Common integrals in quantum field theory: Difference between revisions

Revision as of 16:10, 16 February 2024

Revision as of 16:12, 16 February 2024

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which are found using the [[quadratic equation]]:

<math display="block">\begin{align}

\lambda_{\pm} &= \tfrac{1\over }{2} ( a+b) \pm \tfrac{1\over }{2} \sqrt{(a+b)^2-4(ab - c^2)}. \\

&= \tfrac{1\over }{2} ( a+b) \pm \tfrac{1\over }{2} \sqrt{a^2 +2ab + b^2 -4ab + 4c^2}. \\

&= \tfrac{1\over }{2} ( a+b) \pm \tfrac{1\over }{2} \sqrt{(a-b)^2+4c^2}.

\end{align}</math>

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The eigenvectors can be written as:

<math display="block">\begin{bmatrix} \fracdfrac{1}{\eta} \\ -\fracdfrac{a - \lambda_-}{c\eta} \end{bmatrix},
\qquad
\begin{bmatrix} -\fracdfrac{b - \lambda_+}{c\eta} \\ \fracdfrac{1}{\eta} \end{bmatrix} </math>

for the two eigenvectors. Here {{mvar|η}} is a normalizing factor given by,