A Lie Group Symmetry Across Measurement Categories Induced by a Conserved Scalar Ontology

### **A Lie Group Symmetry Across Measurement Categories Induced by a Conserved Scalar Ontology**

**Abstract**

We propose a fundamental restructuring of the role of physical constants and units of measurement. Traditional frameworks treat constants as empirically derived values and the invariance of physical law under a change of units as a symmetry of description, formalized by a simple Lie group of scaling $(\mathbb{R}^+)^k$. We argue this view is incomplete. We posit a single, conserved, dimensionless scalar, denoted $S_u$, as the ontological basis for all measurable particle properties. Within a categorical framework where measurement axes (`Mass`, `Energy`, `Frequency`, etc.) are Objects, we demonstrate that the physical constants are not mere numbers but are the Morphisms that form a transitive Lie group of transformations, $G_T$, acting *between* these Objects. This "Transformation Group," whose elements include the named constants ($c, h, G, k_B$) as a generator set, describes a fundamental symmetry of *being*, not just description. The existence of this group is made necessary by the axiom of a singular substance $S_u$, as the group's transformations are precisely those that preserve the quantity of $S_u$ when the observational basis is changed. This reframes the constants as operators in a unified symmetry group, unifies conservation laws, and provides a new, more fundamental basis for the dimensional consistency of physical law.

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#### **1. Introduction: The Incompleteness of Constants**

The Standard Model of physics is built upon a set of fundamental constants ($c, h, G, k_B$, etc.) whose values are determined empirically. They are treated as fundamental features of the universe, yet their role remains conceptually ambiguous. The success of dimensional analysis, formalized by the Buckingham π theorem and the Lie group theory of units, demonstrates that physical laws must be invariant under a change of arbitrary base units. This is understood as a symmetry of *description*, where the Lie group of scaling transformations, $G_S \cong (\mathbb{R}^+)^k$, acts independently on each of the `k` fundamental dimensions (e.g., Mass, Length, Time).

This paper argues that this descriptive symmetry is a mere shadow of a deeper, ontological symmetry. We posit that the reason physical laws are scalable is because all physical properties are themselves interchangeable projections of a single, invariant source. We will show that the constants do not just enable unit conversion; they are the elements of a more profound Lie group that transforms the very categories of measurement into one another.

#### **2. The Ontological Axiom: The Conserved Scalar $S_u$**

We begin with a single axiom:

**Axiom:** The complete state of a fundamental particle or localized system can be characterized by a single, dimensionless, conserved scalar quantity, the Universal State or Substance, denoted $S_u$.

All observable physical properties, such as mass ($m$), energy ($E$), and frequency ($f$), are dimensioned projections of this single underlying substance. We can conceptualize a measurement as the action of a projection operator, $\mu_X$, which maps the dimensionless $S_u$ to a value on a specific measurement axis $X$:

$m = \mu_{Mass}(S_u)$

$E = \mu_{Energy}(S_u)$

$f = \mu_{Frequency}(S_u)$

The conservation of $S_u$ is the primordial conservation law from which all familiar conservation laws (of mass, energy, momentum) are derived.

#### **3. A Categorical Framework for Measurement**

To formalize the relationships between these projections, we define a category we will call **Meas**.

*   **Objects:** The Objects in **Meas** are the abstract axes of measurement themselves: `Mass`, `Energy`, `Frequency`, `Length`, `Time`, `Temperature`, `Momentum`, etc. These are types, not values.

*   **Morphisms:** The Morphisms in **Meas** are the physical constants that map one Object to another. A morphism $F: A \to B$ is a conversion operator from axis $A$ to axis $B$.

For example:

*   $c^2: \text{Mass} \to \text{Energy}$

*   $h: \text{Frequency} \to \text{Energy}$

*   $k_B: \text{Temperature} \to \text{Energy}$

*   $1/c: \text{Length} \to \text{Time}$

Crucially, the set of all morphisms is complete. For any two objects $A, B \in \text{Obj(Meas)}$, there exists a morphism $F: A \to B$. Most of these morphisms are unnamed composites of the "famous" constants (the Holy Quadinity). For example, the morphism from `Frequency` to `Mass` is the composition $(1/c^2) \circ h = h/c^2$. The traditionally named constants are merely a convenient **generator set** for the full groupoid of morphisms.

#### **4. Extending the Lie Group: From Scaling to Transformation**

The conventional Lie group of units, $G_S = (\mathbb{R}^+)^k$, acts *on* the objects of **Meas** but not *between* them. It represents the freedom to rescale our rulers. The $S_u$ framework reveals a new, more powerful Lie group, $G_T$, whose elements are the morphisms of **Meas**.

We define the group $G_T$ as the set of all invertible morphisms in **Meas**. This set forms a Lie group under the operation of composition:

1.  **Closure:** The composition of any two transformations is another transformation in the set. As shown, $(1/c^2) \circ h = h/c^2$, which is a valid morphism `Frequency → Mass`.

2.  **Identity:** For every object $A$, the identity morphism $Id_A: A \to A$ exists, corresponding to the dimensionless multiplicative identity, 1.

3.  **Inverse:** Every morphism $F: A \to B$ has an inverse $F^{-1}: B \to A$. The inverse of $c^2: \text{Mass} \to \text{Energy}$ is $1/c^2: \text{Energy} \to \text{Mass}$.

4.  **Associativity:** Composition of these linear operators is associative.

5.  **Manifold Structure:** The group elements are parameterized by the real-valued measures of the constants, giving $G_T$ the continuous manifold structure required for a Lie group

This **Transformation Group** $G_T$ does not describe a symmetry of our equations; it describes a fundamental symmetry of reality itself—the perfect, lossless convertibility between its different manifestations.

#### **5. The Role of $S_u$ in the Group Structure**

The existence of the group $G_T$ is a direct and necessary consequence of the $S_u$ axiom. The transformations of $G_T$ are precisely the operators required to maintain the invariance of $S_u$ across different observational frames (i.e., measurement axes).

Consider the relationship between `Mass`, `Frequency`, and `Energy`. This can be represented by a commutative diagram:

For the diagram to commute, the two paths from `S_u` to `Energy` must be equivalent:

$$ \mu_{Energy} = h \circ \mu_{Frequency} $$

And similarly for the other paths:

$$ \mu_{Energy} = c^2 \circ \mu_{Mass} $$

By equating these, we find that the morphisms must be related in a way that preserves the integrity of the projections from the single source:

$$ h \circ \mu_{Frequency} = c^2 \circ \mu_{Mass} $$

$$ \mu_{Mass} = (h/c^2) \circ \mu_{Frequency} $$

This confirms that the constant $h/c^2$ is the required group element for the transformation `Frequency → Mass`. The group $G_T$ is therefore not an abstract mathematical structure, but the operational manifestation of the unity of $S_u$. It is the group of transformations that leaves $S_u$ invariant.

#### **6. Implications and Future Directions**

This reformulation has profound implications:

1.  **Redefinition of Constants:** Physical constants are not fundamental numbers but are the **generators of a Lie group of physical transformations**. Their values are the parameters of this group in a given system of units.

2.  **Unification of Conservation Laws:** The conservation of energy, mass, and momentum are revealed to be different facets of the single, primordial law of the **Conservation of $S_u$**.

3.  **Reframing the Fine-Tuning Problem:** The question "Why do the constants have these specific values?" is ill-posed. The correct question is, "What is the structure of the Lie algebra of $G_T$?" The values we measure are just the coordinates of these abstract algebraic operators in the arbitrary basis of our SI units.

4.  **A Guide for New Physics:** If a new, apparently fundamental property of nature is discovered, this framework provides a clear path for investigation. We must seek the morphisms (the new "constants") that connect it to the existing objects in **Meas**. If such a morphism exists, we have not discovered a new substance, but a new dimension of projection for $S_u$—a new dimension for the manifold of the group $G_T$.

#### **7. Conclusion**

We have argued that the current understanding of physical constants as numerical values and dimensional analysis as a symmetry of description is insufficient. By positing a single, conserved, dimensionless scalar $S_u$ as the source of all physical properties, we reveal a more fundamental structure. The physical constants are the elements of a Lie group $G_T$ that acts to transform the categories of measurement into one another. This group represents a true symmetry of reality, ensuring that the quantity of underlying substance, $S_u$, is conserved regardless of the observational basis. This framework offers a more unified, coherent, and structurally sound foundation for the mathematical language of physics.