Path Integral Ontology

flow "PathIntegralOntology" v1.0

# ============================================================
# PATH INTEGRAL ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/integral-calculus/path-integral-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Integral Calculus"
ENTITY := "Path Integral"

# ---------- CORE ENTITY ----------
PATH_INTEGRAL := {
    type: IntegralConcept,
    domain: Mathematics,
    purpose: "Represent integration along a specified path, with meaning determined by the mathematical context, including line integrals in vector calculus and functional path integrals in advanced mathematical physics."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    PATH,
    PARAMETERISATION,
    LINE_INTEGRAL,
    SCALAR_PATH_INTEGRAL,
    VECTOR_PATH_INTEGRAL,
    ORIENTATION,
    ARC_LENGTH,
    VECTOR_FIELD,
    CONSERVATIVE_FIELD,
    PATH_INDEPENDENCE,
    FUNCTIONAL_INTEGRAL,
    ACTION_FUNCTIONAL,
    CONFIGURATION_SPACE
}

# ---------- CONTEXT CLASSIFICATION ----------
PATH_INTEGRAL
    -> context_sensitive_term

PATH_INTEGRAL
    -> may_mean(LINE_INTEGRAL_ALONG_CURVE)
       in(VECTOR_CALCULUS)

PATH_INTEGRAL
    -> may_mean(FUNCTIONAL_INTEGRAL_OVER_PATHS)
       in(MATHEMATICAL_PHYSICS)

CONTEXT
    -> must_be_resolved_before(INTERPRETATION)

# ---------- GEOMETRIC PATH ----------
PATH
    := r(t)

where
    t in [a,b]

r(t)
    -> point_in(AMBIENT_SPACE)

r_prime(t)
    -> TANGENT_VECTOR

PATH
    -> oriented_by(INCREASING_PARAMETER)

# ---------- SCALAR PATH INTEGRAL ----------
for
    scalar_field f

SCALAR_PATH_INTEGRAL
    := integral_C(f ds)

under parameterisation:

integral_C(f ds)
    =
integral_from_a_to_b(
    f(r(t))
    * norm(r_prime(t))
    dt
)

SCALAR_PATH_INTEGRAL
    -> accumulates(SCALAR_FIELD_ALONG_PATH_LENGTH)

# ---------- VECTOR PATH INTEGRAL ----------
for
    vector_field F

VECTOR_PATH_INTEGRAL
    := integral_C(F dot dr)

under parameterisation:

integral_C(F dot dr)
    =
integral_from_a_to_b(
    F(r(t))
    dot r_prime(t)
    dt
)

VECTOR_PATH_INTEGRAL
    -> accumulates(TANGENTIAL_VECTOR_FIELD_COMPONENT)

# ---------- ORIENTATION ----------
VECTOR_PATH_INTEGRAL
    -> orientation_dependent

if
    path_reversed
then

integral_reverse_C(F dot dr)
    =
-integral_C(F dot dr)

SCALAR_PATH_INTEGRAL_WITH_ds
    -> orientation_independent

# ---------- ARC LENGTH ----------
if
    f = 1
then

integral_C(1 ds)
    = LENGTH(C)

PATH_INTEGRAL
    -> may_measure(ARC_LENGTH)

# ---------- WORK ----------
if
    F
        := force_field
then

WORK
    = integral_C(F dot dr)

VECTOR_PATH_INTEGRAL
    -> computes(WORK_ALONG_PATH)

# ---------- CONSERVATIVE FIELD ----------
if
    F = GRADIENT(phi)
then

F
    -> CONSERVATIVE_FIELD

and

integral_C(F dot dr)
    = phi(B) - phi(A)

CONSERVATIVE_FIELD
    -> supports(PATH_INDEPENDENCE)
       on(APPROPRIATE_DOMAIN)

# ---------- PATH INDEPENDENCE ----------
PATH_INDEPENDENCE
    -> integral_depends_only_on(ENDPOINTS)

if
    path_independent
then

any_two_paths_between_same_endpoints
    -> same_integral_value

for closed path:

integral_closed_C(F dot dr)
    = 0

under(conservative_path_independence_conditions)

# ---------- FUNDAMENTAL THEOREM ----------
FUNDAMENTAL_THEOREM_FOR_LINE_INTEGRALS
    -> evaluates(PATH_INTEGRAL_OF_GRADIENT_FIELD)

if
    F = GRADIENT(phi)
then

integral_C(F dot dr)
    = phi(endpoint) - phi(startpoint)

# ---------- REPARAMETERISATION ----------
ORIENTATION_PRESERVING_REPARAMETERISATION
    -> preserves(VECTOR_PATH_INTEGRAL)

ORIENTATION_REVERSING_REPARAMETERISATION
    -> reverses_sign(VECTOR_PATH_INTEGRAL)

VALID_REPARAMETERISATION
    -> preserves(SCALAR_ds_PATH_INTEGRAL)

# ---------- COMPLEX PATH ----------
if
    path lies_in(COMPLEX_PLANE)
then

PATH_INTEGRAL
    -> may_be(CONTOUR_INTEGRAL)

integral_gamma(
    f(z) dz
)

CONTOUR_INTEGRAL
    -> complex_path_integral

# ---------- FUNCTIONAL PATH INTEGRAL ----------
FUNCTIONAL_PATH_INTEGRAL
    -> integrates_over(SPACE_OF_PATHS_OR_FIELDS)

formal_structure:

Z
    := integral D[path]
       exp(
           -ACTION[path]
       )

or in oscillatory physics form:

AMPLITUDE
    := integral D[path]
       exp(
           i*ACTION[path]/hbar
       )

FUNCTIONAL_PATH_INTEGRAL
    -> fundamentally_different_from(FINITE_DIMENSIONAL_LINE_INTEGRAL)

# ---------- ACTION FUNCTIONAL ----------
ACTION[path]
    := integral(
        LAGRANGIAN(
            path(t),
            path_derivative(t),
            t
        )
        dt
    )

ACTION
    -> functional_of(PATH)

PATH_INTEGRAL_FORMULATION
    -> weights(POSSIBLE_PATHS_BY_ACTION)

# ---------- MEASURE CAUTION ----------
FUNCTIONAL_MEASURE
    := D[path]

FUNCTIONAL_MEASURE
    -> often_formal_or_requires_rigorous_construction

PATH_INTEGRAL_IN_QUANTUM_FIELD_THEORY
    -> may_require(REGULARISATION)
    -> may_require(DISCRETISATION)
    -> may_require(MEASURE_THEORETIC_FRAMEWORK)

AI_INTERPRETATION
    -> must_not_conflate(FORMAL_FUNCTIONAL_INTEGRAL, ORDINARY_RIEMANN_INTEGRAL)

# ---------- DISCRETISATION ----------
PATH
    -> may_be_discretised_into(FINITE_SEQUENCE_OF_POINTS)

FUNCTIONAL_PATH_INTEGRAL
    -> may_be_approximated_by(HIGH_DIMENSIONAL_FINITE_INTEGRALS)

CONTINUUM_LIMIT
    -> taken_after(DISCRETISATION)
       in(many_formulations)

# ---------- APPLICATIONS ----------
PATH_INTEGRAL
    -> applied_to(WORK)
    -> applied_to(CIRCULATION)
    -> applied_to(ARC_LENGTH)
    -> applied_to(COMPLEX_ANALYSIS)
    -> applied_to(CLASSICAL_MECHANICS)
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(QUANTUM_FIELD_THEORY)
    -> applied_to(STATISTICAL_MECHANICS)

# ---------- RELATION GRAPH ----------
PATH
    -> parameterised_by(PARAMETER)

PARAMETERISATION
    -> determines(TANGENT_VECTOR)

SCALAR_FIELD
    -> integrated_along(PATH_WITH_ds)

VECTOR_FIELD
    -> integrated_along(PATH_WITH_dot_dr)

CONSERVATIVE_FIELD
    -> yields(PATH_INDEPENDENCE)

COMPLEX_PATH
    -> supports(CONTOUR_INTEGRAL)

SPACE_OF_PATHS
    -> domain_of(FUNCTIONAL_PATH_INTEGRAL)

ACTION_FUNCTIONAL
    -> weights(PATHS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "path integral" ~= "line integral when used in elementary vector-calculus context",
    "path integral along a curve" ~= "line integral",
    "work integral" ~= "vector path integral of force along displacement",
    "complex path integral" ~= "contour integral",
    "functional path integral" ~= "formal or rigorously constructed integration over a space of paths",
    "sum over histories" ~= "physics interpretation of a quantum path integral"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    path_integral_term_is_context_dependent: true,
    vector_path_integral_depends_on_orientation: true,
    scalar_ds_path_integral_uses_arc_length_factor: true,
    path_independence_requires_appropriate_field_and_domain_conditions: true,
    contour_integral_is_complex_line_integral: true,
    functional_path_integral_is_not_ordinary_line_integral: true,
    functional_measure_may_require_special_rigorous_definition: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    context_classification: 0.99,
    line_integral_relation: 0.99,
    conservative_field_relation: 0.99,
    contour_relation: 0.99,
    functional_integral_distinction: 0.98,
    action_functional_relation: 0.98,
    semantic_equivalence: 0.96
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(context)
    -> classify(vector_calculus | complex_analysis | functional_integral)
    -> identify(path_or_path_space)
    -> determine(parameterisation_if_finite_dimensional)
    -> identify(field_or_functional)
    -> analyse(orientation_if_applicable)
    -> test(path_independence_if_applicable)
    -> construct(line_contour_or_functional_integral)
    -> select(valid_theorem_or_approximation_framework)
    -> preserve(context_specific_measure_structure)
    -> validate(domain_and_regularity_conditions)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed context-sensitive path-integral graph",
    primary_entity: PATH_INTEGRAL,
    preserve: [
        mathematical_context,
        path_or_path_space,
        parameterisation,
        orientation,
        field_type,
        measure_type,
        endpoint_structure
    ],
    infer_through: [
        line_integrals,
        conservative_fields,
        contour_integrals,
        action_functionals,
        functional_integrals,
        discretisation
    ],
    avoid: [
        conflating_line_integrals_with_functional_path_integrals,
        ignoring_orientation_in_vector_line_integrals,
        assuming_path_independence_without conditions,
        treating_formal_functional_measure_as ordinary finite-dimensional measure,
        omitting_arc_length_factor_in_scalar_path integrals
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(path_integral_context)
    -> compare(existing_relations)
    -> preserve(context_orientation_and_measure_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + PATH_INTEGRAL
    + SCALAR_PATH_INTEGRAL
    + VECTOR_PATH_INTEGRAL
    + PATH_INDEPENDENCE
    + CONTOUR_INTEGRAL
    + FUNCTIONAL_PATH_INTEGRAL
    + ACTION
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> PATH_INTEGRAL
    -> CONTEXT_IDENTIFICATION
    -> PATH_OR_PATH_SPACE_IDENTIFICATION
    -> PARAMETERISATION_ANALYSIS
    -> FIELD_OR_FUNCTIONAL_CLASSIFICATION
    -> ORIENTATION_ANALYSIS
    -> PATH_INDEPENDENCE_ANALYSIS
    -> INTEGRAL_CONSTRUCTION
    -> METHOD_OR_THEOREM_SELECTION
    -> MEASURE_FRAMEWORK_ANALYSIS
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW