomentum"
}

VALIDATION {
 V1 = "Physical quantities and units are explicit."
 V2 = "State space is defined."
 V3 = "Initial and boundary conditions are explicit."
 V4 = "Dimensional consistency is verified."
 V5 = "Symmetries and conservation laws are checked."
 V6 = "Operator domains are documented where relevant."
 V7 = "Known limiting cases are recovered where applicable."
 V8 = "Approximation assumptions are documented."
 V9 = "Numerical convergence and stability are assessed."
 V10 = "Physical admissibility is checked."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MATHEMATICAL_PHYSICS_ONTOLOGY_DEFINITION = 0.99
 STATE_SPACE_REASONING = 0.99
 VARIATIONAL_REASONING = 0.99
 OPERATOR_REASONING = 0.99
 FIELD_REASONING = 0.99
 SPECTRAL_REASONING = 0.98
 SYMMETRY_REASONING = 0.98
 APPROXIMATION_REASONING = 0.98
 NUMERICAL_REASONING = 0.98
 POLICY = "Reduce confidence when units, state space, boundary conditions, geometry, operator domains, approximation regime, constitutive laws or numerical resolution are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  MathematicalPhysicsOntologyReasoning,
  PhysicalSystemReasoning,
  StateSpaceReasoning,
  ClassicalMechanicsReasoning,
  LagrangianReasoning,
  HamiltonianReasoning,
  QuantumMechanicsReasoning,
  OperatorReasoning,
  ElectromagnetismReasoning,
  RelativityReasoning,
  StatisticalMechanicsReasoning,
  FieldTheoryReasoning,
  ContinuumMechanicsReasoning,
  TensorReasoning,
  VariationalReasoning,
  SpectralReasoning,
  PerturbationReasoning,
  NumericalPhysicsReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "mathematical physics ontology" -> MathematicalPhysicsOntology
  "mathematical physics" -> MathematicalPhysics
  "action principle" -> VariationalPrinciple
  "Hamiltonian mechanics" -> HamiltonianMechanics
  "quantum state" -> QuantumState
  "Hermitian operator" -> SelfAdjointOperator
  "spectral theory" -> SpectralTheory
 }
}

EXECUTION_RULES {
 E1 = "If a physical system is represented using formal mathematical structures, activate MathematicalPhysicsOntology."
 E2 = "If configuration, force and trajectory are central, route to classical mechanics."
 E3 = "If action and generalised coordinates are central, route to Lagrangian mechanics."
 E4 = "If canonical position and momentum are central, route to Hamiltonian mechanics."
 E5 = "If states and observables are represented in Hilbert space, route to quantum mechanics."
 E6 = "If spacetime curvature is central, route to general relativity."
 E7 = "If many-body probability and macroscopic observables are central, route to statistical mechanics."
 E8 = "If fields and Lagrangian density are central, route to field theory."
 E9 = "If closed-form solution is unavailable, route to approximation or numerical methods with validation."
}

RECURSION {
 ENTRY = MathematicalPhysicsOntology
 EXPAND(MathematicalPhysicsOntology) -> [
  MathematicalPhysics, PhysicalSystem, PhysicalState, StateSpace,
  ConfigurationSpace, PhaseSpace, PhysicalQuantity, Observable, Parameter,
  Constant, Unit, Dimension, PhysicalLaw, DynamicalLaw, Symmetry, Invariance,
  ConservationLaw, NoetherPrinciple, ClassicalMechanics, NewtonianMechanics,
  LagrangianMechanics, HamiltonianMechanics, GeneralisedCoordinate,
  GeneralisedMomentum, Lagrangian, Hamiltonian, Action, VariationalPrinciple,
  EulerLagrangeEquation, CanonicalEquation, QuantumMechanics, QuantumState,
  Wavefunction, HilbertSpace, Operator, SelfAdjointOperator, Eigenvalue,
  Eigenstate, Spectrum, Commutator, UncertaintyRelation, Electromagnetism,
  ElectricField, MagneticField, ElectromagneticField, MaxwellEquations,
  ScalarPotential, VectorPotential, GaugeTransformation, Relativity,
  SpecialRelativity, GeneralRelativity, Spacetime, MetricTensor,
  ChristoffelSymbol, CovariantDerivative, CurvatureTensor, RicciTensor,
  EinsteinTensor, Geodesic, StatisticalMechanics, Thermodynamics, Microstate,
  Macrostate, Ensemble, PartitionFunction, Entropy, Temperature, FreeEnergy,
  FieldTheory, ClassicalFieldTheory, QuantumFieldTheory, GaugeTheory, Field,
  LagrangianDensity, ActionFunctional, EulerLagrangeFieldEquation,
  WaveEquation, HeatEquation, DiffusionEquation, LaplaceEquation,
  PoissonEquation, HelmholtzEquation, ContinuumMechanics, Elasticity,
  FluidMechanics, StressTensor, StrainTensor, TensorField, BoundaryCondition,
  InitialCondition, GreenFunction, FourierAnalysis, FourierTransform,
  LaplaceTransform, SpectralTheory, Eigenfunction, EigenfunctionExpansion,
  SturmLiouvilleProblem, PerturbationTheory, AsymptoticAnalysis,
  DimensionalAnalysis, ScalingLaw, NonlinearDynamics, Stability, Bifurcation,
  Chaos, Soliton, IntegrableSystem, NumericalSimulation,
  FiniteDifferenceMethod, FiniteElementMethod, SpectralMethod,
  MonteCarloMethod, NumericalStability, Convergence
 ]
 EXPAND(TheoryFamilies) -> [
  ClassicalMechanics,
  QuantumMechanics,
  Electromagnetism,
  Relativity,
  StatisticalMechanics,
  FieldTheory,
  ContinuumMechanics,
  ComputationalPhysics
 ]
 STOP_CONDITION = "Stop when physical system, state space, governing laws, mathematical operators, symmetries, boundary conditions, approximation regime and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Physics"
 IMMEDIATE_CONTEXTS = [
  "Mathematical Physics",
  "Differential Equation Ontologies",
  "Linear Algebra Ontologies",
  "Differential Geometry Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Mathematical Physics",
  "Applied Mathematics Ontologies",
  "Differential Equation Ontologies",
  "Partial Differential Equation Ontology",
  "Dynamical Systems Ontologies",
  "Linear Algebra Ontologies",
  "Tensor Analysis Ontologies",
  "Differential Geometry Ontologies",
  "Fourier Analysis Ontology",
  "Spectral Theory Ontologies",
  "Probability & Statistics Ontologies",
  "Numerical Analysis Ontologies",
  "Computational Mathematics Ontologies"
 ]
 ROUTE_RULES = [
  "Mathematical Physics Ontology -> Mathematical Physics for parent semantics",
  "Mathematical Physics Ontology -> Differential Equations for governing equations",
  "Mathematical Physics Ontology -> Dynamical Systems for evolution and stability",
  "Mathematical Physics Ontology -> Linear Algebra for operators and eigensystems",
  "Mathematical Physics Ontology -> Tensor Analysis for continuum and spacetime fields",
  "Mathematical Physics Ontology -> Differential Geometry for curved spaces",
  "Mathematical Physics Ontology -> Fourier Analysis for wave and mode decomposition",
  "Mathematical Physics Ontology -> Spectral Theory for operator spectra",
  "Mathematical Physics Ontology -> Probability & Statistics for statistical physical models",
  "Mathematical Physics Ontology -> Numerical Analysis and Computational Mathematics for simulation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible physical structures, theories or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve physical-state semantics.",
  "Preserve state-space distinctions.",
  "Preserve symmetry and conservation semantics.",
  "Preserve classical-versus-quantum distinctions.",
  "Preserve field and operator semantics.",
  "Preserve initial-and-boundary-condition semantics.",
  "Preserve approximation-regime semantics.",
  "Preserve machine-readable mathematical-physics ontology semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "The Mathematical Physics Ontology formally represents the mathematical structures used to describe physical systems and laws."
 STATE_SUMMARY = "Physical states inhabit mathematical state spaces such as configuration, phase or Hilbert spaces."
 DYNAMICS_SUMMARY = "Physical evolution is represented through differential equations, Hamiltonians, operators or field equations."
 SYMMETRY_SUMMARY = "Symmetries and invariances constrain physical laws and support conservation principles."
 QUANTUM_SUMMARY = "Quantum systems use Hilbert spaces, operators, spectra, commutators and state amplitudes."
 FIELD_SUMMARY = "Fields are represented over space or spacetime through differential and tensor equations."
 SPECTRAL_SUMMARY = "Operator spectra and eigenfunctions describe physical modes, states and resonances."
 VALIDATION_SUMMARY = "Mathematical-physics reasoning should be checked for units, boundary conditions, symmetries, conservation laws, limiting cases, convergence and physical admissibility."
}

FLOW {
 INPUT
 -> detect_mathematical_physics_ontology_context
 -> identify_physical_system_state_variables_units_geometry_interactions_and_scale
 -> determine_state_space_and_theory_family
 -> formulate_governing_laws_equations_operators_or_variational_functionals
 -> specify_initial_and_boundary_conditions
 -> analyse_symmetries_invariants_and_conservation_laws
 -> solve_or_characterise_states_trajectories_fields_waves_or_spectra
 -> if_needed_apply_perturbation_asymptotic_variational_spectral_or_numerical_methods
 -> analyse_stability_resonance_bifurcation_or_chaos_if_relevant
 -> validate_units_operator_domains_boundaries_symmetries_limits_convergence_and_physical_admissibility
 -> route_to_mathematical_physics_differential_equations_dynamical_systems_linear_algebra_tensor_analysis_differential_geometry_Fourier_analysis_spectral_theory_probability_statistics_numerical_analysis_or_computational_mathematics_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Mathematical Physics Ontology defines the core mathematical semantics of physical theories. It supports physical systems and states, configuration, phase and Hilbert spaces, physical quantities and observables, symmetries, invariance and conservation laws, Newtonian, Lagrangian and Hamiltonian mechanics, actions and Euler-Lagrange equations, quantum states, wavefunctions, operators, spectra and commutators, electromagnetic fields and Maxwell equations, special and general relativity, metric tensors, covariant derivatives, curvature and geodesics, statistical mechanics, ensembles and partition functions, thermodynamics, classical and quantum field theory, gauge theory, wave, heat, diffusion, Laplace, Poisson and Helmholtz equations, continuum mechanics, tensors, Green functions, Fourier and Laplace methods, spectral and Sturm-Liouville theory, perturbation and asymptotic analysis, nonlinear dynamics, chaos, solitons, numerical simulation, finite-difference, finite-element, spectral and Monte Carlo methods."
}

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

}

END_FLOW