Quantum field theory

• 1. Introduction

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  Quantum field theory begins with a question that sounds almost philosophical: what is the universe actually made of? The answer it proposes is startling. Every electron, every photon, every particle you have ever heard of is not a tiny billiard ball floating through empty space. It is a ripple in an invisible field that pervades all of existence. Pull on that thread and the entire fabric of modern physics unravels before you.

  But QFT's story is not a smooth march of progress. It is a saga of brilliance colliding with catastrophe, of infinities threatening to swallow the entire enterprise, and of theorists who nearly abandoned the framework altogether before a handful of physicists found a way through. What drove them forward, what obstacles nearly stopped them, and what QFT still cannot do are the threads this documentary will follow.

• 2. Fields Before Quantum

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  Michael Faraday coined the English word "field" in 1845, and in doing so recast the way physicists think about forces. Before Faraday, the dominant picture was Isaac Newton's "action at a distance," in which gravity reaches across the void instantaneously. Newton himself was uneasy with this. In letters to Richard Bentley he wrote that it was "inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact." Faraday gave the "something else" a name and a shape: fields, properties filling space even when devoid of matter.

  The theory of classical electromagnetism was completed in 1864 through James Clerk Maxwell's equations, which described the relationship between electric fields, magnetic fields, electric current, and electric charge. Those equations implied that electric and magnetic fields propagate from point to point at a finite speed, which turned out to be the speed of light. Action at a distance was, as the source puts it, "conclusively refuted."

  Yet classical electromagnetism failed at the atomic level. It could not explain the discrete spectral lines of atoms, nor the distribution of blackbody radiation. Max Planck's study of blackbody radiation broke the deadlock. He treated atoms as tiny oscillators whose energies could only take discrete values, a process called quantization. Albert Einstein extended this idea in 1905, proposing that light itself consists of individual packets of energy called photons. Electromagnetic radiation was simultaneously a wave in a classical field and a stream of discrete particles. That paradox was the seed from which quantum field theory would eventually grow.

• 3. Building Quantum Electrodynamics

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  Paul Dirac's 1927 paper, titled "The quantum theory of the emission and absorption of radiation," introduced the term quantum electrodynamics and added an interaction term to the free electromagnetic field. Using first-order perturbation theory, Dirac explained spontaneous emission, the phenomenon in which an electron sheds energy and releases a photon without any external trigger. The explanation rested on quantum fluctuations: even in a perfect vacuum, the electromagnetic field retains a non-zero minimum energy and never fully settles, and it is this perpetual jitter that stimulates electrons to radiate.

  In 1928, Dirac wrote down the relativistic wave equation for electrons, now called the Dirac equation. It successfully produced the correct spin of the electron, the correct electron g-factor of 2, and the fine structure of hydrogen predicted by the Sommerfeld formula. Troublingly, it also implied the existence of negative energy states, which would have made atoms unstable. The resolution, worked out by Dirac and others by 1929, required postulating a particle with the same mass as the electron but opposite electric charge. This was the first theoretical proposal of antimatter. Carl David Anderson confirmed the existence of such a particle, the positron, in cosmic ray observations in 1932.

  Between 1928 and 1930, Jordan, Eugene Wigner, Heisenberg, Pauli, and Enrico Fermi arrived at a profound unification: material particles like electrons are not fundamentally different from photons. Both are excited states of underlying quantum fields. Given enough energy, particles could be created from these fields. Fermi used this idea in 1932 to explain beta decay, proposing that in the process of nuclear decay an electron is created out of the surrounding electron field, just as a photon is created from the electromagnetic field.

• 4. The War Against Infinities

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  Robert Oppenheimer demonstrated in 1930 that higher-order perturbative calculations in quantum electrodynamics always produced infinite quantities, such as the self-energy of the electron. The tools available at the time could not handle interactions involving photons with extremely high momenta. The problem was so severe that John Archibald Wheeler and Werner Heisenberg proposed in 1937 and 1943 respectively replacing QFT entirely with the so-called S-matrix theory, which ignored the microscopic details of interactions altogether. In 1945, Richard Feynman and Wheeler went further and suggested abandoning QFT entirely, proposing action at a distance as the mechanism of particle interactions.

  Ernst Stueckelberg worked in relative isolation between 1934 and 1938, publishing a relativistically invariant formulation of QFT and, by 1947, a complete renormalization procedure. The theoretical community largely failed to notice. The breakthrough that did register came in 1947, when Willis Lamb and Robert Retherford measured a minute difference in two energy levels of the hydrogen atom, a discrepancy now called the Lamb shift. Hans Bethe estimated its numerical value. Norman Myles Kroll, Lamb, James Bruce French, and Victor Weisskopf then confirmed it using a method in which infinities cancelled each other, though the method was clumsy and could not be generalized.

  The lasting solution arrived around 1950, built by Julian Schwinger, Richard Feynman, Freeman Dyson, and Shinichiro Tomonaga. Their procedure, renormalization, replaced the unobservable bare values of mass and charge, infinite as they might be, with their finite measured values. Tomonaga described the logic in his Nobel lecture: the mass and charge observed in experiments are not the original mass and charge but those modified by field reactions, so one substitutes experimental values for them. The renewed theory explained the electron's anomalous magnetic moment and vacuum polarization, with calculations matching experimental measurements to a remarkable degree. Feynman also introduced the path integral formulation of quantum mechanics and the Feynman diagram, a visual bookkeeping device for tracking the contributions of virtual particles to any interaction.

• 5. The Second Crisis

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  By 1949 the mood among theorists was euphoric. Freeman Dyson had proved that renormalization works for quantum electrodynamics, and optimism ran high that QFT would soon explain all microscopic phenomena. That optimism did not last. Dyson's 1949 proof also showed that renormalization is only possible for a small class of theories he called renormalizable. The Fermi theory of the weak interaction was not among them. Any perturbative calculation beyond the first order in such a non-renormalizable theory produced infinities that could not be removed by redefining any finite number of physical quantities.

  The strong interaction posed a separate problem. The coupling constant of QED, known as the fine-structure constant, is small enough that only the simplest Feynman diagrams matter in realistic calculations. The coupling constant in the strong interaction is roughly of order one. Complicated, higher-order Feynman diagrams are therefore just as important as simple ones, making the perturbative method useless for producing reliable predictions.

  Faced with this double impasse, many theorists turned away from QFT. Julian Schwinger, by contrast, spent more than a decade developing an alternative approach called source theory, which he summarized in 1966 and expanded across a three-volume set titled "Particles, Sources, and Fields." In source theory there are no divergences and no renormalization. Schwinger used it to calculate the anomalous magnetic moment of the electron without encountering any infinite quantities. He also applied it to a QFT description of gravity, reproducing all four of Einstein's classical results: gravitational redshift, the deflection of light by gravity, the slowing of light by gravity, and the perihelion precession of Mercury. The physics community's indifference to this work was, in Schwinger's own words, "depressing, but understandable."

• 6. The Standard Model Emerges

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  Yang Chen-Ning and Robert Mills generalized the local symmetry of QED in 1954, opening the door to a class of theories called non-Abelian gauge theories, in which the gauge bosons themselves carry charge, unlike the electrically neutral photon. Sheldon Glashow developed a non-Abelian gauge theory unifying electromagnetism and the weak interaction in 1960. Abdus Salam and John Clive Ward reached the same theory independently in 1964. The resulting electroweak theory was non-renormalizable, which made it difficult for other physicists to take it seriously.

  The barrier fell in 1971 when Gerard 't Hooft proved that non-Abelian gauge theories are renormalizable. Weinberg's 1967 electroweak theory was immediately revived. In 1970, Glashow, John Iliopoulos, and Luciano Maiani extended it from leptons to quarks, marking its completion. The mechanism that gives mass to the W and Z bosons without destroying gauge symmetry, spontaneous symmetry breaking via the Higgs boson, had been proposed in separate papers by Peter Higgs, Robert Brout, Francois Englert, Gerald Guralnik, Carl Hagen, and Tom Kibble.

  Harald Fritzsch, Murray Gell-Mann, and Heinrich Leutwyler showed in 1971 that certain strong-interaction phenomena could also be explained by non-Abelian gauge theory, giving birth to quantum chromodynamics, or QCD. The key insight came in 1973, when David Gross, Frank Wilczek, and Hugh David Politzer showed that QCD is asymptotically free: as interaction energy increases, the strong coupling constant decreases. At high energies, the coupling becomes small enough that perturbative methods work after all. The combined electroweak theory and QCD together form the Standard Model. Its last unconfirmed ingredient, the Higgs boson, was detected at CERN in 2012.

• 7. What QFT Still Cannot Do

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  Gravity remains the one fundamental interaction that QFT has not successfully absorbed. Attempts to quantize gravity produce a non-renormalizable theory, meaning infinities arise that cannot be tamed by redefining a finite number of parameters. One response was string theory, a type of two-dimensional QFT with conformal symmetry. Joël Scherk and John Schwarz first proposed in 1974 that string theory might serve as a quantum theory of gravity.

  Supersymmetry is another proposed extension, hypothesizing a symmetry that relates bosons and fermions. The first supersymmetric QFT in four dimensions was built by Yuri Golfand and Evgeny Likhtman in 1970, but their result was largely ignored at the time, partly due to the Iron Curtain. The work of Julius Wess and Bruno Zumino in 1973 brought supersymmetry to wider attention. If supersymmetry were exact, every fermion would have a bosonic superpartner and vice versa. It offers potential resolutions to the hierarchy problem, the grand unification of coupling constants, and the nature of dark matter. Experiments have not yet produced evidence for any supersymmetric particle.

  QFT also lacks a fully rigorous mathematical foundation. Haag's theorem shows that there does not exist a well-defined interaction picture for QFT, which means the perturbation theory underlying every Feynman diagram is fundamentally ill-defined in a strict mathematical sense. Since the 1950s, a subfield called constructive quantum field theory has sought to place QFT on axiomatic footing. One landmark open question in this program is the Yang-Mills existence and mass gap problem, listed among the Millennium Prize Problems: whether a non-trivial quantum Yang-Mills theory exists on four-dimensional space and possesses a mass gap.