cognitive/knowledge_base/mathematics/path_integral_theory.md

type	id	created	modified	tags	aliases
mathematical_concept	path_integral_theory_001	2024-02-05	2024-02-05	path-integrals statistical-mechanics field-theory active-inference	path-theory integral-formulation

Theoretical Foundations of Path Integrals

Mathematical Structure

Measure Theory Foundation

The path integral is defined on an infinite-dimensional measure_space:

\int \mathcal{D}[x(\tau)] \exp(-S[x(\tau)])

where:

• \mathcal{D}[x(\tau)] is the functional_measure
• S[x(\tau)] is the action_functional
• Links to wiener_measure and feynman_kac

Functional Analysis

class FunctionalSpace:
    """Represents space of paths."""
    def __init__(self, metric: Callable):
        self.metric = metric
        self.topology = self._induced_topology()
    
    def action_functional(self, path: Callable) -> float:
        """Compute action of path."""
        return integrate_action(self.lagrangian, path)

Links to:

• banach_spaces
• hilbert_spaces
• sobolev_spaces

Statistical Mechanics Connection

Partition Function

The partition_function in path integral form:

Z = \int \mathcal{D}[x(\tau)] \exp(-\beta H[x(\tau)])

where:

• \beta is inverse temperature (thermodynamic_beta)
• H is the hamiltonian_functional
• Links to free_energy_principle

Fluctuation Theory

def fluctuation_dissipation(response_function, correlation):
    """Implement fluctuation-dissipation theorem."""
    omega = frequency_grid()
    return beta * omega * imaginary(response_function) == \
           fourier_transform(correlation)

Links to:

• fluctuation_theorems
• dissipation_function
• response_theory

Field Theory Aspects

Generating Functionals

The generating_functional formalism:

Z[J] = \int \mathcal{D}[\phi] \exp(-S[\phi] + \int J\phi)

Connections to:

• field_theory
• correlation_functions
• ward_identities

Effective Action

class EffectiveAction:
    """Compute effective action via Legendre transform."""
    
    def legendre_transform(self, 
                          generating_functional: Callable,
                          source: np.ndarray) -> np.ndarray:
        """Perform Legendre transform to get effective action."""
        field = functional_derivative(generating_functional, source)
        return (source * field).sum() - generating_functional(source)

Links to:

• legendre_transform
• effective_field_theory
• renormalization_flow

Active Inference Extensions

Path-Space Free Energy

The path_space_free_energy combines path integrals with Active Inference:

F[q] = \int \mathcal{D}[x(\tau)] q[x(\tau)] \ln \frac{q[x(\tau)]}{p[x(\tau)]}

where:

• q[x(\tau)] is the variational_density
• p[x(\tau)] is the target_density
• Links to variational_calculus

Markov Blanket Dynamics

class MarkovBlanketDynamics:
    """Implement path integral dynamics with Markov blanket."""
    
    def blanket_decomposition(self, 
                            system_trajectory: np.ndarray) -> Dict:
        """Decompose trajectory into blanket components."""
        return {
            'internal': self.get_internal_dynamics(system_trajectory),
            'blanket': self.get_blanket_dynamics(system_trajectory),
            'external': self.get_external_dynamics(system_trajectory)
        }

Links to:

• markov_blanket_theory
• synchronization_dynamics
• information_geometry

Advanced Topics

Stochastic Processes

Connection to stochastic_differential_equations:

dx = f(x)dt + \sqrt{2D}dW

where:

• f(x) is the drift_field
• D is the diffusion_tensor
• dW is wiener_process

Critical Dynamics

class CriticalDynamics:
    """Analyze critical behavior in path integral systems."""
    
    def correlation_length(self, 
                         temperature: float,
                         critical_temp: float) -> float:
        """Compute correlation length near criticality."""
        reduced_temp = (temperature - critical_temp) / critical_temp
        return self.amplitude * np.power(abs(reduced_temp), -self.nu)

Links to:

• critical_phenomena
• scaling_theory
• universality

Geometric Phase

Analysis of geometric_phase effects:

\gamma = \oint A_\mu dx^\mu

where:

• A_\mu is the berry_connection
• Links to holonomy and fiber_bundles

Numerical Implementation

Path Sampling Methods

class PathSamplingMethods:
    """Implement various path sampling algorithms."""
    
    def metropolis_path_sampling(self,
                               action: Callable,
                               num_samples: int) -> List[np.ndarray]:
        """Metropolis sampling in path space."""
        paths = []
        current_path = self.initial_path()
        
        for _ in range(num_samples):
            proposed_path = self.perturb_path(current_path)
            acceptance_ratio = np.exp(action(current_path) - 
                                   action(proposed_path))
            
            if np.random.random() < acceptance_ratio:
                current_path = proposed_path
            paths.append(current_path.copy())
            
        return paths

Links to:

• mcmc_methods
• hybrid_monte_carlo
• langevin_dynamics

Discretization Schemes

class PathDiscretization:
    """Implement path discretization methods."""
    
    def discrete_action(self, 
                       path_points: np.ndarray,
                       dt: float) -> float:
        """Compute discretized action."""
        kinetic = self.discrete_kinetic(path_points, dt)
        potential = self.discrete_potential(path_points)
        return kinetic + potential

Links to:

• finite_difference
• symplectic_integrators
• variational_integrators

Applications

Quantum Systems

• quantum_mechanics
• quantum_field_theory
• quantum_statistics

Statistical Systems

• phase_transitions
• critical_dynamics
• non_equilibrium_processes

Active Matter

• collective_behavior
• self_organization
• pattern_formation

References

• feynman_1965 - Path Integral Formulation of Quantum Mechanics
• zinn_justin_2002 - Path Integrals in Quantum Field Theory
• kleinert_2009 - Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics
• friston_2019 - A Free Energy Principle for a Particular Physics