On the Hopf algebra strucutre of perturbative quantum field theories.

*(English)*Zbl 1041.81087From the summary: We want to demonstrate that renormalization is not an ad hoc procedure, but on the contrary arises in a very natural manner from the properties of (quasi-)Hopf algebras. Our point of departure will be a closer look at the forest formula. This formula is an operation which is concerned with subgraphs \(\gamma_i\in F\Gamma\), and with replacing the graph \(\Gamma\) by a sum of expressions of the form \(R[Z[\gamma_i]\Gamma/\gamma_i]\), where \(\gamma_i\) is a proper subgraph and \(\Gamma/\gamma_i\) is an expression where in the big graph \(\Gamma\) we shrink \(\gamma_i\) to a point. So, renormalization is concerned with replacing a graph \(\Gamma\) by products of graphs, and summation over all possible subgraphs. Both, \(\overline\Gamma\) and the corresponding counterterm \(Z[\overline \Gamma]\) have this form. Their difference is finite.

This finiteness means that we can establish an equivalence relation \[ A\sim B \Leftrightarrow\lim_{h\to 0}\Bigl(\lim_{\varepsilon\to 0}[A-B]\Bigr)=0. \tag{1} \] Thus, we consider expressions which have the same pole part in the regularization parameter \(\varepsilon\), but different finite terms, as equivalent. We are interested in the renormalization procedure per se, and not in the finite differences between various renormalization schemes.

Now consider an equation taken from the theory of Hopf algebras. \[ m \bigl[(S\otimes id)\Delta[X]\bigr]\sim E\circ\overline [X]=0.\tag{2} \] It is our aim to bring the whole theory of renormalization in a form such that

– there exists a counit \(\overline e\) which annihilates Feynman graphs,

– the coproduct \(\Delta\) generates all the terms necessary to compensate the subdivergences,

– in the set \({\mathcal A}\) furnished by all Feynman diagrams we identify graphs without subdivergences as the primitive elements in the Hopf algebra,

– the antipode \(S[R[X]]\) coincides with the counterterm \(Z_{[X]}\) for a given graph \(X\),

– renormalization schemes correspond to maps \(R:{\mathcal A}\to{\mathcal A}\) which fulfil \(R[X]\sim X\),

– \(m[(S\otimes id)\Delta[X]]\) is the finite term \((\sim 0)\) which corresponds to the renormalized Feynman graph.

This finiteness means that we can establish an equivalence relation \[ A\sim B \Leftrightarrow\lim_{h\to 0}\Bigl(\lim_{\varepsilon\to 0}[A-B]\Bigr)=0. \tag{1} \] Thus, we consider expressions which have the same pole part in the regularization parameter \(\varepsilon\), but different finite terms, as equivalent. We are interested in the renormalization procedure per se, and not in the finite differences between various renormalization schemes.

Now consider an equation taken from the theory of Hopf algebras. \[ m \bigl[(S\otimes id)\Delta[X]\bigr]\sim E\circ\overline [X]=0.\tag{2} \] It is our aim to bring the whole theory of renormalization in a form such that

– there exists a counit \(\overline e\) which annihilates Feynman graphs,

– the coproduct \(\Delta\) generates all the terms necessary to compensate the subdivergences,

– in the set \({\mathcal A}\) furnished by all Feynman diagrams we identify graphs without subdivergences as the primitive elements in the Hopf algebra,

– the antipode \(S[R[X]]\) coincides with the counterterm \(Z_{[X]}\) for a given graph \(X\),

– renormalization schemes correspond to maps \(R:{\mathcal A}\to{\mathcal A}\) which fulfil \(R[X]\sim X\),

– \(m[(S\otimes id)\Delta[X]]\) is the finite term \((\sim 0)\) which corresponds to the renormalized Feynman graph.

##### MSC:

81T17 | Renormalization group methods applied to problems in quantum field theory |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |