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A case in point is the notion of an infinitesimal, a non-zero quantity that is smaller than any finite quantity.Infinitesimals were used by Kepler, Galileo, Newton, Leibniz and many others in developing and using their respective physical theories, despite lacking a mathematically rigorous foundation, as Berkeley clearly showed in his famous 1734 treatise criticizing infinitesimals.
Such criticisms did not prevent various 18th Century mathematicians, scientists, and engineers such as Euler and Lagrange from using infinitesimals to get accurate answers from their calculations.
Nevertheless, the pull towards rigor led to the development in the 19th century of the concept of a limit by Cauchy and others, which provided a rigorous mathematical framework that effectively replaced the theory of infinitesimals.
The rigged Hilbert space was used to do so for quantum mechanics (Böhm 1966) and then for quantum field theory (Bogoluliubov et al. The complementary approaches, rigor and pragmatics, which are exhibited in the development of quantum mechanics, later came about in a more striking way in connection with the development of quantum electrodynamics (QED) and, more generally, quantum field theory (QFT).
The emphasis on rigor emerges in connection with two frameworks, algebraic QFT and Wightman’s axiomatic QFT.
A rigorous foundation was eventually provided for infinitesimals by Robinson during the second half of the 20th Century, but infinitesimals are rarely used in contemporary physics.
For more on the history of infinitesimals, see the entry on continuity and infinitesimals.That was done within Schwartz’s theory of distributions, which was later used in developing the notion of a rigged Hilbert space.The theory of distributions was used to provide a mathematical framework for quantum field theory (Wightman 1964).Nevertheless, it has been spectacularly successful in providing numerical results that are exceptionally accurate with respect to experimentally determined quantities, and in making possible expedient calculations that are unrivaled by other approaches.The two approaches to QFT continue to develop in parallel.Fleming (2002, 135–136) brings this into focus in his discussion of differences between Haag’s (1995); Haag’s book presents algebraic QFT, and Weinberg’s book presents Lagrangian QFT.While both books are ostensibly about the same subject, Haag gives a precise formulation of QFT and its mathematical structure, but does not provide any techniques for connecting with experimentally determined quantities, such as scattering cross sections.The emphasis on pragmatics arises most notably in Lagrangian QFT, which uses perturbation theory, path integrals, and renormalization techniques.Although some elements of the theory were eventually placed on a firmer mathematical foundation, there are still serious questions about its being a fully rigorous approach on a par with algebraic and Wightman’s axiomatic QFT.Von Neumann promotes an alternative framework, which he characterizes as being “just as clear and unified, but without mathematical objections.” He emphasizes that his framework is not merely a refinement of Dirac’s; rather, it is a radically different framework that is based on Hilbert’s theory of operators.Dirac is of course fully aware that the \(\delta\) function is not a well-defined expression. First, as long as one follows the rules governing the \(\delta\) function (such as using the \(\delta\) function only under an integral sign, meaning in part not asking the value of a \(\delta\) function at a given point), then no inconsistencies will arise.