The Rigor Gambit: A Formal Inquiry into the Ontological Status of Physical Constants

 

Author: The Persistent Grad Student
Affiliation: Department of Foundational Inquiry, (A Fictional University Where This is Allowed)

Abstract
The common pedagogical practice in theoretical physics of "setting fundamental constants to 1" (e.g., ħ=c=1) is treated as a benign convenience. This paper argues that this practice, while computationally useful, masks a deep ontological ambiguity that is never formally addressed. We demonstrate that the act of selectively setting certain constants to unity is not arbitrary but is instead a powerful informal heuristic that points to a fundamental classification of physical quantities. We propose a rigorous mathematical definition to distinguish between dimensional constants (human artifacts, eligible to be "set to 1") and dimensionless constants (physical invariants, ineligible for such treatment). The resistance to this formalization is examined not as a mathematical problem, but as a sociological one within the physics community.

1. Introduction: The Hand-Wave Heuristic
The initiation rite for any graduate student of theoretical physics involves learning the incantation: "We now adopt units where ħ=c=G=k_B=1." This is typically followed by a wave of the hand and the justification that this "simplifies the algebra" or "removes clutter." The student is expected to accept this without a rigorous definition of the principle that allows this rescaling for some parameters but not for others.

This paper originates from a simple, forbidden question: "What is the formal mathematical criterion that determines whether a physical parameter can be 'set to 1'?"

2. The Gedankenexperiment: Setting Everything to 1
To expose the ambiguity, consider the following gambit. If the principle is mere "convenience," then the ultimate convenience is to set every parameter in the Lagrangian of the Standard Model to 1. The math becomes trivial; the Lagrangian reduces to a collection of terms with unit coefficients. We could then declare physics solved and, as the joke goes, go have a beer.

The immediate, visceral rejection of this proposal by any physicist proves that the "convenience" argument is a superficial gloss. The real, unstated reason is that setting the electron mass m_e or the fine-structure constant α to 1 results in predictions that violently contradict experimental reality. In contrast, setting c or ħ to 1 does not.

This reveals a fundamental dichotomy:

• Category A (The "Set-to-1-able"): Parameters whose numerical value can be changed arbitrarily via a choice of units without altering physical predictions.

• Category B (The "Not-Set-to-1-able"): Parameters whose numerical value is invariant under any change of units and whose value has direct physical consequences.

3. A Proposed Formal Definition: The Dimensionality Criterion
The heuristic distinction can be formalized with mathematical rigor.

Definition 1: Dimensional Constant
A physical quantity Q is a dimensional constant if and only if its value can be changed to any positive number by a choice of system of units. This is equivalent to the statement that Q has non-trivial dimensions (i.e., its dimensions are not the identity element in the group of dimensional analysis). Examples: c ([L][T]⁻¹), ħ ([M][L]²[T]⁻¹), G ([M]⁻¹[L]³[T]⁻²).

Definition 2: Dimensionless Constant
A physical quantity α is a dimensionless constant if and only if its value is invariant under any change of system of units. This requires that its dimensions are trivial ([M]⁰[L]⁰[T]⁰). Examples: the fine-structure constant α, the electron-to-proton mass ratio m_e / m_p.

Theorem: The Eligibility for "Setting to 1"
A physical constant K is eligible to be "set to 1" via a choice of units if and only if K is a dimensional constant.

Proof:

1. (If) Let K be a dimensional constant with dimensions [K]. A new system of units can always be defined where the base units are rescaled such that the numerical value of K becomes 1. This is a coordinate transformation on the space of units. The physics remains unchanged because all measurable quantities are dimensionless ratios, which are invariant under this rescaling.

2. (Only if) Let K be a dimensionless constant. Any attempt to "set K=1" is not a choice of units but a change to the theory itself. Its value governs the strength of interactions or the relative scales of phenomena. Altering it changes the physical predictions of the theory. Therefore, it cannot be "set to 1" by fiat; it must be measured.

4. The Sociological Implication: The Unspoken Truth
This formal definition validates the grad student's intuition: dimensional constants are not profound features of the universe but are human-centric scaling artifacts. They are the conversion factors between the arbitrary rulers we invented (meters, seconds, kilograms) and the natural scales of the universe (Planck length, Planck time, Planck mass).

The "mystery" of why c ≈ 3e8 m/s is a pseudo-mystery. The true, profound fact is that spacetime has a geometry with a fundamental invariant speed at all. The numerical value is merely a measure of how mismatched our human-scale second is to our human-scale meter, relative to the fundamental structure of reality.

The resistance to formalizing this distinction is sociological. To admit that c, ħ, and G are human artifacts is to dismantle a pillar of popular science discourse and to force a reevaluation of what the true "fundamental constants" are: the dimensionless parameters like α. This reorientation would shift the focus of foundational inquiry away from explaining large numbers and toward explaining the values of these pure, dimensionless ratios.

5. Conclusion: A Call for Rigorous Pedagogy
The "hand-wave" is a symptom of a deeper pedagogical failure. The practice of natural units should be taught not as a convenient trick, but as the rigorous mathematical act it is: a coordinate transformation that reveals the invariant core of physical theory.

We call for the explicit teaching of the Dimensionality Criterion as the formal justification for "setting constants to 1." This would:

1. Demystify dimensional constants, correctly identifying them as artifacts of measurement.

2. Re-mystify dimensionless constants, properly focusing attention on the true unknowns of physical theory.

3. Empower students to understand the difference between a choice of coordinates (physics) and a change to the theory itself (not physics).

The grad student who asks for this rigor is not being difficult; they are asking the first truly fundamental question. A field that cannot—or will not—provide a rigorous answer to a question about its most basic procedures risks mistaking its own conventions for deep truths. The first step toward genuine progress is to recognize that the emperor's constants are, in fact, just his clothes, and that the real emperor is the dimensionless structure of the universe itself.