Klein–Gordon

The Klein–Gordon equation is the direct scalar quantization of the quadratic relativistic mass shell. Its covariance is immediate, but its second-order time evolution and indefinite conserved density show why a relativistic field cannot simply be treated as a Schrödinger wavefunction with a new dispersion relation.

Reading sequence

1. Complete Metric and Units → Four-Vectors → Energy–Momentum Relation to fix the shared metric, covariant contractions, and two-sheeted mass shell.
2. Read the Klein–Gordon Equation for the scalar differential equation and its introductory analysis.
3. Work through Plane-Wave Solutions, the Conserved Current, and the Klein–Gordon Inner Product. They connect initial data, gauge-covariant flux, and normalized solution spaces.

The owner derives

$$\left(\Box+\frac{m^2c^2}{\hbar^2}\right)\phi=0,$$

tests both plane-wave branches, identifies the two required initial data, derives the complex-field current, and shows why its time component is not a Born probability density over the full solution space. It also recovers the leading Schrödinger limit. The detailed controlled expansion belongs to Klein–Gordon to Schrödinger, with a Gaussian-packet application in The Scalar Nonrelativistic Limit in Practice.

Choose an application

Task	Pages	Main distinction to keep
Solve in a prescribed field	External Potentials	Covariant derivatives are ordered operators; a scalar mass shift differs from an electrostatic potential.
Find bound levels	The Scalar Relativistic Coulomb Problem	The source boundary and conserved norm are part of the model.
Propagate initial data and forcing	Solving Scalar Initial-Value Problems and its propagator owner	A retarded inverse supplies the source response; initial data supply a homogeneous solution.
Interpret a result	Interpreting the Klein–Gordon Equation	Energy, charge, one-particle norm, and detector probability are different objects.

How to read the result

Keep three claims separate:

• the differential equation is a Lorentz-covariant scalar equation;
• a complex solution has a conserved indefinite current associated with global phase symmetry;
• in QFT that current is robustly interpreted as charge, while the scalar amplitude becomes a field.

A real scalar has no nontrivial continuous $U(1)$ charge current. Negative frequency is a mode label before quantization, not by itself a completed antiparticle interpretation.

Checks to carry forward

Before using a scalar result, identify both Cauchy data or the condition that selects a frequency sector. State the mode normalization and whether a current is being interpreted as charge or probability. In an external field, distinguish canonical energy from $E-q\Phi$, and include boundary conditions at any singular source. For a propagator, name both its source factor and its temporal prescription.

The chapter develops wave mechanics and the structures needed for a field-theory handoff. A quantized many-particle state, local observables, and an interaction model require additional construction; they are not inferred from a successful solution of the scalar differential equation.

References

• J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw–Hill, 1964.
• W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd ed., Springer, 2000.
• W. Pauli and V. Weisskopf, “Über die Quantisierung der skalaren relativistischen Wellengleichung,” Helvetica Physica Acta 7, 709–731, 1934.