In the present work the functional integral technique was applied to the study of the collective excitations of electron gas at a spherical surface. The quanta of these collective excitations are the surface plasmons. Starting from the fundamental principles of the quantum theory, the generating functionals of the many-electron system in a spherical surface were introduced in two different cases, when the electron-electron Coulomb interaction is absent or is present. By means of the Hubbard-Stratonovich transformation, the generating functional of the electron system in the presence of the electron-electron interaction was transformed into that of a scalar field, the effective action of this scalar field was established and the plasmon field was introduced. Then the expression of the Hamiltonian of the quantum field of plasmons was derived and the system of algebraic equations determining the energies of plasmons was established.