Burgin, Mark Hypernumbers and extrafunctions. Extending the classical calculus. (English) Zbl 1253.46050 SpringerBriefs in Mathematics. New York, NY: Springer (ISBN 978-1-4419-9874-3/pbk; 978-1-4419-9875-0/ebook). xvi, 153 p. (2012). Classical physics would be impossible without the passage from rational to real numbers in the 19th century. The first rigorous mathematical concept of a real number is due to Georg Cantor. Until now mathematical models of physical theories are based on real numbers with no exception. Disappointingly experience with modern quantum field theory tells us that beyond finite real numbers infinities are often present in calculations and unavoidable. They were first encountered in quantum electrodynamics. Soon after the basic formulation of QED the presence of infinities was confirmed for instance in calculations of the self-energy of bound electrons and the vacuum energy. Most infinities arise from the high-energy part of Feynman integrals and are thus called ultraviolet divergencies. The struggle with infinities is present until today, though physicists try to eliminate them by a procedure called renormalization, either by dimensional regularization or by introducing counterterms into the Lagrangian. No doubt, on the mathematical side another passage is needed, i.e. from real numbers to hypernumbers in order to deal with infinite quantities, and in a next step to extend the classical calculus. Real functions ought to be replaced by extrafunctions.This is done in the present book by Mark Burgin, however without any attempt to apply the formalism to present-day physics. He has worked in this field since 1990 and has published many articles on the subject. The volume provides a survey over his work. The reader should be aware that the new theory cannot be identified with another subject known as nonstandard analysis. In contrast to nonstandard analysis, there exist no infinitely small hypernumbers. One of the benefits is that functional integrals, often called path integrals, now use hypermeasures. Another claim is that one finds solutions of differential equations for which it has been shown that they do not have ordinary solutions nor in the sense of distribution theory. Any real function can now be differentiated or integrated. Measurable finite physical quantities now appear as differences of two hypernumbers. This is easily demonstrated by a simple well-known example, the Casimir effect. Every physicist interested in mathematical physics will find Burgin’s approach inspiring in many respects. Reviewer: Gert Roepstorff (Aachen) Cited in 3 Reviews MSC: 46G12 Measures and integration on abstract linear spaces 46G05 Derivatives of functions in infinite-dimensional spaces 46F10 Operations with distributions and generalized functions 46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) 81T99 Quantum field theory; related classical field theories Keywords:hypernumbers; infinities in physics; renormalization Citations:Zbl 1118.46040 PDFBibTeX XMLCite \textit{M. Burgin}, Hypernumbers and extrafunctions. Extending the classical calculus. New York, NY: Springer (2012; Zbl 1253.46050) Full Text: DOI