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 TMF, 1989, Volume 81, Number 3, Pages 336–353 (Mi tmf5377)

Combinatorics of the $R$ operation

A. N. Vasil'ev

Abstract: 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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English version:
Theoretical and Mathematical Physics, 1989, 81:3, 1244–1257

Bibliographic databases:

Citation: A. N. Vasil'ev, “Combinatorics of the $R$ operation”, TMF, 81:3 (1989), 336–353; Theoret. and Math. Phys., 81:3 (1989), 1244–1257

Citation in format AMSBIB
\Bibitem{Vas89} \by A.~N.~Vasil'ev \paper Combinatorics of~the $R$ operation \jour TMF \yr 1989 \vol 81 \issue 3 \pages 336--353 \mathnet{http://mi.mathnet.ru/tmf5377} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1035677} \transl \jour Theoret. and Math. Phys. \yr 1989 \vol 81 \issue 3 \pages 1244--1257 \crossref{https://doi.org/10.1007/BF01018954} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1989DP21300002} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. N. Vasil'ev, S. È. Derkachev, N. A. Kivel', A. S. Stepanenko, “Proof of conformal invariance in the critical regime for models of Gross–Neveu type”, Theoret. and Math. Phys., 92:3 (1992), 1047–1054
2. A. N. Vasil'ev, A. S. Stepanenko, “A method of calculating the critical dimensions of composite operators in the massless nonlinear $\sigma$ model”, Theoret. and Math. Phys., 95:1 (1993), 471–481
3. L. Ts. Adzhemyan, N. V. Antonov, T. L. Kim, “Composite operators, short–distance expansion and Galilean invariance in the theory of fully developed turbulence. Infrared corrections to the Kolmogorov's scaling”, Theoret. and Math. Phys., 100:3 (1994), 1086–1099
4. A. N. Vasil'ev, M. I. Vyazovskii, S. È. Derkachev, N. A. Kivel', “On equivalence of renormalizations for standard and dimensional regularizations of $2D$ four-fermion interactions”, Theoret. and Math. Phys., 107:1 (1996), 441–455
5. M. I. Vyazovskii, N. A. Kivel', “Analysis of chiral anomaly in dimensional regularization by means of projection technique”, Theoret. and Math. Phys., 109:3 (1996), 1536–1543
6. Smirnov, VA, “Applied asymptotic expansions in momenta and masses - Introduction”, Applied Asymptotic Expansions in Momenta and Masses, 177 (2002), 1
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