We give a closed-form, finite-polynomial expression for the Foldy-Wouthuysen (FW) transformation ${U}^{\ensuremath{-}1}$ for arbitrary-spin Bhabha fields. Our result is obtained by appropriately normalizing the Lorentz transformation operator, expressing this transformation as a finite polynomial, and then properly interpreting the energy, mass, and especially momentum operators involved. For integer-spin fields the built-in subsidiary components are projected out. An algorithm is given which allows one to easily write the expression for the FW transformation of any Bhabha field. We comment on the properties of ${U}^{\ensuremath{-}1}$. We note that the columns of the FW transformation are the metric-orthonormal eigenvectors of the Hamiltonian, ${\stackrel{^}{u}}_{k}$, and provide the relation of the ${\stackrel{^}{u}}_{k}$ to the solutions of the wave equation, ${\ensuremath{\psi}}_{k}$. Special cases up to $\mathcal{S} = 3$ are listed and investigated. Some physical and mathematical applications of our method and results are also given.

Bhabha first-order wave equations. VI. Exact, closed-form, Foldy-Wouthuysen transformations and solutions

We prove the existence of a Foldy-Wouthuysen (FW) transformation which decouples the mass states of all the Bhabha Poincare 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 Poincare Lie algebra. We observe that although the FW transformation expressed as a power series in c/sup -1/ 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 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/sup -1/. 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 maymore » be implying about the possibility of finding a totally consistent high-spin field theory. In a later paper we present a derivation of the exact, closed-form transformation. (AIP)« less

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

Spinors obeying the Dirac equation also obey the Klein‐Gordon equation, but the converse is not true. In this paper we make a systematic study of four‐component spinors obeying the Klein‐Gordon equation, with special regard for the additional solutions. The starting point is the Lagrangian density L=ψ,μγμγνψ,ν−m2ψψ, and we first develop from it the theory of a classical spinor field. We then proceed to the canonical quantization of this field and are confronted by some anticommutators of creation and annihilation operators equal to −1, and the subsequent need for an indefinite metric in Hilbert space. Quantum electrodynamics can be reformulated, and in spite of a modified fermion propagator, gradient coupling, and vertices with two photon lines, the amplitude for Compton scattering to order e2 is the usual one. Special problems arising for massless fermions are indicated, and we note that the four‐fermion point interaction is now renormalizable. Some interesting variations of strong interactions also be...

Klein‐Gordon Equation for Spinors

We use a powerful matrix theorem to derive the closed-form, finite-polynomial, matrix expression of a Lorentz transformation for the class of relativistic, first-order wave equations described in the preceding paper Combined with the method developed there, this theorem allows the exact, closed-form expression to be given for a Foldy-Wouthuysen (FW) transformation An algorithm is given which is a simple procedural prescription for writing down the FW transformation results we have obtained We discuss two specific examples which illustrate our results, and in a third example show that our method must be modified if the wave equation is not first-order in space (AIP)

Foldy-Wouthuysen transformations in an indefinite-metric space. IV. Exact, closed-form expressions for first-order wave equations

Foldy-Wouthuysen transformations, Lorentz transformations and the De Sitter group

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http://streaming.ictp.it/preprints/P/66/011.pdf

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Equivalent representations of the Dirac equation

Completeness and dispersion relations in the complex angular momentum plane

Proper time formulation of the Dirac electron theory

Completeness, unitarity and dispersion relations in potential scattering