Free Scalar Field: Field Equation and Matrix Mechanics Solution

Field Equation for a Free Scalar Field

The action for a free scalar field φ(x) in four-dimensional spacetime is given by:

S = ∫ d⁴x [ (1/2) ∂μφ ∂μφ – (1/2) m²φ² ],

where:

  • φ(x) is the scalar field.
  • m is the mass of the scalar field.
  • ∂μ = ημν ∂ν, with the metric signature (+, -, -, -).

The Euler-Lagrange equation for this action yields the Klein-Gordon equation:

□φ + m²φ = 0,

where:

□ = ∂μ∂μ = ∂²/∂t² – ∇²

is the d’Alembertian operator.

Matrix Mechanics Representation

1. Discretizing Spacetime

Spacetime is replaced by a finite lattice with points xi (e.g., i = 1, 2, …, N). The field φ(x) is represented as a vector:

φ(x) → 𝐯 = [ φ(x₁), φ(x₂), …, φ(xN) ]T.

2. Representing Derivatives with Matrices

The derivative operators ∂μ and □ are approximated using finite difference methods:

  • The spatial Laplacian ∇² is represented by a matrix 𝐋.
  • The time derivative ∂²/∂t² is represented by another matrix.

The d’Alembertian □ becomes:

□φ → 𝐃𝐯,

where 𝐃 is the discretized representation of □.

3. Equation in Matrix Form

The Klein-Gordon equation in matrix form is:

𝐃𝐯 + m²𝐯 = 0.

4. Solution Using Eigenmodes

The solution can be found by diagonalizing the operator 𝐃 + m²𝐈. Let 𝐔 be the matrix of eigenvectors and Λ the diagonal matrix of eigenvalues:

𝐃 = 𝐔Λ𝐔†.

The solution to the matrix equation is then:

𝐯(t) = 𝐔 e-i√Λt𝐜,

where 𝐜 is determined by the initial conditions.

Physical Interpretation

  • Eigenmodes: Each eigenmode corresponds to a plane wave solution e-i(E t – k⋅x) with the dispersion relation E² = k² + m².
  • Superposition: The field evolution is a superposition of eigenmodes governed by the eigenvalues and eigenvectors of 𝐃.

Applications

This matrix mechanics representation is commonly used in numerical simulations of quantum field theories and lattice field theory computations.

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