Bhabha first-order wave equations. V. Indefinite metric and Foldy-Wouthuysen transformations

We prove the existence of a Foldy-Wouthuysen (FW) transformation which decouples the mass states of all the Bhabha Poincaré generators and diagonalizes the Hamiltonian. Since the Bhabha fields operate in an indefinite-metric space, such an existence is not a priori guaranteed. The FW-transformed generators are given and satisfy the Poincaré Lie algebra. We observe that although the FW transformation expressed as a power series in $c^{-1\mathrm{}}$ more clearly exhibits the physics of the situation, it is only in the Dirac and Duffin-Kemmer-Petiau special cases of the Bhabha fields that the FW-transformation power series can easily be summed to yield a closed-form expression. The general closed-form expression surrenders more readily to another technique. Therefore, as a first calculation, in this paper we present a method of generating the FW transformation as a power series in $c^{-1\mathrm{}}$. Our discussion concentrates on the indefinite metric, the physics which is evident in the power-series form (such as size and types of Zitterbewegung), and on a detailed examination of special cases up to spin 3/2. In all the above, a special handling of the built-in subsidiary components of the integer-spin fields is once again necessary. We also comment on what the indefinite metric may be implying about the possibility of finding a totally consistent high-spin field theory.