ADOH IKECHUKU PROMISE

The objective of this project is to derive the conserved current associated with the Lagrangian density of a scalar field theory. Employing the principles of classical field theory and Noether's theorem, we investigate the consequences of the global U(1) symmetry present in the Lagrangian for a complex scalar field. By applying Noether's systematic procedure, which relates symmetries of the action to conserved currents, we calculate the Noether current corresponding to the global phase invariance of the theory. The conserved current is then explicitly determined by evaluating the functional derivatives with respect to the scalar field and its first derivative. This current plays a crucial role in the quantization of the scalar field via a canonical procedure and elucidates the underlying connection between symmetries and conservation laws as per Noether's profound theorem. The analysis is carried out for both the Klein-Gordon and the complex scalar field, providing insights into the general methodology while highlighting the interpretational aspects in each case. The overarching aim is to exemplify the utility of Noether's
theorem as a powerful tool for unveiling conserved quantities from symmetry considerations within field theories