Ontological Models Framework in Quantum Theory

• The Ontological Models Framework is a mathematically precise, realist architecture that represents quantum theories using distributions over ontic state trajectories.
• It generalizes standard point-like models by introducing a process space that recovers classical results, including the PBR theorem and Born-rule predictions via path integrals.
• The framework clarifies the ontic versus epistemic status of quantum states and supports hybrid constructions that bridge classical and process-based quantum representations.

The Ontological Models Framework provides a mathematically precise, realist architecture for representing probabilistic theories—especially quantum theory—in terms of underlying structures known as ontic states and the processes or distributions over these states. It offers both “point-like” models (standard framework) and a generalized “process-based” paradigm that departs from the classical picture, allowing for a richer representation of quantum phenomena and clarifying the subtleties of ontic versus epistemic interpretations of quantum states. This framework underpins much of contemporary research into quantum foundations, contextuality, and objective-versus-epistemic probabilistic assignments.

1. Ontic State Space and Process Space

Let $\Lambda$ denote the ontic state space. In classical implementations, $\Lambda$ consists of elements $\lambda$ representing complete hypothetical descriptions of a system at an instant. The key generalization is from point-localization in $\Lambda$ to path-localization in its process space:

• Path (Process) space $\Gamma = \Lambda^{[0,T]}$: Each $\lambda(\cdot): [0,T] \to \Lambda$ specifies the evolution of ontic states over the preparation and measurement interval.
• Measure on process space: There exists a reference measure $d\mu[\lambda(\cdot)]$ on $\Gamma$ such that any distribution $P[\lambda(\cdot)]$ is normalized: $$\int_{\Gamma} d\mu[\lambda(\cdot)]\,P[\lambda(\cdot)] = 1$$.

This expansion to process space means that the fundamental building blocks of the theory are entire trajectories rather than instantaneous states, and all operational statistics must be recovered via integrals over paths.

2. Representation of Preparations

In the generalized framework, quantum state preparation does not correspond to a simple distribution over $\Lambda$, but instead to a distribution over paths through $\Lambda$:

• For each quantum pure state $|\psi\rangle$, associate a path distribution $P_\psi[\lambda(\cdot)] \geq 0$ on $\Gamma$, satisfying $$\int_\Gamma d\mu[\lambda(\cdot)] P_\psi[\lambda(\cdot)] = 1$$.
• At no time is the system “in” a single $\lambda \in \Lambda$; it always executes a full trajectory $\lambda(\cdot)$.
• In toy spin-$\tfrac{1}{2}$ models, one may impose a relative-frequency constraint: the fraction of time $\lambda(t)$ visits certain regions must reproduce Born-rule frequencies, but in full generality, all preparation statistics are fully encoded in $P_\psi[\cdot]$ (Yong, 2017).

3. Measurement as Functionals on Paths

Quantum measurement outcomes in the process-based framework are not direct evaluations of the ontic state at a single instant, but are determined probabilistically by the entire path:

• Measurement response functional $\xi_M\bigl(k[\lambda(\cdot)]\bigr)$ maps the entire trajectory to probabilities for outcome $k$.
• The functionals satisfy normalization: $\sum_k \xi_M(k[\lambda(\cdot)]) = 1$ for all $\lambda(\cdot)$.
• Outcome probabilities: Given preparation $P_\psi$ and measurement $M$,

$$p(k | \psi, M) = \int_\Gamma d\mu[\lambda(\cdot)]\,\xi_M\bigl(k[\lambda(\cdot)]\bigr)\;P_\psi[\lambda(\cdot)]$$

• All quantum operational statistics (POVM predictions, Born rule) can be recovered by suitable choices of $P_\psi$ and $\xi_M$ such that $p(k|\psi, M) = \langle\psi|\Pi^M_k|\psi\rangle$, for the relevant POVM elements $\Pi^M_k$ (Yong, 2017).

4. Reduction to Standard Point-Like Models and the PBR Theorem

The process-based formalism strictly generalizes the standard ontological models framework:

• Point-like limit: If the only permitted paths are constant in time, $\lambda(t) \equiv \lambda_0 \in \Lambda$, then $\Gamma$ collapses to $\Lambda$. The preparation distribution $P_\psi[\lambda(\cdot)]$ becomes $$p(\lambda_0|\psi)\,\delta[\lambda(\cdot) - \lambda_0]$$ and $\xi_M(k[\lambda(\cdot)]) = \xi_M(k|\lambda_0)$.
• Standard ontological model formula is recovered:

$$p(k|\psi, M) = \sum_{\lambda_0 \in \Lambda} p(\lambda_0|\psi)\,\xi_M(k|\lambda_0)$$

This is the architecture analyzed in major contextuality and $\psi$-ontic/$\psi$-epistemic theorems, notably the PBR result (Yong, 2017).

The “ontological theorem for processes” asserts that if quantum theory is to be reproduced, the distributions $P_\psi[\cdot]$ for distinct $|\psi\rangle$ must have disjoint support in $\Gamma$, which, in this point-like limit, reduces exactly to the PBR condition: non-overlapping $p(\lambda|\psi)$.

5. Ontic and Epistemic Status of Quantum States

The process-based ontological model reveals crucial subtleties in the status of quantum states:

• No well-defined $p(\lambda|\psi)$: the system never “sits” at a particular $\lambda$.
• A given $\lambda(t)$ can occur along multiple $\psi$-paths—$\lambda$ alone is not sufficient to reconstruct $|\psi\rangle$.
• Quantum state $|\psi\rangle$ is neither ontic nor epistemic about $\Lambda$; rather, $|\psi\rangle$ determines the path-distribution $P_\psi[\cdot]$, which fixes the probability distribution over outcomes for any measurement $M$ via the path-integral formula.
• Quantum probabilities are objective in the sense that all agents with knowledge of $P_\psi[\cdot]$ and $\xi_M[\cdot]$ agree on the assignment $p(k|\psi,M)$ (Yong, 2017).

6. Hybrid and Limiting Constructions

The framework allows for “hybrid” models:

• Hybrid preparation: Preparation may follow standard ontological protocol, while measurements act as path functionals.
• Hybrid measurement: Measurements may respond only to instantaneous $\lambda(t)$, but preparations are path-distributions. These constructions interpolate between the classical ontological and process-based scenarios and provide a spectrum to analyze departures from strict quantum behaviour.

7. Conceptual and Foundational Significance

The ontological models framework—particularly in its process-based generalization—supports key conclusions:

• The operational statistics of quantum theory require a representation in terms of distributions over processes, not merely states.
• The quantum state does not directly encode underlying reality at a fixed time; rather, it encodes objective knowledge about measurement statistics via the ensemble of possible trajectories.
• This architecture dissolves the apparent dichotomy between $\psi$-ontic and $\psi$-epistemic models for quantum theory, showing that quantum states have an epistemic role, but concerning outcome statistics, not the underlying ontic configuration.
• The process-based ontological model further illuminates the assumptions, limits, and possible extensions of quantum realism, and generalizes the PBR and related no-go theorems to a trajectory (path-support) formulation (Yong, 2017).

In summary, the Ontological Models Framework provides a rigorous foundation for representing and interpreting operational quantum statistics in terms of process distributions and measurement functionals. When restricted to point-like paths, it collapses to the conventional ontological models picture and its associated foundational theorems. Its process-based formulation extends the capacity to analyze realist models of quantum phenomena beyond standard representations and clarifies the epistemic/ontic status of quantum states in this context.