A simple means for deriving quantum mechanics

A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle’s environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.

💡 Research Summary

The manuscript proposes a novel framework called “positional mechanics” that aims to reproduce all observable predictions of non‑relativistic quantum mechanics while retaining a classical‑looking ontology. In this picture, a physical system is described by a set of “paths” – time‑parameterized functions that map each instant to a complete system state. The author introduces the notion of an “r‑set”, the collection of paths that respect momentum conservation for any subsystem that does not interact with its environment over a given interval. By allowing two paths that coincide at a particular time to be concatenated – i.e., splicing the past of one with the future of the other – and by repeatedly closing the r‑set under this concatenation operation, the theory generates a larger set S of admissible system histories.

Positional systems consist of particles and fields. Particles follow continuous, piecewise‑differentiable trajectories in space, and their linear momentum is defined as the product of mass and the tangent (velocity) of the trajectory. Crucially, velocity is not part of the particle’s state; instead, velocity can change instantaneously at discrete events called “momentum‑changing events”. These events are of two types: (i) “concatenations”, which are abstract switches between admissible r‑set paths, and (ii) “interactions” with the fields, which also produce stochastic velocity jumps. The rates of these events are denoted Γ_c and Γ_i, respectively, and the conditional probability distributions for the post‑event velocity are Q_c and Q_i. The author models the number of events in a short time interval Δt by Poisson statistics and writes down an evolution equation for the joint probability density η(x,v|e) of finding a particle at position x with velocity v given an environmental state e.

From these ingredients the author claims to recover the hallmark quantum phenomena:

1. Non‑determinism with momentum conservation – Even though velocities jump randomly, the construction guarantees that any path passing through a state where a momentum measurement has been recorded must preserve the measured momentum range, because concatenations always start from an r‑set path that respects conservation.

2. Non‑local correlations – For a pair of free particles with total momentum zero, a concatenation that changes the velocity of one particle forces a compensating change in the other, reproducing the instantaneous correlation characteristic of entangled states.

3. Non‑additive probabilities – By arranging measurement sequences that differ only in whether the environment distinguishes between two outcomes, the set of admissible paths changes, leading to probability assignments that violate classical additivity, mimicking quantum interference effects.

4. Particle‑identity effects – When two particles are indistinguishable, certain concatenations become possible that are forbidden for distinguishable particles, thereby encoding exchange symmetry into the probability structure.

5. Uncertainty – Because the field state carries only partial information about a particle’s path, an observer never has complete knowledge of position and velocity simultaneously, reproducing the uncertainty principle in an epistemic sense.

6. Tunnelling, reflection, refraction – The stochastic, non‑deterministic nature of particle‑field interactions yields qualitative explanations for barrier penetration, back‑scattering, and angle‑dependent deflection without invoking wave‑like amplitudes.

The author then defines “quantum positional systems” as the subclass of positional systems whose parameters (the rates Γ and the distributions Q) are tuned so that the full statistical predictions coincide with those of standard quantum mechanics. Connections to Bohmian mechanics (existence of definite trajectories), stochastic mechanics (random velocity jumps), the many‑worlds interpretation (all concatenated histories are realized), and objective collapse models (discrete events that alter the state) are discussed qualitatively.

A brief sketch of a relativistic extension is offered in the final section, suggesting that field states would now carry four‑momentum information and that concatenations would respect Lorentz invariance, but no explicit covariant formalism or Lagrangian is presented.

Critical assessment
While the proposal is imaginative, several serious shortcomings limit its claim of reproducing quantum mechanics. First, the concatenation operation is introduced as an abstract mathematical rule without a clear physical mechanism; it is unclear how a real system “chooses” a new r‑set path and how this choice could be experimentally observed. Second, the evolution equation for η is highly formal and does not lead to the Schrödinger equation, the Born rule, or the Hilbert‑space structure; consequently, interference phases and the role of complex amplitudes remain unexplained. Third, the treatment of measurements and entanglement relies on the assumption that the environment’s state encodes the entire past, but the paper does not specify how decoherence or pointer states emerge within the framework. Fourth, many definitions (r‑set, concatenation, Q_c, Q_i) are ambiguous, and the notation is inconsistent, making rigorous derivations difficult. Fifth, no concrete predictions that differ from standard quantum mechanics are offered, so the theory is empirically indistinguishable and therefore not falsifiable in its present form. Finally, the relativistic extension is only a sketch; without a covariant action principle or a demonstration of how to recover quantum field theory, the claim of a fully relativistic version remains speculative.

In summary, the manuscript presents a novel deterministic‑plus‑stochastic ontology that attempts to embed quantum statistics in a path‑concatenation framework. It succeeds in highlighting interesting analogies with existing interpretations but falls short of providing a mathematically rigorous, experimentally testable, and complete reconstruction of quantum mechanics. Further work would need to derive the standard wave‑function formalism, clarify the physical meaning of concatenations, and produce distinctive experimental signatures.