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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.
Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. [ 1 ] : 13–15 Other integrals can be approximated by versions of the Gaussian integral.
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
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 ...
The path integral which now remains is (neglecting the corrections in the dots) of Gaussian form and can be integrated exactly: [] ...
In the path integral formulation of quantum field theory the following Gaussian integral of Grassmann quantities is needed for fermionic anticommuting fields, with A being an N × N matrix: ∫ exp [ − θ T A η ] d θ d η = det A {\displaystyle \int \exp \left[-\theta ^{\rm {T}}A\eta \right]\,d\theta \,d\eta =\det A} .
The time-varying Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution identity, + ′ = ′, which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator H , K t ( x ) = e − t H , {\displaystyle K_{t}(x)=e^{-tH}\,,} which is the infinitesimal diffusion ...
Since the theory is just Gaussian, no ultraviolet regularization or renormalization is needed. Therefore, the topological invariance of right hand side ensures that the result of the path integral will be a topological invariant. The only thing left to do is provide an overall normalization factor, and a natural choice will present itself.