The possibility that quantum mechanics is foundationally the same as classical theories in explaining phenomena in space and time is postulated. Such a view is motivated by interpreting the experimental violation of Bell inequalities as resulting from questions of geometry and algebraic representation of variables, and thereby the structure of space, rather than realism or locality. While time remains Euclidean in the proposed new structure, space is described by Projective geometry. A dual geometry facilitates description of a physically real quantum particle trajectory. Implications for the physical basis of Bohmian mechanics is briefly examined, and found that the hidden variables pilot-wave model is local. Conceptually, the consequence of this proposal is that quantum mechanics has common ground with relativity as ultimately geometrical. This permits the derivation of physically meaningful quantum Lorentz transformations. Departure from classical notions of measurability is discussed.

An analysis of both the original and the CHSH Bell inequalities is presented. Two additional mathematical assumptions are identified in the theorem. These are: all variables in the inequalities have a field algebraic structure, and all variables have measurability as a mathematical property. This means the variables are of metric-type, mathematically indistinguishable from those of classical theories. The consequences of attributing the violation of Bell's theorem to these assumptions are examined.

Calculations assuming position indeterminacy in the Dirac equation are reported.Energy shift contributions for low-lying states of hydrogen-like atoms are calculated by treating the position indeterminacy as a perturbation additional to standard quantum mechanics. The results are consistent with the current discrepancy between theory and experiment for Lamb shifts in hydrogen, deuterium and the helium ion.

Position indeterminacy contributions to the decay width of ortho-positronium (o-Ps)are calculated for lowest order. Contributions improve agreement between theory and the Ann Arbor group measurements, while suggesting a value larger than that of the Tokyo group.

The possibility that quantum mechanics is founded on non-metric space has been previously introduced as an alternative consequence of Bell inequalities violation. This work develops the concept further by an analysis of the iconic Heisenberg gedanken experiment. No lower bound is found in the gedanken uncertainly relation for a non-metric spatial background. This result has the fundamental consequence that the quantum particle trajectory is retained in non-metric space and time. Assignment of measurement number-values to unmeasured incompatible variables is found to be mathematically incorrect. The current disagreement between different formulations of the empirically verified error-disturbance relations can be explained as a consequence of the structure of space. Quantum contextuality can likewise be explained geometrically. An alternative analysis of the extended EPR perfect anti-correlation configuration is given. The consensus that local causality is the sole assumption is found to be incorrect. There is also the additional assumption of orientation independence. Inequalities violation does not therefore mandate rejection of local causality. Violation of the assumption of orientation independence implies rejection of metric, non-contextual variables algebraically representing physical quantities.