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The study of relativistic processes is based on collision phenomena at energies much larger than the rest energies of the particles involved. In this regime, a large number of new particles is typically produced, with large momenta, or, equivalently, small wavelengths. For this reason, the scheme of ordinary Quantum Mechanics, based on the Schrödinger equation for wave functions that depend on a fixed number of variables, is no longer applicable. A suitable framework is rather provided by electromagnetism, that describes radiation phenomena, and therefore the production and absorption of photons. This analogy leads in a natural way to field theory, in which the dynamical variables that describe a given physical system are fields, i.e. variables labelled by the space coordinates, and independent of each other.

The dynamical variables that describe relativistic systems are , that is, functions defined in each point of ordinary space. Important examples are the electromagnetic fields, and Dirac and Yukawa fields. The field description of a physical system allows a direct implementation of the principle of covariance, that guarantees the invariance of the equations of motion under changes of reference frame, and of the principle of causality, which is connected to the principle of locality, namely, the independence of variables associated to different points in space at the same time.

Before discussing scattering processes in field theory, we briefly recall a few general results in scattering theory in the traditional Lippman-Schwinger formulation.

The calculation of the scattering amplitude for a given process is greatly simplified by a graphical technique, originally introduced by R. Feynman in the context of quantum electrodynamics. We define the following symbols: Equation (3.89) is represented as while the graphical form of eq. (3.90) is It is understood that each internal vertex corresponds to a space-time integration.

We have discussed in Section 2.1 the importance of relativistic invariance in the formulation of theories of fundamental interactions. We have studied in detail the case of scalar fields, transforming as under a Lorentz transformation . A second well-known example (which will be discussed at length in Chapters 7 and 8) is provided by vector fields: In the general case, we have a system of complex fields , = 1, ..., , with the transformation law where the matrix obeys the condition for any two Lorentz transformations . The vector field is an obvious example; in that case, we have simply ≡ ).

The simplest example of a phenomenological application of the theory of spinor fields is electrodynamics, originally formulated by P.A.M. Dirac to provide a relativistic description of electrons and photons. The theory is based on a pair of spinor fields of opposite chiralities, and ; invariance under phase multiplication of the spinor fields is assumed, and the corresponding conserved current is identified with the electromagnetic current. No scalar field is present. Under these assumptions, the free-field Lagrangian is given by is manifestly invariant under the phase transformations where , are real constants. Notice that we have inserted a Dirac mass term, which is allowed by the assumed invariance properties, while Majorana mass terms are not, as mentioned in Section 5.2. The conserved current is given by and the constant e plays the role of elementary charge.

Weak interaction processes such as nucleon decay, or decay, are correctly described by an effective theory, usually referred to as the Fermi theory of weak interactions. The Fermi Lagrangian density for and decays is given by From the measured values of muon and neutron lifetimes, one obtains while the value can be extracted from the measurement of baryon semi-leptonic decay rates. The most striking feature of weak interactions, correctly taken into account by the Fermi theory, is the violation of parity invariance, that arises from the measurement of neutrino helicities in weak decay processes. [5] Note also that the Fermi constant sets a natural mass scale for weak interactions: We note that the field theory defined by the interaction in eq. (7.1) is non-renormalizable, since it contains operators with mass dimension 6; it is an effective theory, in the sense that it can only be used to compute amplitudes in the semi-classical approximation.

We have seen in Chapter 7 that the neutral vector boson fields coupled to the (2) and (1) charges appear as linear combinations of the massless photon field and of the field; similarly, left-handed quarks with charge −1/3 enter the weak interaction term as linear combinations of the corresponding mass eigenfields. We show here that these mixing phenomena are consequences of the mechanism of mass generation in the model.

Fermion mass terms are forbidden by the gauge symmetry of the standard model. Indeed, a Dirac mass term for a fermion field is not invariant under a chiral transformation, i.e. a transformation that acts differently on left-handed and right-handed components. The gauge transformations of the standard model are precisely of this kind. Again, this difficulty can be circumvented by means of the Higgs mechanism.

We present here the full Lagrangian density of the standard model with one Higgs doublet. We have where