Why Dirac Equation?

Something About Schrödinger Equation

Firstly, let’s write down the Schrödinger equation as an introduction:

Of course, it’s not special relativity covariance. Why? Because in the space-time of relativity, the time and space is dependent. However, we can find that one can separate the Schrödinger equation into two equations: TE and SE.

Also, in special relativity, the mass of an object increases with its velocity, and time and space are relative, not absolute:

So, the Schrödinger equation, is based on the framework of Newtonian physics, where time is absolute, and the variation of mass with speed is not considered.

This leads to the Schrödinger equation’s inability to accurately describe particles moving at high speeds. Of course, we can also see it from the structure of the equation.

Let’s take it a step further:

In quantum mechanics, the Schrödinger equation deals with the wave function, where time and space are treated separately, with time existing as an independent parameter. This means that the Schrödinger equation describes the evolution of a “static” quantum state over time.

In special relativity, time and space are unified and form a four-dimensional spacetime, where the motion of an object involves changes not just in spatial position but also in the temporal dimension. Therefore, any equation attempting to describe the behavior of high-speed particles needs to consider the relativity of both space and time.

In summary, the Schrödinger equation is based on a non-relativistic physical framework, assuming time as absolute, and neglects the relativity of time and space at high speeds. This is the fundamental reason why it is not applicable to special relativity.

In general relativity, which involves complex spacetime curvature and varying gravitational fields, combining the Schrödinger equation with general relativity becomes more complex.

Currently, there is no unified theory that completely integrates quantum mechanics and general relativity. This is a significant challenge in modern physics, known as the problem of Quantum Gravity.

Here are some quick examples of possible directions for combining quantum theory and general relativity. I’m not going to illustrate it further, and their correctness has yet to be researched:

Quantum Field Theory in Curved Spacetime
Loop Quantum Gravity, LQG
String Theory
Gravity Duality

Dirac Equation in Relativistic Situations

As we discussed before, the Schrödinger equation does not apply to relativistic effects. So, in order to fix it and make it satisfy the situation of relativistic situations, we should correct it to the Dirac equation.

The Dirac equation is relativistically covariant. This means that it adheres to the principles of special relativity, particularly the invariance of physical laws across different inertial reference frames.

Here we are going to fix Schrödinger equation, until it satisfy relativistic situation:

To make the derivation easier, I prefer use the natural unit system, fix .

We have an obvious idea, which is begin with making a transformation

Here, normally we would use the mass-energy relationship to solve for. But in this case we will get a negative energy. But Dirac found a special and beautiful equation, which is

And then, we make the transformation before, we get

this is the derivative formation of Dirac equation.

In order to obtain the conditions that and should satisfy, multiply in both sides

and we can perform this operation in the first term

So we get

Let us return to the most initial purpose. We want this equation to satisfy the relativistic case, so it should satisfy the mass-energy relationship

Comparing these two formulas, we get
[
\left{

\right.
]

**Look Out! **This equation tells us some properties:
$Unknown environment 'enumerate'\begin{enumerate} \item $\alpha_i$ and $\beta$ are matrices (otherwise the relation does not hold) \item $\alpha_i$ and $\beta$ are commuted respectively \item $\alpha_i^2$ and $\beta^2$ are unit matrices \item $tr(\alpha_i)=tr(\beta)=0$ $(tr(\alpha_a)=tr(\alpha_a\beta^2)=tr(\beta\alpha_a\beta)=-tr(\alpha_a\beta^2))\Rightarrow (tr(\alpha_i)=tr(\beta)=0)$ (other cases are the same) \item eigenvalue=$\pm1$ $(\alpha_ax_a=\lambda x_a) \Rightarrow (\alpha_a\alpha_ax_a=x_a=\lambda\alpha_ax_a=\lambda^2x_a )\Rightarrow (\lambda^{2}=1 )\Rightarrow (\lambda=\pm1 )$ \item $\alpha_a$ and $\beta$ have even dimensions (trace is 0, and +1 and -1 have same number of eigenvalues) \item $\alpha_i$ and $\beta$ are Hermitian \end{enumerate}$

So, it is easy to associate the Pauli matrix based on the above properties. Theoretically, we can take any four four-dimensional matrices that satisfy the above conditions.

However, we actually tend to take the matrices under Pauli-Dirac Rep. for convenience:

Finally, we seek a more concise four-dimensional vector expression to rewrite the Dirac equation.

Multiply in both sides, we get

And then, combining and into a four-dimensional vector

At this point, the equation is rewritten as

So now, we derive **Dirac equation **successfully. Based on its derivation, it is easy to see that the Dirac equation is relativistically covariant.

Prediction of Electron Spin

Now, we have and , which represented four-dimensional potential and four-dimensional energy-momentum of the electromagnetic field respectively.
So we have Dirac equation

Then consider non-relativistic situation (), we have

and

Thus, we have
$Missing or unrecognized delimiter for \Big\begin{aligned} (-m+E-q\phi)\psi_A& =\vec{\sigma}\cdot(\vec{p}-q\vec{A}) \ &=\frac1{2m}[\vec{\sigma}\cdot(\vec{p}-q\vec{A})]^2 \ &=\frac1{2m}\Big{(\vec{p}-q\vec{A})^2+i\vec{\sigma}[(\vec{p}-q\vec{A})\times(\vec{p}-q\vec{A})]\Big}\psi_A \ &=\frac1{2m}\Big{(\vec{p}-q\vec{A})^2+i\vec{\sigma}\cdot(\nabla\times A)\Big}\psi_A \ &=\frac1{2m}\Big{(\vec{p}-q\vec{A})^2+i\vec{\sigma}\cdot\vec{B}\Big}\psi_A \end{aligned}$
So we get

In this equation, we have the Hamiltonian

where . Look out again! What does it mean? It means the electron does has a spin!

Thus, we get .

In this respect, the Dirac equation perfectly solves the problem that the Schrödinger equation failed to solve, and predict the spin of particle successfully.

Any%

Why Dirac Equation?

Something About Schrödinger Equation

Dirac Equation in Relativistic Situations

Prediction of Electron Spin