Obtaining Dirac equation from symmetries

[tomkeus]

If we consider nonrelativistic QM, we will find Galilean group under the hood. Thanks to this, group theory enables us to find equations of motion directly from the symmetry principles. For example, if we take only geometric symmetries, we will get that the state space is broken into irreducible spaces characterized by momentum and angular momentum. For elementary system, angular momentum is zero so, condition that state lies in k-irreducible space (k being the momentum) is $$|\vec{P}||\Psi\rangle =|\vec{k}||\Psi\rangle$$ or $$P^2|\Psi\rangle=k^2|\Psi\rangle$$ which is Shrodinger equation for free particle $$(\nabla^2+k^2)\Psi(\vec{r})=0$$.

If we now take the Poencare group as group representing space-time symmetries and apply the same reasoning as above we will come to the relation $$P_{\mu}P^{\mu}|\Psi\rangle=m^2|\Psi\rangle$$ being the Klein-Gordon equation for free particle $$(\Box+m^2)\Psi(x)=0$$.

Now, what is common to both equations is that spin is excluded, that is to say, we are for some reason prevented to take into consideration both the rotational and translational symmetry so we cannot adequatly describe spin-$$\frac{1}{2}$$ particles.

My question is, is there a way to get Dirac equation the same way Schrödinger and Klein-Gordnon can be directly obtained from symmetries or we have to follow little more complicated path? If we cannot, what prevents us? Is the reason for this that Dirac equation is first order in derivatives? Is there a second order version of Dirac equation?

 

[George Jones]

My question is, is there a way to get Dirac equation the same way Schrödinger and Klein-Gordnon can be directly obtained from symmetries or we have to follow little more complicated path?

The answers are "yes" and "yes," that is, it is possible to derive the non-interacting Dirac equation from (infinite-dimensional unitary) representations of the Poincare group (special-relativistic version of the Galilean group), and the path is substantially more complicated.

 

[Haelfix]

Yup.

Dirac's eqn has been derived in many different contexts with different starting assumptions.. Some are a little adhoc, others are not (usually in increasing lvls of complexity).

The cleanest derivations imo tend to be algebraic in nature.