Why a standard model? Some fundamentals

Until 1945, particle physics was mostly based on the study of cosmic rays. The turning point came with the advent of accelerators and the subsequent particle proliferation.

Elementary (to our knowledge) particles can be classified according to their spin statistics, and thus in order to whether they participate in strong interactions, as shown in the Table:

Tab:particle_classification

Strongly interactive particles are said hadrons. Electrons and neutrinos that do not interact via the strong force and do not show internal structure are called leptons.
Particle proliferation exclusively concerned hadrons.
In addition, leptons and non-hadronic bosons are supposed to participate only in weak and electromagnetic interactions, while baryons and mesons show strong and weak (as well as electromagnetic) interactions.

This framework is well–described by the Standard Model of Particle Physics (SM) which is a gauge theory and a Lorentz invariant renormalizable quantum field theory that integrates the Quantum Chromo Dynamics (QCD) [*] with the electroweak unification theory [*] according to the GWS–model developed by S.L. Glashow, S. Weinberg and A. Salam.

The full theory was developed in the early 1970s and it has successfully explained almost all experimental results while sometimes it precisely predicted a wide variety of phenomena. The Standard Model does not include the gravitational interaction which is not described by a quantum field theory and it is also negligible in the description of phenomena between elementary particles at the energy scale fitting for the other forces.

 

Particles and fundamental interactions

There are several formulations of relativistic quantum field theories for interacting particles. The most commonly accepted is the Lagrangian approach which leads to a quantum field theory by the customary method of the canonical quantization.
It also provides a series of advantages such as its simplicity in conducting the formulation of gauge theories on what is essentially the classical level, and as the fact that it gives a way to make the relativistic covariance explicit.

In quantum field theories, charges point out any generator of continuous symmetries in a physical system. In correspondence of a symmetry the existence of a conserved current is implied according to the Noether’s theorem. For example, the electric charge is the generator of the U(1) symmetry group of electromagnetism and the corresponding conserved current is the electric current. The charge is then the generator of the local symmetry group.

Particles and the fundamental interactions are mathematically described in terms of quantum fields: matter particles are defined by fermionic fields, while bosons, which are the mediators of the fundamental interactions, are expressed by gauge fields (*).
The construction of these bosonic fields and the way in which the interactions occur start from the analysis of the symmetries of the system.  The application of the gauge invariance allows to find the expression for the densities of the lagrangian that correspond to the known interactions.

 

The Dirac equation

The relativistic quantum-mechanical behavior of a free fermion is described by the Dirac equation

\left(i\hslash\gamma^\mu\partial_{\mu} -mc\right)\psi(x) =0 

that follows from the free Dirac Lagrangian

\mathcal{L} = i\overline{\psi}(x)i\gamma^\mu\partial_\mu\psi(x) -m\psi\overline{\psi}

which describes the evolution of non-interacting particles where:

  • \gamma^\mu are the Dirac matrices;
  • \psi(x) is the Dirac bispinor field of spin–\frac{1}{2} particles with mass m that evolves according to the i\gamma^\mu\partial_\mu term;
  • \overline{\psi}(x), called psi-bar, is the Dirac-adjoint of  \psi (x) that indicate the conjugate field and it is defined as \overline{\psi}(x)= \psi^{\dag}(x)\gamma^0.

The Dirac equation is invariant for a group of global transformations. In order to maintain the invariance also under local transformations, it is needed to mathematically introduce the gauge fields that physically corresponds to the mediators of the interactions.

Free bosons are described by the Klein-Gordon free particle equation, whose Lagrangian is

\mathcal{L}=(\partial_\mu\phi^+)(\partial^\mu\phi)-m^2\phi^+\phi.

It is worthy to note that here the mass term, bilinear respect to the field \phi, is proportional to m^2.

 

References and further readings:
R.H. Mohapatra, Unification and Supersymmetry: the Frontiers of Quark-Lepton Physics – Third Edition, (Springer-Verlag), 2002.
C. Quigg, Gauge Theories of the Strong, Weak and Electromagnetic Interactions, (Addison-
Wesley Publishing Company), 1997.
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