Neutrinos are neutral particles of spin; they are completely relativistic in the massless limit. In order to describe neutrinos and their interactions, we need a relativistic theory of spin 1 particles. The appropriate framework to describe the elementary particles in general and the neutrinos in particular, is relativistic quantum mechanics and quantum field theory. In this chapter and in the next two chapters, we present the essentials of these topics required to understand the physics of the weak interactions of neutrinos and other particles of spin 0, 1, and.
We shall use natural units, in which ћ = c = 1, such that all the physical quantities like mass, energy, momentum, length, time, force, etc. are expressed in terms of energy. In natural units:
The original physical quantities can be retrieved by multiplying the quantities expressed in energy units by appropriate powers of the factors ћ, c, and ћ c. For example, mass m = E/c2, momentum p = E/c, length l = ћc/E, and time t = ћ /E, etc.
In the relativistic framework, space and time are treated on equal footing and the equations of motion for particles are described in terms of space–time coordinates treated as four- component vectors, in a four-dimensional space called Minkowski space, defined by xμ, where μ = 0, 1, 2, 3 and xμ = (x0' = t, x1 = x, x2 = y, x3 = z') in any inertial frame, say S. In another inertial frame, say, whichis moving with a velocity in the positive X direction, the space–time coordinates are related to xμ through
the Lorentz transformation given by:
remains invariant under Lorentz transformations. For this reason, the quantity is called the length of the four-component vector xμ in analogy with the length of an ordinary vector, that is, which is invariant under rotation in three-dimensional Euclidean space. Therefore, the Lorentz transformations shown in Eq. (2.1) are equivalent to a rotation in a four-dimensional Minkowski space in which the quantity defined as, remains invariant, that is, it transforms as a scalar quantity under the Lorentz transformation. This is similar to a rotation in the three-dimensional Euclidean space in which the length of an ordinary vector, defined as remains invariant, that is, transforms as a scalar under rotation.