Mathematical Physics

AI INTERPRETATION NOTE
This Flow Algebra defines the Mathematical Physics layer within the Kireth Research Semantic Library. Mathematical physics applies rigorous mathematical structures and methods to physical theories, systems and laws. This parent ontology connects classical mechanics, quantum mechanics, electromagnetism, relativity, statistical mechanics, thermodynamics, field theory, wave mathematics, continuum mechanics, spectral theory, differential geometry, tensor analysis, variational methods, partial differential equations, symmetry, conservation laws and computational physics.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = MathematicalPhysics
 ENTITY_CLASS = MathematicalPhysicalSystemsFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Mathematical physics is the branch of applied mathematics concerned with formulating, analysing and solving physical theories and systems using rigorous mathematical structures."
 SYMBOLIC_FORMS = [
  "F=ma",
  "i*hbar*partial psi/partial t=H psi",
  "partial_mu F^(mu nu)=mu_0 J^nu",
  "G_(mu nu)=8*pi*G*T_(mu nu)/c^4"
 ]
 SEMANTIC_ROLE = "Provides the parent semantic framework for mathematically formulated physical laws, fields, particles, waves, continua, symmetries, conservation principles, operators, variational systems and physical PDEs."
}

SCOPE {
 INCLUDE = [
  MathematicalPhysics, PhysicalSystem, PhysicalState, PhysicalLaw, Observable,
  Measurement, Parameter, Constant, Symmetry, ConservationLaw, Invariance,
  ClassicalMechanics, NewtonianMechanics, LagrangianMechanics,
  HamiltonianMechanics, AnalyticalMechanics, QuantumMechanics, QuantumState,
  Wavefunction, Operator, HilbertSpace, Eigenvalue, Eigenstate, Commutator,
  Electromagnetism, ElectricField, MagneticField, MaxwellEquations,
  Relativity, SpecialRelativity, GeneralRelativity, Spacetime, MetricTensor,
  Curvature, StatisticalMechanics, Thermodynamics, PartitionFunction,
  Entropy, Temperature, FieldTheory, ClassicalFieldTheory, QuantumFieldTheory,
  GaugeTheory, WaveEquation, DiffusionEquation, HeatEquation,
  HelmholtzEquation, LaplaceEquation, PoissonEquation, ContinuumMechanics,
  Elasticity, FluidMechanics, StressTensor, StrainTensor, TensorAnalysis,
  DifferentialGeometry, Manifold, CoordinateSystem, CovariantDerivative,
  Geodesic, VariationalPrinciple, ActionFunctional, EulerLagrangeEquation,
  HamiltonPrinciple, PartialDifferentialEquation, BoundaryCondition,
  InitialCondition, GreenFunction, FourierAnalysis, FourierTransform,
  LaplaceTransform, SpectralTheory, EigenfunctionExpansion,
  SturmLiouvilleTheory, PerturbationTheory, AsymptoticAnalysis,
  DimensionalAnalysis, Scaling, NonlinearPhysics, Chaos, Soliton,
  IntegrableSystem, ComputationalPhysics, NumericalSimulation,
  FiniteDifferenceMethod, FiniteElementMethod, SpectralMethod, MonteCarloMethod
 ]
 EXCLUDE_AS_PRIMARY = [
  ExperimentalPhysicsWithoutMathematicalModel,
  PureMathematicsWithoutPhysicalInterpretation,
  EngineeringDesignWithoutPhysicalTheoryFocus,
  DescriptivePhysicsWithoutFormalMathematicalStructure
 ]
 DISAMBIGUATION_RULE = "When physical laws, states, fields or observables are represented and analysed primarily through formal mathematical structures, interpret the context as mathematical physics."
}

CORE_CONCEPTS {
 MathematicalPhysics = "Mathematical formulation and analysis of physical theories."
 PhysicalSystem = "Defined collection of physical entities, states and interactions."
 PhysicalState = "Mathematical representation of a system configuration or condition."
 PhysicalLaw = "Mathematical rule describing relationships among physical quantities."
 Symmetry = "Transformation leaving specified physical structure or laws invariant."
 ConservationLaw = "Invariant physical quantity associated with system dynamics."
 Field = "Physical quantity defined over space, spacetime or another domain."
 Operator = "Mathematical mapping acting on states, functions or observables."
 VariationalPrinciple = "Principle determining physical evolution through extremisation of a functional."
 SpectralTheory = "Study of eigenvalues and eigenfunctions of physical operators."
}

RELATIONS {
 MathematicalPhysics --is_branch_of--> AppliedMathematics
 MathematicalPhysics --formalises--> PhysicalTheory
 PhysicalSystem --has--> PhysicalState
 PhysicalLaw --governs--> PhysicalSystem
 Symmetry --implies_or_supports--> ConservationLaw
 ClassicalMechanics --uses--> DifferentialEquations
 LagrangianMechanics --uses--> VariationalPrinciple
 HamiltonianMechanics --uses--> SymplecticStructure
 QuantumMechanics --uses--> HilbertSpace
 QuantumObservable --is_represented_by--> Operator
 Electromagnetism --uses--> MaxwellEquations
 GeneralRelativity --uses--> DifferentialGeometry
 StatisticalMechanics --uses--> Probability
 Thermodynamics --relates_to--> StatisticalMechanics
 FieldTheory --uses--> VariationalPrinciple
 WaveEquation --is_type_of--> PartialDifferentialEquation
 ContinuumMechanics --uses--> TensorAnalysis
 SpectralTheory --supports--> QuantumMechanics
 FourierAnalysis --supports--> WaveAndFieldAnalysis
 GreenFunction --solves--> LinearDifferentialOperatorProblem
 PerturbationTheory --approximates--> PhysicalSystem
 ComputationalPhysics --numerically_solves--> 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
 HAMILTONIAN(p,q,L) := H=p*qdot-L
 SCHRODINGER(psi,H) := i*hbar*partial psi/partial t=H*psi
 COMMUTATOR(A,B) := [A,B]=A*B-B*A
 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
 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 requires explicit physical interpretation of variables and parameters."
 A2 = "Physical equations must preserve dimensional consistency."
 A3 = "Boundary and initial conditions are part of a complete physical model."
 A4 = "Symmetries and invariances constrain admissible physical laws."
 A5 = "Continuous and discrete physical models must not be conflated."
 A6 = "Approximation methods require explicit asymptotic or perturbative assumptions."
 A7 = "Operator domains and boundary conditions affect spectra and physical predictions."
 A8 = "Mathematical-physics results require physical, analytical and numerical validation."
}

REASONING {
 R1_CONTEXT = identify(physical_system,state_variables,parameters,geometry,forces_or_fields)
 R2_THEORY = classify(classical,quantum,electromagnetic,relativistic,statistical,field,continuum)
 R3_FORMULATION = construct(equations,operators,functionals,initial_conditions,boundary_conditions)
 R4_SYMMETRY = analyse(invariance,conservation_laws,generators)
 R5_ANALYSIS = solve_or_characterise(equilibria,waves,spectra,fields,trajectories)
 R6_APPROXIMATION = select(perturbation,asymptotic,spectral,variational_or_numerical_method)
 R7_STABILITY = analyse(stability,resonance,bifurcation,chaos_or_dispersion)
 R8_SPECTRAL = compute(eigenvalues,eigenfunctions,modes)
 R9_NUMERICAL = solve(discretised_model,convergence,error,stability)
 R10_VALIDATION = check(dimensions,limits,symmetries,conservation,boundaries,physical_consistency)
}

CONSTRAINTS {
 C1 = "Mathematical solutions are physically admissible only when they satisfy the model's physical constraints."
 C2 = "Boundary conditions can alter spectra and qualitative behaviour."
 C3 = "Classical approximations can fail at quantum, relativistic or strongly nonlinear scales."
 C4 = "Perturbation expansions can fail outside their asymptotic regime."
 C5 = "Coordinate singularities must not automatically be interpreted as physical singularities."
 C6 = "Gauge-dependent quantities must not be confused with gauge-invariant observables."
 C7 = "Numerical discretisation can violate conservation or symmetry if poorly constructed."
 C8 = "Dimensional consistency is necessary but not sufficient for physical correctness."
 C9 = "Linearisation can hide nonlinear instability and multiple solution branches."
 C10 = "Agreement with one limiting case does not fully validate a physical theory."
}

SEMANTIC_EQUIVALENCE {
 "MathematicalPhysics" <=> "MathematicsOfPhysics"
 "AnalyticalMechanics" <=> "ClassicalAnalyticalMechanics"
 "Wavefunction" <=> "QuantumStateAmplitude"
 "ActionFunctional" <=> "Action"
}

VALIDATION {
 V1 = "Physical quantities and units are explicit."
 V2 = "Initial and boundary conditions are explicit."
 V3 = "Dimensional consistency is verified."
 V4 = "Symmetry and conservation properties are checked."
 V5 = "Known limiting cases are recovered where applicable."
 V6 = "Operator domains are documented for spectral problems."
 V7 = "Approximation assumptions are explicit."
 V8 = "Numerical convergence and stability are assessed."
 V9 = "Physical admissibility constraints are enforced."
 V10 = "Results are compared with established analytical or empirical behaviour where possible."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MATHEMATICAL_PHYSICS_DEFINITION = 0.99
 THEORY_CLASSIFICATION_REASONING = 0.99
 VARIATIONAL_REASONING = 0.99
 PDE_REASONING = 0.99
 SPECTRAL_REASONING = 0.98
 SYMMETRY_REASONING = 0.98
 APPROXIMATION_REASONING = 0.98
 NUMERICAL_REASONING = 0.98
 POLICY = "Reduce confidence when physical scale, units, boundary conditions, geometry, approximation regime, operator domain, constitutive law or numerical resolution are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, MathematicalPhysicsReasoning,
  ClassicalMechanicsReasoning, LagrangianReasoning, HamiltonianReasoning,
  QuantumMechanicsReasoning, ElectromagnetismReasoning, RelativityReasoning,
  StatisticalMechanicsReasoning, ThermodynamicsReasoning, FieldTheoryReasoning,
  WaveEquationReasoning, ContinuumMechanicsReasoning, TensorAnalysisReasoning,
  DifferentialGeometryReasoning, VariationalReasoning, SpectralTheoryReasoning,
  PerturbationReasoning, ComputationalPhysicsReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "mathematical physics" -> MathematicalPhysics
  "mathematics of physics" -> MathematicalPhysics
  "Lagrangian mechanics" -> LagrangianMechanics
  "Hamiltonian mechanics" -> HamiltonianMechanics
  "quantum mechanics" -> QuantumMechanics
  "field theory" -> FieldTheory
  "spectral theory" -> SpectralTheory
  "variational principle" -> VariationalPrinciple
 }
}

EXECUTION_RULES {
 E1 = "If a physical system is represented through formal mathematics, activate MathematicalPhysics."
 E2 = "If trajectories and forces dominate, route to classical mechanics."
 E3 = "If action principles or canonical variables dominate, route to Lagrangian or Hamiltonian mechanics."
 E4 = "If state vectors and operators dominate, route to quantum mechanics."
 E5 = "If fields and spacetime equations dominate, route to electromagnetism, relativity or field theory."
 E6 = "If many-particle statistics dominate, route to statistical mechanics."
 E7 = "If continua, stress or flow dominate, route to continuum mechanics."
 E8 = "If eigenvalues or modes dominate, route to spectral theory."
 E9 = "If closed-form analysis is unavailable, route to computational physics with convergence validation."
}

RECURSION {
 ENTRY = MathematicalPhysics
 EXPAND(MathematicalPhysics) -> [
  PhysicalSystem, PhysicalState, PhysicalLaw, Observable, Measurement,
  Parameter, Constant, Symmetry, ConservationLaw, Invariance,
  ClassicalMechanics, NewtonianMechanics, LagrangianMechanics,
  HamiltonianMechanics, AnalyticalMechanics, QuantumMechanics,
  QuantumState, Wavefunction, Operator, HilbertSpace, Eigenvalue, Eigenstate,
  Commutator, Electromagnetism, ElectricField, MagneticField,
  MaxwellEquations, Relativity, SpecialRelativity, GeneralRelativity,
  Spacetime, MetricTensor, Curvature, StatisticalMechanics, Thermodynamics,
  PartitionFunction, Entropy, Temperature, FieldTheory, ClassicalFieldTheory,
  QuantumFieldTheory, GaugeTheory, WaveEquation, DiffusionEquation,
  HeatEquation, HelmholtzEquation, LaplaceEquation, PoissonEquation,
  ContinuumMechanics, Elasticity, FluidMechanics, StressTensor, StrainTensor,
  TensorAnalysis, DifferentialGeometry, Manifold, CoordinateSystem,
  CovariantDerivative, Geodesic, VariationalPrinciple, ActionFunctional,
  EulerLagrangeEquation, HamiltonPrinciple, PartialDifferentialEquation,
  BoundaryCondition, InitialCondition, GreenFunction, FourierAnalysis,
  FourierTransform, LaplaceTransform, SpectralTheory, EigenfunctionExpansion,
  SturmLiouvilleTheory, PerturbationTheory, AsymptoticAnalysis,
  DimensionalAnalysis, Scaling, NonlinearPhysics, Chaos, Soliton,
  IntegrableSystem, ComputationalPhysics, NumericalSimulation,
  FiniteDifferenceMethod, FiniteElementMethod, SpectralMethod,
  MonteCarloMethod
 ]
 EXPAND(TheoryFamilies) -> [
  ClassicalMechanics,
  QuantumMechanics,
  Electromagnetism,
  Relativity,
  StatisticalMechanics,
  FieldTheory,
  ContinuumMechanics,
  ComputationalPhysics
 ]
 STOP_CONDITION = "Stop when physical system, governing mathematics, geometry, initial and boundary conditions, symmetries, approximation regime and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"
 RELATED_ONTOLOGIES = [
  "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 -> Applied Mathematics for parent semantics",
  "Mathematical Physics -> Differential Equations for governing physical equations",
  "Mathematical Physics -> Dynamical Systems for stability and nonlinear evolution",
  "Mathematical Physics -> Linear Algebra for operators, states and eigensystems",
  "Mathematical Physics -> Tensor Analysis for fields, continua and relativity",
  "Mathematical Physics -> Differential Geometry for curved spaces and spacetime",
  "Mathematical Physics -> Fourier Analysis for waves and spectral decomposition",
  "Mathematical Physics -> Spectral Theory for operator eigenstructure",
  "Mathematical Physics -> Probability & Statistics for statistical mechanics and quantum probability",
  "Mathematical Physics -> Numerical Analysis and Computational Mathematics for simulation"
 ]
}

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

OUTPUT {
 CANONICAL_DEFINITION = "Mathematical physics formulates and analyses physical theories using rigorous mathematical structures."
 MECHANICS_SUMMARY = "Classical, Lagrangian and Hamiltonian mechanics describe physical motion through forces, actions and canonical variables."
 QUANTUM_SUMMARY = "Quantum mechanics represents physical states in Hilbert spaces and observables as operators."
 FIELD_SUMMARY = "Electromagnetism, relativity and field theory represent physical fields over space or spacetime."
 STATISTICAL_SUMMARY = "Statistical mechanics links microscopic states to macroscopic thermodynamic behaviour using probability."
 CONTINUUM_SUMMARY = "Continuum mechanics models distributed matter through fields such as stress, strain, velocity and density."
 SPECTRAL_SUMMARY = "Spectral theory analyses physical modes through eigenvalues and eigenfunctions of operators."
 VALIDATION_SUMMARY = "Mathematical-physics models should be checked for dimensions, boundary conditions, symmetries, limits, conservation, numerical convergence and physical admissibility."
}

FLOW {
 INPUT
 -> detect_mathematical_physics_context
 -> identify_physical_system_state_variables_parameters_units_geometry_and_interactions
 -> classify_classical_quantum_electromagnetic_relativistic_statistical_field_or_continuum_theory
 -> formulate_governing_equations_operators_functionals_initial_conditions_and_boundary_conditions
 -> analyse_symmetries_invariants_and_conservation_laws
 -> solve_or_characterise_trajectories_fields_waves_spectra_or_equilibria
 -> if_exact_solution_unavailable_select_perturbation_asymptotic_variational_spectral_or_numerical_method
 -> assess_stability_resonance_dispersion_bifurcation_or_chaos_if_relevant
 -> validate_dimensions_limits_symmetries_boundary_conditions_conservation_and_physical_admissibility
 -> route_to_applied_mathematics_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 layer defines rigorous mathematical formulation of physical systems. It supports classical, Newtonian, Lagrangian and Hamiltonian mechanics, quantum states, operators and Hilbert spaces, electromagnetism and Maxwell equations, special and general relativity, spacetime geometry, statistical mechanics and thermodynamics, classical and quantum field theory, wave, heat, diffusion, Helmholtz, Laplace and Poisson equations, continuum mechanics, elasticity and fluid mechanics, stress and strain tensors, differential geometry, variational principles and action functionals, Green functions, Fourier and Laplace transforms, spectral and Sturm-Liouville theory, perturbation and asymptotic methods, dimensional analysis, nonlinear physics, chaos, solitons, integrable systems and computational physics."
}

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

}

END_FLOW