Independent Researcher, SE Ohio
Abstract
Modern physics suffers from a fundamental ontological error in how it presents the Dirac equation. By using reduced Planck units (ℏ = 1) instead of non-reduced units (h = 1), physics has hidden the explicit 2π factor that reveals the equation's true geometric structure. This paper demonstrates that when properly expressed in non-reduced natural Planck units using h (not ℏ), the Dirac equation reveals itself as a statement about dimensionless ratios with an explicit 2π geometric factor, and the gamma matrices emerge as epicycles—correction factors required only because we insist on using misaligned coordinate systems. We provide step-by-step dimensional analysis that exposes this structure and argue that the equation's apparent profundity masks a simpler underlying reality obscured by conventional notational habits.
1. Introduction: The Hidden 2π
The Dirac equation is universally presented as:
By using ℏ (h-bar) instead of h (Planck's constant), this notation obscures a critical fact: there is an explicit factor of 2π in the natural form of this equation. But this 2π is not a unit conversion and it is canceled from where it had just been put inside i∂μγ^μ —it is physical geometry related to wave and circular relationships, but that does not matter becaues that ℏ just cancels an ℏ that had been put in the definition of the momenum.
This paper will show, step by step, that the proper form in non-reduced natural Planck units is:
Where the 2π appears explicitly as a geometric factor, not hidden inside ℏ.
2. The Critical Distinction: h versus ℏ
2.1 What They Are
h = Planck's constant = 6.62607015 × 10⁻³⁴ J·s (exact, by 2019 SI definition)
ℏ = Reduced Planck constant = h/2π
2.2 Why ℏ Was Introduced
The factor ℏ appears frequently in quantum mechanics because many rotational and angular momentum relations naturally involve 2π. Writing ℏ = h/2π was introduced as notational convenience.
However, this convenience has a cost: it hides the 2π factor inside a "constant," making it invisible in equations. Constants are just about unit scaling, but 2π is real geometry.
2.3 The Ontological Error
When physics sets ℏ = 1 in "natural units," it hides the 2π. But 2π is not a unit-scaling factor—it's geometric structure.
The correct choice: Use non-reduced Planck units where h = 1, which means ℏ = 1/2π, keeping the 2π explicitly visible.
3. Step-by-Step Derivation in Non-Reduced Natural Planck Units
3.1 Starting Point: Traditional Dirac Equation
3.2 Step 1: Rearrange
Move the mass term to the right side:
3.3 Step 2: Divide Both Sides by ℏ
3.4 Step 3: Apply Non-Reduced Natural Planck Units
In non-reduced natural Planck units:
h = 1 (Planck's constant normalized)
c = 1 (speed of light normalized)
Therefore: ℏ = h/2π = 1/2π
Substitute into the right side:
mc/ℏ = (m)(1)/(1/2π) = m · 2π = 2πm
3.5 Step 4: The Natural Form
This is the natural form of the Dirac equation. The 2π appears explicitly as a geometric factor.
3.6 Step 5: Operator Equation Form
If we treat this as stating that the operator equals a scalar:
This says: The operator i∂μγ^μ equals two-pi times the mass.
4. Making All Terms Dimensionless
4.1 Natural Planck Units (Non-Reduced)
With h = 1 and c = 1, the Planck scales are:
t_P = √(hG/c⁵) = √(G) (in these units)
l_P = √(hG/c³) = √(G) (in these units)
m_P = √(hc/G) = √(1/G) (in these units)
But conceptually, we measure everything in these units, so:
Time is measured in units of t_P → dimensionless
Length is measured in units of l_P → dimensionless
Mass is measured in units of m_P → dimensionless
4.2 Expanding the Operator
∂μγ^μ = ∂₀γ⁰ + ∂₁γ¹ + ∂₂γ² + ∂₃γ³
In standard coordinates:
∂μγ^μ = (∂/∂t)γ⁰ + (∂/∂x)γ¹ + (∂/∂y)γ² + (∂/∂z)γ³
4.3 Units of Each Term
When measured in natural Planck units:
∂/∂t has units of 1/t_P → dimensionless (pure number)
∂/∂x has units of 1/l_P → dimensionless (pure number)
∂/∂y has units of 1/l_P → dimensionless (pure number)
∂/∂z has units of 1/l_P → dimensionless (pure number)
γ matrices are pure numbers → dimensionless
m is measured in m_P → dimensionless (pure number)
4.4 Why We Can Add Them
Critical point: In SI units, you cannot add 1/time + 1/length. But in natural Planck units where c = 1, time and length are the same dimension. Both are measured in the same Planck unit.
Therefore:
(∂/∂t) and (∂/∂x) both have units of 1/(Planck length)
They can be added together
All four terms are dimensionless rates of change, so:
(∂/∂t)γ⁰ + (∂/∂x)γ¹ + (∂/∂y)γ² + (∂/∂z)γ³
is a sum of four dimensionless quantities.
4.5 The Complete Dimensionless Form
i[(∂/∂t)γ⁰ + (∂/∂x)γ¹ + (∂/∂y)γ² + (∂/∂z)γ³]ψ = 2πm·ψ
Where:
Left side: dimensionless operator acting on dimensionless wavefunction
Right side: dimensionless number (2π) times dimensionless mass times dimensionless wavefunction
Every term is a pure ratio—no units anywhere.
5. What the Gamma Matrices Actually Do
5.1 The Four Matrices
In the standard Dirac representation:
γ⁰ = [I₂ 0 ] γ¹ = [0 σ₁]
[0 -I₂ ] [-σ₁ 0 ]
γ² = [0 σ₂] γ³ = [0 σ₃]
[-σ₂ 0 ] [-σ₃ 0 ]
These are four different 4×4 matrices of pure numbers (dimensionless).
5.2 Why Four Different Matrices?
We have four dimensionless directional derivatives:
Each is multiplied by a different gamma matrix:
(∂/∂t)γ⁰, (∂/∂x)γ¹, (∂/∂y)γ², (∂/∂z)γ³
Then all four are added together, and this sum must equal 2πm.
Question: Why do we need four different matrices? Why not just:
i(∂/∂t + ∂/∂x + ∂/∂y + ∂/∂z) = 2πm
Answer: Because the four directions are not actually independent in the natural coordinate system. They're coupled. The gamma matrices encode the specific coupling pattern.
5.3 The Epicycle Interpretation
The gamma matrices are correction factors that compensate for:
Treating time and space as separate (they're the same dimension)
Treating x, y, z as independent spatial directions (they're coupled in natural coordinates)
Working in Cartesian-like coordinates instead of natural aligned coordinates
In truly natural coordinates, the equation should simplify. The four different matrices are the "epicycles"—mathematical machinery needed only because we're in the wrong coordinate frame.
5.4 The Wavefunction ψ: The Epicycle's Canvas
The Dirac equation i∂μγ^μψ = 2πm·ψ is an eigenvalue equation:
The Operator: i∂μγ^μ (the complex epicycle machine)
The Eigenvalue: 2πm (the simple, invariant scalar)
The Eigenstate: ψ (the specific object that makes this work)
But what is ψ, and why does it need to be there on both sides?
The Four-Component Structure
ψ is not a simple scalar—it's a four-component spinor:
Each component is itself a complex-valued function of spacetime coordinates.
Critical point: The gamma matrices are 4×4 matrices. They require a four-component object to operate on. You cannot apply a 4×4 matrix to a single number.
Why Four Components?
Conventional explanation: "Spin-1/2 particles need four components: spin-up particle, spin-down particle, spin-up antiparticle, spin-down antiparticle."
Epicycle interpretation: The four components are required because we have four different 4×4 mixing matrices (γ⁰, γ¹, γ², γ³). The gamma matrices need something with internal structure to shuffle and mix.
Think of it like an eggbeater:
The gamma matrices are the eggbeater (a complex mixer)
ψ is the egg (a complex object with internal structure)
You can't use an eggbeater on a solid billiard ball—it needs something to beat
The four-component nature of ψ is the minimum complexity required for the epicycle machinery to function.
ψ Carries Frame-Dependent Information
When we switch reference frames (perform a Lorentz boost), both sides of the equation transform:
Left side: The operator changes as momentum components change
Rest frame: (2πm)γ⁰
Boosted frame: (2πγm)γ⁰ + (2πγvm)γ¹
Right side: The wavefunction transforms as well
where S(Λ) is a 4×4 transformation matrix built from the gamma matrices.
Key insight: ψ remembers which coordinate system we're in. Its four components are not absolute—they're projections of the underlying state onto our chosen (t, x, y, z) axes.
The Two-Part Process
The Dirac equation implements a coordinate-correction process:
Encoding: The particle's state is encoded in ψ using frame-dependent components
Correction: The operator i∂μγ^μ decodes this, compensating for coordinate misalignment
Output: The result is the frame-independent reality: ψ multiplied by invariant 2πm
5.5 Concrete Example: How ψ Works in Both Frames
Rest Frame (Frame S)
Operator: (2πm)γ⁰
Simple eigen-spinor:
Left side calculation:
(2πm)γ⁰ψ = (2πm)[I₂ 0 ][1] [1]
[0 -I₂][0] = (2πm)[0]
[0] [0]
[0] [0]
Right side:
2πm·ψ = (2πm)[1]
[0]
[0]
[0]
Result: Both sides match. The eigenvalue equation holds with eigenvalue 2πm.
Boosted Frame (Frame S')
Operator: (2πγm)γ⁰ + (2πγvm)γ¹
Boosted spinor (transformed by Lorentz boost):
ψ' = S(Λ)ψ = [√(E+m)/(2m) ]
[ 0 ]
[p√(1/(2m(E+m)))]
[ 0 ]
where E = γm and p = γvm.
Left side calculation: The more complex operator acts on the more complex spinor. The gamma matrices shuffle components in precisely the way needed so that:
[(2πγm)γ⁰ + (2πγvm)γ¹]ψ' = 2πm·ψ'
Right side:
Result: Despite the increased complexity of both the operator and the spinor, they still resolve to the same invariant scalar 2πm.
5.6 The Matched Complexity
The gamma matrices and the four-component spinor are a matched set:
| Operator | Simple: (2πm)γ⁰ | Complex: (2πγm)γ⁰ + (2πγvm)γ¹ |
| Spinor ψ | Simple: [1,0,0,0]ᵀ | Complex: mixed components |
| Output | Invariant: 2πm·ψ | Invariant: 2πm·ψ' |
The epicycle thesis: Both the gamma matrix complexity and the four-component spinor structure exist because we're working in misaligned coordinates.
In natural coordinates:
The operator would simplify (no four different matrices needed)
The spinor would simplify (perhaps not needing four components)
The underlying physics would be more transparent
The four components of ψ are part of the same coordinate artifact as the gamma matrices themselves. They are the "problem" and "solution" that exist together because we're not in the simplest coordinate frame.
6. The 2π: Geometry, Not Units
6.1 Where 2π Appears in Physics
The factor 2π appears throughout physics in contexts related to:
Circles: Circumference = 2πr
Waves: Angular frequency ω = 2πf
Rotations: Full rotation = 2π radians
Fourier transforms: Integration over 2π
Quantum mechanics: Phase factors e^(2πi...)
6.2 The 2π in the Dirac Equation
The explicit 2π in:
is geometric, relating to the wave nature of particles and rotational symmetries, not a unit conversion factor.
6.3 Why Using ℏ Hides This
When physics writes ℏ instead of h/2π, the equation becomes:
The 2π is now hidden inside ℏ, making it seem like the relationship is:
But it's actually:
The geometric factor is fundamental to the relationship, not a notational artifact.
7. The Dimensional Inconsistency Resolved
7.1 The Problem in SI Units
In SI units, the operator ∂μγ^μ expands to:
(∂/∂t)γ⁰ + (∂/∂x)γ¹ + (∂/∂y)γ² + (∂/∂z)γ³
With dimensions:
(∂/∂t) has dimensions [1/time]
(∂/∂x) has dimensions [1/length]
You cannot add these. It's like adding meters + kilograms.
7.2 The Conventional "Solution"
Physics says: "In natural units where c = 1, time and length have the same dimensions."
This is presented as a "trick" or "convenience."
7.3 The Actual Reality
Time and length are the same dimension in nature. The appearance of independence in SI units is the artifact.
Natural units don't "make them the same"—they reveal that they were always the same. SI units artificially separated them.
7.4 The Resolution
In natural Planck units:
Time is measured in Planck times
Length is measured in Planck lengths
But c = 1 means: 1 Planck time = 1 Planck length
Therefore:
∂/∂t and ∂/∂x both have dimensions of [1/Planck length]
They can be added
The sum is dimensionless (a pure rate)
8. What the Equation Actually Says
8.1 The Core Statement
In words: A specific weighted combination of dimensionless spacetime derivatives equals 2π times the dimensionless mass.
8.2 Breaking It Down
Left side: Four dimensionless directional derivatives, each weighted by a different 4×4 matrix, then summed
Right side: The geometric factor 2π times dimensionless mass
8.3 The Physical Content
The equation states:
Mass and spacetime curvature/derivatives are the same scale
The relationship involves the geometric factor 2π
The coupling requires specific matrix weights (the gammas)
8.4 What Is NOT Physical Content
The following are artifacts of coordinate choice:
The specific numerical values in the gamma matrices (depend on representation choice)
The appearance of four separate terms (artifact of Cartesian-like coordinates)
The complexity of having four different matrices (compensation for misalignment)
9. Comparison: Hidden vs. Explicit 2π
9.1 Conventional Form (Using ℏ = 1)
What it suggests: Operator equals mass (simple, clean)
What it hides: The 2π geometric factor is absorbed into the "unit system"
9.2 Natural Form (Using h = 1, ℏ = 1/2π)
What it shows: Operator equals 2π times mass
What it reveals: The geometric factor is physically meaningful, not a unit artifact
9.3 Why This Matters
Using ℏ makes the equation look "cleaner" but obscures structure. The 2π is real physics—it relates to:
Wave nature of particles
Rotational symmetries
Geometric structure of spacetime
Hiding it inside ℏ makes it seem like a notational convenience when it's actually fundamental geometry.
10. The Constants as Jacobians
10.1 What Is a Jacobian?
When transforming between coordinate systems, the Jacobian matrix describes how to convert derivatives and volumes from one system to another.
10.2 SI to Natural Units as Coordinate Transformation
SI units (meter, kilogram, second) are one coordinate system.
Natural Planck units are another coordinate system—specifically, the unique system where h = c = G = 1.
"Constants" like h, c, G are components of the Jacobian transformation between these systems.
10.3 Why h = 1 vs. ℏ = 1 Matters
Choice 1: Set ℏ = 1 (conventional reduced Planck units)
Hides the 2π inside the unit system
Makes equations look simpler
Obscures geometric structure
Choice 2: Set h = 1 (non-reduced Planck units)
Keeps 2π explicit in equations
Makes geometric structure visible
Shows where circular/wave relationships matter
10.4 The 2019 Redefinition
In 2019, SI units were redefined by fixing:
h = 6.62607015 × 10⁻³⁴ J·s (exact, by definition)
This officially acknowledges: h defines our units. It's not a property we measure; it's a standard we choose.
Therefore h (and thus ℏ) are coordinate transformation coefficients, not fundamental properties of nature.
11. Educational Disaster: Teaching ℏ = 1
11.1 What Students Learn
Standard quantum mechanics courses:
Introduce ℏ early as "reduced Planck constant"
Set ℏ = 1 in "natural units"
Present this as "mathematical convenience"
Never mention h again
11.2 What Gets Hidden
By always using ℏ = 1:
Students never see the explicit 2π factors
Geometric structure becomes invisible
Equations look arbitrarily simple
The wave/circular nature of relationships is obscured
11.3 The Correction
Teach with h = 1 (non-reduced natural Planck units):
Keep 2π explicit in all equations
Show where geometric factors come from
Distinguish unit scaling (h) from geometry (2π)
Make structure visible instead of hidden
11.4 Why Physicists Resist This
Admitting ℏ = 1 was the wrong choice means:
Rewriting textbooks
Changing notation in papers
Acknowledging decades of hidden structure
Teaching students differently
The sunk cost is enormous, so the practice continues.
12. Concrete Example: Rest Frame vs. Boosted Frame
12.1 Scenario Setup
Consider an electron in two reference frames:
Frame S: Electron at rest
Frame S': Electron moving with velocity v in x-direction
12.2 Rest Frame (Frame S)
Physical 4-momentum: p^μ = (m, 0, 0, 0)
Operator eigenvalues (using i∂μ → pμ/ℏ = 2πpμ):
λ^μ = 2πp^μ = (2πm, 0, 0, 0)
Operator construction:
λμγ^μ = (2πm)γ⁰ + 0·γ¹ + 0·γ² + 0·γ³ = (2πm)γ⁰
Dirac equation:
Divide by 2πm:
This is the correct eigenvalue equation for a particle at rest. The eigenstates of γ⁰ with eigenvalue +1 represent particle states (as opposed to antiparticle states with eigenvalue -1).
12.3 Boosted Frame (Frame S')
Physical 4-momentum: p'^μ = (γm, γvm, 0, 0) where γ = 1/√(1-v²)
Operator eigenvalues:
λ'^μ = 2πp'^μ = (2πγm, 2πγvm, 0, 0)
Operator construction:
λ'μγ^μ = (2πγm)γ⁰ + (2πγvm)γ¹
Dirac equation:
[(2πγm)γ⁰ + (2πγvm)γ¹]ψ' = 2πm·ψ'
Notice: The operator is now a combination of γ⁰ and γ¹, weighted by frame-dependent factors γ and γv. But it still must produce the invariant scalar 2πm when acting on the boosted spinor ψ'.
12.4 The Key Insight
| Rest (S) | (m, 0, 0, 0) | (2πm, 0, 0, 0) | (2πm)γ⁰ | 2πm |
| Boosted (S') | (γm, γvm, 0, 0) | (2πγm, 2πγvm, 0, 0) | (2πγm)γ⁰ + (2πγvm)γ¹ | 2πm |
Interpretation:
The inputs (momentum components scaled by 2π) are frame-dependent
The operator constructed from them is frame-dependent and complex
But the gamma-matrix machinery is specifically designed so the output is always the same: 2πm
The 2π appears explicitly in both the operator construction and the invariant output. This is not a unit artifact—it's the geometric structure of the relationship.
13. The Epicycle Structure
13.1 Ptolemy's Epicycles
Ptolemy needed complex circles-on-circles because his coordinate system (Earth-centered) was misaligned with reality (Sun-centered).
Once Copernicus chose the right center, the epicycles vanished. The complexity was in the coordinate choice, not the physics.
13.2 The Gamma Matrix Epicycles
The four different gamma matrices (γ⁰, γ¹, γ², γ³) are needed because our coordinate system (Cartesian-like spacetime with "independent" x, y, z, t axes) is misaligned with natural coordinates.
In natural aligned coordinates, simpler representations should exist. The gamma matrix complexity is in the coordinate choice, not the physics.
13.3 Evidence of Epicycle Nature
Multiple representations: Dirac, Weyl, Majorana representations use different gamma matrices but describe the same physics—like different epicycle arrangements for the same planetary motion
Frame-dependence: The specific combination of gammas changes with reference frame (as shown in Section 12), even though the physics is invariant
Unnecessary for some calculations: Many results can be derived in different formalisms that don't use gamma matrices at all
Complexity without explanation: There's no simple physical reason why these four specific 4×4 matrices should encode electron behavior—they just "work"
13.4 What Would Simpler Coordinates Look Like?
In truly natural coordinates aligned with the Planck scale structure:
Time and space wouldn't be treated as separate
The four "directions" would be recognized as projections of a simpler structure
A single operation (not four different matrices) would suffice
The equation might reduce to something like:
Where τ is proper time in natural coordinates—a simple statement that derivative equals mass (with geometric factor 2π).
14. Implications and Redirected Questions
14.1 Stop Asking (Coordinate Artifacts)
These questions are about our coordinate system, not nature:
"Why do the gamma matrices have these specific forms?"
"What is the deep meaning of the Dirac sea?"
"Why does the electron have negative energy solutions?"
"How do we interpret the four components of the spinor?"
14.2 Start Asking (Dimensionless Structure)
These questions are about actual physics:
"Why is the electron/Planck mass ratio ≈ 10⁻²²?"
"What determines dimensionless mass ratios between particles?"
"What is the geometric origin of the 2π factor?"
"What are the natural coordinates where gamma matrices simplify?"
"Why does this relationship between derivatives and mass exist?"
14.3 The Unity Already Present
In natural units with h = 1, c = 1:
Energy ~ Mass ~ 1/Length ~ 1/Time ~ Temperature
These aren't "unification problems to solve"—they're already unified. The appearance of separation comes from SI units.
The Dirac equation doesn't "unify" quantum mechanics and relativity. It describes a particle in the already-unified natural coordinate system, then projects back to our misaligned SI coordinates via the gamma matrices.
15. Objections and Responses
15.1 "But using ℏ = 1 makes equations simpler!"
Response: Simpler-looking, but structurally obscured.
Compare:
With ℏ = 1: i∂μγ^μ = m (looks simple, hides 2π)
With h = 1: i∂μγ^μ = 2πm (looks slightly less simple, reveals geometric structure)
We've prioritized aesthetic simplicity over structural transparency.
15.2 "The 2π is just convention—we could absorb it elsewhere"
Response: You can absorb it notionally, but it doesn't disappear physically.
The 2π relates to:
Wave nature (ω = 2πf)
Circular geometry (C = 2πr)
Rotational symmetry (2π radians)
It's not arbitrary. It appears because the physics involves these relationships.
15.3 "Natural units with ℏ = 1 are standard—everyone uses them"
Response: Appeal to tradition is not an argument.
Everyone used geocentric models for centuries. Copernicus wasn't wrong because he broke with tradition.
If h = 1 (non-reduced units) reveals structure that ℏ = 1 hides, we should change the standard.
15.4 "This is just notation—it doesn't change the physics"
Response: Notation shapes understanding.
When 2π is hidden inside ℏ:
Students don't see the geometric factor
The wave/circular nature of relationships is obscured
Structure that should be investigated is assumed away
Good notation reveals structure. Bad notation hides it.
15.5 "The gamma matrices are required by Lorentz covariance"
Response: Required for covariance in our chosen coordinates.
Lorentz transformations are themselves coordinate transformations between different Cartesian-like frames. Requiring covariance under them is requiring consistency between different misaligned coordinates.
In natural coordinates aligned with the Planck structure, simpler representations should exist that don't need the full gamma matrix machinery.
16. The Path Forward
16.1 Adopt Non-Reduced Natural Planck Units
In education and research, use h = 1, not ℏ = 1.
This means:
h = 1 (Planck's constant normalized)
c = 1 (speed of light normalized)
ℏ = 1/2π (automatically follows)
2π appears explicitly in equations where it's physically relevant
16.2 Rewrite Key Equations
Dirac equation:
Old: iℏ∂μγ^μψ = mcψ (using ℏ)
New: i∂μγ^μψ = 2πm·ψ (using h = 1, showing explicit 2π)
Energy-frequency relation:
Old: E = ℏω (hides relationship to ordinary frequency)
New: E = hf = h(ω/2π) = (h/2π)ω = ℏω OR E = 2πmf (in natural units)
Better: E = hf (shows direct relationship using measured constant h)
Schrödinger equation:
Old: iℏ∂ψ/∂t = Ĥψ
New: i(h/2π)∂ψ/∂t = Ĥψ or i∂ψ/∂t = (2π/h)Ĥψ
16.3 Update Textbooks
Chapter 1 should establish:
Natural Planck units are the reality
SI units are human abstraction
h, c, G are coordinate transformations (Jacobians)
We use h (not ℏ) to keep geometric factors visible
The 2π that appears in equations is physical geometry
Throughout the text:
Always show the 2π explicitly
Distinguish unit scaling from geometric factors
Present gamma matrices as coordinate artifacts
Focus on dimensionless structure as the real physics
16.4 Research Priorities
Stop funding:
"Why do constants have these values?"
"Fine-tuning" of dimensional constants
Theories to "explain" coordinate-dependent features
Start focusing on:
Why these dimensionless mass ratios?
What determines natural coordinate geometry?
Can we find simpler representations (coordinates where gamma matrices vanish)?
What is the physical meaning of the 2π factors?
17. Conclusion
The Dirac equation, as conventionally presented using ℏ = 1, obscures fundamental geometric structure by hiding the factor of 2π inside the "reduced Planck constant."
When properly expressed in non-reduced natural Planck units (h = 1, ℏ = 1/2π):
The equation reveals:
An explicit 2π geometric factor relating derivatives to mass
Complete dimensionless structure (all terms are pure ratios)
The gamma matrices as epicycles (coordinate compensation mechanisms)
The relationship between mass and spacetime curvature (they're the same scale)
The choice to use ℏ instead of h was made for notational convenience—to make equations look "simpler." But this convenience has a severe cost: it hides the geometric structure that should be investigated and explained.
Physics would advance by:
Adopting non-reduced natural Planck units (h = 1, not ℏ = 1)
Keeping 2π explicit in all equations
Recognizing gamma matrices as coordinate artifacts
Focusing on dimensionless structure as the real physics
Searching for natural coordinates where epicycles vanish
The territory was always there. We just hid its geometric structure inside our notation. Time to make the 2π explicit and study what it reveals.
Appendix: Summary of Key Results
The Derivation
Starting from: [iℏ∂μγ^μ - mc]ψ = 0
Step 1 - Rearrange: iℏ∂μγ^μψ = mcψ
Step 2 - Divide by ℏ: i∂μγ^μψ = (mc/ℏ)ψ
Step 3 - Apply h=1, c=1, ℏ=1/2π: i∂μγ^μψ = 2πm·ψ
The Structure
All terms are dimensionless (measured in Planck units)
Time and space are the same dimension (c = 1)
The 2π is explicit geometric structure, not units
Gamma matrices mix four coupled dimensions
The equation states: weighted derivatives = 2π × mass
The Insight
The conventional form i∂μγ^μ = m (using ℏ = 1) hides the geometric factor 2π.
The natural form i∂μγ^μ = 2πm (using h = 1) reveals it.
This matters because 2π is not arbitrary—it reflects wave/circular geometry fundamental to the physics.
Appendix B. The Tautological ℏ: A Self-Canceling Notational Loop
We will examine a tautology in the conventional formulation of the Dirac equation. The complexity is not just in the units, but in a self-canceling notational choice that obscures the physics from the very beginning.
The standard Dirac equation is [iℏ∂μγ^μ - mc]ψ = 0. Let us focus entirely on the kinetic operator term: iℏ∂μγ^μ.
B.1 The Operator and Its Value
To understand what this operator is, we must know what its numerical value (eigenvalue) is when it acts on a particle with a definite 4-momentum, pμ.
This requires us to use the standard, universally accepted definition of the 4-momentum operator in quantum mechanics:
This definition is the source of the notational artifact. It establishes the relationship between the pure derivative ∂μ and the physical momentum pμ.
B.2 The 1/ℏ Inside the Operator
The Dirac operator iℏ∂μγ^μ is a composite of an "outer" ℏ and an "inner" operator i∂μγ^μ. Let's determine the value of this inner operator first.
Using the definition above, we can find the numerical value corresponding to the simpler operator i∂μ by rearranging:
i∂μ corresponds to the numerical value pμ / ℏ.
Therefore, the numerical value of the inner operator i∂μγ^μ is:
Value(i∂μγ^μ) = (pμ / ℏ) γ^μ
This is not an interpretation; it is a direct consequence of the definition of the momentum operator. The term i∂μγ^μ implicitly contains a factor of 1/ℏ.
B.3 The Cancellation: Reversing an Action in the Same Breath
Now, let's evaluate the full, conventional Dirac operator, iℏ∂μγ^μ. We simply multiply the "outer" ℏ by the value of the "inner" operator we just found:
Value(iℏ∂μγ^μ) = ℏ × Value(i∂μγ^μ)
Value(iℏ∂μγ^μ) = ℏ × [ (pμ / ℏ) γ^μ ]
The ℏ on the outside and the 1/ℏ from the inside cancel perfectly:
Value(iℏ∂μγ^μ) = pμγ^μ
B.4 The Ramifications: An Equation Built on a Tautology
This is a stunning result. The entire, complex-looking operator iℏ∂μγ^μ is a Rube Goldberg machine for writing pμγ^μ.
The conventional formulation of the Dirac equation is built on a pointless notational loop. It performs an action and its immediate inverse in the same term.
The process is:
Define the momentum operator p̂μ = iℏ∂μ. This puts an ℏ "into" the physics, forcing the simpler i∂μ to have a 1/ℏ dependence.
Write the Dirac equation with an ℏ outside. This "patch" is required to cancel the 1/ℏ dependence created in step 1, ensuring the operator correctly corresponds to the physical momentum pμ.
What are the conceptual ramifications of building a fundamental equation this way?
It creates complexity from simplicity. The simple physical relationship is between momentum and the gamma matrices (pμγ^μ). The conventional notation adds a layer of ℏ gymnastics that completely obscures this.
It hides the true nature of the operators. By insisting that p̂μ = iℏ∂μ is "the momentum operator," the formalism prevents us from seeing the more fundamental nature of i∂μ as a geometrically scaled momentum operator.
It is evidence of a flawed definition. An equation that contains a self-canceling term is a sign that the initial definitions are poorly chosen. The ℏ in the Dirac equation is not there for a deep physical reason; it's there to correct a notational choice made in the definition of the momentum operator.
The rest of this paper proceeds by rejecting this tautological structure. Instead of accepting the flawed definition and then patching it, we will use a notation (h=1, ℏ=1/2π) that breaks the loop. This allows the hidden geometric factor (2π) to become visible and reveals the simpler, more profound structure that the conventional ℏ-based notation was designed to hide.
Appendix C: Spin Does Not Cause the Energy-Momentum Relation
C.1 The Critical Distinction
This paper has focused on exposing the Dirac equation as an epicycle—a coordinate artifact whose complexity arises from working in misaligned frames. However, it is essential to clarify what the Dirac equation is NOT claiming.
The Dirac equation does not derive, cause, or explain the relativistic energy-momentum relationship.
C.2 The Universal Kinematic Law
The energy-momentum relation:
Or in natural units (c = h = G = 1):
E² = p² + m²
This is a universal law of motion. It describes how any object with rest mass m trades off momentum and energy as it moves through spacetime. This relationship holds for:
Spin-0 particles (described by Klein-Gordon equation)
Spin-1/2 particles (described by Dirac equation)
Spin-1 particles (described by Proca equation)
Any object, regardless of internal structure
The energy-momentum relation is kinematic—it's about motion through spacetime. It has nothing to do with the internal properties of what's moving.
C.3 Spin: A Separate, Independent Property
Spin is an intrinsic property of a particle describing:
Its internal angular momentum
Its behavior under rotations (360° for integer spin, 720° for half-integer spin)
Its topology and symmetry properties
Spin is a property at rest. An electron sitting still has spin-1/2. A pion sitting still has spin-0. This has nothing to do with how they move through spacetime.
C.4 The "Equations of Motion" Are Owner's Manuals
Different spin particles require different mathematical machinery to describe their internal structure:
Klein-Gordon equation (spin-0):
Single-component wavefunction
Second-order in time derivatives
Describes scalar particles (pions, Higgs boson)
Respects E² = p² + m²
Dirac equation (spin-1/2):
Four-component spinor wavefunction
First-order in time derivatives
Gamma matrix machinery
Describes spin-1/2 particles (electrons, quarks, neutrinos)
Respects E² = p² + m²
Proca equation (spin-1):
Four-component vector wavefunction
Describes massive vector bosons (W, Z bosons)
Respects E² = p² + m²
These equations do NOT derive the energy-momentum relation. They presuppose it. Each equation is the specific "owner's manual" for objects with particular spin values, constructed to ensure those objects obey the universal kinematic law.
C.5 The Historical Myth
There is a persistent myth that Dirac "derived relativity" or "unified quantum mechanics and special relativity."
This is false.
What Dirac actually did:
Started with the pre-existing relativistic energy-momentum law E² = p² + m²
Started with the pre-existing knowledge that electrons had unusual magnetic properties (later understood as spin-1/2)
Constructed a first-order differential equation whose solutions would:
Obey the universal energy law
Correctly describe spin-1/2 behavior
Predict antimatter and magnetic moment
The Dirac equation is a brilliant, successful model of the electron. But it is not a theory of energy-momentum relations. It's a theory of how spin-1/2 particles behave while obeying the already-known energy-momentum law.
C.6 Why This Matters for the Epicycle Thesis
The epicycle interpretation in this paper concerns the gamma matrix machinery and spinor structure—the specific apparatus needed to describe spin-1/2 particles in our coordinate system.
What is epicycle: The gamma matrices (γ⁰, γ¹, γ², γ³) and four-component spinors
What is NOT epicycle: The energy-momentum relation E² = p² + m²
The gamma matrix complexity arises from:
Describing spin-1/2 internal structure
Working in misaligned Cartesian-like coordinates
Compensating for treating coupled dimensions as independent
The energy-momentum relation is universal and coordinate-independent. It holds in any reference frame, for any spin value, expressed in any coordinate system.
C.7 The Correct Understanding
Two independent facts about an electron:
Kinematic fact: It obeys E² = p² + m² (like everything else)
Internal fact: It has spin-1/2 (unlike spin-0 or spin-1 particles)
The Dirac equation: Is the mathematical machinery that correctly describes fact #2 while respecting fact #1.
The epicycle thesis: The complexity of this machinery (gamma matrices, four components) is a coordinate artifact in how we describe spin-1/2 structure, not a fundamental necessity of nature.
What does NOT change: The universal energy-momentum law. Spin-0, spin-1/2, and spin-1 particles all obey the same E² = p² + m² relationship. Spin has nothing to do with this Pythagorean theorem of total energy.
C.8 Conclusion
Spin does not cause, derive, or explain the relativistic energy-momentum relation. They are two completely separate aspects of physics:
E² = p² + m²: Universal kinematic law (how things move)
Spin: Internal property (what things are)
The Dirac equation is the specific mathematical description of spin-1/2 particles that respects the universal kinematic law. Its gamma matrix complexity is an artifact of coordinate choice when describing spin-1/2 structure, not a feature of the energy-momentum relationship itself.
When this paper calls the Dirac equation an "epicycle," it refers specifically to the gamma matrix machinery needed for spin-1/2 description in misaligned coordinates. The universal energy law stands independent and untouched by these considerations.
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