[Paper Review] The Euler Legacy to Modern Physics

G. Dattoli, M. Franco|arXiv (Cornell University)|Sep 20, 2010

Quantum Electrodynamics and Casimir Effect11 references3 citations

TL;DR

This paper explores the foundational role of Euler's mathematical innovations—particularly the gamma and zeta functions—in shaping modern theoretical physics. It demonstrates how Euler’s work on divergent series and special functions underpinned key developments in quantum field theory and string theory, with modern physicists like Feynman and Schwinger later rediscovering and applying these 'mathematical curiosities' to solve fundamental physical problems.

ABSTRACT

Particular families of special functions, conceived as purely mathematical devices between the end of XVIII and the beginning of XIX centuries, have played a crucial role in the development of many aspects of modern Physics. This is indeed the case of the Euler gamma function, which has been one of the key elements paving the way to string theories, furthermore the Euler-Riemann Zeta function has played a decisive role in the development of renormalization theories. The ideas of Euler and later those of Riemann, Ramanujan and of other, less popular, mathematicians have therefore provided the mathematical apparatus ideally suited to explore, and eventually solve, problems of fundamental importance in modern Physics. The mathematical foundations of the theory of renormalization trace back to the work on divergent series by Euler and by mathematicians of two centuries ago. Feynman, Dyson, Schwinger... rediscovered most of these mathematical "curiosities" and were able to develop a new and powerful way of looking at physical phenomena.

Motivation & Objective

• To trace the historical and mathematical lineage from Euler’s work to modern theoretical physics.
• To highlight the critical role of special functions—especially the gamma and Riemann zeta functions—in advancing quantum field theory and string theory.
• To demonstrate that foundational ideas in renormalization theory originated in Euler’s investigations of divergent series.
• To illustrate how 18th- and 19th-century mathematics provided the essential tools for solving contemporary problems in high-energy physics.
• To emphasize the rediscovery of Euler’s insights by 20th-century physicists such as Feynman, Dyson, and Schwinger as pivotal to modern physics.

Proposed method

• Analyzes historical mathematical developments by Euler, Riemann, and Ramanujan, focusing on the gamma and zeta functions.
• Examines the role of divergent series in Euler’s work and their relevance to renormalization in quantum field theory.
• Traces the conceptual evolution from Euler’s formal manipulations of infinite series to modern regularization techniques.
• Connects the analytic continuation of the Riemann zeta function to physical applications in string theory and quantum field theory.
• Uses historical and mathematical analysis to show the continuity between classical special functions and modern theoretical physics.
• Highlights the conceptual bridge between abstract mathematics and physical theory through case studies in renormalization and string theory.

Experimental results

Research questions

• RQ1 How did Euler’s work on the gamma and zeta functions contribute to the development of modern quantum field theory?
• RQ2 In what ways did Euler’s treatment of divergent series anticipate modern renormalization techniques?
• RQ3 What is the significance of the Riemann zeta function in the context of string theory and quantum field theory?
• RQ4 How did later physicists like Feynman and Schwinger rediscover and apply Euler’s mathematical insights?
• RQ5 To what extent can the foundations of modern theoretical physics be traced back to 18th-century mathematical innovations?

Key findings

• Euler’s gamma function played a crucial role in the development of string theory, particularly in the formulation of the Veneziano amplitude.
• The Euler-Riemann zeta function provided the mathematical framework essential for renormalization in quantum field theory.
• Euler’s early work on divergent series anticipated modern regularization and renormalization procedures used in quantum electrodynamics.
• The analytic continuation of the zeta function, originally explored by Euler and later formalized by Riemann, underpins critical calculations in string theory.
• Feynman, Dyson, and Schwinger independently rediscovered and applied Euler’s mathematical tools, leading to the formulation of modern quantum field theory.
• The mathematical apparatus developed by Euler and his contemporaries proved ideally suited for addressing fundamental problems in modern physics, despite being initially considered purely theoretical or abstract.