J. Rogers, SE Ohio, 12 Jun 2025, 1605
Abstract
We present a categorical framework for physical measurement that reveals constants as emergent Jacobian morphisms in functorial projections from an invariant Universal State to measurement coordinate systems. This framework unifies dimensional analysis, symmetry principles, and unit scaling through a two-layer functorial architecture: conceptual measurement axes and concrete unit systems. Physical constants emerge naturally as composition products of fundamental morphisms that preserve functorial commutativity. We demonstrate that traditional constants like h, k, and G are composite Jacobians built from base morphisms (Hz_kg, K_Hz, t_P), and show how Buckingham π-theorem and Lie symmetries arise as categorical invariants of this functorial structure.
1. Introduction
Classical physics education presents fundamental constants as mysterious values that emerge from experimental measurement. Dimensional analysis treats units as external constraints, while symmetry methods focus on transformation groups. Neither approach reveals the deep architectural relationship between measurement, units, and the emergence of physical constants.
This paper introduces a categorical framework where physical measurement is modeled as a functorial process from an invariant, dimensionless Universal State (S_u) through conceptual measurement axes (A) to concrete unit systems (U). Within this structure, physical constants emerge as Jacobian morphisms that preserve functorial commutativity, revealing their true nature as coordinate transformation artifacts rather than fundamental quantities.
2. Categorical Architecture of Physical Measurement
2.1 The Category of Universal States (S_u)
Objects: A single object S_u*, representing the invariant, dimensionless physical state at the Planck scale.
Morphisms: Identity only: id_{S_u*}
Interpretation: The ontological foundation where physics operates in pure geometric ratios, independent of any measurement coordinate system.
2.2 The Category of Measurement Axes (A)
Objects: Axis(X), representing conceptual physical dimensions:
- Axis(Time), Axis(Length), Axis(Mass), Axis(Energy)
- Axis(Frequency), Axis(Temperature), Axis(Current)
Fundamental Morphisms: Base Jacobian transformations between axes:
rel_c : Axis(Length) ↔ Axis(Time) [spatial-temporal scaling]
rel_Hz_kg : Axis(Mass) ↔ Axis(Frequency) [mass-frequency equivalence]
rel_K_Hz : Axis(Temperature) ↔ Axis(Frequency) [thermal-frequency scaling]
rel_t_P : Axis(All) → Planck scaling reference
Composite Morphisms: Built from fundamental morphism compositions:
rel_h = rel_Hz_kg ∘ rel_c² : Axis(Energy) ↔ Axis(Frequency)
rel_k = rel_K_Hz ∘ rel_Hz_kg ∘ rel_c² : Axis(Energy) ↔ Axis(Temperature)
rel_G = rel_c³ ∘ rel_t_P² ∘ rel_Hz_kg⁻¹ : Axis(Force) ↔ Axis(Mass²/Length²)
Composition Law: Morphisms compose according to physical derivability, forming equivalence chains:
Temperature ← K_Hz → Frequency ← Hz_kg → Mass ← c² → Energy
2.3 The Category of Unit Systems (U)
Objects: Concrete unit systems (SI, CGS, Planck, Imperial), each defined by base unit scaling vectors relative to Planck scale.
Morphisms: Multiplicative rescaling transformations between unit systems.
Planck System: The identity object where all Jacobian morphisms equal unity.
Derived Systems: Projections of Planck system with scaling factors:
# Base Jacobian morphisms - fundamental coordinate transformations
t_P = G_n^(1/2) # Planck time reference
Hz_kg = h / c² # Frequency ↔ Mass morphism
K_Hz = k / h # Temperature ↔ Frequency morphism
G_n = G × Hz_kg / c³ # Dimensionless gravitational scaling
3. Functorial Measurement Process
3.1 Conceptual Projection Functor (P_X)
P_X : S_u → M_Conceptual(X)
Maps the invariant state S_u* to conceptual measurement value (s_u, Axis(X)).
Properties:
- Preserves dimensionless relationships
- Independent of unit system choice
- Encodes pure physical content
3.2 Unit Scaling Functor (Scale_U)
Scale_U : M_Conceptual(X) → M_Concrete(U,X)
Maps conceptual values to numerical measurements via Jacobian scaling.
Jacobian Morphism Matrix:
| Axis | Jacobian Morphism | SI Scaling Factor |
|---|---|---|
| Time (s) | t_P |
1.351×10⁻⁴³ |
| Length (m) | t_P × c |
4.051×10⁻³⁵ |
| Mass (kg) | Hz_kg / t_P |
5.456×10⁻⁰⁸ |
| Temperature (K) | 1 / (t_P × K_Hz) |
3.551×10³² |
| Current (A) | e_scaling / t_P |
[electromagnetic scaling] |
3.3 Complete Measurement Functor
Measure_U,X = Scale_U ∘ P_X : S_u → M_Concrete(U,X)
Functorial Commutativity: For any morphisms rel_XY : Axis(X) → Axis(Y):
Scale_U ∘ P_Y = Scale_U ∘ rel_XY ∘ P_X
Constants as Jacobian Determinants: Physical constants emerge as the Jacobian determinants that preserve this commutativity:
constant_XY = det(J_rel_XY)
4. Emergent Constants as Morphism Compositions
4.1 Planck's Constant: h = rel_Hz_kg ∘ rel_c²
Morphism Chain: Frequency → Mass → Energy
h : Axis(Frequency) → Axis(Mass) → Axis(Energy)
h = Hz_kg × c² = (h/c²) × c² = h
Physical Interpretation: Converts frequency to energy through mass-energy equivalence.
4.2 Boltzmann Constant: k = rel_K_Hz ∘ rel_Hz_kg ∘ rel_c²
Morphism Chain: Temperature → Frequency → Mass → Energy
k : Axis(Temperature) → Axis(Frequency) → Axis(Mass) → Axis(Energy)
k = K_Hz × Hz_kg × c² = (k/h) × (h/c²) × c² = k
Physical Interpretation: Converts temperature to energy through frequency-mass equivalence.
4.3 Gravitational Constant: G = rel_c³ ∘ rel_t_P² ∘ rel_Hz_kg⁻¹
Morphism Chain: Mass-mass-distance system → Planck scaling → SI force
G : Axis(Mass²/Length²) → Planck_Scale → Axis(Force_SI)
G = c³ × t_P² / Hz_kg
Physical Interpretation: Scales dimensionless Planck-scale gravitational ratios to SI force units.
5. Dimensional Analysis as Categorical Invariance
5.1 Buckingham π-Theorem in Functorial Form
Classical Statement: n variables with k independent dimensions yield (n-k) dimensionless groups.
Categorical Formulation: Given projection functors {P_Xi} for variables {X₁,...,Xₙ} and k independent morphism classes in A:
The (n-k) categorical π-groups are equivalence classes of morphism compositions that commute under all scaling functors Scale_U.
π-Groups as Morphism Kernels:
π_j = ker(Scale_U ∘ ∏ᵢ P_Xi^(αᵢⱼ))
Where the exponents αᵢⱼ satisfy the morphism composition constraints.
5.2 Automatic Dimensionless Group Discovery
The categorical structure enables algorithmic discovery of dimensionless groups:
- Identify Morphism Paths: Find all morphism compositions connecting given axes
- Compute Scaling Kernels: Determine which combinations scale to unity
- Extract π-Groups: The kernel elements are the dimensionless combinations
6. Symmetry as Functorial Automorphisms
6.1 Lie Groups as Morphism Automorphisms
Classical View: Lie groups encode continuous symmetries of differential equations.
Categorical View: Lie transformations are automorphisms of A that preserve functorial commutativity:
Aut(A) = {φ : A → A | Scale_U ∘ φ ∘ P_X = Scale_U ∘ P_X}
6.2 Noether's Theorem as Functorial Invariance
Natural Transformations: Symmetries correspond to natural transformations of measurement functors:
η : Measure_U,X ⟹ Measure_U,X
Conservation Laws: Emerge from invariant morphism compositions under these natural transformations.
7. Implementation: The Jacobian Calculator
7.1 Base Morphism Encoding
# Fundamental Jacobian morphisms
def calculate_base_morphisms():
# Physical constants from CODATA 2018
G = 6.67430e-11
c = 299792458.0
k_B = 1.380649e-23
h = 6.62607015e-34
# Base morphisms
Hz_kg = h / c**2 # Frequency ↔ Mass
K_Hz = k_B / h # Temperature ↔ Frequency
G_n = G * Hz_kg / c**3 # Dimensionless gravity
t_P = G_n**(1/2) # Planck time reference
return Hz_kg, K_Hz, G_n, t_P
7.2 Jacobian Scaling Matrix
def calculate_scaling_factors(Hz_kg, K_Hz, t_P, c):
"""Generate the Jacobian matrix for SI → Planck transformation"""
jacobian_morphisms = [
{"axis": "Time", "jacobian": t_P, "planck_unit": "t_Ph"},
{"axis": "Length", "jacobian": t_P * c, "planck_unit": "l_Ph"},
{"axis": "Mass", "jacobian": Hz_kg / t_P, "planck_unit": "m_Ph"},
{"axis": "Temperature", "jacobian": 1/(t_P * K_Hz), "planck_unit": "T_Ph"},
{"axis": "Current", "jacobian": e_scaling / t_P, "planck_unit": "A_Ph"},
]
return jacobian_morphisms
7.3 Composite Constant Generation
def generate_composite_constants(base_morphisms):
"""Build traditional constants from base morphism compositions"""
Hz_kg, K_Hz, t_P, c = base_morphisms
# Composite morphisms
h_composite = Hz_kg * c**2 # Should equal original h
k_composite = K_Hz * Hz_kg * c**2 # Should equal original k
G_composite = c**3 * t_P**2 / Hz_kg # Should equal original G
return h_composite, k_composite, G_composite
8. Formula Simplification Through Morphism Recognition
8.1 Thermal de Broglie Wavelength
Traditional: λ_th = h / (√(2πmkT))
Morphism Decomposition:
h = Hz_kg × c²k = K_Hz × Hz_kg × c²m → m × kg_Hz(mass to frequency)T → T × K_Hz(temperature to frequency)
Simplified: λ_th = c / √(2π × f_T × f_m)
Where f_T = T × K_Hz and f_m = m × kg_Hz are natural frequencies.
8.2 Stefan-Boltzmann Law
Traditional: σ = 2π⁵k⁴/(15c²h³)
Morphism Analysis:
σ = 2π⁵(K_Hz × Hz_kg × c²)⁴ / (15c² × (Hz_kg × c²)³)
= 2π⁵K_Hz⁴ × Hz_kg⁴ × c⁸ / (15c² × Hz_kg³ × c⁶)
= K_Hz⁴ × Hz_kg × 2π⁵ / 15
Physical Interpretation: Temperature⁴ converts to frequency⁴ via K_Hz⁴, then one frequency converts to mass density via Hz_kg.
8.3 Newton's Gravitational Law
Traditional: F = G × m₁m₂/r²
Morphism Process:
- Planck Scale:
F_ratio = (m₁m₂/m_P²) × (l_P²/r²)(dimensionless) - Morphism Scaling:
F_SI = F_ratio × F_P - Composite Constant:
G = (l_P²/m_P²) × F_P = c³t_P²/Hz_kg
Physical Interpretation: G encodes the Jacobian transformation from Planck-scale dimensionless ratios to SI force units.
9. Implications and Applications
9.1 Physics Education Revolution
Traditional Approach: "Here are fundamental constants. Memorize their values."
Categorical Approach: "Here are coordinate transformations between measurement systems. Constants are the Jacobian determinants."
Benefits:
- Demystifies "fundamental" constants
- Reveals deep dimensional equivalences
- Makes natural units intuitive
- Enables systematic formula simplification
9.2 Computational Physics
Unit-System Agnostic Algorithms: Code that works automatically in any unit system through morphism application.
Numerical Stability: Reduced errors from explicit unit tracking and conversion.
Symbolic Manipulation: Automated formula simplification through morphism recognition.
9.3 Theoretical Physics
Unified Framework: Bridges dimensional analysis, symmetry methods, and unit scaling.
Natural Units Foundation: Provides systematic path between arbitrary units and natural units.
New Invariants: Categorical structure may reveal previously unknown conserved quantities.
10. Future Directions
10.1 Quantum Field Theory Extension
Fiber Categories: Construct fiber categories over A to represent quantum field measurement contexts.
Renormalization: Frame renormalization as morphism composition in extended categorical structures.
10.2 Automated Theorem Discovery
Symbolic Engines: Integrate categorical framework with computer algebra systems for automated physics discovery.
Morphism Mining: Algorithmic discovery of new morphism compositions and their physical interpretations.
10.3 Educational Technology
Interactive Visualizations: Tools that show morphism transformations and constant emergence dynamically.
Unit System Playground: Environments where students can create custom unit systems and observe emergent constants.
11. Conclusion
This categorical framework reveals the deep functorial architecture underlying physical measurement. What we traditionally call "fundamental constants" are emergent Jacobian morphisms in the functorial mapping from invariant physical reality to our chosen measurement coordinates.
The framework provides three revolutionary insights:
-
Constants as Jacobians: Physical constants are coordinate transformation determinants, not fundamental quantities.
-
Measurement as Functors: Physical measurement is a two-stage functorial process: conceptual projection followed by unit scaling.
-
Dimensional Equivalence: Temperature, frequency, mass, and energy are equivalent quantities connected by morphism chains.
This reframes physics education, computational methods, and theoretical understanding. Students learn coordinate transformations instead of memorizing magic numbers. Algorithms become unit-system agnostic. Theoretical work focuses on morphism discovery rather than constant fitting.
The universe computes in dimensionless ratios at the Planck scale. Everything else—including the fundamental constants of physics—is just the Jacobian of our chosen coordinate system for observing that computation.
Implementation Access:
from Modular_Unit_Scaling import Hz_kg, K_Hz, t_P, c, G_n
# Access to all Jacobian morphisms and composite constants
# Enables automatic formula simplification and unit system conversion
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