Free Field Operator: Building Quantum Fields
Free Field Operator: Building Quantum Fields How Quantum Fields Evolve Without Interactions π― Our Goal We aim to construct the free scalar field operator \( A(x,t) \), which describes a quantum field with no interactions—just free particles moving across space-time. π§ Starting Expression This is the mathematical formula for our field \( A(x,t) \): \[ A(x, t) = \frac{1}{(2\pi)^{3/2}} \int_{\mathbb{R}^3} \frac{1}{\sqrt{k_0}} \left[ e^{i(k \cdot x - k_0 t)} a(k) + e^{-i(k \cdot x - k_0 t)} a^\dagger(k) \right] \, dk \] x: Spatial position t: Time k: Momentum vector k₀ = √(k² + m²): Relativistic energy of the particle a(k): Operator that removes a particle (annihilation) a†(k): Operator that adds a particle (creation) 𧩠What Does This Mean? The field is made up of wave patterns (Fourier modes) linked to momentum \( k \). It behaves like a system that decides when and where ...
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