Everything about Feynman Path Integral totally explained
» This article is about a formulation of quantum mechanics. For integrals along a path, also known as line or contour integrals, see line integral.
The
path integral formulation of
quantum mechanics is a description of quantum theory which generalizes the
action principle of
classical mechanics. It replaces the classical notion of a single, unique history for a system with a sum, or
functional integral, over an infinity of possible histories to compute a
quantum amplitude.
The path integral formulation was developed in 1948 by
Richard Feynman. Some preliminaries were worked out earlier, in the course of his
doctoral thesis work with
John Archibald Wheeler.
This formulation has proved crucial to the subsequent development of
theoretical physics, since it provided the basis for the grand synthesis of the 1970s called the
renormalization group which unified
quantum field theory with
statistical mechanics. If we realize that the
Schrödinger equation is essentially a
diffusion equation with an imaginary diffusion constant, then the path integral is a method for the enumeration of random walks. For this reason path integrals had also been used in the study of
Brownian motion and
diffusion before they were introduced in quantum mechanics.
Formulating quantum mechanics
The path integral method is an alternative formulation of
quantum mechanics. The canonical approach, pioneered by
Erwin Schrödinger,
Werner Heisenberg and
Paul Dirac paid great attention to
wave-particle duality and the resulting
uncertainty principle by replacing
Poisson brackets of
classical mechanics by
commutators between
operators in quantum mechanics. The
Hilbert space of
quantum states and the
superposition law of quantum amplitudes follows. The path integral starts from the superposition law, and exploits
wave-particle duality to build a generating function for quantum amplitudes.
Abstract formulation
Feynman proposed the following
postulates:
- The probability for any fundamental event is given by the square modulus of a complex amplitude.
- The amplitude for some event is given by adding together the contributions of all the histories which include that event.
- The amplitude a certain history contributes is proportional to
The path integral in quantum-mechanical interpretation
In one philosophical interpretation of quantum mechanics, the "sum over histories" interpretation, the path integral is taken to be fundamental and reality is viewed as a single indistinguishable "class" of paths which all share the same events. For this interpretation, it's crucial to understand what exactly an event is. The sum over histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin (see the reference below) claim the interpretation explains the Einstein-Podolsky-Rosen paradox without resorting to nonlocality.
Some advocates of interpretations of quantum mechanics emphasizing decoherence have attempted to make more rigorous the notion of extracting a classical-like "coarse-grained" history from the space of all possible histories.
References
Further Information
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