Rational Quantum Mechanics (RaQM) Overview

• Rational Quantum Mechanics (RaQM) is a reconstruction framework that reformulates quantum theory using logical, Bayesian, epistemic, and discrete approaches.
• It employs methodologies such as Carnapian rational reconstruction, Bayesian coherence with desirable gambles, and epistemic estimation to derive key quantum predictions.
• RaQM challenges traditional axiomatizations by offering alternative interpretations of wave functions, measurement updates, and probability structures in quantum mechanics.

Searching arXiv for papers on Rational Quantum Mechanics and closely related formulations.

Rational Quantum Mechanics (RaQM) is a family of reconstruction programs in the foundations of quantum theory that treats quantum mechanics as rationally reconstructible rather than merely axiomatic. Across the literature, the label has been used for philosophically distinct projects: a Carnapian program of rational reconstruction and inferentialist semantics, Bayesian and coherence-based derivations from desirable gambles on Hermitian matrices, epistemic-restriction and relational formalisms, and discretized or local-realistic alternatives that aim to recover standard quantum predictions from rational or arithmetic constraints. This suggests that RaQM is not a single settled interpretation, but a research family unified by the demand that the logical, semantic, probabilistic, or ontological structure of quantum mechanics be made explicit (Toader, 2024, Benavoli et al., 2016, Budiyono, 2020, Palmer, 18 Feb 2026).

1. Scope, nomenclature, and principal usages

The term “Rational Quantum Mechanics” has been applied to several non-equivalent enterprises. In some papers it denotes a semantic-reconstructive program, in others a generalized Bayesian theory, and in still others a deterministic or discretized ontology. A concise map of the main usages is therefore indispensable.

Strand	Core claim	Representative papers
Carnapian reconstruction	QM should be formulated as a reconstructed formal language whose logic and semantics can be assessed	(Toader, 2024)
Bayesian/coherence RaQM	QM is the Bayesian theory generalized to Hermitian matrices; rationality is coherence of gambles	(Benavoli et al., 2016, Benavoli et al., 2016)
Epistemic/relational reformulation	QM is an estimation calculus under epistemic restriction or a theory of relational amplitudes/facts	(Budiyono, 2020, Yang, 2017, Martin-Dussaud et al., 2022)
Discrete or local-realistic RaQM	QM is recovered from discretized Hilbert space, path-based mechanics, or discrete stochastic models	(Palmer, 21 Jan 2026, Palmer, 18 Feb 2026, Tesse, 4 Feb 2026, Sciarretta, 2017)

These usages share a reconstructionist ambition but differ over what is being reconstructed. The central object may be a formal language, a space of acceptable gambles, an agent’s estimator, a relational amplitude network, a discretized state space, or a hidden-variable path ensemble. A plausible implication is that the term “RaQM” is best treated as umbrella terminology rather than as the name of a single canonical formalism.

2. Carnapian rational reconstruction and inferentialist semantics

In the Carnapian strand, the central thesis is that quantum mechanics cannot be properly philosophically assessed until it is rationally reconstructed. The motivating claim is that questions about whether the logic of physics should be revised cannot be answered until the theory is stated in a systematic form that includes formal logic. Within this framework, RaQM is the proposal that QM be presented as a Carnapian language: a formal calculus together with rules that determine meaning and inferential use. The paper distinguishes several articulations of “rational reconstruction”: the early constructional-system model of the Aufbau; the later conception of an empirical theory as a partially interpreted formal language with semantic or correspondence rules; and further developments involving Ramsey sentences and the Hilbert epsilon operator. The intended payoff is metasemantic evaluation, especially the test of categoricity: if a reconstruction is categorical, semantics is unique up to isomorphism; if not, the rules do not uniquely determine meaning (Toader, 2024).

This program is framed against two objections. One, associated with Healey, holds that reconstructing a theory in formal language with correspondence rules is impracticable and not useful for revealing scientific structure. The other, associated with Wallace, holds that Carnapian reconstruction fails because the observational/theoretical split is destabilized by theory-ladenness. The inferentialist reply is that both objections assume a representationalist semantics. If the reconstructed language is given inferential rather than representational articulation, and if inferentialism is global rather than restricted to logical vocabulary, then correspondence rules and the observational/theoretical distinction cease to bear the central semantic burden. In that setting, Carnap’s distinction between L-rules and P-rules becomes less fundamental, and QM can in principle be treated as a language whose meanings are fixed by inferential roles. The conclusion is limited but explicit: there is no principled reason to deny that QM could be rationally reconstructed in this manner, but such a reconstruction has not yet been produced; standard axiomatizations such as von Neumann’s are not obviously Carnapian reconstructions, and more recent reconstructions such as Hardy’s are not formulated as languages with explicit consequence relations (Toader, 2024).

3. Bayesian coherence, desirable gambles, and quantum probability

A second major usage identifies RaQM with a generalized Bayesian theory on the space of Hermitian matrices. In this formulation, the analogue of a classical gamble is a Hermitian matrix

$G \in C := \mathbb{C}_h^{n\times n},$

and rationality is captured by a coherent set of strictly desirable gambles $K\subseteq C$. The core requirements are that $K$ be a non-pointed convex cone, that every positive semidefinite nonzero matrix be desirable,

$H^+ := \{G\in C: G\gneq 0\}\subseteq K,$

and that an openness or Archimedean condition hold. Via the pairing $G\cdot R=\operatorname{Tr}(G^\dagger R)$, the dual objects are density matrices, and the exact representation theorem identifies coherent gamble sets with closed convex quantum credal sets. In this framework, Born expectations are coherent lower and upper previsions, measurement update is Bayes’ rule on density matrices, partial trace is marginalization, tensor product is independence, and unitary evolution follows from strong temporal coherence. The paper states the programmatic slogan explicitly: “Quantum mechanics is the Bayesian theory in the complex numbers” (Benavoli et al., 2016).

A closely related result extends Gleason’s theorem by formulating quantum probability as rational betting coherence. The setting is a gambling scenario in which a bookmaker announces a projective measurement

$\Pi^*=\{\Pi_1^*,\dots,\Pi_n^*\},$

and a gambler accepts or rejects Hermitian-matrix gambles. The same four coherence axioms appear in quantum form: accept partial gain, avoid partial loss, positive homogeneity, and additivity. The main theorem is that for every finite dimension $n>0$, including $n=2$, a probability measure on the lattice of orthogonal projectors is coherent if and only if it has the Born form

$p(\Pi_\ell)=\mathrm{Tr}(\Pi_\ell\rho),$

for a unique density matrix $\rho$. A central claim is therefore that dispersion-free assignments are not merely nonstandard but incoherent: the paper constructs gambles $K\subseteq C$0 and $K\subseteq C$1 that are each individually desirable under a dispersion-free assignment, while their sum $K\subseteq C$2 is a negative gamble, producing a Dutch book. In this version of RaQM, density matrices are rational belief states and Born’s rule is a rationality theorem rather than an added postulate (Benavoli et al., 2016).

4. Epistemic restriction and relational reformulations

An epistemic-restriction version reconstructs nonrelativistic quantum mechanics as a calculus of optimal estimation. The starting point is a statistical model in which position distributions are not freely assignable but are constrained by an underlying fluctuating momentum field $K\subseteq C$3, where the global nonseparable random variable $K\subseteq C$4 has mean zero and variance of order $K\subseteq C$5. An agent observes positions sampled from $K\subseteq C$6 and seeks a weakly unbiased, classically consistent, minimum-MSE estimator of the local momentum field. The estimator is

$K\subseteq C$7

and the central epistemic decomposition is

$K\subseteq C$8

Defining

$K\subseteq C$9

the formal rules of quantum mechanics emerge from averaging over the restricted ensemble. In this construction, the wave function is not an objective agent-independent field but an epistemic summary of the best estimate of momentum given position data; unitary Schrödinger evolution is a normative update rule when no new selection information is acquired, collapse is Bayesian updating after trajectory selection, and uncertainty follows from a Cramér–Rao trade-off between momentum-field estimation and mean-position estimation (Budiyono, 2020).

A relational strand instead takes relations, not monadic properties, as primitive. In one formulation, a system $K$0 is completely described relative to an apparatus $K$1 by a relational probability amplitude matrix $K$2, and measurement is a bidirectional process $K$3. Joint probabilities arise from the product of two conjugate directional amplitudes,

$K$4

so that

$K$5

Superposition and entanglement are then interpreted as distinct rules for counting alternatives: in the absence of entanglement, one sums amplitudes and then squares; in the entangled case, one squares first and then sums. Wave functions and reduced density matrices are secondary objects derived from $K$6, and Schrödinger evolution appears as the special case in which the relational matrix factorizes and has zero entanglement entropy (Yang, 2017).

A more radical relational reformulation is given by fact-nets. Here facts are “binary entities involving two systems,” represented graphically by a multigraph whose vertices are systems and whose edges are facts. For each system $K$7, the basic quantitative object is an amplitude

$K$8

with conditional probabilities

$K$9

Hermiticity, incompatibility of parallel facts, and the chain property provide the structural constraints. In this formalism, Hilbert spaces are derived from relational data rather than postulated; chain-completeness corresponds to commuting triangles of amplitude maps and yields unitary transformations as derived consistency conditions. Measurement is modeled as restriction of a fact-net relative to a reference system, and time is encoded by local orderings of neighborhoods rather than by intrinsic evolution of an observer-independent state (Martin-Dussaud et al., 2022).

5. Discretized Hilbert space, impossible counterfactuals, and Bell’s theorem

A sharply different use of RaQM denotes a deterministic, locally realistic, Schrödinger-equation-preserving but discretized alternative to standard quantum mechanics. In this approach, Hilbert space is gravitationally discretized, and states are defined only in bases satisfying rationality conditions such as

$H^+ := \{G\in C: G\gneq 0\}\subseteq K,$0

If squared amplitudes or phases are irrational in a given basis, the state is undefined there. Qubits are represented by bit strings of length $H^+ := \{G\in C: G\gneq 0\}\subseteq K,$1, with Born weights encoded by the frequencies of $H^+ := \{G\in C: G\gneq 0\}\subseteq K,$2 and $H^+ := \{G\in C: G\gneq 0\}\subseteq K,$3, and a hidden permutation $H^+ := \{G\in C: G\gneq 0\}\subseteq K,$4 plays the role of an outcome-selecting variable. Complementarity and noncommutativity are then reinterpreted through number-theoretic incompatibility of counterfactual bases, often using Niven’s theorem and an “impossible triangle” argument on the Bloch sphere. In the Bell context, the proposal distinguishes nominal measurement settings, which experimenters freely choose, from exact settings, which are not fully controllable because of unavoidable perturbations; measurement independence is therefore denied only for exact settings, not for nominal settings. On this view, Bell inequalities fail because the required counterfactual worlds are not jointly well-defined, not because of nonlocal influence (Palmer, 21 Jan 2026).

A closely related development formulates the same family of ideas through a discretized Riemann sphere and a constructive treatment of complex structure. The basic rationality rule is

$H^+ := \{G\in C: G\gneq 0\}\subseteq K,$5

and the imaginary unit is introduced at $H^+ := \{G\in C: G\gneq 0\}\subseteq K,$6 as a permutation/negation operator on a 2-bit string,

$H^+ := \{G\in C: G\gneq 0\}\subseteq K,$7

The paper argues that interference, complementarity, uncertainty, non-commutativity, and violation of Bell’s inequality all derive from the same hidden number-theoretic property of the cosine function once the unphysical continuum of Hilbert space is abandoned. The proposed metaphysical lesson is holism rather than nonlocality: subsystem descriptions are said to be defined only relative to a globally constrained state-space structure (Palmer, 18 Feb 2026).

6. Path-based, stochastic, and classicality-oriented rationalizations

Another set of approaches seeks a rational underpinning for quantum mechanics through explicit trajectory models. In positional mechanics, particles always occupy points in space, follow continuous, piecewise differentiable paths, and satisfy the literal kinematic relation $H^+ := \{G\in C: G\gneq 0\}\subseteq K,$8 along each path. The central probabilistic object is a conditional phase-space density $H^+ := \{G\in C: G\gneq 0\}\subseteq K,$9, and momentum-changing events occur through concatenations of compatible paths and particle-field interactions. The basic kinetic equation is

$G\cdot R=\operatorname{Tr}(G^\dagger R)$0

with $G\cdot R=\operatorname{Tr}(G^\dagger R)$1. In the large-$G\cdot R=\operatorname{Tr}(G^\dagger R)$2 regime this yields Madelung-like hydrodynamic equations, from which Schrödinger dynamics is recovered for a tuned subclass. The same framework claims to reproduce quantum position and momentum probabilities, tunnelling, entanglement, spin, and particle-identity effects, while treating Bohmian mechanics, stochastic mechanics, many worlds, and physical collapse as special or derived descriptions (Tesse, 4 Feb 2026).

A related but not explicitly named RaQM model reconstructs nonrelativistic quantum mechanics from a discrete spacetime random walk on a Euclidean lattice. Space and time are integer lattices,

$G\cdot R=\operatorname{Tr}(G^\dagger R)$3

with $G\cdot R=\operatorname{Tr}(G^\dagger R)$4, and lattice units chosen at Compton scales. Particles carry source momentum and phase, lattice nodes store local traces left by earlier particles, and quantum-force effects arise from local exchange rules when a particle encounters a node with a mismatched stored trace. Ensemble distributions are then shown to recover free-particle spreading, constant-force motion, the harmonic oscillator, particle-in-a-box spectra, the Delta potential, ring and sphere quantization, and Bell-CHSH-type correlations. The paper is explicit that it is “not about ‘Rational Quantum Mechanics’ by name,” but it presents a discrete, local, arithmetic reconstruction that is close in spirit to RaQM (Sciarretta, 2017).

A further associated use concerns the quantum-to-classical boundary. The “Quantum Ratio” paper introduces the criterion

$G\cdot R=\operatorname{Tr}(G^\dagger R)$5

where $G\cdot R=\operatorname{Tr}(G^\dagger R)$6 is the quantum fluctuation range of the center-of-mass wave function and $G\cdot R=\operatorname{Tr}(G^\dagger R)$7 is the linear size of the body defined from the internal bound-state wave function. The proposed rule of thumb is $G\cdot R=\operatorname{Tr}(G^\dagger R)$8 quantum behavior and $G\cdot R=\operatorname{Tr}(G^\dagger R)$9 classical behavior for the center of mass of an isolated body. Elementary particles satisfy $\Pi^*=\{\Pi_1^*,\dots,\Pi_n^*\},$0, so they remain quantum mechanical even when decohered; decoherence, mixed state, and classical state are explicitly distinguished. The authors connect this to a broader line of work sometimes called RaQM and use it to argue that classicality pertains to certain collective variables, not to a wholesale abandonment of quantum description (Konishi et al., 17 Feb 2025).

7. Comparative assessment, controversies, and current status

The surveyed literature shares several recurring motifs. First, RaQM programs typically treat the wave function as non-fundamental or at least non-self-sufficient: it becomes a credal object, an estimator, a bookkeeping device for relations, or an emergent summary of deeper structure. Second, probabilities are often recast as rational constraints: coherence of desirable gambles, Cramér–Rao efficiency, conditional amplitudes between facts, or admissibility under rationally defined bases. Third, measurement is typically reconstrued as update or interaction rather than as primitive collapse. These commonalities are real, but they do not eliminate the deep divergence between agent-relative, semantic, relational, and hidden-variable versions of the program.

The tensions are correspondingly sharp. Some RaQM formulations are explicitly non-ontic about the wave function and emphasize epistemic or normative structure (Budiyono, 2020); others are explicitly ontological and deterministic (Palmer, 21 Jan 2026). Some inherit the continuum Hilbert-space formalism and reinterpret it; others reject the continuum as unphysical (Palmer, 18 Feb 2026). Some reconstruct standard quantum mechanics from rationality constraints on Hermitian matrices (Benavoli et al., 2016); others propose alternative state spaces, graph grammars, or path ontologies (Martin-Dussaud et al., 2022, Tesse, 4 Feb 2026). This suggests that agreement on the word “rational” does not imply agreement on semantics, ontology, or mathematical architecture.

Several papers are also explicit about incompleteness. The Carnapian reconstruction program argues only that RaQM is possible in principle; it does not claim that such a reconstruction already exists (Toader, 2024). Fact-nets are presented as a program rather than a completed theory, with open questions including infinite fact-sets, time and causality, Wigner’s friend, nonlocality, entropy and information-theoretic structure, general POVMs, and category-theoretic refinement (Martin-Dussaud et al., 2022). Even in adjacent probabilistic reconstructions, the role of complex numbers remains a live issue: the QBist analysis of real-vector-space quantum theory finds that the elegant SIC-based Born-rule rewriting of complex quantum mechanics largely disappears in the real case, where the optimal informationally complete reference devices are asymmetric and the resulting formula is “genuinely ugly.” A plausible implication is that, for at least some rationalist reconstructions, the complex field is not merely convenient notation but part of the structural specialness of ordinary quantum theory (Fuchs et al., 2022).

Taken together, these works portray Rational Quantum Mechanics as a broad reconstructionist domain rather than a single doctrine. Its unifying question is not whether quantum mechanics is true, but what sort of rational structure makes it intelligible: formal-inferential, Bayesian, epistemic, relational, arithmetic, or ontological. The answer remains unsettled, and that unsettledness is itself a defining feature of the contemporary RaQM literature.