Resolution of constraints and gauge equivalence in algebraic Schrödinger representation of quantum electrodynamics

Abstract

Abstract The algebraic formalism of QED is expounded in order to demonstrate both the resolution of constraints and to verify gauge equivalence between temporal gauge and Coulomb gauge on the quantum level. In the algebraic approach energy eigenstates of QED in temporal gauge are represented in an algebraic GNS basis. The corresponding Hilbert space is mapped into a functional space of generating functional states. The image of the QED-Heisenberg dynamics becomes a functional energy equation for these states. In the same manner the Gauß constraint is mapped into functional space. By suitable transformations the functional image of the Coulomb forces is recovered in temporal gauge. The equivalence of this result with the functional version of QED in Coulomb gauge is demonstrated. The meaning of the various transformations and their relations are illustrated for the case of harmonic oscillators. If applied to QCD this method allows an exact derivation of effective color "Coulomb" forces, in addition it implies a clear conception for the incorporation of various algebraic representations into the formal Heisenberg dynamics and establishes the algebraic "Schrödinger" equation for quantum fields in functional space.