This chapter investigates a quantum gas in the grand-canonical ensemble with the grand canonical Hamiltonian ensemble. The path integral formalism pioneered by Feynman has proven to be an extremely useful tool for understanding and handling quantum mechanics, quantum field theory, and statistical mechanics. The path integral approach is a powerful tool that considers nonperturbative calculations for the investigation of many-particle systems. It is worth noting that while the path integral approach seems to be quite cumbersome in quantum mechanics, it does provide a powerful tool in quantum field theory. The entire aforementioned derivation is calculated in order to justify the evaluation of the path integral at equal times. The chapter examines the example of an atomic Fermi gas by considering an electron gas. The Fermi liquid concept is the deviation of the electron gas from the ideal gas behavior. It utilizes the so-called diagram technique to calculate some quantities.