The Nobel Prize in Physics honors studies that reveal the basic laws of nature. Maxwell’s equations unified electricity and magnetism in the 19th century, but they are not sufficient for the tiny realm smaller than an atomic nucleus. Quantum electrodynamics (QED) describes the electromagnetic field as particles called photons and the interaction with electrons and positrons using quantum mechanics and relativity. Early calculations, however, produced many infinities and seemed to break the theory. Tomonaga, Schwinger and Feynman overcame this obstacle with a technique called renormalization, which extracts finite, measurable quantities. As a result they predicted, with spectacular accuracy, the electron’s magnetic strength (g-2) and the Lamb shift of hydrogen, both later confirmed experimentally. The mathematical framework and computational methods they built have become the standard for modern physics. QED indirectly supports the design of transistors, lasers and even the precise time control of GPS satellites. In ongoing research on quantum computers or the search for new particles, the notion of renormalization remains indispensable. Their achievements therefore continue to influence both science and society.

Quantum electrodynamics (QED) is a four-dimensional Abelian gauge theory that couples the Dirac electron field to a U(1) gauge field. When one performs a perturbative expansion with a covariant gauge fixing, ultraviolet divergences appear in self-energies and vertex functions. Tomonaga introduced the independent-parameter method, systematically organizing the divergences by defining the fields on spatial hypersurfaces and transforming the parameters. The Tomonaga-Schwinger equation, which he derived, is a covariant evolution equation for many-body systems, using a three-dimensional hypersurface as its time parameter. Schwinger employed the Green-function method based on functional variations of the action to derive n-point functions and ensured gauge invariance through the Ward-Takahashi identities. He was the first to calculate the one-loop electron anomalous magnetic moment a_e=α/2π, matching the experimental value within its uncertainty. Feynman rewrote the path integral as a time-ordered exponential and introduced the graphical bookkeeping device now called the Feynman diagram. This made it intuitive to handle phase-space integrals and symmetry factors even for high-loop corrections. The process of redefining the renormalization constants Z1, Z2 and Z3 in terms of observables is called renormalization, which renders the predictions finite. Renormalization-group equations describe how these redefinitions vary with the energy scale and were later applied to asymptotic freedom and critical phenomena. QED was extended to the renormalizable SU(2)L×U(1)Y gauge theory of the Standard Model and is indispensable for precision electroweak calculations. State-of-the-art QED computations now reach the four- and five-loop levels, pushing the theoretical uncertainty of a_e down to the 10^−13 level. Yet the 4.2σ discrepancy in the muon g-2 remains an important clue in the search for physics beyond the Standard Model. The same renormalization techniques are applied to quantum chromodynamics, to critical phenomena in condensed-matter systems, and even to statistical models such as the Ising model. Thus the 1965 Nobel Prize marks not merely the completion of one theory but the establishment of analytic methods for quantum field theory as a whole.

Tomonaga’s independent-parameter method regularized field operators on a spatial Cauchy hypersurface, eliminating the ambiguity of time ordering. By redefining the renormalization constants Z2 and Z3 on the mass shell, he illustrated how divergences can be absorbed while keeping external legs on-shell. Schwinger introduced functional differential operators and manipulated the vacuum-to-vacuum generator Z[J] directly, deriving the Ward-Takahashi identities non-perturbatively. In Phys. Rev. 82 (1951) 664-679 he discussed in detail the Lorentz structure of the photon self-energy and the scheme dependence of charge renormalization. Feynman’s path-integral formalism organized vertex combinatorics graphically and laid the ground for Cutkosky cuts and dispersion-relation analyses. Together their work provided the conceptual backbone for renormalization algorithms that later evolved into the BPHZ and dimensional-regularization MS schemes. The Tomonaga-Schwinger equation later underpinned the Schwinger-Keldysh formalism, widely used for covariant treatments of non-equilibrium fields. The positive sign of the QED4 β-function ensures perturbative control in the infrared and supports high-precision many-body electronic calculations. The Laporta algorithm for IBP reduction combined with complex-plane master integrals has enabled multi-loop (g-2) computations with uncertainties down to 10^−13. Vacuum-polarization tensor soft-hard separation is incorporated via NRQED into evaluations of the muonic-hydrogen Λ shift, improving bound-state QED. Tomonaga’s field re-phasing idea is mathematically isomorphic to bosonization in Luttinger liquids. Quantum-simulator experiments now emulate lattice QED in cold-atom systems, observing RG flows in real time. Schwinger-Dyson studies of dynamical mass generation in QED3 and QED4 are applied to high-Tc superconductors and graphene. Thus the 1965 achievements continue to furnish a technical platform for field theory, supporting developments from high-energy physics to condensed matter and mathematical QFT.