Lectures on Quantum Field Theory. Ashok Das

      Since the contravariant and the covariant vectors transform in an inverse manner, such a product is easily seen to be invariant under Lorentz transformations. This is the generalization of the scalar product of the three dimensional Euclidean space (1.7) to the four dimensional Minkowski space and is invariant under Lorentz transformations which are the analogs of rotations in Minkowski space. In fact, any product of Lorentz tensors defines a scalar if all the Lorentz indices are contracted, namely, if there is no free Lorentz index left in the resulting product. (Two Lorentz indices are said to be contracted if a contravariant and a covariant index are summed over all possible values.)

      Given this, we note that the length of a (four) vector in Minkowski space can be determined to have the form (compare with (1.8))

      Unlike the Euclidean space, however, here we see that the length of a vector need not always be positive semi-definite (recall (1.9)). In fact, if we look at the Minkowski space itself, we find that

      This is the invariant length (of any point from the origin) in this space. The invariant length between two points infinitesimally close to each other follows from this to be

      where τ is known as the proper time.

      For coordinates which satisfy (see (1.19), we will set c = 1 from now on for simplicity)

      we say that the region of space-time is time-like for obvious reasons. On the other hand, for coordinates which satisfy

      the region of space-time is known as space-like. The boundary of the two regions, namely, the region for which

      defines trajectories for light-like particles and is, consequently, known as the light-like region. (Light-like vectors, for which the invariant length vanishes, are nontrivial unlike the case of the Euclidean space in (1.9).)

Figure 1.1: Different invariant regions of Minkowski space.

      Thus, we see that, unlike the Euclidean space, the Minkowski space-time manifold separates into four invariant cones (namely, regions which do not mix under Lorentz transformations), which in a two dimensional projection has the form of wedges shown in Fig. 1.1. The different invariant cones (wedges) are known as

      All physical processes are assumed to take place in the future light cone or the forward light cone defined by

      Given the contravariant and the covariant coordinates, we can define the contragradient and the cogradient respectively as (c = 1)

      From these, we can construct the Lorentz invariant quadratic operator

      which is known as the D’Alembertian. It is the generalization of the Laplacian to the four dimensional Minkowski space.

      Let us note next that energy and momentum also define four vectors in this case. (Namely, they transform like four vectors under Lorentz transformations.) Thus, we can write (remember that c = 1, otherwise, we have to write

)

      Given the energy-momentum four vectors, we can construct the Lorentz scalar

      The Einstein relation for a free particle (remember c = 1)

      where m represents the rest mass of the particle, can now be seen as the Lorentz invariant condition

      In other words, in this space, the energy and the momentum of a free particle must lie on a hyperbola satisfying the relation (1.31).

      We already know that the coordinate representations of the energy and the momentum operators take the forms

      We can combine these to write the coordinate representation for the energy-momentum four vector operator as

      Finally, let us note that in the four dimensional space-time, we can construct two totally anti-symmetric fourth rank tensors ϵ^µνλρ, ϵ_µνλρ, the four dimensional contravariant and covariant Levi-Civita tensors respectively. We will choose the normalization ϵ⁰¹²³ = 1 = −ϵ₀₁₂₃ so that

      where ϵ_ijk denotes the three dimensional Levi-Civita tensor with ϵ₁₂₃ = 1. An anti-symmetric tensor such as ϵⁱ_jk is then understood to denote

      and so on. This completes the review of all the essential

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