The functional integral
Consider the functional integral for a scalar field. Formulate the rules for obtaining Feynman diagrams from the generating functional.
Obtain the expressions for the vacuum energy (the energy density of the zero-point oscillations) and draw the corresponding Feynman diagrams for the \(\lambda\phi^4\) model.
Show that vacuum loops (“bubbles”) do not contribute to scattering amplitudes.
Compute the vacuum energy density for scalar, fermion, and massive and massless vector fields. Write the final expression for the mean vacuum energy density including quantum corrections.