Common integrals in quantum field theory: Difference between revisions

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

<math display="block">\int_{-\infty}^{\infty} \exp\left[ -{1 \over \hbar} \left( f\left( q_0 \right) + {1\over 2} \left( q-q_0\right)^2ff^{\prime \prime} \left( q-q_0\right) + \cdots \right ) \right] d^nq.</math>Here <math> f^{\prime \prime} </math> 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.