The field theory behind QED was so accurate and successful in predictions that efforts were made to apply the same basic concepts for the other forces of nature.
In particular, de Broglie in 1924 introduced the idea of a wave description of elementary systems in the following way: "we proceed in this work from the assumption of the existence of a certain periodic phenomenon of a yet to be determined character, which is to be attributed to each and every isolated energy parcel".
(Fermi's breakthrough was somewhat foreshadowed in the abstract studies of Soviet physicists, Viktor Ambartsumian and Dmitri Ivanenko, in particular the Ambarzumian–Ivanenko hypothesis of creation of massive particles (1930).
The Dirac equation accommodated the spin-1/2 value of the electron and accounted for its magnetic moment as well as giving accurate predictions for the spectra of hydrogen.
A subtle and careful analysis in 1933 by Niels Bohr and Léon Rosenfeld[12] showed that there is a fundamental limitation on the ability to simultaneously measure the electric and magnetic field strengths that enter into the description of charges in interaction with radiation, imposed by the uncertainty principle, which must apply to all canonically conjugate quantities.
Their analysis was crucial to showing that the limitations and physical implications of the uncertainty principle apply to all dynamical systems, whether fields or material particles.
The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease.
[18][19] In 1928, Jordan and Eugene Wigner found that the quantum field describing electrons, or other fermions, had to be expanded using anti-commuting creation and annihilation operators due to the Pauli exclusion principle (see Jordan–Wigner transformation).
What made the situation in the 1940s so desperate and gloomy, however, was the fact that the correct ingredients (the second-quantized Maxwell–Dirac field equations) for the theoretical description of interacting photons and electrons were well in place, and no major conceptual change was needed analogous to that which was necessitated by a finite and physically sensible account of the radiative behavior of hot objects, as provided by the Planck radiation law.
[23] While he was traveling by train from the conference to Schenectady he made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by Lamb and Retherford.
[40] Great progress was made after realizing that all infinities in quantum electrodynamics are related to two effects: the self-energy of the electron/positron, and vacuum polarization.
Renormalization requires paying very careful attention to just what is meant by, for example, the very concepts "charge" and "mass" as they occur in the pure, non-interacting field-equations.
These directly corresponded (through the Schwinger–Dyson equation) to the measurable physical processes (cross sections, probability amplitudes, decay widths and lifetimes of excited states) one needs to be able to calculate.
By 1965 James D. Bjorken and Sidney David Drell observed: "Quantum electrodynamics (QED) has achieved a status of peaceful coexistence with its divergences ...".
In the 1950s Yang and Mills, following the previous lead of Hermann Weyl, explored the impact of symmetries and invariances on field theory.
The weak interactions require the additional feature of spontaneous symmetry breaking, elucidated by Yoichiro Nambu and the adjunct Higgs mechanism, considered next.
The electroweak interaction part of the Standard Model was formulated by Sheldon Glashow, Abdus Salam, and John Clive Ward in 1959,[43][44] with their discovery of the SU(2)xU(1) group structure of the theory.
The Goldstone and Higgs idea for generating mass in gauge theories was sparked in the late 1950s and early 1960s when a number of theoreticians (including Yoichiro Nambu, Steven Weinberg, Jeffrey Goldstone, François Englert, Robert Brout, G. S. Guralnik, C. R. Hagen, Tom Kibble and Philip Warren Anderson) noticed a possibly useful analogy to the (spontaneous) breaking of the U(1) symmetry of electromagnetism in the formation of the BCS ground-state of a superconductor.
The electroweak theory of Weinberg and Salam was shown to be renormalizable (finite) and hence consistent by Gerardus 't Hooft and Martinus Veltman.
Gravitation is a tensor field description of a spin-2 gauge-boson, the "graviton", and is further discussed in the articles on general relativity and quantum gravity.
[46] There are technical problems underlain by the fact that the Newtonian constant of gravitation has dimensions involving inverse powers of mass, and, as a simple consequence, it is plagued by perturbatively badly behaved non-linear self-interactions.
Parallel breakthroughs in the understanding of phase transitions in condensed matter physics led to novel insights based on the renormalization group.
The gauge field theory of the strong interactions, quantum chromodynamics, relies crucially on this renormalization group for its distinguishing characteristic features, asymptotic freedom and color confinement.