Geodesic
Combinatorics of the $R$ operation
Teoretičeskaâ i matematičeskaâ fizika, Tome 81 (1989) no. 3, pp. 336-353 Cet article a éte moissonné depuis la source Math-Net.Ru

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By using the functional language the new proof is given for the fundamental combinatorial statement in the renormalization theory [1], i. e. the application of $R$-operation to the diagrams of the initial theory is equivalent to the addition to the initial interaction $V(\varphi)$ the counterterms $\Delta V(\varphi)=-LH(\varphi)$, where $L$ defines $R=R(L)$ counter term operation on the diagrams such that the counter term $L\gamma$ corresponds with the graph $\gamma$, and $H(\varphi)$ is the $S$-matrix functional represented by the diagrams. (In the quantum field theory the operator of $S$-matrix is given by $T\exp V(\hat\varphi)=NH(\hat\varphi)$, where $T$ is a Wick chronological product, $N$ is a normal product, $\hat\varphi$ is a free field operator, $V(\hat\varphi) = iS_\mathrm{int}(\hat\varphi)$ is an interaction quantum operator.) The statement is proved for any $V$ and for an arbitrary operation $L$. The composite operators and the Wilson expansion are also considered. 
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     author = {A. N. Vasil'ev},
     title = {Combinatorics of~the $R$ operation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {336--353},
     year = {1989},
     volume = {81},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1989_81_3_a1/}
}
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A. N. Vasil'ev. Combinatorics of the $R$ operation. Teoretičeskaâ i matematičeskaâ fizika, Tome 81 (1989) no. 3, pp. 336-353. http://geodesic.mathdoc.fr/item/TMF_1989_81_3_a1/

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