A Comment on Background Independence in Quantum Theory. Issue 1 (14th July 2015) - Brändas, Erkki J.
Abstract: In this communication we take up the significance and purpose of selecting the proper coordinate system from the flat space‐time of non‐relativistic theories to the quantum theoretic formulation of general relativity. The universal background problem is straight forwardly framed as a momentum‐energy portrait in nexus with its space‐time conjugates. The description is based on operator matrix algebra, where the related analogue of the secular equation yields a Klein‐Gordon type equation and the associated Minkowski eigentime element. The energy‐momentum and their conjugate partners are represented by spaces that have (+, −) signatures. The general theory implicates both non‐zero‐ and zero rest‐mass entities, and it is proved that the conjugate relationship between energy and time provide a simple derivation of the Schwarzschild line element for the case of a gravitational field outside a spherical non‐rotational uncharged mass. This result, indicating the appearance of a black hole as a true singularity in the energy‐time formulation, and obtained as a direct consequence of their conjugate relationship, manifests background independence in concert with Einstein's equivalence principle. Inducing a reformulation of the Lorentz Transformation respecting the indefinite Minkowski metric, displays an interesting relation between complex dilations and indefinite metric spaces, validating the complex symmetric ansatz. Abstract : It is proven that a basic quantum theoretic ansatz of momentum‐energy and their conjugates, space time, impart the celebrated Schwarzschild metric, showing contrary to the classical case, the Schwarzschild radius to be a true theoretical singularity. An abstract linear mapping of the Lorentz transformation in the Minkowski space displays an interesting relation between complex dilation and indefinite metric spaces.