Klein-Gordon Field

Table of Contents

1. Introduction
2. Historical Background
3. Relativistic Foundations
4. Derivation of the Klein-Gordon Equation
5. Lagrangian Formulation
6. Euler-Lagrange Equation for Fields
7. Plane Wave Solutions
8. Canonical Quantization
9. Creation and Annihilation Operators
10. Commutation Relations
11. Energy-Momentum Tensor
12. Feynman Propagator
13. Complex Klein-Gordon Field
14. Symmetries and Conserved Currents
15. Applications and Relevance
16. Limitations and Further Developments
17. Conclusion

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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.

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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.

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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.

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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 \( m \).

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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.

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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:

\[
\Box \phi + m^2 \phi = 0
\]

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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 \( E_p = \sqrt{p^2 + m^2} \).

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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:

\[
[\phi(\vec{x}, t), \pi(\vec{y}, t)] = i\delta^3(\vec{x} – \vec{y})
\]

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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 \( p \).

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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.

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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.

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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.

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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:

\[
j^\mu = i(\phi^* \partial^\mu \phi – \phi \partial^\mu \phi^*)
\]

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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.

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15. Applications and Relevance

• Toy model in QFT
• Describes neutral mesons
• Appears in cosmology (inflaton field)
• Foundation for perturbation theory and renormalization

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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

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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.

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