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Algebraic Schrödinger Representation of Quantum Chromodynamics in Temporal Gauge and Resolution of Constraints

Abstract The algebraic formalism of QCD is expounded in order to demonstrate the resolution of Gauß constraints on the quantum level. In the algebraic approach energy eigenstates of QCD 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 QCD-Heisenberg dynamics becomes a functional energy equation for these states. In the same manner the Gauß constraints are mapped into functional space. In functional space the Gauß constraints can be exactly resolved. The resolutions are defined by nonperturbative recurrence relations. The longitudinal color electric energy can be expressed by means of these resolvents, which leads to "dressed" color Coulomb forces in temporal gauge. Although present in the system, the longitudinal vector potentials do not affect its energy eigenvalues. This leads to a selfconsistent subsystem within the functional energy equation in temporal gauge which has to be identified with a functional energy equation in Coulomb gauge. In addition this procedure implies a clear conception for the incorporation of various algebraic representations into the formal Heisenberg dynamics and establishes the algebraic “Schrödinger” equation for QCD in functional space.