3.5.4  Renormalization

In 1947, Hans Bethe (1906-2005) introduced the idea of renormalization, by attaching infinities to corrections of mass and charge that were actually fixed to a finite value by experiments. In this way, infinities were absorbed in those constants and yield a finite result in good agreement with experiments. This method was developed further by many other physicists, and it is now possible to get fully covariant formulations that were finite at any order in a perturbation series of Quantum Electrodynamics. This covariant and gauge invariant formulations of Quantum Electrodynamics allowed computations of observables at any order of perturbation theory.

In 1948, Feynman introduced sophisticated pictorial diagrams which depicted all possible interactions pertaining to a given interaction. These diagrams showed that the electromagnetic force is the interaction of photons between interacting particles. We shall also talk about these diagrams in section 5.8. Feynman’s mathematical technique initially seemed very different from the operator-based approach, but it was shown later that the two approaches were equivalent.

Renormalization, as the need to attach a physical meaning at certain divergences appearing in the theory through integrals, has subsequently become one of the fundamental aspects of Quantum Field Theory and has come to be seen as a criterion for a theory’s general acceptability. Even though renormalization works very well in practice, Feynman was never entirely comfortable with its mathematical validity, even referring to renormalization as a “shell game” and “hocus pocus”.

The rationale behind renormalization is to avoid divergences that appear in physical predictions by shifting them into a part of the theory where they do not influence empirical statements. A Quantum Field Theory is called renormalizable if all infinities can be absorbed into a redefinition of a finite number of coupling constants and masses. A consequence for QED is that the physical charge and mass of the electron must be measured and cannot be computed from first principles.

In order to define a theory on a continuum, one may first place a cutoff on the fields, by postulating that quanta cannot have energies above some extremely high value. This has the effect of replacing continuous space by a structure where very short wavelengths do not exist, as on a lattice. Lattices break rotational symmetry, and one of the crucial contributions made by Feynman, Pauli and Villars, and modernized by ’t Hooft and Veltman, is a symmetry-preserving cutoff for perturbation theory, which is called regularization. There is no known symmetrical cutoff outside of perturbation theory, so for rigorous or numerical work people often use an actual lattice.

On a lattice, every quantity is finite but depends on the spacing. When taking the limit to zero spacing, one makes sure that the physically observable quantities like the observed electron mass stay fixed, which means that the constants in the Lagrangian defining the theory depend on the spacing. By allowing the constants to vary with the lattice spacing, all the results at long distances become insensitive to the lattice, defining a continuum limit.

The renormalization procedure only works for a certain limited class of quantum field theories, called renormalizable quantum field theories. A theory is perturbatively renormalizable when the constants in the Lagrangian only diverge at worst as logarithms of the lattice spacing for very short spacings. The continuum limit is then well defined in perturbation theory, and even if it is not fully well defined non-perturbatively, the problems only show up at distance scales that are exponentially small in the inverse coupling for weak couplings. The Standard Model of particle physics is perturbatively renormalizable, and so are its component theories: Quantum Electrodynamics, electroweak theory and Quantum Chromodynamics. Of the three components, Quantum Electrodynamics is believed to not have a continuum limit by itself, while the asymptotically freeandweak and strong color interactions are nonperturbatively well defined.

The renormalization group as developed along Wilson’s breakthrough insights relates effective field theories at a given scale to such at contiguous scales. It thus describes how renormalizable theories emerge as the long distance low-energy effective field theory for any given high-energy theory. As a consequence, renormalizable theories are insensitive to the precise nature of the underlying high-energy short-distance phenomena. This is a blessing in practical terms, because it allows physicists to formulate low energy theories without detailed knowledge of high-energy phenomena. It is also a curse, because once a renormalizable theory such as the standard model is found to work, it provides very few clues to higher-energy processes.

The only way high-energy processes can be seen in the standard model is when they allow otherwise forbidden events, or else if they reveal predicted compelling quantitative relations among the coupling constants of the theories or models.

On account of renormalization, the couplings of Quantum Field Theory vary with scale, thereby confining quarks into hadrons, allowing the study of weakly-coupled quarks inside hadrons, and enabling speculation on ultra-high energy behavior.