non-relativistic physical system and promote each of the observables to a Hermitian operator. The time evolution of the quantum mechanical system (state), in this case, is given by the time dependent Schrödinger equation which has the form
Here ψ(x, t) represents the wave function of the system which corresponds to the probability amplitude for finding the particle at the coordinate x at a given time t and the Hamiltonian, H, has the generic form
with p denoting the momentum of the particle and V (x) representing the potential through which the particle moves. (Throughout the book we will use a bold symbol to represent a three dimensional vector.)
This formalism is clearly non-relativistic (non-covariant) which can be easily seen by noting that, even for a free particle, the dynamical equation (1.1) takes the form
In the coordinate basis, the momentum operator has the form
so that the time dependent Schrödinger equation, in this case, takes the form
This equation is linear in the time derivative while it is quadratic in the space derivatives. Therefore, space and time are not treated on an equal footing in this case and, consequently, the equation cannot have the same form (covariant) in different Lorentz frames. A relativistic equation, on the other hand, must treat space and time coordinates on an equal footing and remain form invariant in all inertial frames (Lorentz frames). Let us also recall that, even for a simple fundamental system such as the Hydrogen atom, the ground state electron is fairly relativistic (
1.2Notations
Before proceeding any further, let us fix our notations. We note that in the three dimensional Euclidean space, which we are all familiar with, a vector is labelled uniquely by its three components. (We denote three dimensional vectors in boldface.) Thus,
where x and J represent respectively the position and the angular momentum vectors of a particle (system) while A stands for any arbitrary vector. In such a space, as we know, the scalar product of any two arbitrary vectors is defined to be
where repeated indices are assumed to be summed. The scalar product of two vectors is invariant under rotations of the three dimensional space which is the maximal symmetry group of the Euclidean space that leaves the origin invariant. This also allows us to define the length of a vector simply as
The Kronecker delta, δij, in this case, represents the metric of the Euclidean space and is trivial (in the sense that all the nonzero components are positive and simply unity). Consequently, it does not matter whether we write the indices “up” or “down”. Let us note from the definition of the length of a vector in Euclidean space that, for any nontrivial vector, it is necessarily positive definite, namely,
When we treat space and time on an equal footing and enlarge our three dimensional Euclidean manifold to the four dimensional space-time manifold, we can again define vectors in this manifold. However, these would now consist of four components. Namely, any point in this manifold 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
