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.

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

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