Towards a Minimal Computational Foundation for Modern Physics

A Research Problem Statement

1. Motivation and Context

Modern physics rests on two principal theoretical frameworks: quantum mechanics (QM) and general relativity (GR), with special relativity (SR) embedded in both. These theories have achieved extraordinary empirical success, yet they carry substantial conceptual and terminological apparatus whose relationship to actual experimental practice is often unclear.

Consider the current state of quantum foundations. Multiple interpretations—Copenhagen, Many-Worlds, Bohmian mechanics, and more recently Barandes' indivisible stochastic reformulation—offer radically different ontological pictures while generating identical empirical predictions. The scientific method, strictly construed, cannot adjudicate between them: only the mapping from experimental configurations to probability distributions over outcomes is testable. The ontological superstructure is, from the standpoint of empirical science, underdetermined.

A similar situation arguably obtains in general relativity. Concepts such as spacetime curvature, frame-dragging, and the geometric interpretation of gravity are powerful heuristics that guided Einstein to correct predictions. Yet one may ask: to what extent is the geometric language essential, and to what extent is it an interpretive layer atop a more austere computational core? The predictions of gravitational lensing angles, perihelion precession, and gravitational wave strain amplitudes are what get tested; the claim that "spacetime is curved" is a story we tell around those predictions.

This research programme proposes to systematically investigate how far the theoretical machinery of modern physics can be reduced to bare computational and experimental essentials, stripping away terminology that does not directly participate in the chain from experimental setup to probabilistic prediction.

2. Central Research Question

To what minimal formal structure can quantum mechanics, special relativity, and general relativity be reduced while preserving their full predictive content, and what does this reduction reveal about the epistemic status of the remaining theoretical vocabulary?

This question decomposes into several sub-questions:

1. What is the irreducible computational content of each theory—the algorithms that take experimental specifications as input and produce probability distributions over measurement outcomes as output?

2. Which theoretical concepts (wave function, spacetime metric, quantum state collapse, geodesic, etc.) are essential to these algorithms, and which are eliminable interpretive glosses?

3. Can a unified formal language be constructed that encompasses QM, SR, and GR at the level of their computational cores, without presupposing ontological commitments that exceed empirical warrant?

4. What are the implications of such a reduction for the search for quantum gravity and other extensions of current physics?

3. Methodological Framework

The proposed research adheres to a strict operationalist and computational methodology, focusing exclusively on elements that participate in the experimental pipeline:

3.1 The Experimental Pipeline

The chain from theory to empirical test comprises the following stages, each of which must be explicitly formalised:

1. Experimental Specification: A complete description of the laboratory setup, including apparatus geometry, initial state preparation procedures, and environmental controls. This description must be finitary and communicable.

2. Measurement Protocol: A specification of which observables are to be measured, including the physical mechanism of measurement (detector physics, calibration procedures, timing protocols).

3. Calibration and Systematics: Procedures for establishing measurement scales, quantifying systematic uncertainties, and ensuring reproducibility.

4. Data Acquisition: The raw output of detectors—typically discrete counts, voltage traces, or timing signals—prior to theoretical interpretation.

5. Statistical Model: The probabilistic model that maps theoretical parameters to distributions over raw data, including noise models and background subtraction.

6. Inference: The extraction of parameter estimates and confidence intervals from data, using well-defined statistical procedures.

Any theoretical concept that does not enter into this pipeline at some stage is a candidate for elimination or reclassification as heuristic rather than essential.

3.2 Computational Reduction Criterion

A theoretical term is deemed computationally essential if and only if it appears in the minimal specification of the algorithm that computes probability distributions over measurement outcomes from experimental specifications. A term is interpretive if it can be removed from this specification without altering the computed distributions.

For example: in quantum mechanics, the Hilbert space formalism and Born rule are computationally essential (they constitute the algorithm). The claim that the wave function "really exists" or "collapses" is interpretive—it does not change what gets computed.

4. Application to Quantum Mechanics

4.1 The Computational Core

The predictive content of non-relativistic quantum mechanics can be stated as follows:

Given a configuration space $\mathcal{C}$, a Hilbert space $\mathcal{H} = L^2(\mathcal{C})$, and a Hamiltonian operator $\hat{H}$, the algorithm for computing measurement predictions is:

1. Represent the initial preparation as a density operator $\hat{\rho}_0$ on $\mathcal{H}$.
2. Evolve: $\hat{\rho}(t) = e^{-i\hat{H}t/\hbar} \hat{\rho}_0 e^{i\hat{H}t/\hbar}$.
3. For observable $\hat{A}$ with spectral decomposition $\hat{A} = \sum_a a \hat{P}_a$, compute outcome probabilities as $P(a) = \mathrm{Tr}(\hat{P}_a \hat{\rho}(t))$.

This is the algorithm. Everything else—interpretations of what the wave function "means," debates about collapse versus branching versus stochastic trajectories—is commentary that does not alter steps 1–3.

4.2 Research Tasks

• Formalise this algorithm in a manner that makes no reference to ontological vocabulary (no "particles," "waves," or "observers" in the formal specification—only preparation procedures, evolution rules, and measurement protocols).
• Investigate whether the complex Hilbert space structure is the unique mathematical setting for this algorithm, or whether equivalent formulations exist (cf. real Hilbert spaces, quaternionic quantum mechanics, operational frameworks).
• Characterise precisely which features of the formalism are conventional (choice of basis, phase conventions) versus which carry physical content.

5. Application to Relativity

5.1 Special Relativity: The Computational Core

The predictive content of special relativity concerns the transformation of measured quantities (lengths, time intervals, frequencies, momenta) between reference frames in relative motion, and the kinematics of high-velocity processes.

The minimal computational content is:

1. Measured spacetime coordinates in one frame are related to those in another by linear transformations preserving the interval $\Delta s^2 = c^2 \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2$.
2. Kinematic quantities (energy, momentum) transform covariantly under these transformations.
3. Predictions for detection rates, Doppler shifts, and time dilations follow algebraically.

The interpretation of this as a "geometry of spacetime" is a heuristic that organises the algebra. Whether spacetime "really is" a four-dimensional manifold is not tested by any experiment; what is tested is whether the transformation laws hold.

5.2 General Relativity: The Computational Core

General relativity predicts deviations from Newtonian gravity and special-relativistic kinematics in the presence of mass-energy. The computational content includes:

1. Predictions for light deflection angles near massive bodies.
2. Predictions for perihelion precession rates.
3. Predictions for gravitational redshifts.
4. Predictions for gravitational wave strain signals at detectors.

The claim that these effects arise because "spacetime is curved" is a geometric interpretation of the field equations. The field equations themselves—and their solutions in particular spacetimes—constitute the computational core. One could, in principle, treat the Einstein equations as a set of coupled nonlinear PDEs whose solutions yield the required predictions, without committing to the ontological claim that spacetime is a substantival curved manifold.

5.3 Research Tasks

• Reconstruct the empirically tested predictions of SR and GR from minimal postulates concerning measurement transformations and dynamical equations, avoiding ontological vocabulary where possible.
• Investigate the extent to which "geometric" language (curvature, geodesics, manifolds) is heuristically useful versus formally essential to the predictive algorithms.
• Compare this reductive approach to existing operational reconstructions of relativity (e.g., radar-based formulations).

6. Towards a Unified Computational Framework

A more ambitious goal is to develop a unified formal language that encompasses QM, SR, and GR at the level of their computational cores. This would involve:

6.1 Common Structure

Both quantum mechanics and relativistic physics share a common logical structure at the predictive level:

• Input: Specification of an experimental arrangement (apparatus, preparation, protocol).
• Algorithm: Application of mathematical rules (Hilbert space evolution, tensor calculus, symmetry constraints).
• Output: Probability distributions over measurement outcomes, or deterministic predictions with quantified uncertainties.

A unified computational framework would formalise this shared structure explicitly, treating the specific mathematical content of QM and GR as instances of a more general schema.

6.2 Finitist and Constructivist Considerations

Taking inspiration from finitist and constructivist mathematics, one may impose additional discipline on the formal framework:

• All experimental specifications should be finitary (describable in finite terms).
• All algorithms should be effective (implementable as computer programmes in principle).
• Existence claims should be constructive (accompanied by procedures for constructing or approximating the objects in question).

This would rule out certain mathematical idealisations (exact real numbers, completed infinities, non-measurable sets) whose physical relevance is questionable.

6.3 Research Tasks

• Develop a formal metalanguage for specifying physical theories as algorithms from experimental inputs to probabilistic outputs.
• Apply this metalanguage to QM, SR, and GR, identifying the common structure and the theory-specific content.
• Investigate whether this framework suggests natural constraints on candidate theories of quantum gravity (e.g., by requiring that any proposed theory be specifiable within the same computational schema).

7. Case Study: Gravitational Wave Detection

To ground this research in concrete experimental practice, consider gravitational wave astronomy as a case study.

7.1 The Experimental Reality

What LIGO/Virgo actually measures is differential displacement of test masses, registered as shifts in an interference fringe pattern. The raw data is a time series of photodetector outputs, contaminated by thermal noise, seismic noise, quantum shot noise, and numerous other systematics.

The claim that these signals represent "ripples in spacetime curvature from merging black holes" is an interpretation. What is tested is whether the template waveforms derived from numerical relativity match the observed signals with acceptable statistical confidence.

7.2 Computational Content

The predictive chain is:

1. Model the source (binary inspiral, merger, ringdown) using numerical or analytical solutions of the field equations.
2. Propagate the wave to the detector.
3. Compute the expected detector response (strain sensitivity, antenna pattern).
4. Generate template waveforms.
5. Perform matched-filter analysis against the data.
6. Assess statistical significance and infer source parameters.

Every step involves computation. The ontological claim that gravitational waves "are" curvature oscillations does not enter the calculation except as an organising metaphor.

7.3 Research Task

• Provide a complete specification of the computational pipeline for gravitational wave detection that makes explicit the role of each theoretical concept.
• Identify which concepts are doing computational work versus interpretive work.
• Consider alternative formulations of the same pipeline that use different (or less) theoretical vocabulary.

8. Implications and Significance

8.1 Epistemic Hygiene

A clear separation of computational content from interpretive overlay promotes epistemic hygiene in physics. It clarifies what is being tested in any experiment and prevents confusion between confirmed predictions and speculative ontology.

8.2 Guidance for Theory Construction

When constructing new theories (e.g., quantum gravity), a computational-first approach focuses attention on what the theory should predict and how those predictions are to be computed, rather than on what ontological picture the theory suggests.

8.3 Educational Value

A reduced, algorithm-centric presentation of physics could be pedagogically valuable, providing students with a clearer view of the logical structure of theories before introducing interpretive layers.

8.4 Connections to Computation Theory

This programme has natural connections to quantum computing, computational complexity, and the physics of information. If the computational content of a theory is what matters, then questions about computational complexity of physical predictions become physically meaningful.

9. Relation to Existing Work

This research programme intersects with, but is distinct from, several existing research traditions:

• Operational Quantum Mechanics: Frameworks such as those of Ludwig, Davies-Lewis, and more recent categorical approaches (Abramsky, Coecke) that reconstruct QM from operational primitives. The present proposal is broader in scope, encompassing relativity, and more focused on experimental practice.

• Quantum Reconstruction Programmes: Attempts to derive quantum mechanics from information-theoretic axioms (Hardy, Chiribella et al., Masanes-Müller). These provide alternative axiomatisations but often remain at the level of abstract structure rather than experimental procedures.

• Finitist and Ultrafinitist Physics: Proposals to reformulate physics in finite terms (e.g., Fredkin, Wolfram, 't Hooft on cellular automaton interpretations). The present proposal does not commit to discreteness of spacetime but does insist on finiteness of experimental specifications.

• Philosophy of Scientific Representation: Work by philosophers (van Fraassen, Giere, Suárez) on the representational content of scientific theories. This provides conceptual background but does not usually engage with computational specifics.

10. Proposed Research Outputs

A successful execution of this research programme would produce:

1. A formal specification language for physical theories as algorithms mapping experimental inputs to probabilistic outputs.

2. Complete specifications in this language for: (a) non-relativistic quantum mechanics, (b) quantum field theory at the level needed for Standard Model predictions, (c) special relativity, (d) general relativity at the level needed for solar system tests and gravitational wave predictions.

3. An analysis of the irreducible formal content of each theory, distinguishing computational essentials from interpretive overlays.

4. A comparative study of alternative formulations (e.g., Hilbert space vs. path integral vs. stochastic for QM) from the standpoint of computational equivalence.

5. Recommendations for how this framework might constrain or guide the search for quantum gravity.

6. Pedagogical materials presenting physics from a computation-first perspective.

11. Conclusion

The accumulation of interpretive vocabulary in theoretical physics has outpaced the growth of empirically tested content. While conceptual heuristics have sometimes led to genuine discoveries, the underdetermination of ontology by experiment means that much of the interpretive superstructure is, strictly speaking, optional.

This research programme proposes to take seriously the idea that what physics actually provides is a set of algorithms for computing predictions from experimental specifications. By systematically reducing QM, SR, and GR to their computational cores, we may achieve greater clarity about what is known, greater rigour in theory construction, and perhaps new insights into the structure of physical law.

The goal is not to eliminate theoretical physics but to discipline it—to distinguish sharply between what is computed and what is commented, and to ensure that the commentary does not obscure the computation.

References (Indicative)

The following areas of literature are relevant to this programme:

• Operational and axiomatic quantum mechanics (Ludwig, Hardy, Chiribella et al.)
• Categorical quantum mechanics (Abramsky, Coecke, Heunen)
• Computational complexity and physics (Aaronson, Watrous)
• Constructive and finitist foundations (Bishop, Martin-Löf, Troelstra)
• Philosophy of scientific representation (van Fraassen, Giere)
• Foundations of relativity (Brown's dynamical approach, radar-based formulations)
• Operational approaches to spacetime (Malament, Ehlers-Pirani-Schild)

Document prepared as a starting point for theoretical research. December 2025.