Abstract

This paper develops a unified framework in which relativistic worldlines, quantum path‑space, and subjective experience arise from a single geometric and functional structure. A massless, nonlocal functional Ψ is defined over the space of future‑directed timelike curves Γ₊(M). A massive organism corresponds to a single worldline γ ⊂ M. Experience is modelled as the action of a bounded operator Π on Ψ, projecting the global path‑space functional onto the restricted perceptual channel defined by γ. The framework offers a coherent way to interpret the measurement problem, the relationship between quantum superposition and relativistic worldlines, and the emergence of first‑person experience. Several open mathematical problems are identified for future development.

 

1. Introduction

Physics describes matter and spacetime with extraordinary precision, but it does not explain why experience is first‑person, unified, or restricted to a single classical trajectory. Quantum mechanics describes superpositions of possible paths; relativity describes a single worldline. Conscious experience appears to track the latter, not the former.

Panpartism proposes a simple idea: treat awareness as a massless, nonlocal functional defined over the entire space of possible worldlines. A massive body follows one worldline; a massless functional does not. Experience arises when a projection operator Π restricts the global functional Ψ to the organism’s worldline γ.

This approach does not alter established physics. It provides a vantage point from which quantum superposition, relativistic worldlines, and subjective experience can be understood within a single structure.

 

2. Conceptual motivation

2.1 Masslessness and nonlocality

In relativity, massless entities have no rest frame and no proper time. They do not “move through” spacetime in the way massive bodies do. If awareness is modelled as massless—not as a particle, but as a functional—then it is naturally nonlocal and atemporal. It belongs on path‑space, not spacetime.

2.2 Dimensional recursion

A simple geometric recursion motivates the structure:

0D: a point  

1D: infinite points → a line  

2D: infinite lines → a plane  

3D: infinite planes → a space  

4D: infinite spaces → a timeline  

5D: infinite timelines → the space of possible worldlines  

This identifies the “fifth dimension” with Γ₊(M), the space of all possible 4D worldlines.

2.3 Decoherence and worldline restriction

Quantum mechanics allows superpositions of paths; relativity gives each massive body a single worldline. Decoherence explains why macroscopic objects do not behave quantum‑mechanically: they are effectively pinned to one classical trajectory.

Thus:

- Ψ describes the global structure (all possible worldlines).  

- γ is the local structure (one worldline).  

- Π is the mechanism that links the two.

 

3. Spacetime and path‑space

3.1 Spacetime

Let (M, g) be a smooth 4‑dimensional Lorentzian manifold with signature (−,+,+,+).

3.2 Path‑space

Γ₊(M) = { γ : [0,1] → M | γ smooth, timelike, future‑directed }.

3.3 Proper‑time action

For a body of mass m:

S[γ] = −m ∫₀¹ √(−g(γ̇(t), γ̇(t))) dt.

3.4 Topology

Equip Γ₊(M) with the C¹ topology or a Sobolev H¹ topology.

 

4. Measure on path‑space

4.1 Cylindrical approximation

Construct finite‑dimensional approximations Γ₊⁽ⁿ⁾(M) via partitions of [0,1], with cylindrical measures μₙ.

4.2 Projective limit

If the limit exists:

μ = lim μₙ.

This yields a σ‑additive measure on Γ₊(M).

 

5. Awareness functional Ψ

5.1 Definition

Ψ[γ] = exp(i S[γ] / ħ).

This is mathematically identical to the Feynman weight, but interpreted here as a nonlocal awareness functional.

5.2 Hilbert‑like space

H = L²(Γ₊(M), μ).  

Ψ ∈ H.

5.3 Inner product

⟨Ψ₁, Ψ₂⟩ = ∫ Ψ₁*[γ] Ψ₂[γ] dμ(γ).

 

6. Projection operator Π

6.1 Definition

Π : H → H is a bounded linear operator satisfying:

1. 0 ≤ (ΠΨ)[γ] ≤ 1  

2. ‖ΠΨ‖ ≤ ‖Ψ‖  

3. continuity with respect to the topology of Γ₊(M)

6.2 Kernel representation

(ΠΨ)(γ) = ∫ Π(γ, γ′) Ψ(γ′) dμ(γ′).

6.3 Interpretation

Π acts as a restriction or weighting over path‑space. It is the mathematical mechanism by which a global functional becomes a localised experiential channel.

 

7. Experience as projection

Let γ be the organism’s worldline.

Define:

E(γ) = (ΠΨ)(γ).

Here:

- Ψ encodes the global structure,  

- Π defines the restriction,  

- E(γ) is the experienced reality along γ.

This provides a minimal mathematical model for first‑person experience.

 

8. Interpretation and implications

8.1 Measurement problem

Quantum mechanics describes superpositions of paths; relativity describes a single worldline. In this framework:

- Ψ contains all paths,  

- Π selects a restricted channel,  

- E(γ) is the experienced outcome.

Collapse becomes a projection, not a physical discontinuity.

8.2 Unity of consciousness

A single worldline γ and a single projection Π naturally produce a unified experiential stream E(γ).

8.3 Free will and worldline variation

The dimensional recursion suggests:

4D = one timeline  

5D = the space of possible timelines  

Movement through path‑space corresponds to structured variation among possible worldlines.

8.4 Phenomenological parallels

Reports of multi‑angle perception, atemporality, or expanded awareness can be interpreted as partial access to path‑space. These are not treated as evidence, only as parallels the model can accommodate.

 

9. Open mathematical problems

1. Construct a rigorous σ‑additive measure μ on Γ₊(M).  

2. Prove completeness of H.  

3. Characterise bounded operators Π.  

4. Determine when Π is compact, trace‑class, or self‑adjoint.  

5. Formalise the equivalence between a 5‑dimensional configuration space and Γ₊(M).  

6. Identify representations of Π that correspond to cognitive or phenomenological structures.  

7. Explore categorical realisations of the recursion R(n+1) = F(R(n)).

 

10. Conclusion

Panpartism offers a unified structure in which quantum superposition, relativistic worldlines, and subjective experience arise from a single architecture based on:

- dimensional recursion,  

- path‑space geometry,  

- a massless functional Ψ,  

- projection via Π onto a worldline γ.

 

The framework is speculative but mathematically coherent and compatible with established physics. It identifies clear mathematical problems for future work and provides a compact way to relate nonlocal structure, worldline dynamics, and first‑person experience.