QFT: Lay of the land

\begin{align} \mathcal{L} &= -\frac{1}{2} \sum_{μ, ν} η^{μν} \frac{\partial φ}{x^μ} \frac{\partial φ}{x^ν} - \frac{1}{2} κ^2 φ^2 \\ &= \frac{1}{2} \left[ \Box φ - κ^2 φ^2 \right] \end{align}

The equations of motion for a spin 0 particle is given by

\begin{equation} \Box φ = κ^2 φ \end{equation}

• E.g.: Higgs Boson

\begin{equation} \mathcal{L} = i \sum_{μ} \bar{Ψ}\, γ^{μ} \frac{\partial}{\partial X^μ} Ψ - m\bar{Ψ}\, Ψ \end{equation}

The equations of motion for a spin 1/2 particle is given by:

\begin{equation} i \sum_{μ} γ^{μ} \frac{\partial}{\partial X^μ} Ψ = m Ψ \end{equation}

• E.g.: Electron

\begin{equation} \mathcal{L} = - \frac{1}{4} \sum_{μ, ν} F^{μν} F_{μν} \end{equation}

The equations of motion for a spin 1 particle is given by:

\begin{equation} \sum_{μ, ν} F^{μν} = 0 \end{equation}

• E.g.: Photon

\begin{equation} \mathcal{L} = \frac{1}{16 π G} g^{μν} R_{μν} \end{equation}

The equations of motion for a spin 2 particle is given by:

\begin{equation} R_{μν} = 0 \end{equation}

• E.g.: Graviton(?)