Gell-Mann-Low equation

The Gell-Mann-Low equation is a differential equation that underlies the concept of the renormalization group in quantum field theory . Interconnects the square of the seed charge $e_{0}^{2}$ ${\ displaystyle e_ {0} ^ {2}}$ ${\ displaystyle e_ {0} ^ {2}}$ trimming parameter $\Lambda$ ${\ displaystyle \ Lambda}$ $\ Lambda$ and function $d$ ${\ displaystyle d}$ $d$ , determining the ratio between the values of the square of the seed charge for two values of the cutoff parameter.

Definition

${\frac {de_{0}^{2}}{d\ln \xi }}{}=\psi (e_{0}^{2})$ ${\ displaystyle {\ frac {de_ {0} ^ {2}} {d \ ln \ xi}} {} = \ psi (e_ {0} ^ {2})}$ , here $e_{0}^{2}$ ${\ displaystyle e_ {0} ^ {2}}$ - squared seed charge, $\xi$ ${\ displaystyle \ xi}$ - value connecting two values of the square of the parameter of the cutoff $\Lambda _{1}^{2}=\xi$ ${\ displaystyle \ Lambda _ {1} ^ {2} = \ xi}$ and $\Lambda _{2}^{2}=\xi +d\xi$ ${\ displaystyle \ Lambda _ {2} ^ {2} = \ xi + d \ xi}$ , $\psi (e_{0}^{2})$ ${\ displaystyle \ psi (e_ {0} ^ {2})}$ - Gell-Mann-Low function $\psi (e_{0}^{2})=e_{0}^{2}{\Bigl [}{\frac {\partial d(e_{0}^{2},x)}{\partial x}}{\Bigr ]}_{x=1}$ ${\ displaystyle \ psi (e_ {0} ^ {2}) = e_ {0} ^ {2} {\ Bigl [} {\ frac {\ partial d (e_ {0} ^ {2}, x)}} { \ partial x}} {\ Bigr]} _ {x = 1}}$ function $d$ ${\ displaystyle d}$ determined from the equation $e_{0}^{2}(\Lambda _{2}^{2})=e_{0}^{2}(\Lambda _{1}^{2})d(e_{0}^{2}(\Lambda _{1}^{2}),{\frac {\Lambda _{2}^{2}}{\Lambda _{1}^{2}}})$ ${\ displaystyle e_ {0} ^ {2} (\ Lambda _ {2} ^ {2}) = e_ {0} ^ {2} (\ Lambda _ {1} ^ {2}) d (e_ {0} ^ {2} (\ Lambda _ {1} ^ {2}), {\ frac {\ Lambda _ {2} ^ {2}} {\ Lambda _ {1} ^ {2}}}}$ .

Literature

Sadovsky M. V. Lectures on quantum field theory. - Moscow-Izhevsk: Institute for Computer Studies, 2003. - p. 247. - ISBN 5-93972-241-5 .

Gell-Mann-Low equation

Definition

See also

Literature

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