That said, one can argue for the form of the free field Lagrangian starting from the assumption that it should be possible to write using only the field itself and its (first) derivatives and be a Lorentz scalar. Not leading to non-linearities requires the Lagrangian to be quadratic in the field and Lorentz invariance pretty much nails down the kinetic term up to a constant that can be normalized away.Demystifier said:Usually KG and Dirac equation are first "derived" without the Lagrangian, then the Lagrangian which leads to this equation is just guessed, and finally one shows that this Lagrangian really leads to this equation. See e.g. Ryder, Quantum Field Theory.
Lagrangian density is a concept in theoretical physics that generalizes the Lagrangian function to fields. It is a function that depends on the field variables and their derivatives with respect to space and time.
A scalar field is a physical quantity that has a single value at each point in space and time. It is described by a scalar function, which is a function that assigns a scalar value to each point in space-time.
A vector field is a physical quantity that has both magnitude and direction at each point in space and time. It is described by a vector function, which is a function that assigns a vector to each point in space-time.
A Spinor field is a mathematical object used in quantum field theory to describe particles with half-integer spin. It is a complex-valued function that transforms in a specific way under rotations in space-time.
Lagrangian densities are used to derive the equations of motion for fields in physics. By varying the action defined by the Lagrangian density, one can obtain the Euler-Lagrange equations that describe how the field evolves in space and time.