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Their contributions to mathematical physics beyond quantum mechanics are then considered, and the focus will be on the influence that these contributions had on subsequent developments in quantum theorizing, particularly with regards to quantum field theory and its foundations.The entry quantum field theory provides an overview of a variety of approaches to developing a quantum theory of fields.
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.
There are two competing mathematical strategies that are used in connection with physical theory, one emphasizes rigor and the other pragmatics.
The pragmatic approach often compromises mathematical rigor, but offers instead expediency of calculation and elegance of expression.
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.The purpose of this article is to provide a more detailed discussion of mathematically rigorous approaches to quantum field theory, as opposed to conventional approaches, such as Lagrangian quantum field theory, which are generally portrayed as being more heuristic in character.The current debate concerning whether Lagrangian quantum field theory or axiomatic quantum field theory should serve as the basis for interpretive analysis is then discussed.In short, when pragmatics and rigor lead to the same conclusion, pragmatics trumps rigor due to the resulting simplicity, efficiency, and increase in understanding.As in the case of the notion of an infinitesimal, the Dirac \(\delta\) function was eventually given a mathematically rigorous foundation.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.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.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.