Vladimir Fock

Vladimir Aleksandrovich Fock was born in St. Petersburg on December 22, 1898, into a family of foresters and engineers in the last years of imperial Russia. He came of age in a city that would, before he finished his studies, be renamed twice (Petrograd in 1914, Leningrad in 1924) and would live through war, revolution, civil war, and famine. He served briefly as an artillery officer in the First World War, then returned to the university in Petrograd to study mathematical physics. He graduated in 1922, three years into the new Soviet state, and stayed where he had been born to become one of the few quantum theorists who would build an entire scientific life inside the USSR.

The Leningrad physics community in the 1920s was small but extraordinarily lively. Abram Ioffe ran the Physico-Technical Institute (the “Phystech”), the seedbed of nearly every important Soviet physicist of the next generation. Fock joined the orbit immediately, splitting his time between the Phystech, the Vavilov State Optical Institute, and Leningrad University, where he was made a professor in 1932. Unlike his European contemporaries, he never had the option of moving between Göttingen, Copenhagen, Munich, and Cambridge to soak up the new mechanics in real time. He had to read the papers as they crossed the border, often months late, and rebuild the arguments himself with the small group of theorists around him. That discipline (working out the foundations on his own rather than receiving them from a teacher) shaped his lifelong style: every result of Fock’s reads as if it were derived from scratch.

Vladimir Aleksandrovich Fock (or Fok; ) (December 22, 1898 – December 27, 1974) was a Soviet physicist, who did foundational work on quantum mechanics and quantum electrodynamics.

His first major contribution came in 1926, the same year Schrödinger published the wave equation. Fock, working independently and almost simultaneously with Oskar Klein and Walter Gordon, wrote down what we now call the Klein-Gordon equation, the first relativistic wave equation for a scalar particle. The equation has a complicated history (Schrödinger had derived it earlier and rejected it because it gave the wrong fine structure for hydrogen, then settled for his non-relativistic equation), but Fock arrived at it from the Soviet side of the iron curtain with no contact with the others. It is the simplest relativistic generalization of Schrödinger’s equation, and it sits at the foundation of every modern quantum field theory of bosons. In a fairer naming convention it would carry his name too.

Then came 1930, and the work that fixed his place in the quantum story for good. Douglas Hartree in Cambridge had proposed a self-consistent-field method for many-electron atoms: guess a wavefunction for each electron, compute the average potential they all create, solve for new wavefunctions in that potential, and iterate until nothing changes. The method worked. It gave usable atomic structure calculations for the first time. But it had a flaw. Hartree treated each electron as moving in the mean field of the others as if the electrons were distinguishable, ignoring the Pauli exclusion principle that Wolfgang Pauli had introduced five years earlier. Fock saw what was missing. He rebuilt Hartree’s method with the wavefunction written as an antisymmetrized product, a Slater determinant (named for John Slater, who arrived at the same construction the same year), so that the exchange of any two electrons flipped the sign of the total wavefunction, as it must for fermions. The result is the Hartree-Fock method, the single most-used numerical procedure in computational chemistry for the next ninety years and counting.

The Hartree-Fock equations look intimidating at first, but the idea is simple. Hartree said: each electron sees the average cloud of the others. Fock said: yes, but the electron also feels a quantum interference effect from the indistinguishability of identical fermions, which lowers the energy whenever two electrons of the same spin avoid each other. That extra term is the exchange interaction. It is what makes lithium’s outer electron sit higher than a classical mean-field calculation would predict. It is what gives ferromagnetism its sign. It is the reason chemistry is what it is. Every density-functional theory code today carries Fock’s exchange term either explicitly (in hybrid functionals) or as the thing the functional is trying to approximate.

While doing all of this, Fock invented a piece of mathematics so useful that physicists now use it without remembering whom it came from. To handle systems with variable numbers of particles, like the photon field, he constructed the direct sum of zero-particle, one-particle, two-particle, three-particle, and higher Hilbert spaces, with creation and annihilation operators connecting the floors of this infinite hotel. We call it Fock space. Its basis states (the Fock states with definite particle number) are the natural language of quantum field theory. When you read about photon number states in quantum optics, or particle number conservation in a Bose-Einstein condensate, or about n-photon laser states in a quantum computing paper, you are using the framework Fock built in his Leningrad office in the late 1920s and early 1930s.

The 1930s were a terrible time to be a Soviet intellectual. Stalin’s Great Purge ran from 1936 to 1938; physicists were not spared. Matvei Bronstein, a brilliant young quantum gravity theorist at the Phystech and a close friend of Lev Landau’s, was shot in 1938. Lev Landau himself was arrested in 1938 and spent a year in the Lubyanka. Vavilov, the great geneticist (brother of the physicist Sergei Vavilov, who ran the optical institute where Fock worked), died in a Saratov prison in 1943. Fock was arrested too, twice. The first time, in 1935, was brief; the second, in 1937, was longer and more dangerous. What saved him, by his own account and by the account of his friends, was a combination of stubbornness and the timely intervention of Pyotr Kapitsa, the physicist who had returned from Cambridge and whose direct letters to Stalin and to Molotov pulled several colleagues out of the camps. Fock walked back into his Leningrad office and continued working as if nothing had happened. Then, in 1941, came the German invasion. The siege of Leningrad would kill roughly a million people. Fock was evacuated to Kazan with most of the academy and lived out the war there, returning in 1944 to a city that had been transformed by starvation and shelling. He never left again.

His primary scientific contribution lies in the development of quantum physics and the theory of gravitation, although he also contributed significantly to the fields of mechanics, theoretical optics, and physics of continuous media. In 1926, he derived the Klein–Gordon equation. He gave his name to Fock space, the Fock representation and Fock state, and developed the Hartree–Fock method in 1930. He made many subsequent scientific contributions during the rest of his life. Fock developed…

The second half of his career turned increasingly toward general relativity. Fock had been one of relativity’s earliest and most enthusiastic Soviet exponents at a time when official Marxist-Leninist philosophy was suspicious of Einstein’s theory as “idealistic.” He argued, in books and papers and lecture halls, that relativity was in fact compatible with dialectical materialism, that spacetime was an objectively real entity rather than a frame-dependent construction, and that the theory described the genuine physics of gravitation, not just a coordinate convention. He paid for this stance with criticism from the philosophers but he held the line, and through the 1950s and 1960s his book The Theory of Space, Time and Gravitation (1955) was the standard Russian-language relativity text and a major influence on the Soviet generation that produced Yakov Zel’dovich and Andrei Sakharov.

He also developed his own characteristic critique of Einstein. Fock argued that the equivalence principle, properly understood, held only locally and not globally, and that Einstein’s general principle of relativity was without unique physical content because no system of coordinates was truly preferred. His position was idiosyncratic and provoked decades of debate; modern relativity has, in essence, accepted the substance of his point about the local nature of equivalence while continuing to call the theory “general relativity” out of historical inertia. He worked on the many-body problem in general relativity, on radio-wave propagation around the Earth (a problem with obvious Soviet defense applications), and on geophysical methods for prospecting oil and rocks. He kept his lectures open and his manner gentle, and a generation of Soviet theoretical physicists passed through his classroom.

He died in Leningrad on December 27, 1974, five days after his seventy-sixth birthday, having outlived Stalin, the war, the purges, and most of his European contemporaries. He had survived a system that destroyed many of his colleagues, and he had built, while surviving, a body of work that touches almost every corner of modern theoretical physics: Klein-Gordon, Hartree-Fock, Fock space, the Fock representation, the Fock state. The infrastructure of the field is Fock-shaped, even where the name has been worn off the dial.

In the arc of this story Fock occupies a particular role. Heisenberg, Schrödinger, Dirac, and Born built the mechanics; Pauli supplied the exclusion principle that organizes electrons in atoms; Fock supplied the mathematics that lets you actually compute with that principle when there are more than two of them, and the formal language (Fock space) in which the modern field-theoretic version of all of this lives. Without Hartree-Fock there is no computational chemistry, no density-functional theory, no honest ab initio prediction of molecular structure. Without Fock space there is no quantum field theory in any tractable form. He is the quantum-many-body engineer of the early twentieth century, the man who built the workbench on which everyone else’s particles get assembled.