---
title: "The Klein-Gordon and Dirac Equations"
description: "Demanding a wave equation that treats space and time the way relativity does gives a second-order equation with negative-energy solutions and no positive probability density. Insisting on first order "
canonical: https://lightmysky.com/learn/science/the-klein-gordon-and-dirac-equations-mt_njCW9OEevW
source: https://lightmysky.com/learn/science/the-klein-gordon-and-dirac-equations-mt_njCW9OEevW.md
retrieved: 2026-09-12
---

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# The Klein-Gordon and Dirac Equations

Demanding a wave equation that treats space and time the way relativity does gives a second-order equation with negative-energy solutions and no positive probability density. Insisting on first order in time forces the wavefunction to carry four components, and spin and antiparticles arrive as consequences rather than as additions.

Subject: Science · Area: Quantum & Modern Physics · Ages 23 to 24
Page: https://lightmysky.com/learn/science/the-klein-gordon-and-dirac-equations-mt_njCW9OEevW

## Ready when they can

- Builds the Klein-Gordon equation from the relativistic energy-momentum relation and states the two problems it brings
- Explains why first order in time forces matrices, and what the four components of a Dirac spinor stand for
- Reads spin one half and the existence of antiparticles out of the equation rather than assuming either

## Lesson: Why relativity forces spin and antiparticles

Relativity says energy and momentum obey E squared equals p squared c squared plus m squared c to the fourth. Quantizing that relation directly gives the Klein-Gordon equation, which is second order in time.

Klein-Gordon brings two problems. Its solutions include negative energies with no bottom, and the probability it defines can turn negative, so it cannot serve as a one-particle equation.

Demanding first order in time forces the coefficients to anticommute, which plain numbers cannot do, so they must be matrices. The wavefunction then carries four components: particle spin up, particle spin down, antiparticle spin up, antiparticle spin down.

**Example.** Spin one half falls out of those matrices, matching the two Stern-Gerlach bands, and the negative roots describe the positron found in 1932. Both arrived as consequences, never as extra assumptions.

**Recap.** Relativity plus first order in time gives matrices, four components, spin one half and antiparticles.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Field Quantisation: Creation Operators and Particle Number](https://lightmysky.com/learn/science/field-quantisation-creation-operators-and-particle-number-mt_BJyZWQA5cb)
- [Spacetime Intervals and Four-Vectors](https://lightmysky.com/learn/science/spacetime-intervals-and-four-vectors-mt_CYdi7Qpvrf)
- [Angular Momentum from Its Commutation Relations](https://lightmysky.com/learn/science/angular-momentum-from-its-commutation-relations-mt_u0y20XJh-v)