Abstract
The standard treatment of natural units in physics education relies on informal procedures like "setting ℏ=c=1" without rigorous mathematical justification. We demonstrate that natural units are simply the application of a change-of-basis transformation (Jacobian) from arbitrary human unit systems to the unique, dimensionless physical scale of reality. This reframing exposes that dimensional analysis is linear algebra, physical constants are Jacobian matrix entries, and "Planck units" are not a separate unit system but rather the diagonal components of the transformation matrix itself. The widespread confusion stems from treating coupled basis vectors as if they were independently meaningful scalars.
1. Introduction: The Conceptual Nightmare
Physics students are taught to "set constants to 1" when working in natural units:
- "Let c = 1"
- "Work in units where ℏ = 1"
- "Set G = 1 for simplicity"
When pressed for justification, the response is vague: "It has something to do with units." This hand-waving obscures what is actually a straightforward mathematical operation. But these are not natural ratios, you are still in a defined unit chart, just with a couple axis unified to each other. You still have units.
The purpose of this paper is to provide the rigorous framework that physics education systematically omits.
2. The Fundamental Distinction
2.1 Physical Reality is Dimensionless
The universe exists independent of human observation. Physical quantities have definite ratios to one another. These ratios are pure numbers—dimensionless scalars.
There are no "meters" in the fabric of spacetime. There are no "seconds" in the passage of time. There are no "kilograms" in the property of inertia. These are human inventions. They exist only in our heads. We picked arbitrary convenient scales for our perceived reality at the size we are.
2.2 Unit Charts are Arbitrary Human Constructs
To measure and communicate about physical quantities, humans create unit systems:
- SI (Système International): {meter, kilogram, second, ampere, ...}
- CGS: {centimeter, gram, second, ...}
- Imperial: {foot, pound, second, ...}
These systems exist entirely in our imagination. They are coordinate charts we place over physical reality for convenience. We can create infinitely many such charts—each is equally valid (or invalid) as a description of nature. Their basis rotations can all be rotated between each other perfectly.
2.3 The Critical Error
The standard framework confuses:
- Unit charts (arbitrary, infinite, imaginary)
- Physical reality (unique, dimensionless, invariant)
This confusion leads to the false notion that there are "different natural units" (Planck units, Stoney units, etc.). In truth, there is only ONE natural scale—physical reality itself. What vary are the arbitrary unit charts from which we begin.
All unit charts must also align to the singular scale of the universe to describe that universe.
3. The Mathematical Structure
3.1 Dimensional Analysis is Linear Algebra
Every unit chart defines a basis in a dimensional vector space. For mechanical systems, this space is typically 3-dimensional, spanned by {Mass, Length, Time}.
Any physical quantity Q can be expressed as:
[Q] = M^α L^β T^γ
This is a vector in log-space:
log[Q] = (α, β, γ)
Dimensional analysis is vector addition in this space. Unit conversions are linear transformations.
3.2 Physical Constants as Coupling Vectors
The fundamental constants h, c, G are not independent scalars. They are coupled vectors in dimensional space:
[c] = L^1 M^0 T^(-1) → (0, 1, -1)
[h] = L^2 M^1 T^(-1) → (1, 2, -1)
[G] = L^3 M^(-1) T^(-2) → (-1, 3, -2)
These form the dimensional matrix D:
h c G
⎡ 1 0 -1 ⎤ M
D = ⎢ 2 1 3 ⎥ L
⎣ -1 -1 -2 ⎦ T
This matrix encodes how the constants {h, c, G} are constructed from the basis {M, L, T}.
3.3 The Jacobian Transformation
To transform from an arbitrary unit chart (e.g., SI) to the natural physical scale, we must solve:
⎛ M_Jacobian ⎞ ⎛ h ⎞
⎜ L_Jacobain ⎟ = D^(-1) ⎜ c ⎟
⎝ T_Jacobain ⎠ ⎝ G ⎠
The inverse matrix D^(-1) is the Jacobian that maps from our arbitrary basis to physical reality.
Computing this inverse yields:
M_Jacobian = √(hc/G) ≈ 5.46 × 10^(-8) kgL_Jacobian = √(hG/c³) ≈ 4.05 × 10^(-35) m
T_Jacobian = √(hG/c⁵) ≈ 1.35 × 10^(-43) s
These are the entries traditionally called "Planck units."
3.4 "Planck Units" are Jacobian Entries
What physics calls "Planck length," "Planck mass," and "Planck time" are not units. They are the components of the transformation matrix D^(-1).
They tell us: "The SI meter is 10^35 times larger than the one natural unit free physical scale" (and similarly for mass and time against that same unified scale).
The name "Planck units" is deeply misleading. A more accurate name would be: "SI-to-Natural Ratio Conversion Factors" or simply "The Jacobian from SI to Unit Free Physical Reality."
4. The Inseparability Problem
4.1 Cannot Isolate Components
The expression L_p = √(hG/c³) appears to define a "natural length." But examine what this actually says:
- G has dimensions [L³M^(-1)T^(-2)]
- h has dimensions [L²M T^(-1)]
- c has dimensions [LT^(-1)]
To define "length," we use constants that themselves contain mass, length, and time. The definitions are circular and coupled. This is a tautology. But when you are inside the tautology it appears to be doing heavy lifting.
You cannot define L_natural without simultaneously defining M_natural and T_natural. They form a system of equations, not independent quantities.
4.2 Basis Vectors vs Scalars
The standard treatment makes a category error: it treats the components (L_p, M_p, T_p) as if they were independent scalars that could be "chosen" separately.
But these are basis vectors. They form a coordinate system. Using "Planck length" while keeping SI seconds is like rotating your x-axis but leaving your y-axis unchanged—you destroy the orthogonality of your coordinate system. You are not choosing a "natural unit system," you are just choosing a different unit chart that is like SI but that makes specific problems easier to work.
4.3 The Planck Basis
There is no such thing as "a Planck unit" in isolation. They do not define their own scale, they define the si unit chart's scale agains a physical reality that is a unified single physical scale that is unit free. It is not about measurement in that natural basis, it is about relationships and unity.
There is only:
The SI Planck Basis = the complete coordinate system obtained by applying D^(-1) to {h, c, G}.
The natural unit free physical reality is an all-or-nothing transformation. You either rotate your entire basis to align with physical reality, or you don't. Partial transformations are mathematically incoherent.
5. The One Natural Scale
5.1 Terminal Object in Category Theory
In category-theoretic terms, physical reality is the terminal object in the category of unit charts.
- Objects: All possible unit systems any intelligence might invent
- Morphisms: Change-of-basis transformations (Jacobians)
- Terminal object: The single unified dimensionless physical scale
Every unit chart has a unique morphism (Jacobian) to this terminal object. The physical scale itself has no "units"—it simply is.
5.2 No Multiple Natural Scales
The standard framework speaks of "different natural units":
- Planck units (using h, c, G)
- Stoney units (using e, c, G)
- Atomic units (using e, m_e, ℏ)
This is category confusion. These are not different physical realities. They are different conventional choices of defining any one of the other unit charts against a unified reality that only has a single unit free physical reality. You are just defining unit charts similar to si, just that simplies an area for conceptual clarity and calculation convenience.
Physical reality is unique, unit free, and unified. The Jacobian from any given unit chart to physical reality is unique. What varies is which arbitrary unit chart you're starting from, not what you're mapping to.
5.3 The ℏ Error
Standard treatments often use ℏ (h-bar) instead of h:
ℏ = h/(2π)
This creates the illusion of two different constants, leading to claims of "different natural units."
But ℏ is not a distinct physical constant. It is a notational convenience—a pre-factored form of h.
When you write E = ℏω, you have:
E = (h/2π)(2πf) = hf
The 2π cancels immediately. You haven't done anything physical—you've just shuffled symbols on paper.
The Jacobian from SI to physical reality has one entry for action, not two. Whether you write it as h or as ℏ is arbitrary bookkeeping, not physics.
The correct Jacobian uses h, c, G—the actual measured constants, not notational variants.
When you calcualte a reduced Planck unit all you did was divide the actual SI Jacobian by 1/sqrt(2pi), that is not defining a different scale that is just dividing both sides of a formula by the same constant.
If I said "x = 7" and then divided both sides by a fixed constant, it would not change the value of x.
If I say "m_P = √(hc/G)" and then divide both sides by √(2pi) it does not change the value of m_P, same SI Jacobian.
6. What "Natural Units" Actually Means
6.1 The Correct Procedure
When a physicist says "work in natural units," what they actually mean is:
- Start with equations written in an arbitrary unit chart (usually SI)
- Apply the Jacobian D^(-1) to transform all quantities to the dimensionless physical scale
- Work with dimensionless ratios (the actual physics)
- If needed for communication, transform back to SI using D
This is a standard change of basis—nothing more, nothing less.
6.2 Why Constants "Disappear"
In the natural basis, the constants h, c, G are no longer needed because you're already working in the scale they defined in your unit chart
It's like measuring room dimensions: if you use "one room-length" as your unit, you don't need to keep writing "1 room = 1 room." The conversion factor is absorbed into your choice of basis.
The constants don't "become 1." They become unnecessary because you've eliminated the mismatch between your coordinate system and physical reality.
Constants vanish because in natural rations there are no units, so you don't have unit chart artifacts. You are working in a unified single scale of reality.
6.3 The Invalid Operation
What is mathematically invalid is the informal procedure: "set c = 1."
You cannot "set" a dimensional quantity equal to a dimensionless number. Did you change the length, the time, or both? That's a type error. It's like saying "let 5 meters = 1." One is a length, one is a number—they're not the same kind of thing.
What you actually mean is: "transform to coordinates where we can see space and time are really the same thing we see as two diffrent effects in our perceptions."
What you are really doing is just fixing your error bars
c= l_P/t_P
l_P is how badly we miscalled length to physical reality.
t_P is how badly we miscalled time to physical reality.
c just rotates one unified thing between two of our perceptual categories,
7. Why the Obfuscation Persists
7.1 Sacred Symbol Manipulation
Physics education has built entire pedagogical structures around manipulating symbols in the SI chart. Expertise is demonstrated by knowing:
- Which constants to "set to 1" in which order
- Which "natural units" to "choose" for which problems
- How to navigate between "different natural unit systems"
If the truth were taught clearly—"compute the Jacobian from your arbitrary chart to physical reality"—this edifice of mystification would collapse.
7.2 The Threat of Trivialization
Admitting that natural units are just a basis transformation would trivialize decades of expertise. Students could understand it immediately with basic linear algebra.
The obfuscation is load-bearing. It maintains the priesthood's control over "sophisticated physical intuition" that is, in reality, just confused notation.
The physical reality is that the universe is trivially unified and only has one physical scale.
7.3 Dimensional Analysis as Philosophy
When critics point out the mathematical incoherence, they are dismissed: "You don't understand the subtleties." "This is just dimensional analysis—not real physics." "Do you think you're smarter than 120 years of the greatest scientists?"
But the critics are correct. The emperor has no clothes. The "subtleties" are: there are none in reality. There's your confused arbitrary chart in your own head, there's physical reality, there's the Jacobian matrix that rotates between measurement and ratios. Done.
The resistance to this clarity is not scientific—it's sociological.
8. Conclusion
Natural units, properly understood, are:
- A change of basis from arbitrary human coordinates to dimensionless physical reality
- Encoded by a Jacobian matrix whose entries are combinations of h, c, G
- An all-or-nothing transformation—you cannot use components separately
- A map to a unique terminal object—physical reality has one scale, not many
The standard framework obscures this with informal procedures and mystical language about "setting constants to 1" or "choosing appropriate units."
This is not physics. This is notational confusion masquerading as profundity.
The rigorous statement is simple: Dimensional analysis is linear algebra. Natural units are applying the Jacobian.
Everything else is obfuscation.
Appendix: Catalog of Dogma and Corrections
Dogma 1: "Set c = 1"
Correction: You cannot set a dimensional quantity equal to a dimensionless number. You are applying a Jacobian transformation to eliminate the mismatch between your arbitrary units and physical reality.
Dogma 2: "Choose units where ℏ = c = 1"
Correction: You do not choose physical reality. Physical reality is invariant relationships between things. You are computing how your arbitrary unit chart relates to that invariant reality. Also, use h, not ℏ—the latter is just notational pre-cancellation of 2π. Because ℏw = hf, no 2pi on either side.
Dogma 3: "Planck units are a system of units"
Correction: "Planck units" are not units. They are the Jacobian entries—the transformation matrix components that relate SI to dimensionless physical reality.
Dogma 4: "Work in Planck units vs Stoney units vs atomic units"
Correction: There is ONE physical reality. These are different arbitrary starting points (different conventional choices of constants), not different natural scales. The terminal object is unique.
Dogma 5: "Planck length is the natural unit of length"
Correction: "Planck length" by itself is meaningless. It is one component of a coupled transformation. You cannot use L_p while keeping SI seconds—that's like rotating one axis but not the others.
Dogma 6: "[length] = [time] in natural units"
Correction: Dimensions are basis vectors in your arbitrary chart. In physical reality, there are no separate "dimensions"—only dimensionless ratios. Saying [length] = [time] is human bookkeeping, not physics.
Dogma 7: "Different natural units for different problems"
Correction: Physical reality doesn't change based on your problem. You have one Jacobian from your chart to reality. How you choose to express that Jacobian (which constants to make explicit) is notation, not physics.
Dogma 8: "Natural units require physical intuition to use properly"
Correction: Natural units require knowing how to invert a 3×3 matrix. Any claim of deeper "subtlety" is gatekeeping.
Dogma 9: "Using ℏ vs h gives different natural units"
Correction: ℏ = h/(2π). When you write ℏω = (h/2π)(2πf) = hf, the 2π cancels. There is one Jacobian entry for action, not two. Notational choices don't create different physical scales.
Dogma 10: "This is just dimensional analysis, not rigorous math"
Correction: Dimensional analysis IS rigorous mathematics—it is linear algebra over the reals. The dimensional matrix D and its inverse D^(-1) are well-defined linear transformations. The standard treatment violates basic linear algebra by treating coupled basis vectors as independent scalars.
Dogma 11: "We've done it this way for 120 years"
Correction: Length of tradition does not validate incoherence. Epicycles lasted 1400+ years. For 120 years, physics has taught students to manipulate symbols without understanding the underlying coordinate transformation. This is pedagogical failure, not accumulated wisdom.
Dogma 12: "Multiple natural unit systems exist"
Correction: Unit systems are human inventions that exist in our heads. Physical reality is unique and dimensionless. What exist are: (1) infinitely many arbitrary unit charts we could invent, (2) one physical reality, (3) the unique Jacobian from each chart to reality.
The bottom line: Stop setting constants to 1. Start applying Jacobians. The universe has one scale. We made up coordinates. The math that connects them is undergraduate linear algebra.
That's the whole story.
No comments:
Post a Comment