Common Integrals in Quantum Field Theory - Integrals That Can Be Approximated By The Method of Steepest Descent | Integrals Approximated Method Steepest Descent

Integrals That Can Be Approximated By The Method of Steepest Descent

In quantum field theory n-dimensional integrals of the form

appear often. Here is the reduced Planck's constant and f is a function with a positive minimum at . These integrals can be approximated by the method of steepest descent.

For small values of Planck's constant, f can be expanded about its minimum

Here is the n by n matrix of second derivatives evaluated at the minimum of the function.

If we neglect higher order terms this integral can be integrated explicitly.

$\int_{-\infty}^{\infty} \exp\left d^nq \approx \exp\left \sqrt{ (2 \pi \hbar )^n \over \det f^{\prime \prime} }$ .

Read more about this topic: Common Integrals In Quantum Field Theory

Famous quotes containing the words method and/or descent:

“Argument is conclusive ... but ... it does not remove doubt, so that the mind may rest in the sure knowledge of the truth, unless it finds it by the method of experiment.... For if any man who never saw fire proved by satisfactory arguments that fire burns ... his hearer’s mind would never be satisfied, nor would he avoid the fire until he put his hand in it ... that he might learn by experiment what argument taught.”
—Roger Bacon (c. 1214–1294)

“In the world of the celebrity, the hierarchy of publicity has replaced the hierarchy of descent and even of great wealth.”
—C. Wright Mills (1916–1962)