Nomological Impossibilities in Quantum Physics

• Nomological impossibilities are constraints imposed by fundamental laws that rule out certain states or operations even if they are conceptually conceivable.
• They play a key role in quantum measurement theory by enforcing limits through principles like microcausality, locality, and discrete Hilbert space structures.
• These constraints challenge interpretations such as hidden-variable models and counterfactual reasoning by mandating law-level restrictions in quantum systems.

Nomological impossibilities are constraints imposed by the fundamental laws of physics that strictly forbid the existence, definition, or realization of certain states, operations, or measurement scenarios, regardless of technological capacity. These impossibilities delineate the boundary between what is mathematically or conceptually conceivable and what is permitted by the governing dynamics and symmetries of physical theories.

1. Formal Definition and Theoretical Foundations

A nomological impossibility is a scenario, state, or operation that is mathematically or logically conceivable, but whose existence is strictly forbidden by the dynamical or structural laws of a physical theory. Such impossibilities contrast with mere practical limitations or technological obstacles. The term “nomological” refers to constraints imposed by the nomoi (laws) of nature, in distinction to “anomic” (lawless or unconstrained) cases.

A classic taxonomy, as articulated in the context of quantum foundations, distinguishes:

• Ontic vs. Epistemic: Pertains to whether the wavefunction (or state) refers to an element of physical reality or merely encodes knowledge/statistics.
• Nomic (Nomological) vs. Anomic: Nomic scenarios are regulated by law-level constraints; anomic ones allow maximal freedom and variability in behavior not governed by explicit law (Drezet, 2021).

Nomological impossibilities typically arise when the joint application of theoretical assumptions (such as microcausality, locality, Hilbert space structure, or the properties of quantum observables) forbids a class of quantum measurements, states, or interpretations that would otherwise seem possible.

2. Nomological Impossibility in Quantum Measurement Theory

A key locus for nomological impossibilities is quantum measurement theory in relativistic and field-theoretic settings.

Sorkin's “Impossible Measurement” Scenario

The protocol introduced by Sorkin considers a sequence of interventions on quantum fields in overlapping but non-causally related spacetime regions. Applied naively, this protocol appears to allow instantaneous signaling between spacelike regions by correlating the outcomes of measurements executed in causally disjoint domains.

However, the imposition of law-level (nomological) constraints derived from relativistic quantum field theory rescinds this possibility:

• Microcausality: Operator commutation relations $[A(x),B(y)] = 0$ whenever $(x-y)^2 < 0$ enforce that measurements in spacelike separated regions cannot influence each other.
• Localization and No-Overlap: State preparations and measurements are constrained such that projectors associated with distinct spacetime regions $O_i$ are orthogonal, precluding any overlap $P_i^{O_1}U_tP_i^{O_3} = 0$.
• Symmetrization: For identical fermions (or bosons), the global wavefunction must be (anti)symmetric, and only one branch of the wavefunction can propagate to any given detector.
• Locality of Interaction: Detector–field couplings represented in the Hamiltonian cannot simultaneously satisfy commutation for measurements on two distinct operators in the same local apparatus (Huhtala et al., 9 Jul 2025).

Collectively, these constraints ensure that FTL signaling protocols constructed from such joint measurement sequences are non-realizable: the necessary operations are not just technologically unavailable, they are forbidden by the physical theory—classic nomological impossibility.

3. Discrete Hilbert Space, Impossible Counterfactuals, and Bell’s Theorem

In the Rational Quantum Mechanics (RaQM) framework, a profound example of nomological impossibility emerges from gravitationally induced discretization of Hilbert space. Here, only basis states expressible with rational probability amplitudes and rational phase increments are defined on the physical state space—a large but discrete set forming a fractal, $p$-adic Cantor-like geometry (Palmer, 21 Jan 2026).

Rationality Constraints and State Nonexistence

• For any putative state $|\psi\rangle = \sum_{i=1}^N \psi_i |i\rangle$ to exist in a given basis, it must satisfy $|\psi_i|^2 \in \mathbb{Q}$ and $\phi_i/(2\pi)\in \mathbb{Q}$ for all $i$.
• Bases where any amplitude-squared or phase is irrational correspond to non-existent (nomologically impossible) counterfactual states.

Implications for Counterfactual Reasoning and Locality

• In Bell-type experiments, joint counterfactual assignments (such as hypothetical outcomes under alternative simultaneous settings) generally require simultaneous rationality among multiple angular variables (measurement directions), impossible except on a set of measure zero due to Niven’s theorem.
• Bell’s inequalities, which presume summability of all such counterfactual combinations, cannot even be formulated: some terms correspond to impossible counterfactuals absent from the rational state space.
• This rescues local causality: Bell violations arise not from signaling or nonlocality, but from the nomological impossibility of certain counterfactual joint settings (Palmer, 21 Jan 2026).

4. Nomological Impossibility in Ontological Models

Constraints at the law-dynamical level also rule out entire classes of hidden-variable models for quantum theory.

No-Go Theorems for $\Psi$-Anomic Models

A “$(x-y)^2 < 0$0-anomic” model is one where the response functions (which specify outcome probabilities for measurements given a hidden-variable $(x-y)^2 < 0$1) do not depend on the quantum state $(x-y)^2 < 0$2 except through $(x-y)^2 < 0$3. That is, $(x-y)^2 < 0$4 for all $(x-y)^2 < 0$5.

Drezet’s (Drezet, 2021) H-ROI(II) theorem shows:

• Any model that is both $(x-y)^2 < 0$6-anomic, satisfies Preparation Independence (PIP), and restricted ontic indifference (ROI—a locality-type constraint for specific quantum paths) is inconsistent with quantum predictions.
• No $(x-y)^2 < 0$7-anomic model survives PIP + ROI: the wavefunction must enter fundamentally into the law governing outcomes.
• This result extends and strengthens previous no-go results by Pusey-Barrett-Rudolph (PBR) and Hardy, which variously restrict $(x-y)^2 < 0$8-epistemic and partial anomic models under broader or different assumptions.

This constitutes a nomological impossibility for any interpretation seeking to treat the quantum state as mere epistemic or anomic structure; law-level constraints enforce nomological (lawlike) status for the wavefunction in viable hidden-variable accounts.

5. Nomological Impossibility and the Ontology of Bohmian Mechanics

A paradigmatic debate in the ontology of Boh