Counting the Uncountable: A Narrative Review of Second Quantization and Feynman Diagrams, from Dirac's Oscillators to the Many-Body Problem

Abstract

Second quantization and its diagrams are field theory's working language: the occupation-number formalism—Dirac's 1927 quantized radiation, Jordan and Wigner's anticommuting fermions, Fock's space—that makes variable particle number natural, and the Feynman diagrams—the space-time pictures of 1949 whose algebra Dyson unified—that make perturbation theory drawable. This article presents a narrative review of the primary literature that built that language, from Dirac's 1927 emission and absorption quantization, Jordan and Wigner's 1928 Pauli equivalence, and Fock's 1932 configuration space, through the QED reconstruction of Tomonaga's 1946 invariant formulation, Schwinger's 1948 magnetic moment, Feynman's 1949 theory of positrons and space-time approach, and Dyson's 1949 unification, Feynman's 1962 lecture volume, Mattuck's 1976 guide to the many-body diagrams, Peskin and Schroeder's 1995 introduction, and Coleman's 2015 many-body synthesis. The synthesis is organized around three themes: the occupation-number settlement, in which variable particle number became formalism—bosonic commutators, fermionic anticommutators, Fock's space; the diagrammatic revolution, in which QED's infinities were organized into pictures with rules—and measured; and the many-body expansion, in which the same diagrams described solids, fluids, and superconductors. It is concluded that second quantization's genius is bookkeeping—particles as excitations—and that the diagrams are the bookkeeping's pictures: perturbation theory made visible, its rules the field's grammar.