II.1.2 Quantum Field Theories

In trying to merge Quantum Mechanics with Special Relativity, Klein-Gordon and Dirac equations, replaced Schroedinger equation with a covariant form by including the mass-energy equivalence relation . This was developed further by applying the quantization to a field, rather than a fixed set of particles. The first complete Quantum Field Theory was Quantum Electrodynamics (QED), which provides a quantum description of the electromagnetic interaction. Based on his new equation, Dirac posited the existence of an anti-electron, or the positron, which was experimentally confirmed after several years. The basic idea behind Quantum Electrodynamics is that the electromagnetic field should be represented by matrices in the same way that position and momentum were represented in Quantum Mechanics. The ideas of Quantum Mechanics were thus extended to systems having an infinite number of degrees of freedom, as infinite array of quantum oscillators. Therefore, Quantum Electrodynamics is based on the quantization of the electromagnetic field, in addition to the relativistic theory of the electron from the Dirac wave equation.

In the beginning, physicists believed that with QED it is possible to perform any computation for any physical process involving photons and charged particles. However, further studies in 1937 and 1939, revealed that such computations were reliable only at a first order of perturbation theory. At higher orders in the series infinities emerged, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself. More difficulties with QED theory were realized in the 1940s, due to further discrepancies after the improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom, now known as the Lamb shift, and the magnetic moment of the electron.

In 1947, the new concept of re-normalization was introduced by Hans Bethe (1906-2005), 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. Feynman s mathematical technique initially seemed very different from the operator-based approach, but it was shown later that the two approaches were equivalent.

Due to the need to attach a physical meaning at certain divergences appearing in the theory through integrals, re-normalization 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 re-normalization works very well in practice, Feynman was never entirely comfortable with its mathematical validity, even referring to re-normalization as a shell game and hocus pocus . The rationale behind re-normalization 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 calledregularization. There is no known symmetrical cutoff outside of perturbation theory, so for rigorous or numerical work people often use an actual lattice.

After its remarkable success as the first Quantum Field Theory, QED became a model for other interactions, such as the Quantum Chromodynamics (QCD) of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion.

In developing a gauge theory for the weak force in the 1960s, it was discovered that it must also incorporate the electromagnetic force, thus it was called the electroweak theory, which was the first unification between different fundamental forces. This indicated that, though outwardly diverse, the fundamental interactions are in fact separate facets of one single underlying force. Therefore, the search for a unified field theory of all the four fundamental forces, is one of the major goals of particle physics.

In the beginning of the 1950s, QED become a reliable theory, but it took more time until Quantum Field Theory could be applied successfully to important physical problems in a systematic way. The theories explored relied on a rich variety of symmetries that was able to predict new particles and explain their structure and interactions.

As we shall discuss in section II.2, the combined re-normalizable theory associated with the gauge group  is dubbed the Standard Model of Elementary Particles, which includes six types of leptons and six types of quarks, which are spin-half particles, in addition to four spin-one bosons, that mediate the interactions between the fermions, and the symmetry breaking mechanism of the theory is the spin-zero Higgs boson, which was discovered in 2012, after 40 years of its prediction.