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Physics Notes

Here are some derivations and notes from my coursework.

Quantum Field Theory

Definition: The Lagrangian Density

For a scalar field \(\phi\), the Lagrangian density is given by:

\[ \mathcal{L} = \frac{1}{2} (\partial_\mu \phi)(\partial^\mu \phi) - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{4!} \phi^4 \]

This describes a massive scalar field with a quartic interaction.

Tip: Metric Signature

In these notes, I strictly use the mostly minus metric signature \((+,-,-,-)\).

General Relativity

Important

Remember that covariant derivatives \(\nabla_\mu\) do not commute when acting on vectors in a curved spacetime. This gives rise to the Riemann curvature tensor:

\[ [\nabla_\mu, \nabla_\nu] V^\rho = R^\rho_{\sigma\mu\nu} V^\sigma \]