mathematical objects

Path integrals are mathematical objects that can be considered as generalizations to an infinite number of variables, represented by paths, of usual integrals.

an integral where the function to be integrated is evaluated along a path or curve

In mathematics, a path integral is an integral where the function to be integrated is evaluated along a path or curve.

an analytic continuation of a method for summing up all possible random walks

The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks.

as being the sum of all paths through an infinite space–time

The path integrals are usually thought of as being the sum of all paths through an infinite space–time.

powerful tools for the study of quantum mechanics

Path integrals are powerful tools for the study of quantum mechanics.

the mathematical expression of a core quantum mechanical principle

Known as the path integral, the path integral is the mathematical expression of a core quantum mechanical principle: Anything that can happen does happen.

A line integral

A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve.

the generalization of the integral above to all quantum mechanical problems

The path integral is just the generalization of the integral above to all quantum mechanical problems

defines a probabilistic algorithm

The path integral defines a probabilistic algorithm to generate a Euclidean scalar field configuration.

a canonical vector-based navigation strategy used by a diverse array of both flying and walking animals, potentially including Drosophila

Path integration is a canonical vector-based navigation strategy used by a diverse array of both flying and walking animals, potentially including Drosophila.

a sum over all possible literal paths

a chain should now be clear that the path integral is simply a sum over all possible literal paths the polymer can form between two fixed endpoints.

a highly conserved, self-motion-based navigation strategy

Path integration (PI) is a highly conserved, self-motion-based navigation strategy.

a staple ingredient in quantum field theories

The path integral is a staple ingredient in quantum field theories, but they’re essentially impossible to evaluate exactly.

a mathematical formalization of Hughyen's principle

Path integrals are really a mathematical formalization of Hughyen's principle.

an integral (in the general sense that appears in topics like 'integration theory') over a pretty pathological looking infinite dimensional LCSC function space

But the path integral, formally (i.e., to mathematicians) is an integral (in the general sense that appears in topics like 'integration theory') over a pretty pathological looking infinite dimensional LCSC function space.

a probabilistic algorithm to generate a Euclidean scalar field configuration

The path integral defines a probabilistic algorithm to generate a Euclidean scalar field configuration.

The integral around

The integral around is called a path integral or line integral.

the natural formulation of the laws which rule the global and stochastic evolutions

Differential equations constitute the appropriate expressions of some deterministic and local physical laws, while the path integrals (belonging to Quantum Mechanics) are the natural formulation of the laws which rule the global and stochastic evolutions.

natural language for describing force carriers

The path integral formulation is the natural language for describing force carriers.

a functor from a cobordism category to C*-algebras, associating to each object of the cobordism category (i.e. each manifold) an operator-algebra for that specific space and to each morphism in the cobordism category (i.e. each cobordism) a morphism of operator-algebras that encodes time evolution

There, the path integral is a functor from a cobordism category to C*-algebras, associating to each object of the cobordism category (i.e. each manifold) an operator-algebra for that specific space and to each morphism in the cobordism category (i.e. each cobordism) a morphism of operator-algebras that encodes time evolution.