Lectures on Quantum Field Theory. Ashok Das
will be specified uniquely by four coordinates and, consequently, any vector would also have four components. However, unlike the case of the Euclidean space, there are now two distinct four vectors that we can define on this manifold, namely, (µ = 0, 1, 2, 3 and we are being a little sloppy in representing the four vector by what may seem like its component)
Here c represents the speed of light (necessary to give the same dimension to all the components) and we note that the two four vectors simply represent the two distinct possible ways space and time components can be embedded into the four vector. On a more fundamental level, the two four vectors have distinct transformation properties under Lorentz transformations (in fact, one transforms inversely with respect to the other) and are known respectively as contravariant and covariant vectors.
The contravariant and the covariant vectors are related to each other through the metric tensor of the four dimensional manifold, commonly known as the Minkowski space, namely,
From the forms of the contravariant and the covariant vectors in (1.10) as well as using (1.11), we can immediately read out the components of the metric tensors for the four dimensional Minkowski space which are diagonal with the signature (+, −, −, −). Namely, we can write them in the matrix form as
The contravariant metric tensor, ηµν, and the covariant metric tensor, ηµν, are inverses of each other, since they satisfy
Furthermore, each is symmetric as they are expected to be, namely,
This particular choice of the metric is conventionally known as the Bjorken-Drell metric and this is what we will be using throughout these lectures. Different authors, however, use different metric conventions and you should be careful in reading the literature. (As is clear from the above discussion, the nonuniqueness in the choice of the metric tensors reflects the nonuniqueness of the embedding of space and time components into a four vector. Physical results, however, are independent of the choice of a metric.)
Given two arbitrary four vectors
we can define an invariant scalar product of the two vectors as
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
