Adding the Hermetian conjugate is necessary in order to ensure that the Lagrangian, which is a mathematical function used to describe the dynamics of a system, remains real and physically meaningful. This is important for making accurate predictions and calculations in physics.
The Hermetian conjugate is added to the Lagrangian in order to account for the complex nature of certain physical quantities, such as quantum wavefunctions. By taking the complex conjugate of these quantities, the Lagrangian can accurately describe the system's dynamics in a real and consistent manner.
No, the Hermetian conjugate can only be added to Lagrangians for systems that involve complex quantities. If the system is purely real, then there is no need to add the Hermetian conjugate.
Adding the Hermetian conjugate does not change the form of the Lagrangian equations of motion, but it does ensure that these equations result in real and physically meaningful solutions. This allows for more accurate predictions and calculations in physics.
Yes, there are alternative methods such as using a gauge symmetry, which involves introducing additional fields to the Lagrangian. However, adding the Hermetian conjugate is often the simplest and most straightforward way to ensure the Lagrangian remains real and consistent.