Quantum Field Theory: An Intuitive Approach - L04 | Dirac Equation as Consistent Relativistic Theory

Quantum Field Theory: An Intuitive Approach - Lecture 04 | Dirac Equation as a Consistent Relativistic Theory

In Lecture 04 of the Quantum Field Theory: An Intuitive Approach series, we continue our discussion of the Dirac equation and show how it resolves the fundamental problems encountered in relativistic quantum mechanics.

Building on the previous lecture, we analyze the probability current and probability density associated with the Dirac equation and demonstrate why it is positive definite, in contrast to the Klein–Gordon theory. This resolves the issue of negative probability densities that prevented the Klein–Gordon equation from being interpreted as a consistent single-particle equation of motion.

We explain how the structure of the Dirac equation naturally leads to a well-defined conserved current and discuss the physical interpretation of its solutions. The lecture clarifies why the Dirac equation represents a decisive improvement over the Klein–Gordon equation and why it became a cornerstone in the development of quantum field theory.

This lecture completes the conceptual transition from relativistic quantum mechanics to a field-theoretic description of particles.

Closes the logical gap left by Klein–Gordon theory
Reinforces the necessity of spinor structure
Naturally motivates second quantization and fields

Key Points Covered:
.1. Problem with the Klein–Gordon equation — appearance of negative probability densities.
2. Dirac equation as a relativistic theory — linear in time and space derivatives.
3. Positive-definite probability density in Dirac theory.
4. Conserved probability current and continuity equation for Dirac spinors.
5Why Dirac theory is consistent and its role as a bridge to quantum field theory.