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Vector Fields, Flows and Lie Groups of Diffeomorphisms | | A. Peterman
; | | Date: |
15 Dec 1999 | | Journal: | Eur.Phys.J. C14 (2000) 705-708 | | Subject: | hep-th | | Abstract: | The freedom in choosing finite renormalizations in quantum field theories (QFT) is characterized by a set of parameters ${c_i }, i = 1 ..., n >...$, which specify the renormalization prescriptions used for the calculation of physical quantities. For the sake of simplicity, the case of a single $c$ is selected and chosen mass-independent if masslessness is not realized, this with the aim of expressing the effect of an infinitesimal change in $c$ on the computed observables. This change is found to be expressible in terms of an equation involving a vector field $V$ on the action’s space $M$ (coordinates x). This equation is often referred to as ``evolution equation’’ in physics. This vector field generates a one-parameter (here $c$) group of diffeomorphisms on $M$. Its flow $sigma_c (x)$ can indeed be shown to satisfy the functional equation $$ sigma_{c+t} (x) = sigma_c (sigma_t (x)) equiv sigma_c circ sigma_t $$ $$sigma_0 (x) = x,$$ so that the very appearance of $V$ in the evolution equation implies at once the Gell-Mann-Low functional equation. The latter appears therefore as a trivial consequence of the existence of a vector field on the action’s space of renormalized QFT. | | Source: | arXiv, hep-th/9912131 | | Services: | Forum | Review | PDF | Favorites | |
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