2 Gauge Theories and the Standard Model
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of matter and radiation, including molecular, atomic, nuclear and subnuclear
physics, can be understood in terms of three classes of fundamental interactions:
strong, electromagnetic and weak interactions. In atoms the electrons are bound to
nuclei by electromagnetic forces and the properties of electron clouds explain the
complex phenomenology of atoms and molecules. Light is a particular vibration
of electric and magnetic fields (an electromagnetic wave). Strong interactions bind
the protons and neutrons together in nuclei, being so intensively attractive at short
distances that they prevail over the electric repulsion due to the equal sign charges of
protons. Protons and neutrons, in turn, are composites of three quarks held together
by strong interactions to which quarks and gluons are subject (hence these particles
are called “hadrons” from the Greek word for “strong”). To the weak interactions
are due the beta radioactivity that makes some nuclei unstable as well as the nuclear
reactions that produce the enormous energy radiated by the stars and by our Sun
in particular. The weak interactions also cause the disintegration of the neutron, the
charged pions, the lightest hadronic particles with strangeness, charm, and beauty
(which are “flavour” quantum numbers) as well as the decay of the quark top and of
the heavy charged leptons (the muon μ − and the tau τ − ). In addition all observed
neutrino interactions are due to weak forces.
All these interactions are described within the framework of quantum mechanics
and relativity, more precisely by a local relativistic quantum field theory. To each
particle, described as pointlike, is associated a field with suitable (depending on
the particle spin) transformation properties under the Lorentz group (the relativistic
space-time coordinate transformations). It is remarkable that the description of all
these particle interactions is based on a common principle: “gauge” invariance. A
“gauge” symmetry is invariance under transformations that rotate the basic internal
degrees of freedom but with rotation angles that depend on the space-time point.
At the classical level gauge invariance is a property of the Maxwell equations of
electrodynamics and it is in this context that the notion and the name of gauge
invariance were introduced. The prototype of all quantum gauge field theories,
with a single gauged charge, is QED, Quantum Electro-Dynamics, developed in
the years from 1926 until about 1950, which indeed is the quantum version of
Maxwell theory. Theories with gauge symmetry, at the renormalizable level, are
completely determined given the symmetry group and the representations of the
interacting fields. The whole set of strong, electromagnetic and weak interactions
is described by a gauge theory, with 12 gauged non-commuting charges, which is
called “the Standard Model” of particle interactions (SM). Actually only a subgroup
of the SM symmetry is directly reflected in the spectrum of physical states. A part of
the electroweak symmetry is hidden by the Higgs mechanism for the spontaneous
symmetry breaking of a gauge symmetry.
For all material bodies on the Earth and in all geological, astrophysical and cosmological phenomena a fourth interaction, the gravitational force, plays a dominant
role, while it is instead negligible in atomic and nuclear physics. The theory of
general relativity is a classic (in the sense of non quantum mechanical) description
of gravitation that goes beyond the static approximation described by Newton law
and includes dynamical phenomena like, for example, gravitational waves. The
11
of matter and radiation, including molecular, atomic, nuclear and subnuclear
physics, can be understood in terms of three classes of fundamental interactions:
strong, electromagnetic and weak interactions. In atoms the electrons are bound to
nuclei by electromagnetic forces and the properties of electron clouds explain the
complex phenomenology of atoms and molecules. Light is a particular vibration
of electric and magnetic fields (an electromagnetic wave). Strong interactions bind
the protons and neutrons together in nuclei, being so intensively attractive at short
distances that they prevail over the electric repulsion due to the equal sign charges of
protons. Protons and neutrons, in turn, are composites of three quarks held together
by strong interactions to which quarks and gluons are subject (hence these particles
are called “hadrons” from the Greek word for “strong”). To the weak interactions
are due the beta radioactivity that makes some nuclei unstable as well as the nuclear
reactions that produce the enormous energy radiated by the stars and by our Sun
in particular. The weak interactions also cause the disintegration of the neutron, the
charged pions, the lightest hadronic particles with strangeness, charm, and beauty
(which are “flavour” quantum numbers) as well as the decay of the quark top and of
the heavy charged leptons (the muon μ − and the tau τ − ). In addition all observed
neutrino interactions are due to weak forces.
All these interactions are described within the framework of quantum mechanics
and relativity, more precisely by a local relativistic quantum field theory. To each
particle, described as pointlike, is associated a field with suitable (depending on
the particle spin) transformation properties under the Lorentz group (the relativistic
space-time coordinate transformations). It is remarkable that the description of all
these particle interactions is based on a common principle: “gauge” invariance. A
“gauge” symmetry is invariance under transformations that rotate the basic internal
degrees of freedom but with rotation angles that depend on the space-time point.
At the classical level gauge invariance is a property of the Maxwell equations of
electrodynamics and it is in this context that the notion and the name of gauge
invariance were introduced. The prototype of all quantum gauge field theories,
with a single gauged charge, is QED, Quantum Electro-Dynamics, developed in
the years from 1926 until about 1950, which indeed is the quantum version of
Maxwell theory. Theories with gauge symmetry, at the renormalizable level, are
completely determined given the symmetry group and the representations of the
interacting fields. The whole set of strong, electromagnetic and weak interactions
is described by a gauge theory, with 12 gauged non-commuting charges, which is
called “the Standard Model” of particle interactions (SM). Actually only a subgroup
of the SM symmetry is directly reflected in the spectrum of physical states. A part of
the electroweak symmetry is hidden by the Higgs mechanism for the spontaneous
symmetry breaking of a gauge symmetry.
For all material bodies on the Earth and in all geological, astrophysical and cosmological phenomena a fourth interaction, the gravitational force, plays a dominant
role, while it is instead negligible in atomic and nuclear physics. The theory of
general relativity is a classic (in the sense of non quantum mechanical) description
of gravitation that goes beyond the static approximation described by Newton law
and includes dynamical phenomena like, for example, gravitational waves. The