In 1921 Weyl studied the invariance properties of quantum mechanics. In particular, he noted that the absolute phase of the wave function is not observable. In a formal and generic way, we consider a global transformation of the phase of a Dirac spinor. This can be expressed with the following unitary global gauge transformation 𝑈(𝛼)

Quantum field theory is complex. Ask ourselves why we bother with all these quantum fields in the first place? QED is a field theory of well-defined perturbation expansion and in principle any physical prediction can be calculated with practically infinite precision. So, in this chapter we explore the techniques associated with computing “higher-order” or “purely quantum” effects of electromagnetism. The 𝑆-matrix was written as the Dyson expansion (see Section 9.2), where the factor in the expansion is the electric elementary charge 𝑒.

The classical period of Roman law is conventionally taken to have ended in ad 235 with the death of the Emperor Severus Alexander. It is true that the line of independent classical jurists breaks off there. But this was not a collapse but a change of direction. The leading jurists increasingly became involved in the process of imperial law-making; and their works were the constitutions they composed in the name of their emperor. The constitutions of Diocletian in particular (ad 284–305) show that half a century after the end of the classical period the standards of classical jurisprudence had been maintained. But this was not a period in which new original juristic work appeared; instead, the trend was towards the production of anthologies or epitomes of leading classical works. It therefore seems appropriate to refer to the period from about ad 235 to 305 as the ‘epiclassical’ period of Roman law and to date the decisive break between the classical and the post-classical to about ad 300 (Wieacker 1971).

Roman law divided free citizens into two classes: those who were independent (sui iuris) and those who were dependent on someone else (alieni iuris). The Roman family was patriarchal: all power was vested in the paterfamilias, who was the senior living male. So, a child (at least as long as he or she was legitimate) was subject to the power of his or her paterfamilias, whether father, grandfather, or great-grandfather. Paternal power (patria potestas) was lifelong, so that in principle a man who had already become a grandfather might still be subject to his father’s power and become independent only late in life.

In contradistinction to quantum electrodynamics, the Fermi theory is not renormalizable. This difficulty could not be solved by smoothing the point-like interaction by a massive, and therefore short-range, charged vector particle exchange (the so-called 𝑊+ and 𝑊− bosons): theories with massive charged vector bosons are not renormalizable either.

The first speculations about “charm” were made in the mid-1960s [230], and full attention to it was given in the 1970s with the advent of the Cabibbo–GIM mechanism, as discussed in Section 23.13. In 1970 Drell and Yan discussed the production of massive lepton pairs in hadron–hadron collisions.

The effective Lagrangian method was developed by Weinberg [463] and independently by Wilczek and Zee [464]. It can be seen as a general, powerful method which allows us to quantitatively describe the effects of physics beyond the SM. The idea is that the SM is very effective at describing with high precision all experimental observations up to the tera-electronvolt scale, i.e., at “low energy.”

The equation, developed by Dirac as the union of quantum mechanics and relativity, historically led to the prediction of the existence of a new form of matter – the antimatter – previously unsuspected and unobserved and which was experimentally confirmed several years later with the discovery of the positron. The equation also entailed the explanation of spin. Altogether it represented one of the great triumphs of theoretical physics. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-1/2 particles. In the Standard Model all fundamental building blocks of matter – the quarks and leptons – are represented with such Dirac fields.

During the 1940s and 1950s, the studies continued on the 𝛽 decays. It was found that not all 𝛽 decays occur between nuclear states with identical angular momenta, so the Fermi allowed transitions defined in Section 21.3, which represent a 𝛥𝐽 = 0 operator (see Eq. (21.31)), could not be a complete description.

The successful development of QED represented a great achievement: the theory was very useful, it handled matter and antimatter (electrons and positrons), it introduced the technique of renormalization, and it proved to be extremely useful and precise (for example in computing the anomalous magnetic moments). Nonetheless, QED could not simply explain even the existence of the nucleus of atoms! Indeed, what holds the nucleus together?