Interaction channels can be organized as symmetries of local comparison and completion. Their shared origin becomes physically explanatory when it selects the gauge structure, matter representations and normalized couplings together.
What a unified theory should explain
Electromagnetic, weak and strong interactions are described through gauge fields, while general relativity describes gravity through spacetime geometry. A unified account should explain why these structures coexist and how their parameters are related. Putting their equations beside one another is a useful starting point; a deeper result would derive them from common degrees of freedom and an action whose physical normalization is fixed.
The task includes the charges and representations of matter, symmetry breaking, and the variation of interaction strengths with energy. A relation among couplings must specify the energy at which it holds and the particle thresholds used to compare it with measurements. Gravity adds a further requirement: the same construction must determine how physical lengths, durations and stress couple to the emergent metric.
The original proposal: interactions as temporal geometry
The original unification article identifies all entities as products of inner temporal formation. It proposes that the four familiar interactions correspond to four effective levels involved in forming the physical world, with complementary interaction structures in the author’s psychical-world interpretation. It also extends the idea toward higher dimensional forms of interaction and more complex mass structures. The unifying element is the origin of all these responses in one complex temporal geometry.
The discussion belongs to Chapter III of Duality of Time and points toward the later geometrical and symmetry chapters. The four interaction levels are a proposed classification of physical responses; they should be distinguished from the six nested stages of inner generation and the seventh outward completion. Their counts describe different roles. A constructive physical development asks which source transformations survive each stage, how they act on matter, and how their responses become the forces measured in outward spacetime. This retains the original geometric mechanism while expressing the physical sector in a form that can be calculated.
Interactions as comparisons between local records
Suppose each completed region carries a local state space. The choice of basis used to describe that state is partly conventional. Comparing neighboring regions then requires a transport map Uxy, which takes a state at y into the convention used at x. Under local changes of basis gx and gy, consistency requires
Uxy → gx Uxy gy−1.
This is the central gauge transformation rule in a discrete form. It gives an interaction a concrete role: transport maintains meaningful comparison as local descriptions change. The source selects the admissible transformations and the matter carrier; completion supplies the observable transport statistics. A phase convention gives an Abelian channel, while matrix-valued transformations can give non-Abelian channels.
Yang and Mills’ original paper establishes how local internal symmetry requires a gauge field. In the SMM application, the additional task is to derive the relevant local symmetry from the proposed inner organization.
A loop turns transport into field strength
Transport around a small closed loop compares the final state with its starting convention. Define U□ as the ordered product of link maps around that loop. Because the intermediate basis changes cancel, the trace of U□ is independent of those choices. A source action can therefore contain a gauge-invariant loop term.
U□ ≈ exp(i a2Fμν), Re Tr(I − U□) ≈ (a4/2) Tr(Fμν2).
Here a is the small spatial resolution, the generators are Hermitian, and coupling factors have been included in the connection. The ordinary complex unit satisfies i2 = −1; it represents phase and is distinct from the split-complex unit used for Lorentz boosts. Expanding the loop response gives the familiar quadratic field-strength term.
This worked candidate explains how one rule for local comparison can yield several gauge channels. It does not determine their groups by counting temporal stages. Their selection must follow from the source algebra, allowed carriers and symmetries of the completion map.
From common origin to common normalization
An effective unification target, written schematically with ℏ = c = 1, is
Seff = ∫ d4x √−g [R/(16πG) − Σa Tr(FaμνFaμν)/(4ga2) + ℒmatter].
The important SMM question is how one source fixes the coefficients G and ga after the fields and measurement units have been normalized. Raw counts of internal states become physical couplings only through their contribution to this response. A proposed integer relation is therefore most useful when accompanied by the associated kinetic terms and matter interactions.
Chamseddine and Connes’ spectral-action construction provides a concrete comparison: specified algebraic and spectral data generate gauge and gravitational terms through one action. It demonstrates a mathematical route of the required kind. An SMM realization would need its own source principle to select those data and explain why the resulting outward theory has the observed content.
The decisive calculations
A useful finite model should output its local symmetry group, matter representation and transport action together. The next calculation checks consistency of the quantum gauge symmetry, including anomaly cancellation. Then derive the symmetry-breaking pattern and evolve the couplings to the energy scales at which measurements are compared. These steps connect a common origin to concrete particle physics.
Gravity should enter through the same readout that calibrates matter propagation and clocks. Its weak-field limit must give the established relation between stress and curvature, with a universal coupling across matter species. That common normalization is where the unification proposal meets the hierarchy and equivalence-principle questions.
The most informative outcome would be a relation among several independently measurable quantities, fixed before comparison with data. Couplings, masses, decay rates or deviations in gravitational response could then test the same underlying source structure. Such a calculation would turn the original claim of geometrical unity into a connected physical theory.
Books, papers and further reading
- Mohamed Haj Yousef — Duality of Time
Chapter III: original unification proposal; Chapter V, Complex-Time Hyperbolic Geometry; Chapter VII, Alchemy and Eternal Symmetry. Native edition and contents record.
- C. N. Yang and R. L. Mills — Conservation of Isotopic Spin and Isotopic Gauge Invariance
Original 1954 local non-Abelian gauge construction.
- Ali H. Chamseddine and Alain Connes — The Spectral Action Principle
Primary construction of a common spectral action with gauge and gravitational terms; used as a mathematical comparison.