Abstract
The equivalence postulate approach to quantum mechanics aims to formulate quantum mechanics from a fundamental geometrical principle. Underlying the formulation there exists a basic cocycle condition which is invariant under D-dimensional Mobius transformations with respect to the Euclidean or Minkowski metrics. The invariance under global Mobius transformations implies that spatial space is compact. Furthermore, it implies energy quantisation and undefinability of quantum trajectories without assuming any prior interpretation of the wave function. The approach may be viewed as conventional quantum mechanics with the caveat that spatial space is compact, as dictated by the Möbius symmetry, with the classical limit corresponding to the decompactification limit. Correspondingly, there exists a finite length scale in the formalism and consequently an intrinsic regularisation scheme. Evidence for the compactness of space may exist in the cosmic microwave background radiation.
Highlights
The mathematical modelling of observational data on the smallest and largest distance scales currently rests on two main theories, quantum mechanics and general relativity
In the one dimensional case the Mobius symmetry uniquely fixes the functional form of the inhomogeneous term to be given by the Schwarzian derivative, i.e. ∼ {qa; qc},where the Schwarzian derivative is defined by f 3f 2
An alternative approach to quantising general relativity is to formulate a geometric approach to quantum mechanics
Summary
The mathematical modelling of observational data on the smallest and largest distance scales currently rests on two main theories, quantum mechanics and general relativity. The string consistency conditions dictate that additional extra degrees of freedom beyond the Standard Model, are needed to obtain a consistent theory of perturbative quantum gravity, which is an unambiguous prediction of the theory These may be interpreted as extra spacetime dimensions, and/or as additional gauge symmetries beyond those observed in the Standard Model. As long as the low scale data does not indicate departure from perturbative Standard Model parameterisation of the experimental observations, string theory continues to provide the most detailed framework to calculate the Standard Model parameters from a more fundamental theory. String theory provides a viable perturbative framework to explore how the Standard Model parameters may arise from a fundamental theory and to develop a phenomenological approach to quantum gravity. In the one dimensional case the Mobius symmetry uniquely fixes the functional form of the inhomogeneous term to be given by the Schwarzian derivative, i.e. (qa; qc) ∼ {qa; qc} ,where the Schwarzian derivative is defined by f 3f 2
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