A Lagrangian density , for scalar field , vector field and Spinor field

[zaman786]

TL;DR Summary

how do we get starting Lagrangian density of scalar field or Dirac (Spinor_) field

hi, I have go through many books - they derive Dirac equation from Dirac Lagrangian, KG equation from scalar Lagrangian - but my question is how do we get Dirac or scalar Lagrangian at first place as our starting point - kindly help in this regard or refer some book - which clearly elaborate this-
thanks

 

[Demystifier]

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.

 

[Orodruin]

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.

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.

Then for the interacting theory we write down pretty much all renormalizable terms allowed by Lorentz and gauge invariance.

In short, there are not many other forms the Lagrangian could take based on a few basic and relatively reasonable assumptions.