Relativistic Notation

Neutrinos are neutral particles of spin; they are completely relativistic inthe massless limit. In order to describe neutrinos and their interactions,we need a relativistic theory of spin 1 particles. The appropriate frameworkto describe the elementary particles in general and the neutrinos inparticular, is relativistic quantum mechanics and quantum field theory. Inthis chapter and in the next two chapters, we present the essentials ofthese topics required to understand the physics of the weak interactions ofneutrinos and other particles of spin 0, 1, and.

We shall use natural units, in which ћ = c = 1, such that all thephysical 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 thequantities expressed in energy units by appropriate powers of the factorsћ, c, and ћc. For example, mass m =E/c2, momentump = E/c, lengthl = ћc/E, andtime t = ћ /E,etc.

Metric tensor

In the relativistic framework, space and time are treated on equal footingand the equations of motion for particles are described in terms ofspace–time coordinates treated as four- component vectors, in afour-dimensional space called Minkowski space, defined byxμ, whereμ = 0, 1, 2, 3 andxμ =(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 Xdirection, the space–time coordinates are related toxμ through

the Lorentz transformation given by:

such that

remains invariant under Lorentz transformations. For this reason, thequantity is called the length of the four-component vectorxμ in analogy with the length ofan ordinary vector, that is, which is invariant under rotation inthree-dimensional Euclidean space. Therefore, the Lorentz transformationsshown in Eq. (2.1) are equivalent to a rotation in a four-dimensionalMinkowski space in which the quantity defined as, remains invariant, thatis, it transforms as a scalar quantity under the Lorentz transformation.This is similar to a rotation in the three-dimensional Euclidean space inwhich the length of an ordinary vector, defined as remains invariant, thatis, transforms as a scalar under rotation.