Table of Contents
- Introduction
- Historical Background
- Relativistic Foundations
- Derivation of the Klein-Gordon Equation
- Lagrangian Formulation
- Euler-Lagrange Equation for Fields
- Plane Wave Solutions
- Canonical Quantization
- Creation and Annihilation Operators
- Commutation Relations
- Energy-Momentum Tensor
- Feynman Propagator
- Complex Klein-Gordon Field
- Symmetries and Conserved Currents
- Applications and Relevance
- Limitations and Further Developments
- Conclusion
1. Introduction
The Klein-Gordon (KG) field is the simplest model of a relativistic quantum field. It describes spin-0 bosonic particles and is foundational in quantum field theory (QFT). The KG equation extends classical field theory into the quantum realm, accounting for special relativity.
2. Historical Background
Originally developed as a relativistic analog of the Schrödinger equation, the Klein-Gordon equation was among the first attempts to describe scalar particles. Though inadequate for electrons (spin-1/2), it successfully models spin-0 particles and fields like the Higgs boson.
3. Relativistic Foundations
From special relativity, the energy-momentum relation is:
\[
E^2 = p^2 + m^2
\]
Using operator substitutions:
\[
E \rightarrow i\partial_t, \quad \vec{p} \rightarrow -i\vec{\nabla}
\]
we get the Klein-Gordon equation:
\[
\left( \Box + m^2 \right)\phi(x) = 0
\]
where \( \Box = \partial^\mu \partial_\mu = \partial_t^2 – \nabla^2 \) is the d’Alembertian.
4. Derivation of the Klein-Gordon Equation
Apply the operator form of energy and momentum to the relativistic relation:
\[
\left( \partial_t^2 – \nabla^2 + m^2 \right)\phi(x) = 0
\]
This second-order PDE is the field equation for a scalar particle with mass \( m \).
5. Lagrangian Formulation
The Lagrangian density for a real scalar field is:
\[
\mathcal{L} = \frac{1}{2} \partial^\mu \phi \partial_\mu \phi – \frac{1}{2} m^2 \phi^2
\]
It yields the KG equation via the Euler-Lagrange formalism.
6. Euler-Lagrange Equation for Fields
\[
\frac{\partial \mathcal{L}}{\partial \phi} – \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \right) = 0
\]
Plugging in the KG Lagrangian gives:
\[
\Box \phi + m^2 \phi = 0
\]
7. Plane Wave Solutions
General solutions are superpositions of plane waves:
\[
\phi(x) = \int \frac{d^3p}{(2\pi)^3 2E_p} \left( a_p e^{-ipx} + a_p^* e^{ipx} \right)
\]
with \( E_p = \sqrt{p^2 + m^2} \).
8. Canonical Quantization
Quantize the field \( \phi(x) \) and its conjugate momentum:
\[
\pi(x) = \frac{\partial \mathcal{L}}{\partial \dot{\phi}(x)} = \dot{\phi}(x)
\]
Impose equal-time commutation relations:
\[
[\phi(\vec{x}, t), \pi(\vec{y}, t)] = i\delta^3(\vec{x} – \vec{y})
\]
9. Creation and Annihilation Operators
Fourier expand:
\[
\phi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}} \left( a_p e^{-ipx} + a_p^\dagger e^{ipx} \right)
\]
Here \( a_p^\dagger \) creates, and \( a_p \) annihilates particles with momentum \( p \).
10. Commutation Relations
\[
[a_p, a_{p’}^\dagger] = (2\pi)^3 \delta^3(p – p’)
\]
These define a bosonic Fock space and the structure of the quantum theory.
11. Energy-Momentum Tensor
From Noether’s theorem:
\[
T^{\mu\nu} = \partial^\mu \phi \partial^\nu \phi – \eta^{\mu\nu} \mathcal{L}
\]
This gives conserved quantities such as total energy and momentum.
12. Feynman Propagator
The two-point function:
\[
\Delta_F(x – y) = \langle 0 | T\{ \phi(x) \phi(y) \} | 0 \rangle
\]
is the Green’s function of the KG equation. It appears in Feynman diagram computations.
13. Complex Klein-Gordon Field
Define \( \phi \) as complex. The Lagrangian becomes:
\[
\mathcal{L} = \partial^\mu \phi^* \partial_\mu \phi – m^2 \phi^* \phi
\]
This field supports a conserved U(1) current:
\[
j^\mu = i(\phi^* \partial^\mu \phi – \phi \partial^\mu \phi^*)
\]
14. Symmetries and Conserved Currents
Noether’s theorem links global symmetries to conserved charges. The complex KG field has a conserved particle number:
\[
Q = \int d^3x\, j^0(x)
\]
This conservation is central to particle physics.
15. Applications and Relevance
- Toy model in QFT
- Describes neutral mesons
- Appears in cosmology (inflaton field)
- Foundation for perturbation theory and renormalization
16. Limitations and Further Developments
- Does not describe spin-1/2 particles
- Second-order time equation makes interpretation of probability density ambiguous
- Replaced by Dirac and gauge field theories for fermions and interactions
17. Conclusion
The Klein-Gordon field exemplifies the transition from classical to quantum and non-relativistic to relativistic physics. It is mathematically elegant and pedagogically powerful, serving as a stepping stone to deeper insights in quantum field theory.
