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Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory. (English) Zbl 1263.81245
During the years 1981-83 Igor Batalin and Grigori Vilkovisky published four articles on gauge algebra and the quantization of gauge theories avoiding the use of gauge fixing. The new formalism was immediately seen as an extension of the widely known BRST approach to the renormalization of gauge theories with linearly dependent generators and thus was the starting point of further research and many articles in mathematical physics. The BV formalism, as it was called, now uses an extended notion of the renormalized time-ordered product of fields. In purely mathematical terms, the BV algebra is a graded supercommutative algebra with a unit and an operator $$\Delta$$ satisfying $$\Delta^2=0$$. In a previous paper, Fredenhagen and Rejzner applied the BV formalism to the functional approach to classical field theory. In the present paper the authors extend the concept of a renormalized time-ordered product and show the existence of a quantum sub-algebra closed with respect to time-ordering. Surprisingly, the standard path integral (often Euclidean) approach and regularizations are completely avoided here, though Batalin and Vilkovisky proposed a regularization scheme to deal with the divergencies. Hence, the most important results of the present paper concern the novel approach to renormalization. The standard time-ordered product is replaced by a renormalized product and so the algebraic structure is determined by two products, the star product of the BV algebra and the new version of the time-ordered product. Another important result concerns the use of the Master Ward Identity due to Brennecke and Dütsch allowing the so-called Quantum Master Equation and the quantum BV operators being written in a more detailed and explicit form. The proposed formalism can be applied the Yang-Mills model and General Relativity Theory, hopefully solving the problem of quantum gravity in the future.

##### MSC:
 81T15 Perturbative methods of renormalization applied to problems in quantum field theory 81T08 Constructive quantum field theory 81T13 Yang-Mills and other gauge theories in quantum field theory 81T75 Noncommutative geometry methods in quantum field theory
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