Fields with physical significance are those that verify the Euler–Lagrange equations, or that satisfy the Hamilton’s principle.

For such systems, a general and systematic procedure is available to establish conservation theorems and constants of motion, as a consequence of invariance properties. Thus, conservation laws and selection rules observed in Nature may be imposed as symmetries of the Lagrangian.

The general framework for this purpose is provided by **Noether’s theorem**, that connects conservation laws to continuous symmetry transformations under which the Lagrangian is invariant in form.

There are different kinds of symmetry transformations and therefore many of invariance principles such as continuous or discrete, geometrical or internal, and global or local that can be considered to build a lagrangian.

The requirement of a local gauge invariance can serve as a dynamical principle to guide the construction of interacting field theories.

- Global gauge invariances:
- imply the existence of conserved currents according to Noether’s theorem.

- Local gauge invariances:
- require the introduction of massless vector gauge bosons;
- prescribe (or, more properly, restrict) the form of the interactions of gauge bosons with sources;
- generate interactions among the gauge bosons if the symmetry is non–Abelian.

A lagrangian for a physical system may be invariant under a given set of symmetry transformations but how the symmetry is realized in Nature depends on the properties of the ground state.

In field theories, the ground state is the vacuum state, i.e. the minimum energy state described by the lagrangian.

To understand the symmetries it is necessary to identify the basic **degrees of freedom** according to which they operate.

In the case of quark–lepton symmetry, quarks carry two independent degrees of freedom, flavor and colour, while leptons carry only the flavor.

The flavour is a *manifest* degree of freedom responsible for the variety and richness of the spectrum in the baryon–meson world, while the colour is a *hidden* coordinate responsible for the binding of quarks and anti–quarks in baryons and mesons, as for their ways of interactions.

It is worthy to note that the flavour degrees of freedom of quarks and leptons are identical: this is known as the *quark–lepton symmetry*.

The existence of the *charm* quark and of a new set of hadrons, *charmhadrons*, was first inferred on the basis of this symmetry. The electroweak interactions operate along the flavor degree of freedom, whereas the strong interactions operate along the colour degree of freedom.

A more ambitious program could be to introduce new kinds of interactions operating between quarks and leptons, thus providing a framework for an unified treatment.

Despite of the mathematical elegance achieved in the theory, the connection between exact symmetries and conservation laws is not always found suitable for the description of reality.

In fact, there are many situations in Physics in which the exact symmetry of an interaction is hidden by circumstances, like for instance, the canonical example of the Heisenberg ferromagnet; on the other hand, the subsequent construction of gauge theories would be effective only in the case of exact symmetries.

Nature, instead, exhibits numerous symmetries that are only approximate among which several different realizations are possible.

The Lagrangian may display an imperfect or explicitly broken symmetry, or it may happen that the Lagrangian is symmetric but the physical vacuum does not with respect the symmetry. In the latter case, the symmetry of the Lagrangian is said to be *spontaneously* broken. Therefore, the model leads to theories in which all interactions are mediated by massless vector bosons, whereas only a single massless vector boson, the photon, is evident in Nature.

Indeed, if the Lagrangian of a theory is invariant under an exact continuous symmetry which does not corresponds to a symmetry of the physical vacuum, one or more massless spin–0 particles, known as ** Nambu–Goldstone bosons**, must occur.

Goldstone bosons are then associated with every generator of the gauge group that does not leave the vacuum invariant.

In particular, from the **Goldstone theorem**, the number of Goldstone bosons is precisely the same as the number of charges that do not annihilate the vacuum, or the number of broken–symmetry generators.

It is commonly accepted that there may be no real Nambu–Goldstone bosons in Nature because their existence would give rise to long–range forces in classical Physics and would lead to new effects in scattering and decay processes.

Starting from the early 1980s, however, it was pointed out that any possible non-relativistic classical long-range forces arising from the existence of massless Goldstone particles are spin dependent and would therefore be extremely difficult to observe.

So, gauge theories lead to undesirable massless vector bosons and the spontaneous breakdown of continuous symmetries imply the existence of unwanted spinless particles.

However, if the spontaneously broken symmetry is a local gauge symmetry, the interaction between normally massless gauge bosons and the Goldstone boson endows the gauge bosons with mass and removes the Goldstone boson from the spectrum.

This interaction is known as the *Brout–Englert–Higgs–Guralnik–Hagen–Kibble mechanism of Spontaneous Symmetry Breaking*, or the *Higgs mechanism* for short.