Relativistic Notation
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.
Metric tensor
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:
such that
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.