# Lagrangian density for the spinor fields

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• wasi-uz-zaman
In summary, Lagrangian densities come from classical fields, from a quantum equation, or from the symmetries that should exist. You need to figure out what invariant terms you can write down that are quadratic in the fields ##\psi## and ##\bar\psi##, and you need to figure out what spin rep the eigenstates should have.

#### wasi-uz-zaman

TL;DR Summary
lagrangian density for the spinor fields
hi, i have seen lagrangian density for spin 0 , spin 1/2, spin 1 , but i am not getting from where these langrangian densities comes in at a first place. kindly give me the hint.
thanks

Lagrangian densities do not ”come from” anywhere. They define their respective models and you can typically argue for particular forms based on different symmetry principles. Then it is a matter of asking if the model is useful for describing something observable.

zaman786, Astronuc, vanhees71 and 2 others
What if you give a conctrete example and then we can work it out, what symmetries are present.

vanhees71 and topsquark
i am new in QFT , wants to write lagrangain density - how should i proceed , do i strat from Dirac equation and its adjoint - if so than i have seen derivation of Dirac equation from the Dirac lagrangian density-here confusion lies- so my point is seeing symmetries alone how can we model lagrangian density for spin 1/2 fields .
thanks

Being a Dirac field already tells you the representation of the Lorentz group. Then for the free theory you will have to look at what invariant terms you can write down that are quadratic in the fields ##\psi## and ##\bar\psi##. You will essentially end up with the Dirac Lagrangian from there.

topsquark, vanhees71 and malawi_glenn
The Lagrangian comes from a few places.

One is, you take a classical field that has a Lagrangian, and write the corresponding equivalent for quantum field theory. That only takes you so far, since there isn't really a good classical field theory for electrons. You can get the E&M part of QED this way, but you get stuck fairly quickly for the electron parts.

You also get stuck doing this with gravity, but a different way. The General Relativity Lagrangian converted to quantum formalism gives you a system that is not renormalizable.

Another place Lagrangians come from is starting with the quantum equation that seems to work. So the Dirac equation is a starting place for electrons. You write the Dirac equation and then you try to figure out what Lagrangian it corresponds to.

Another source is, you start with the interactions you know exist from experiments. So you go "backwards" from the interactions, and try to figure out what the Lagrangian should be that gives you the appropriate Feynman diagrams. So, for example, if you have an interaction where a photon comes off a charged particle line, then that gives you a psi-bar-A-psi term.

Another is the symmetries that should exist. If you have a gauge symmetry, then you write down a generic object that has that symmetry. For relativistic invariance, you write down things that are scalars under relativistic transformations. For particle numbers or other quantum numbers that are conserved, you write down generic mathematical objects that conserve those. This brings in such things as SU(3) matrix reps, or unitary matrix reps, and so on.

Another thing is, you want the correct spin rep. Photons are spin-1, for example, and sit in a vector rep. Electrons are spin-1/2 and sit in a spinor rep.

Then there are few things around the edges. For example, you want to make sure that the eigenstates don't have inconvenient instabilities. If your vacuum is unstable, for example, the theory can exhibit things like spontaneous production of infinite particles out of the vacuum.

vanhees71 and topsquark