Leading invariant subspace

All of the results of Section 6.1 apply, mutatis mutandis, to irreducible Lie algebra representations. For example, if T is a homomorphism of Lie algebra representations, then the kernel of T and the image of T are both invariant subspaces. This leads to Schur s Lemma for Lie algebra representations. 

A decomposition similar to Eq. (622) for the advanced dynamics of a system may be accomplished by using a projection operation P that projects onto a dynamically invariant subspace GP- This leads to... 

From the present perspective, an obvious weakness of these calculations is the restriction to the one-photon subspace of Fock space. Because every gauge (choice of g(x,x )) leads to its own commutation relations (42), each has its own Fock space so projection on a one-photon subspace is not gauge-invariant. We have previously shown [21] that the one-photon part ofHmt (i.e linear in the charge e) in an arbitrary gauge is related to the Coulomb gauge interaction (g1 = 0) by... 

This requirement rules out the standard position operator (multiplication by x), because it does not commute with the sign of the energy. The most prominent operator that leaves the positive energy subspace invariant is the Newton-Wigner position operator. Together with any other position operator that has the same property, it leads to inconsistencies with relativistic causality (see [9],... 

---