The manifestly gauge invariant relativistic electrodynamics of neutral atoms is derived from a path-dependent Lagrangian. A second quantised spinor field is used for the description of electrons, while the infinitely heavy nucleus is represented by a static Coulomb field. Classical Lagrangian formalism is constructed. Generalised Poisson brackets are obtained from canonical formalism. Canonical quantisation is performed. The whole description is without any reference to the electromagnetic potentials. A comparison with other formulations is given.