Covariance of the Dirac equation

It is obtained by multiplying (3) with the Dirac matrix /3 = 70 and dividing by 

From 7q = I4 we see immediately that V = cjoVcov- The equation (83) is called the Dirac equation in covariant form. It is best suited for investigations concerning relativistic invariance, because it me is a scalar (which by definition of a scalar is invariant under Lorentz transformations) and the term (7,5) is written in the form of a Minkowski scalar product (if 7 and d were ordinary vectors in Minkowski space, the invariance of this term would be already guaranteed by (81). 

We are going to prove the invariance of the free Dirac equation under Poincare transformations in the form of the following statement Whenever ip ct,x) = i x) is a solution of the free Dirac equation, then (j x) — M (A (x - a)) is also a solution of the free Dirac equation. Here it is assumed that M and A are related by (82). 

In the presence of an external field the Dirac equation will not be invariant, because an external field is not invariant under all Poincare transformations (unless it is a constant). But at least we can expect that the Poincare transformed spinor (x) — M t/j(A (x — a)) is a solution of the Dirac equation with an appropriately transformed potential matrix Here it has to be assumed that 

The proof is done by a simple computation We denote x = A (x — a) and apply the corresponding Poincare transformation to both sides of (83). Hence we multiply both sides by M and replace all x by x. This gives 

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In order to obtain Poincare-covariance of the Dirac equation, Apy must behave as an electromagnetic vector potential, as far as proper Poincare transformations are concerned. The right behavior under a parity transformation would be... 

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