Relativistic Quantum Mechanics

Here, we will try to find a relativistic version of the Schrödinger equation, but without using fields. Then, we will see why this ultimately fails.

Table of Contents

• Table of Contents
• The Klein-Gordon Equation
• The Dirac Equation

The Klein-Gordon Equation

Historically speaking, the first attempt to make quantum mechanics compatible with special relativity was the Klein-Gordon equation, named after Walter Gordon and Oskar Klein (although it was actually first discovered by Erwin Schrödinger). However, it has a major flaw.

First, let's analyze the Schrödinger wave equation

where is the position space wavefunction. We can obviously see that this is not Lorentz invariant, as it is not invariant under the Lorentz transformation. For one, there is one time derivative and two spatial derivatives.

We will try to make this equation relativistic by treating the wavefunction as a scalar field of 4-dimensional spacetime. We can write the wavefunction as a function of spacetime coordinates ,

Note that this is ultimately a flawed approach. Scalar fields are functions of physical spacetime, but wavefunctions are functions of coordinates in the configuration space of the system. If, for instance, we added another particle, the wavefunction would become a function of the spatiotemporal coordinates of both particles, and so on.

Recall that the Schrödinger equation can be derived from the first quantization of the classical Hamiltonian,

By applying and , we get the Schrödinger equation. We can try to do the same thing but instead, we begin with a relativistic equation. Using the energy-momentum relation,

we can use the substitutions and to get

First, the time term can be simplified to

Next, we can factor out the from the right-hand side to get

We can (incorrectly) apply both sides to a wavefunction (field) to get

Of course, the term inside the parantheses can be rewritten as

where is the Minkowski metric. This operator is known as the d'Alembertian operator. Thus, the equation becomes

or

This is the Klein-Gordon equation. Unlike the Schrödinger equation, this is Lorentz invariant. For one, there are equal numbers of spatial and temporal derivatives. This ensures that the equation respects the principles of relativity and can be applied consistently across different inertial frames.

In natural units, we express mass in terms of energy—essentially we absorb the factor of into the mass term. We also take , so the Klein-Gordon equation simplifies to

Note that Klein-Gordon equation is not a specific equation that pertains to a particular particle. We saw in the previous chapter that it also appeared in the context of classical field theory. As such, the equation is instead a general equation that describes the dynamics of a relativistic scalar field.

The solution to the Klein-Gordon equation is a wavefunction that is a superposition of plane waves. First, we take the ansatz

This does indeed satisfy the Klein-Gordon equation, so we can take a linear combination of these solutions (with different and ) to get the general solution

where is the relativistic energy-momentum relation.

The Dirac Equation

As we just saw, the Klein-Gordon equation has major flaws. Dirac sought to fix these flaws by introducing the Dirac equation. He did this by attempting to take the "square root" of both sides of the Klein-Gordon equation. In other words, he tried to factor the Klein-Gordon equation into two first-order equations;

What is the square root of the d'Alembertian operator? We know that in the time coordinate, the d'Alembertian operator is the positive second derivative, and in the spatial coordinates, it is the negative second derivative. This means that if we square this new operator, in the time coordinate, we get the positive second derivative, and in the spatial coordinates, we get the negative second derivative;

Let's write this operator as

From our conditions, we know that corresponds to and corresponds to . This is precisely the rules that govern the spacetime algebra , whose basis is the Dirac gamma matrices. Anyways, we can thus rewrite the Klein-Gordon equation as

Finally, just taking the second factor to be zero, we get the Dirac equation,

In natural units, this simplifies to

and if we use the slash notation , we can write the Dirac equation as