# I Are special relativity rules encoded in the Dirac equation?

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1. Jun 22, 2017

### raeed

This may seem like a stupid question, but i can't get my head around this so please bear with me.
I just looked at the derivation of Dirac equation and my question is:
do the solutions for a free particle obey special relativity? because if yes why? I mean I thought using E2=(mc2)2+(pc)2 would just give us more accurate energy levels. this energy momentum relation doesn't seem to include in itself the rules of special relativity, for example, where does it state that nothing goes faster than light? to state that don't we need the following relations:
E=γmc2 and P=γmv?
if this relation does in fact have SP rules encoded in it, then that means i can derive all the rules of SP from this simple relation right?

2. Jun 22, 2017

### Physics Footnotes

You are absolutely correct that choosing a relativistic-looking Hamiltonian to represent the energy observable does not guarantee that the resulting differential equation will be relativistic. This is just a heuristic argument that reflects the way in which Dirac arrived at his equation, thereby providing a motivation for it. To establish its relativistic character we need to understand what it means for a theory to be relativistic.

The characteristic you need for a theory to be relativistic is that the theory works (without change) in any inertial reference frame, under the assumption that inertial reference frames are connected by Lorentz Transformations. The way we check this is to show that the relevant equation is Lorentz Covariant. This is a bit of a laborious exercise but you'll find it carried out in any decent exposition of the Dirac Equation.

Note also that the condition of Lorentz Covariance will guarantee that a speed measured as $c$ in one frame will also be measured as $c$ in another frame, because this fact follows directly from the Lorentz Transformation connecting the frames. However, it does not guarantee that nothing travels faster than light, or indeed that other strange shenanigans won't happen concerning this speed. If you do a search for the term Zitterbewegung, for example, you'll see just such a puzzling speed-of-light effect one can derive from the Dirac Equation.