The Pre-Physical State Space

Ontic State Space Beyond Spacetime

To unify quantum theory and gravity at a foundational level, it is necessary to introduce a state space that is not defined within spacetime, but rather underlies it. We refer to this as the pre-physical state space. Its elements are ontic states representing complete physical configurations prior to any notion of geometry, causality, or temporal ordering. Spacetime, when it appears, is understood as an emergent structure derived from relations within this deeper space.

Formally, let $\mathcal{O}$ denote the set of ontic states. No a priori metric, causal structure, or dimensionality is assumed on $\mathcal{O}$. Instead, $\mathcal{O}$ is defined by relational and structural properties sufficient to encode both quantum and gravitational degrees of freedom. This avoids presupposing the very structures whose incompatibility motivates unification.

Hilbert–Geometric Hybrid Structures

The pre-physical state space must be rich enough to reproduce the linear structure of quantum theory and the geometric structure of gravity. We therefore consider hybrid structures combining elements of Hilbert space geometry with generalized geometric or topological features.

Concretely, local sectors of $\mathcal{O}$ admit Hilbert-like structures, allowing superposition, inner products, and unitary transformations. Globally, however, $\mathcal{O}$ may possess non-linear or non-manifold features that encode gravitational constraints. The usual Hilbert space of quantum mechanics is recovered as an effective description valid in regimes where geometric backreaction is negligible. Conversely, classical spacetime geometry emerges when quantum interference between sufficiently distinct sectors is suppressed.

This hybrid viewpoint treats neither Hilbert space nor spacetime as fundamental, but instead regards both as limiting representations of a more primitive relational structure.

Measures on State Space and Physical Meaning

A crucial ingredient of the pre-physical framework is the introduction of a measure on $\mathcal{O}$. This measure assigns weights to regions of the ontic state space and plays a dual role: it underlies probabilistic predictions in quantum theory and selects dynamically stable configurations in gravitational contexts.

Importantly, the measure is not interpreted as subjective probability. Rather, it quantifies structural volume or typicality within $\mathcal{O}$. Physical predictions correspond to properties that are stable with respect to this measure, in the sense that they dominate almost all admissible realizations. In the quantum limit, this reduces to the squared-norm measure on Hilbert space, while in the gravitational limit it constrains the admissible geometric configurations by suppressing singular or degenerate regions.

Constraints of Consistency and Non-Degeneracy

Not all measures or structures on $\mathcal{O}$ are physically acceptable. Structural stability imposes two fundamental constraints. First, consistency: the induced physical predictions must be internally coherent and invariant under allowed reparameterizations or decompositions of the state space. Second, non-degeneracy: the measure must not collapse onto sets of zero structural dimension or assign equal weight to mutually incompatible configurations.

Degenerate measures lead to either singular geometries in gravity or arbitrary outcome weights in quantum theory. By excluding such cases, the framework enforces a unique class of admissible measures compatible with both theories. These constraints ensure that the pre-physical state space supports a well-defined emergence of spacetime, dynamics, and probability without internal contradiction.