J. Rogers, SE Ohio
1. Motivation
The original fibration framework models physical law as Cartesian liftings of dimensionless base morphisms through a total category of measurement. This explains the origin of constants as fiber transition (Jacobian) data.
We now show that both dimensional analysis (Buckingham π groups) and continuous symmetry methods (Lie groups) are natural categorical consequences of this structure.
2. Buckingham π Groups as Functorial Projections
2.1 Base Category Perspective
Let B be the base category of dimensionless conceptual axes (Mass, Time, Length, etc.).
Morphisms in B are unit-independent relationships, e.g.,
φ: Force → Mass·Length/Time²
2.2 π Groups as Dimensionless Sections
Define a functor:
Π: Hom(B) → Sect(π)
which assigns to each morphism in B a dimensionless ratio invariant under fiber scaling.
Each π-group represents a canonical section of the fibration, where the section "forgets" the units in E and preserves pure relational structure.
Example: The Stefan–Boltzmann law in Planck units:
ρ/ρ_P ∼ (T/T_P)⁴
is exactly the image of the base morphism ρ ∼ T⁴ under the functor Π, showing that π-groups compile dimensionless invariants automatically.
3. Lie Groups as Symmetry Functors on the Base
3.1 Continuous Symmetries
Let G be a Lie group representing continuous symmetries of the substrate S_u.
Morphisms in B now include infinitesimal transformations, X ↦ X + ε·δX, preserving relational structure.
The fibration π lifts these symmetries into E, producing coordinate-dependent manifestations (e.g., conserved quantities via Noether's theorem).
3.2 Categorical View of Lie Theory
Consider a functor:
L: G → Aut(π)
mapping Lie algebra elements to automorphisms of the fibration, i.e., fiber-preserving transformations that encode unit-consistent symmetry actions.
Conservation laws emerge naturally: the Lie algebra of the base category generates dimensionless invariants, which are lifted into E with constants appearing as fiber Jacobians.
4. Unified Picture: π Groups + Lie Groups in the Fibration
Base category (B):
Pure relational structure of the universe.
Dimensionless invariants (Π):
Sections of π mapping base morphisms to unit-free laws.
Continuous symmetries (Lie/G):
Automorphisms of π preserving fiber structure.
Measurement fibers (E):
Concrete numerical laws with constants as transition maps.
Diagrammatically:
B (dimensionless substrate)│ ├─ Π (dimensionless invariants / π groups)│ └─ L (Lie symmetries / conservation)│ ↓ (Cartesian lifting / fiber projection)E (unit-dependent measurable laws with constants)
The appearance of constants (h, c, G, k_B) is unified: both π-groups and Lie symmetry-derived laws inherit them from fiber projections.
No new assumptions beyond the single physical scale and the base category are required.
5. Implications
Explains why dimensional analysis "works"
π-groups are just canonical lifts of base morphisms.
Explains why Lie symmetry methods "work"
Continuous symmetries act on the substrate, lifted to fibered measurements with constants as Jacobians.
Unifies methods
Dimensional invariants and conserved quantities both arise from the same categorical substrate, showing a deep reason for the empirical success of seemingly unrelated techniques.
Predictive power
Any law that cannot be lifted as a Cartesian section of B (via Π or L) is impossible in physical reality, providing a falsifiable criterion.
6. Conclusion
Buckingham π groups and Lie symmetries are extensions of the fibration framework, not independent techniques. They formalize the same underlying principle: laws are projections of dimensionless base relations.
Constants emerge as fiber transition data, and all measurable complexity is the Jacobian shadow of the substrate.
This categorical unification resolves decades of confusion in theoretical physics and explains why attempts to derive constants or fundamental scales (String Theory, LQG) fail: they confuse fiber artifacts for base truths.
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