Classical Field Theory

Table of Contents

1. Introduction
2. Motivation and Overview
3. Fields as Physical Quantities
4. Types of Classical Fields
5. Lagrangian Formulation of Field Theory
6. Euler-Lagrange Equations for Fields
7. Example: Scalar Field
8. Electromagnetic Field as a Classical Field
9. Energy-Momentum Tensor
10. Noether’s Theorem and Conservation Laws
11. Hamiltonian Formulation of Fields
12. Canonical Quantities
13. Classical Field Theory and Special Relativity
14. Limitations and Transition to Quantum Field Theory
15. Conclusion

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1. Introduction

Classical field theory provides a mathematical framework to describe physical quantities distributed over space and time. It underpins our understanding of electromagnetism, gravitation, fluid dynamics, and sets the stage for quantum field theory. Unlike particle-based mechanics, field theory emphasizes continuous fields rather than point particles.

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2. Motivation and Overview

Fields describe the configuration of physical systems at every point in space and time. They are vital when:

• The interaction is long-range (e.g., electromagnetic fields).
• The medium has continuous degrees of freedom.
• The goal is to be compatible with special relativity.

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3. Fields as Physical Quantities

A field is a quantity defined at every point in space and time:

• Scalar field \( \phi(x, t) \): single-valued function (e.g., temperature)
• Vector field \( \vec{A}(x, t) \): directional quantity (e.g., velocity field)
• Tensor fields: higher-order generalizations

Mathematically, a field is a map from spacetime to a set of physical values.

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4. Types of Classical Fields

• Scalar fields: mass density, temperature, potential.
• Vector fields: electric and magnetic fields.
• Tensor fields: stress tensors, gravitational field in general relativity.
• Spinor fields (in relativistic settings): describe particles with spin-½.

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5. Lagrangian Formulation of Field Theory

Like classical mechanics, field theory can be formulated using the principle of least action. The action is defined as:

\[
S = \int \mathcal{L}(\phi, \partial_\mu \phi, x^\mu)\, d^4x
\]

where \( \mathcal{L} \) is the Lagrangian density.

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6. Euler-Lagrange Equations for Fields

The field equations are obtained by extremizing the action:

\[
\frac{\partial \mathcal{L}}{\partial \phi} – \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \right) = 0
\]

This generalizes the Euler-Lagrange equations from particle mechanics to fields.

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7. Example: Scalar Field

Consider a real scalar field \( \phi(x, t) \) with Lagrangian density:

\[
\mathcal{L} = \frac{1}{2} \partial^\mu \phi \partial_\mu \phi – \frac{1}{2} m^2 \phi^2
\]

The Euler-Lagrange equation yields the Klein-Gordon equation:

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

where \( \Box = \partial^\mu \partial_\mu \) is the d’Alembertian operator.

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8. Electromagnetic Field as a Classical Field

The electromagnetic field is described by the vector potential \( A^\mu \), and the Lagrangian is:

\[
\mathcal{L} = -\frac{1}{4} F^{\mu\nu} F_{\mu\nu}
\]

where \( F^{\mu\nu} = \partial^\mu A^\nu – \partial^\nu A^\mu \). The Euler-Lagrange equations yield Maxwell’s equations in vacuum.

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9. Energy-Momentum Tensor

Defined as:

\[
T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial^\nu \phi – \eta^{\mu\nu} \mathcal{L}
\]

This tensor:

• Encodes energy and momentum densities.
• Is conserved for Lagrangians invariant under spacetime translations.

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10. Noether’s Theorem and Conservation Laws

Noether’s theorem links symmetries of the Lagrangian to conservation laws:

• Time translation → energy conservation.
• Space translation → momentum conservation.
• Phase symmetry → charge conservation.

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11. Hamiltonian Formulation of Fields

The canonical momentum is:

\[
\pi(x) = \frac{\partial \mathcal{L}}{\partial \dot{\phi}(x)}
\]

The Hamiltonian density is:

\[
\mathcal{H} = \pi(x)\dot{\phi}(x) – \mathcal{L}
\]

The total Hamiltonian:

\[
H = \int d^3x\, \mathcal{H}
\]

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12. Canonical Quantities

The Poisson brackets for fields are:

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

This structure is carried over into quantum field theory through commutators.

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13. Classical Field Theory and Special Relativity

The field Lagrangian is constructed to be Lorentz invariant, ensuring that the theory respects the principles of special relativity. Fields transform according to representations of the Lorentz group (scalar, vector, tensor).

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14. Limitations and Transition to Quantum Field Theory

Classical field theory fails to:

• Describe quantum phenomena (e.g., discrete energy levels, entanglement).
• Account for particle creation/annihilation.
• Include Planck’s constant \( \hbar \).

These limitations are addressed in quantum field theory, which quantizes the classical fields.

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15. Conclusion

Classical field theory provides a powerful and elegant framework for describing systems with infinitely many degrees of freedom. It plays a foundational role in physics, underlying everything from electromagnetism to general relativity. Its structure — based on Lagrangians, symmetries, and conservation laws — carries over directly into quantum field theory, making it essential for any serious study of modern theoretical physics.

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