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The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics.It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
Line integral, the integral of a function along a curve; Contour integral, the integral of a complex function along a curve used in complex analysis; Functional integration, the integral of a functional over a space of curves; Path integral formulation, Richard Feynman's formulation of quantum mechanics using functional integration
This expression actually defines the manner in which the path integrals are to be taken. The coefficient in front is needed to ensure that the expression has the correct dimensions, but it has no actual relevance in any physical application. This recovers the path integral formulation from Schrödinger's equation.
A common integral is a path integral of the form ((, ˙)) where (, ˙) is the classical action and the integral is over all possible paths that a particle may take. In the limit of small ℏ {\displaystyle \hbar } the integral can be evaluated in the stationary phase approximation .
The Feynman–Kac formula resulted, which proves rigorously the real-valued case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still an open question. [2] It offers a method of solving certain partial differential equations by simulating random paths of a stochastic process.
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. [1] The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral formalism.They are the imaginary time versions of statistical mechanics partition functions, giving rise to a close connection between these two areas of physics.
Functional integrals arise in probability, in the study of partial differential equations, and in the path integral approach to the quantum mechanics of particles and fields. In an ordinary integral (in the sense of Lebesgue integration ) there is a function to be integrated (the integrand) and a region of space over which to integrate the ...