Poincare transformations

Non-Abelian electrodynamics has been presented in considerable detail in a nonrelativistic setting. However, all gauge fields exist in spacetime and thus exhibits Poincare transformation. In flat spacetime these transformations are global symmetries that act to transform the electric and magnetic components of a gauge field into each other. The same is the case for non-Abelian electrodynamics. Further, the electromagnetic vector potential is written according to absorption and emission operators that act on element of a Fock space of states. It is then reasonable to require that the theory be treated in a manifestly Lorentz covariant manner. 

The group of Poincare transformations consists of coordinate transformations (rotations, translations, proper Lorentz transformations...) linking the different inertial frames that are supposed to be equivalent for the description of nature. The free Dirac equation is invariant under these Poincare transformations. More precisely, the free Dirac equation is invariant under (the covering group of) the proper orthochronous Poincare group, which excludes the time reversal and the space-time inversion, but does include the parity transformation (space reflection). 

Poincare transformations are implemented as unitary transformations in the Hilbert space of the Dirac equation. Their general structure is the following... 

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... 

We note that the time-evolution according to the free Dirac equation is a special Poincare transformation (translation in the time-direction of Minkowski space). It is a unitary transformation generated by the free Dirac operator Hq ... 

The behavior of the covariant potential K,ov under Poincare transformations immediately implies that the potential V (x) is transformed into... 

It has to be assumed that V x) describes the external field at the space-time point X after a Poincare transformation. This requires a certain behavior of the coefficient functions Vj under Poincare transformations. Let me illustrate this with an example. 

If we perform a Poincare transformation of the electromagnetic potentials according to (84), then the corresponding new field strengths E and B will again satisfy Maxwell s equations. 

The last expression gives the potential matrix in the standard representation. The transformation law (89) gives precisely the Poincare transformation of the electromagnetic field strengths E and B, which can be combined into a tensor field on Minkowski space. 

The transformation law of electromagnetic fields under Poincare transformations (as it follows from Maxwell s equations) is almost compatible with the potential transformation law (89). There is a slight mismatch concerning the behavior under the parity transformation. The matrix structure of (91) would require that the fields E and B change their sign under a space reflection, but the electromagnetic field strengths don t. Therefore, the Dirac equation with this potential matrix is not covariant with respect to a parity transformation. 

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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