Dirac Equation and Plane-Wave Solutions

1. The Search for a Relativistic Wave Equation

Let's start by understanding why the Klein-Gordon equation, which we used for scalar fields, is not the whole story. While it's a perfectly valid relativistic equation for spin-0 particles, it has historical "problems" that motivated Paul Dirac to search for an alternative. His primary goal was to find an equation, similar to the Schrödinger equation, that was first-order in the time derivative but also compatible with special relativity.

A great way to understand Dirac's starting point is to see his equation as an attempt to take the "square root" of the relativistic energy-momentum relation, $E^2=p^2{c}^2+m^2{c}^4$.

As the video explained, Dirac proposed a Hamiltonian of the form $H=c(\vec{\alpha }\cdot \vec{p})+\beta m{c}^2$. For this to be consistent with relativity, squaring this Hamiltonian must yield the Klein-Gordon relation $H^2=p^2{c}^2+m^2{c}^4$. This requirement imposes a set of strict algebraic constraints on the objects $\vec{\alpha }$ and $\beta$:

1. ${\alpha }_i^2={\beta }^2=1$ (the identity matrix)
2. $\{{\alpha }_i,{\alpha }_j\}={\alpha }_i{\alpha }_j+{\alpha }_j{\alpha }_i=0$ for $i\ne j$
3. $\{{\alpha }_i,\beta \}=0$

These conditions cannot be satisfied if ${\alpha }_i$ and $\beta$ are numbers; they must be matrices. Furthermore, it can be shown that the smallest matrices that satisfy these anti-commutation relations are 4x4. This is the origin of the four-component nature of the Dirac field.

The Covariant Form and the Dirac Lagrangian

While the Hamiltonian form is intuitive, a more elegant and manifestly Lorentz-covariant form is achieved by defining the gamma matrices (${\gamma }^{\mu }$).

As you've just read, the Lagrangian for a free Dirac field is:

This compact expression is the foundation for everything we will do with fermions. It is Lorentz invariant by construction, and as we saw, it correctly yields the Dirac equation upon variation.

2. Plane-Wave Solutions and the Emergence of Antimatter

Now that we have the equation, let's find its solutions. We will follow the standard physicist's approach: look for plane-wave solutions.

As the video demonstrated, for any given momentum $\vec{p}$, the Dirac equation has four independent solutions.

• Two positive-energy solutions, which we will label $u^{(1)}(p)$ and $u^{(2)}(p)$.
• Two negative-energy solutions, which we will label $u^{(3)}(p)$ and $u^{(4)}(p)$.

The existence of negative-energy solutions was deeply troubling. In classical physics, they can be ignored. But in quantum mechanics, a particle in a positive-energy state could, in principle, radiate a photon and fall into a negative-energy state, and then another, and so on, spiraling down an infinite ladder of negative energy. This would make all atoms unstable.

Dirac's Brilliant Solution: The Dirac Sea and Antimatter

Dirac's resolution to this crisis was a moment of pure genius, leading directly to the prediction of antimatter.

The Modern Interpretation: Antiparticle Spinors

The Dirac Sea is a beautiful and intuitive picture, but it has its own conceptual difficulties (e.g., how does it work for bosons?). The modern approach, rooted in Quantum Field Theory, uses the Feynman-Stückelberg interpretation: a negative-energy particle traveling backward in time is physically indistinguishable from a positive-energy antiparticle traveling forward in time.

This leads us to redefine our solutions. We keep the two positive-energy solutions, $u^{(1)}$ and $u^{(2)}$, to describe our particle (e.g., an electron). We then use the two negative-energy solutions to construct two new positive-energy solutions, which we call $v^{(1)}$ and $v^{(2)}$, to describe the corresponding antiparticle (e.g., a positron).

We seek plane-wave solutions of the form:

• Particles: $\psi (x)=u(p)e^{-ip\cdot x}$, where $p_0=E>0$. Inserting this into the Dirac equation gives the equation for u-spinors:
  $({\gamma }^{\mu }{p}_{\mu }-m)u(p)=0\text{or}(/p-m)u(p)=0$
• Antiparticles: $\psi (x)=v(p)e^{+ip\cdot x}$, where $p_0=E>0$. Inserting this gives the equation for v-spinors:
  $(-{\gamma }^{\mu }{p}_{\mu }-m)v(p)=0\text{or}(/p+m)v(p)=0$

The video you watched shows how to explicitly construct these $v$-spinors from the original negative-energy solutions.

To find explicit solutions, one often works in a specific basis for the gamma matrices. A common choice is the Dirac-Pauli basis:
${\gamma }^0=\left(\begin{array}{cc}I& 0\\ 0& -I\end{array}\right),{\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$
where ${\sigma }^i$ are the 2x2 Pauli matrices. In this basis, the four standard solutions for a particle/antiparticle with four-momentum $p$ are constructed by choosing basis spinors for the spin (e.g., ${\chi }_1=\left(\begin{array}{c}1\\ 0\end{array}\right)$, ${\chi }_2=\left(\begin{array}{c}0\\ 1\end{array}\right)$) and solving the coupled equations. For normalization to $2E$ particles per volume (a convention we'll appreciate during quantization), the solutions are:

• Particle spinors ($u$-spinors):
  $u^{(s)}(p)=\sqrt{E+m}\left(\begin{array}{c}{\chi }_s\\ \frac{\vec{\sigma }\cdot \vec{p}}{E+m}{\chi }_s\end{array}\right)(s=1,2)$
• Antiparticle spinors ($v$-spinors):
  $v^{(s)}(p)=\sqrt{E+m}\left(\begin{array}{c}\frac{\vec{\sigma }\cdot \vec{p}}{E+m}{\chi }_s\\ {\chi }_s\end{array}\right)(s=1,2)$

You don't need to memorize these forms, but you should recognize their structure: they are four-component objects built from two-component Pauli spinors, ${\chi }_s$, which encode the spin information.

3. The Final Puzzle: The Origin of Spin

We are now left with one final question: why are there two particle solutions ($u^{(1)},u^{(2)}$) and two antiparticle solutions ($v^{(1)},v^{(2)}$)? Dirac did not set out to explain spin; he was simply looking for a relativistic wave equation. In one of the most remarkable results in theoretical physics, the property of spin-1/2 emerged naturally and unavoidably from his equation.

The two solutions correspond to the two possible spin orientations of a spin-1/2 particle: spin-up and spin-down.

Conclusion

This lesson has been a deep dive into one of the most important equations in physics. We have seen how Dirac's quest for a first-order relativistic equation led to a structure rich with physical meaning.

Key Takeaways:

• The Dirac equation, $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$, is a relativistic wave equation for spin-1/2 fermions.
• Its mathematical consistency requires the wave function $\psi$ to be a four-component spinor and the ${\gamma }^{\mu }$ to satisfy the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$.
• The equation possesses both positive and negative-energy solutions. The negative-energy solutions are reinterpreted as describing positive-energy antiparticles.
• For any given momentum, there are four independent plane-wave solutions:
  • Two particle solutions ($u^{(1)},u^{(2)}$), corresponding to spin-up and spin-down states. They satisfy $(/p-m)u=0$.
  • Two antiparticle solutions ($v^{(1)},v^{(2)}$), also corresponding to spin-up and spin-down states. They satisfy $(/p+m)v=0$.
• The intrinsic spin-1/2 nature of the particle is not an ad-hoc addition but an emergent property of the equation itself.

Preview of the Next Lesson:
We now have the classical Dirac field and its solutions. The next step, mirroring our work with the Klein-Gordon field, is to perform canonical quantization. We will promote the Dirac field $\psi$ to an operator and impose quantization conditions. However, because fermions obey the Pauli exclusion principle, we will need to use anti-commutation relations instead of commutation relations. This will lead us to creation and annihilation operators for fermions and antifermions, and the construction of the fermionic Fock space.