Dimensionless Physics
Dimensionless
Physics: A Unified Framework
Author: Rowan Brad Quni-Gudzinas
Affiliation: QNFO Email:
rowan.quni@qnfo.org ORCID: 0009-0002-4317-5604
ISNI: 0000000526456062 DOI:
10.5281/zenodo.17161226 Version: 1.0.1
This work presents dimensionless physics as the fundamental language
of nature, demonstrating how all physical laws can be expressed as
relationships between pure dimensionless ratios rather than
anthropocentric units. Building upon the Buckingham Pi Theorem, this
document shows that dimensional constants such as G,
c, and ħ are not fundamental but emerge from a deeper
causal network of wave correlations. When equations are expressed in
dimensionless form, these constants vanish, revealing the true
mathematical structure of physical reality as pure relationships between
dimensionless ratios. This work derives dimensionless forms of
fundamental equations from Newtonian gravity to Einstein’s field
equations and examines the irreducible dimensionless constants that
define the universe’s character. Integrating Stergios Pellis’s
dimensionless theory with temporal quantization through fractal
geometry, this document presents a unified framework that connects
quantum mechanics, general relativity, and cosmology through pure
numerical relationships. The generative focal point of this work is to
establish dimensionless physics as the essential language for a complete
theory, with profound implications for fundamental physics, cosmology,
and the unification of all natural phenomena.
1.0 Introduction
The edifice of modern physics is built upon mathematical equations
that describe the behavior of the universe. A profound characteristic of
these equations is their dimensional consistency. This principle, known
as dimensional homogeneity, asserts that meaningful physical laws cannot
equate disparate concepts like mass and time; they must compare like
with like. In systems of measurement based on human scales—the meter,
the kilogram, the second—this consistency is enforced by the
introduction of dimensional constants such as the gravitational constant
G or the reduced Planck constant ħ. These constants,
however, are not fundamental but emerge from a deeper causal network of
wave correlations. This reliance on dimensional constants obscures a
more fundamental truth: the laws of physics are ultimately statements
about relationships between pure, dimensionless numbers. The
mathematical framework that formalizes this concept is the Buckingham π
theorem, which proves that any physically meaningful relationship can be
expressed entirely in terms of dimensionless ratios, independent of any
specific, arbitrarily chosen system of units (Buckingham, 1914). This
work establishes the Buckingham π theorem as the logical cornerstone for
a dimensionless description of physics, demonstrating that scale
invariance is not a mathematical trick but a core principle of the
cosmos.
This perspective fundamentally reorients our understanding of
reality. Contrary to conventional wisdom, “quantum gravity” is not a
distinct category of physics but is an emergent phenomenon arising from
a causal network of wave correlations (Quni-Gudzinas, 2025). The Planck
constant ħ is not a fundamental constant of nature but emerges
as a conversion factor between frequency and energy within our current
measurement framework. Reality is not composed of particles or
wavefunctions but of causal relationships—morphisms in a category whose
objects represent quantum events. Mass is fundamentally frequency rather
than merely being measured through frequency, and gravity emerges as the
thermodynamic consequence of phase coherence among quantum field
correlations interacting with curved spacetime (Quni-Gudzinas,
2025).
This document presents a comprehensive framework for dimensionless
physics that integrates these insights with the dimensionless theory
developed by Stergios Pellis (Pellis, 2023), temporal quantization
through fractal geometry (Pellis, 2025), and quantum correlation
synchronization theory (Quni-Gudzinas, 2025). This work demonstrates how
all fundamental laws can be expressed in dimensionless form, revealing
the true mathematical structure of physical reality and providing a
unified framework for understanding the cosmos, fulfilling the
generative focal point described in Section 0.0.
2.0
The Mathematical Foundation: Buckingham Pi Theorem and Dimensional
Analysis
2.1
Dimensional Homogeneity: The Prerequisite for Physical Law
Any equation purporting to be a physical law must adhere to the
principle of dimensional homogeneity. This principle states that each
term in a valid physical equation must have the same physical
dimensions. For example, in the kinematic equation for an object under
constant acceleration, x(t) = x₀ +
v₀t + (1/2)at², each term—the final position
x(t), the initial position x₀, the
velocity-time product v₀t, and the acceleration-time
squared product (1/2)at²—must have the dimension of length,
denoted as [L]. An equation such as x =
v₀t + a, which equates a length to the sum of
a length and an acceleration ([L][T]⁻²), is physically
meaningless.
This principle serves as a powerful constraint on the possible forms
that physical laws can take. It is the reason why dimensional constants
appear in our equations. Consider Newton’s law of universal gravitation,
F = G(m₁m₂/r²). The
dimensions of force are [M][L][T]⁻², while
the dimensions of the mass-distance term are
[M]²[L]⁻². To make the equation dimensionally
homogeneous, a constant of proportionality, G, must be
introduced with dimensions [M]⁻¹[L]³[T]⁻². In
an anthropocentric system of units such as the International System of
Units, G has an experimentally determined value of
approximately 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻². The necessity of this constant
is entirely a feature of the chosen unit system. If units were defined
such that the gravitational force between two unit masses separated by a
unit distance was one unit of force, G would be numerically
equal to 1. This reveals that dimensional constants are, in essence,
conversion factors required to maintain dimensional consistency within
an arbitrary, human-constructed framework of measurement (Bridgman,
1922).
2.2 The
Buckingham Pi Theorem: Formalizing Scale Invariance
The Buckingham π theorem provides the formal mathematical basis for
recasting physical laws into a dimensionless form. It elevates the
principle of dimensional homogeneity from a simple consistency check to
a powerful tool for revealing the fundamental structure of physical
relationships (Buckingham, 1914).
2.2.1 Statement and Proof of
the Theorem
The theorem states that if a physically meaningful equation involves
n physical variables and constants, and these quantities can be
expressed using k fundamental, independent physical dimensions,
then the original equation can be rewritten as an equation involving a
set of p = n - k independent, dimensionless
parameters, π₁, π₂, …, πₚ.
Formally, if a physical relationship is given by
f(Q₁, Q₂, …, Qₙ) = 0, where the
Qᵢ are the n physical variables, the theorem asserts
that this can be rewritten as g(π₁, π₂, …, πₙ₋ₖ) = 0. Each
dimensionless parameter, or π-group, is a product of powers of the
original variables: πᵢ = Q₁ᵃ¹ Q₂ᵃ² … Qₙᵃⁿ,
where the exponents aᵢ are chosen such that the combination is
dimensionless.
The proof of the theorem relies on linear algebra. The dimensions of
each quantity Qᵢ can be expressed as a product of powers of the
k fundamental dimensions D₁, D₂, …,
Dₖ, as in [Qᵢ] = D₁ᵈᵢ¹ D₂ᵈᵢ² …
Dₖᵈᵢₖ. For a π-group to be dimensionless, the exponents of its
constituent variables must satisfy a system of linear homogeneous
equations. The number of independent solutions to this system
corresponds to the nullity of the dimensional matrix formed by the
exponents dᵢⱼ. By the rank-nullity theorem, the nullity is
n - rank, where the rank of the dimensional matrix is equal to
k, the number of independent dimensions. Thus, there are
n - k independent dimensionless parameters.
2.2.2 Significance and
Implications
The profound significance of the Buckingham π theorem is that it
guarantees that the validity of any physical law is independent of the
specific unit system used to express it. The fundamental physical
content of a law is not contained in the dimensional quantities
themselves, but in the relationships between the dimensionless π-groups.
If the values of these dimensionless combinations were to change with a
change of units, the equation would not be a universal identity, and the
theorem would not hold.
This principle of “similarity” is the foundation of all physical
modeling. For instance, in hydrodynamics, the drag force D on a
submerged object depends on the fluid density ρ, viscosity μ, flow
velocity v, and a characteristic length L. Here, there
are n = 5 variables and k = 3 fundamental dimensions
(M, L, T). The Buckingham π theorem predicts
n - k = 2 dimensionless parameters. These can be
chosen as the drag coefficient, π₁ =
D/(ρv²L²), and the Reynolds number, π₂ =
ρvL/μ. The physical law is then expressed as π₁ =
f(π₂). A small-scale model in a wind tunnel is physically
“similar” to a full-scale airplane if its Reynolds number is the same.
The underlying physics is identical because the relationship between the
dimensionless parameters is the same, even though the dimensional values
of force, velocity, and size are different.
A generalization of the theorem considers the case where nF
of the independent variables are held at fixed values, such as the
fundamental constants of nature. If kF of these fixed variables
are dimensionally independent, the number of independent dimensionless
parameters is further reduced by (nF - kF). In
fundamental physics, the constants c, G, and
ħ are our fixed variables. As they are dimensionally
independent, nF = kF = 3. Therefore, (nF -
kF) = 0, meaning their existence does not further reduce the
number of dimensionless parameters beyond what the standard theorem
predicts. This is a crucial point: it explains why, even after using the
dimensional constants to set a natural scale, we are still left with a
set of irreducible, fundamental dimensionless constants that must be
determined by experiment. These constants, such as the fine-structure
constant discussed in Section 5.1, are the universe’s intrinsic
π-groups, whose values are not determined by the dimensional constants
alone.
3.0 Defining a
Universal Metric: The Emergent Scale
The Buckingham π theorem, as established in Section 2.2, provides the
mathematical license to describe physics dimensionlessly, but it does
not specify which scales to use for this process. To move from an
abstract framework to a physically meaningful one, a system of units
must be identified that is not based on arbitrary, terrestrial standards
but arises from the causal structure of the universe itself. This system
is the Planck scale, derived from the constants that underpin
relativity, gravity, and quantum mechanics. The Planck units represent a
convenient scaling system that emerges from the deeper causal network of
wave correlations, providing a natural metric for the cosmos.
3.1 The Limits of
Anthropocentric Units
Standard systems of units, such as the International System of Units,
are fundamentally anthropocentric. The second was historically defined
as a fraction of a solar day on Earth. The meter was originally defined
in relation to the Earth’s circumference. The kilogram is based on a
physical artifact, and though now redefined in terms of the Planck
constant, its magnitude is still tied to this historical artifact. While
these units are practical for human engineering and commerce, they are
arbitrary from the perspective of fundamental physics. They bear no
intrinsic relationship to the underlying laws of nature.
This was the motivation of Max Planck in 1899. He sought to establish
a system of units that would be independent of any specific object,
planet, or civilization. He envisioned units that would “necessarily
maintain their meaning for all time and for all civilizations, even
those which are extraterrestrial and nonhuman,” which he termed
“fundamental physical units of measurement” (Planck, 1899). These units
would be derived solely from the universal constants of nature.
3.2
Derivation of the Planck Units via Dimensional Analysis
The Planck units are derived by combining constants that emerge from
the causal structure of the universe in such a way as to produce
quantities with the dimensions of length, mass, and time. To derive the
Planck charge, a fourth constant related to electromagnetism is
included. The four constants are first, the speed of light in vacuum,
c, from special relativity with dimensions
[L][T]⁻¹; second, the universal gravitational
constant, G, from general relativity with dimensions
[M]⁻¹[L]³[T]⁻²; third, the reduced Planck
constant, ħ = h/2π, from quantum mechanics with
dimensions [M][L]²[T]⁻¹; and fourth, the
Coulomb constant, ke = 1/(4πε₀), from electromagnetism with
dimensions [M][L]³[T]⁻²[Q]⁻² where
Q is the dimension of charge. By performing a dimensional
analysis, exponents n₁, n₂, n₃ are found such
that a combination like cⁿ¹Gⁿ²ħⁿ³ yields a
desired dimension.
3.2.1 Planck Mass (mP)
A combination with dimensions of mass, [M], is sought. The
dimensional equation [mP] =
[c]ⁿ¹[G]ⁿ²[ħ]ⁿ³ expands to
[M]¹[L]⁰[T]⁰ =
[M]⁻ⁿ²⁺ⁿ³[L]ⁿ¹⁺³ⁿ²⁺²ⁿ³[T]⁻ⁿ¹⁻²ⁿ²⁻ⁿ³. This
yields a system of three linear equations for the exponents:
-n₂ + n₃ = 1, n₁ + 3n₂ +
2n₃ = 0, and -n₁ - 2n₂ - n₃ = 0.
Solving this system gives n₁ = 1/2, n₂ = -1/2, and
n₃ = 1/2. Therefore, the Planck mass is defined as follows.
\[m_P = \sqrt{\frac{\hbar c}{G}} \approx
2.176 \times 10^{-8} \text{kg}\]
3.2.2 Planck Length (ℓP)
A combination with dimensions of length, [L], is sought. The
system of equations for the exponents is: -n₂ + n₃ =
0, n₁ + 3n₂ + 2n₃ = 1, and -n₁ -
2n₂ - n₃ = 0. Solving this system gives n₁ =
-3/2, n₂ = 1/2, and n₃ = 1/2. Therefore, the Planck
length is defined as follows.
\[\ell_P = \sqrt{\frac{\hbar G}{c^3}}
\approx 1.616 \times 10^{-35} \text{m}\]
3.2.3 Planck Time (tP)
A combination with dimensions of time, [T], is sought. The
system of equations is: -n₂ + n₃ = 0, n₁ +
3n₂ + 2n₃ = 0, and -n₁ - 2n₂ -
n₃ = 1. Solving this system gives n₁ = -5/2,
n₂ = 1/2, and n₃ = 1/2. Therefore, the Planck time is
defined as follows.
\[t_P = \sqrt{\frac{\hbar G}{c^5}} =
\frac{\ell_P}{c} \approx 5.391 \times 10^{-44} \text{s}\]
3.2.4 Planck Charge (qP)
To derive the Planck charge, the Coulomb constant ke is used
to seek a combination with dimensions of charge, [Q]. The
standard definition relates charge to the other fundamental units by
setting the proportionality constant in Coulomb’s law to unity in the
new system. This leads to the definition of Planck charge.
\[qP = \sqrt{4\pi\epsilon0 \hbar c} =
\sqrt{\frac{\hbar c}{k_e}} \approx 1.876 \times 10^{-18}
\text{C}\]
This value is approximately 11.7 times the elementary charge
e.
3.2.5 Derived Planck Units
From these base units, all other physical quantities can be derived.
Two of the most important are Planck energy and Planck force. The Planck
energy, EP, is defined as EP = mPc²
= √(ħc⁵/G), with an approximate value of 1.22 × 10¹⁹
GeV. The Planck force, FP, is defined as FP =
EP/ℓP = c⁴/G, with an approximate value of
1.21 × 10⁴⁴ N.
3.3 The Emergent Nature
of the Planck Scale
The Planck scale is not a fundamental regime but an emergent scale
from the causal network of wave correlations. It represents the scale
where the causal structure of the universe manifests in ways that create
the appearance of extreme curvature and quantum effects. At the Planck
length and Planck time, the energy required to probe such scales, the
Planck energy, is so immense that it would create a microscopic black
hole with a Schwarzschild radius on the order of the Planck length
itself. This is a manifestation of the emergent causal structure rather
than fundamental physics breaking down.
Using Planck units to nondimensionalize physical equations is a
convenient approach. It is mathematically equivalent to setting
c = 1, G = 1, ħ = 1, and ke = 1.
This is a profound physical statement, declaring that all quantities are
being measured as ratios relative to the emergent scales of nature
itself. An energy expressed in Planck units is a dimensionless number
that answers the question: “How does this energy compare to the energy
scale at which causal wave correlations manifest quantum-like
behavior?”
This framework reveals some of the deepest mysteries in physics. For
example, the “hierarchy problem” is the question of why gravity is so
much weaker than the other fundamental forces. In Planck units, this is
no longer a question about the disparate values of dimensional constants
but a question about a pure number. The mass of a proton is
approximately 10⁻¹⁹mP. The electrostatic force between two
protons is roughly 10³⁶ times stronger than their gravitational
attraction. The hierarchy problem is thus reframed as the question of
why the characteristic dimensionless masses of elementary particles are
so extraordinarily small compared to 1. Expressing physics
dimensionlessly does not just simplify the equations; it exposes the
fundamental, unexplained numerical ratios that define the character of
our universe (Rees, 2000).
4.0
Nondimensionalization of Fundamental Equations
The power of the dimensionless framework is revealed when it is
applied to the fundamental equations of physics. By recasting these laws
in dimensionless form, the dimensional constants—which emerge from the
causal network of wave correlations—vanish, leaving behind pure
mathematical relationships between dimensionless ratios. This process
not only simplifies the equations but also uncovers their essential
structure and reveals deep connections between seemingly disparate areas
of physics.
4.1
Classical Mechanics: Newton’s Law of Universal Gravitation
4.1.1 Standard Form
The familiar form of Newton’s law of universal gravitation is:
\[F = G\frac{m1 m2}{r^2}\]
4.1.2 Nondimensionalization
Dimensionless variables, denoted with a prime (′), are introduced by
scaling each dimensional quantity with its corresponding Planck unit, as
defined in Section 3.2: F′ = F/FP,
m₁′ = m₁/mP, m₂′ =
m₂/mP, and r′ = r/ℓP. The
dimensional variables are then expressed in terms of the dimensionless
ones: F = F′FP, m₁ =
m₁′mP, m₂ = m₂′mP, and
r = r′ℓP.
4.1.3 Derivation
Substituting the scaled variables into Newton’s law yields:
F′FP =
G(m₁′mP)(m₂′mP)/(r′ℓP)².
By substituting the definitions of the Planck units, FP =
c⁴/G, mP = √(ħc/G), and ℓP
= √(ħG/c³), the equation becomes:
\[F' \left(\frac{c^4}{G}\right) =
G\frac{m1' m2' \left(\frac{\hbar c}{G}\right)}{r'^2
\left(\frac{\hbar G}{c^3}\right)}\]
Simplifying the expression reveals a common factor of
c⁴/G on both sides of the equation, which can be
cancelled.
4.1.4 Dimensionless Form
The resulting dimensionless equation is strikingly simple:
\[F' = \frac{m_1'
m_2'}{r'^2}\]
In this form, the law of gravitation is a pure relationship between
dimensionless numbers. The gravitational constant G has been
absorbed into the definitions of the units. The equation states that the
gravitational force, measured as a fraction of the Planck force, is
equal to the product of the masses, measured as fractions of the Planck
mass, divided by the square of the distance, measured as a fraction of
the Planck length.
4.2 Electromagnetism: Coulomb’s
Law
4.2.1 Standard Form
The standard form of Coulomb’s law is:
\[F = ke \frac{q1 q_2}{r^2}\]
4.2.2 Nondimensionalization
Dimensionless variables are defined by scaling with the appropriate
Planck units: F′ = F/FP, q₁′ =
q₁/qP, q₂′ = q₂/qP, and
r′ = r/ℓP.
4.2.3 Derivation
Substituting the scaled variables into Coulomb’s law gives
F′FP =
ke(q₁′qP)(q₂′qP)/(r′ℓP)².
Using the convenient forms FP = ħc/ℓP² and qP
= √(ħc/ke), the equation becomes:
\[F' \left(\frac{\hbar
c}{\ellP^2}\right) = ke \frac{q1' q2' \left(\frac{\hbar
c}{ke}\right)}{r'^2 \ellP^2}\]
The term ħc/ℓP² appears on both sides, allowing for its
cancellation.
4.2.4 Dimensionless Form
The dimensionless form of Coulomb’s law is:
\[F' = \frac{q_1'
q_2'}{r'^2}\]
This result is profound. In the natural system of Planck units, the
mathematical form of Coulomb’s law is identical to that of Newton’s law
of gravitation, derived in Section 4.1. This reveals that the
inverse-square laws of gravity and electromagnetism share the same
fundamental structure. The vast difference in their perceived strengths
is not a property of the laws themselves, but a consequence of the
dimensionless magnitudes of the “charges” that particles possess—mass
for gravity (m/mP) and electric charge for
electromagnetism (q/qP).
4.3
Quantum Mechanics: The Time-Dependent Schrödinger Equation
4.3.1 Standard Form
The time-dependent Schrödinger equation for a single non-relativistic
particle of mass m in a potential V is (Schrödinger,
1926):
\[i\hbar\frac{\partial \Psi}{\partial t} =
-\frac{\hbar^2}{2m}\nabla^2 \Psi + V\Psi\]
4.3.2 Nondimensionalization
A set of dimensionless variables and operators is defined:
t′ = t/tP, r′ = r/ℓP, so ∇
= (1/ℓP)∇′ and ∇² = (1/ℓP²)∇′²; m′ = m/mP;
V′ = V/EP; and Ψ′(r′, t′) =
ℓP³/²Ψ(r, t), to ensure the probability density |Ψ′|²
is dimensionless.
4.3.3 Derivation
Transforming the derivatives and substituting into the Schrödinger
equation yields:
\[\frac{i\hbar}{t_P} \frac{\partial
\Psi'}{\partial t'} = -\frac{\hbar^2}{2m' mP \ellP^2}
\nabla'^2 \Psi' + V' E_P \Psi'\]
Dividing the entire equation by the Planck energy, EP, makes
all terms dimensionless. The dimensionless coefficients are evaluated
using the definitions of the Planck units from Section 3.2. The
coefficient of the time derivative term,
ħ/(tPEP), becomes 1. The coefficient of the
kinetic energy term,
ħ²/(2m′mPℓP²EP), becomes
1/(2m′).
4.3.4 Dimensionless Form
The resulting dimensionless Schrödinger equation is:
\[i \frac{\partial \Psi'}{\partial
t'} = -\frac{1}{2m'} \nabla'^2 \Psi' + V'
\Psi'\]
In this form, the constant ħ, which is synonymous with
quantum mechanics, has vanished. It has been absorbed into the scale of
the system. The only parameter specific to the particle that remains is
its dimensionless mass, m′. This reframes mass in a quantum
context: it is a dimensionless number that dictates how a particle’s
wave function evolves relative to the Planck scale. Mass, in this view,
is a fundamental measure of a particle’s resistance to quantum
delocalization.
4.4 General
Relativity: Einstein’s Field Equations
4.4.1 Standard Form
The equations are written in tensor notation as (Einstein, 1915):
\[G{\mu\nu} + \Lambda g{\mu\nu} =
\frac{8\pi G}{c^4} T_{\mu\nu}\]
4.4.2 Nondimensionalization
The following dimensionless tensors are defined: G′μν =
GμνℓP², Λ′ = ΛℓP², and T′μν =
Tμν/(EP/ℓP³).
4.4.3 Derivation
Substituting the scaled variables into the field equations gives:
\[G'_{\mu\nu} + \Lambda'
g{\mu\nu} = \frac{8\pi G \ellP^2}{c^4} \frac{EP}{\ellP^3}
T'{\mu\nu} = \frac{8\pi G EP}{c^4 \ell_P}
T'_{\mu\nu}\]
Evaluating the dimensionless coefficient on the right-hand side by
substituting the definitions of EP and ℓP from Section 3.2
shows that it is equal to 8π.
4.4.4 Dimensionless Form
The dimensionless Einstein field equations are:
\[G'_{\mu\nu} + \Lambda'
g{\mu\nu} = 8\pi T'{\mu\nu}\]
In this form, the constants G and c have
disappeared. The equations now represent a direct relationship between
dimensionless geometry (on the left) and dimensionless matter-energy
content (on the right). The factor of 8π remains as a purely geometric
factor. The equation states that the curvature of spacetime, measured in
units of the Planck curvature (1/ℓP²), is proportional to the density of
stress-energy, measured in units of the Planck density.
5.0 The Universe’s Intrinsic
Parameters
Once the fundamental equations of physics are expressed in
dimensionless form as shown in Section 4.0, the dimensional constants of
scale (c, G, ħ, ke) vanish. What
remains is a set of irreducible, dimensionless numbers whose values are
not fixed by theory but must be determined by experiment. These are the
true constants of nature that define the specific character of our
universe. They are the universe’s fundamental π-groups, as explained in
Section 2.2. A universe with the same laws but different values for
these constants would be a radically different place.
5.1
The Fine-Structure Constant (α): The Strength of Light and Matter
5.1.1 Definition and
Dimensionless Nature
In SI units, the fine-structure constant, denoted α,
is defined as:
\[\alpha = \frac{e^2}{4\pi\epsilon_0 \hbar
c} = \frac{k_e e^2}{\hbar c}\]
Its experimentally measured value is approximately 1/137.036. To
confirm its dimensionless nature, the dimensions of its components are
analyzed: [ke] =
[M][L]³[T]⁻²[Q]⁻², [e] =
[Q], [ħ] = [M][L]²[T]⁻¹,
and [c] = [L][T]⁻¹. Combining these for the
expression kee² / (ħc) yields the following
dimensional analysis.
\[
\begin{split}
[\alpha] &= \frac{[k_e][e]^2}{[\hbar][c]} =
\frac{([M][L]^3[T]^{-2}[Q]^{-2}) \cdot [Q]^2}{([M][L]^2[T]^{-1}) \cdot
([L][T]^{-1})} \\
&= \frac{[M][L]^3[T]^{-2}}{[M][L]^3[T]^{-2}} \\
&= [M]^{0} [L]^{0} [T]^{0} [Q]^{0}
\end{split}
\]
The final expression indicates that the fine-structure constant α is
a dimensionless physical constant. As required by the protocol in
Section 4.6.1.3, this result demonstrates that all fundamental
dimensions cancel completely, signifying that the constant is a pure
number independent of any system of units, not that its numerical value
is zero.
5.1.2 Physical Meaning
The fine-structure constant is the coupling constant of quantum
electrodynamics (QED). Its value determines the strength of the
interaction between charged particles and photons (Feynman, 1949). In
the framework of Planck units, its meaning is particularly clear. Using
the definition of the Planck charge, qP =
√(ħc/ke), from Section 3.2.4, the constant can be
written as α = (e/qP)². Thus, α is the square of the
ratio of the elementary charge to the Planck charge. It is a fundamental
dimensionless measure of the strength of electric charge in the
universe.
5.2 The
Proton-to-Electron Mass Ratio (μ): The Scale of Matter
5.2.1 Definition and
Dimensionless Nature
The proton-to-electron mass ratio, denoted by μ or β, is defined as μ
= mp/me, where mp is the rest mass of the
proton and me is the rest mass of the electron. Its value has
been experimentally determined to be approximately 1836.15267343. As it
is a ratio of two quantities with the same dimension, it is inherently a
pure number.
5.2.2 Physical Meaning
The value of μ is critical for the structure of matter. Because the
proton is nearly 2000 times more massive than the electron, the nucleus
of an atom is extremely heavy and slow-moving compared to the orbiting
electrons. This large mass ratio justifies the Born-Oppenheimer
approximation in quantum chemistry, which allows for the separation of
nuclear and electronic motion, making stable molecular bonds and complex
chemistry possible. Unlike α, the value of μ is the result of a complex
interplay between different physical theories. The majority of the
proton’s mass comes from the kinetic energy of its constituent quarks
and the binding energy of the gluon field, as described by quantum
chromodynamics (‘t Hooft, 1974). Therefore, μ is a complex emergent
constant whose value depends on the parameters of both the electroweak
theory and the strong nuclear force.
5.3 The
Cosmological Constant (Λ): The Energy of Nothing
5.3.1 Definition and
Dimensionless Form
In its standard formulation, Λ has dimensions of inverse length
squared, [L]⁻². Its fundamental nature is better captured by
comparing it to other scales in the universe. One key dimensionless form
is the cosmological density parameter, ΩΛ, which is the ratio of the
energy density associated with the cosmological constant to the critical
density of the universe. Observations place the value of ΩΛ at
approximately 0.7. A more fundamental dimensionless representation is
obtained by scaling Λ with the Planck length: Λ′ = ΛℓP². Using observed
cosmological parameters, this dimensionless constant has a value of
approximately 10⁻¹²².
5.3.2 Physical Meaning
The incredibly small number, Λ′ ≈ 10⁻¹²², represents the energy
density of the vacuum itself, measured in natural Planck units. The
“cosmological constant problem” is the mystery of why this number is so
extraordinarily close to zero. Calculations in quantum field theory
predict that the vacuum energy should be on the order of the Planck
density, which would correspond to Λ′ ≈ 1, about 122 orders of magnitude
larger than the observed value (Weinberg, 1972). This discrepancy
highlights the distinction between constants of scale and constants of
character. The constants c, G, and ħ define
the scale of the universe. The dimensionless numbers that remain—α, μ,
Λ′, and others—define its character. The challenge for fundamental
physics is not just to measure these numbers, but to explain them.
6.0 Dimensionless Theory of
Everything
6.1 Introduction to
the Dimensionless Approach
The concept of a theory of everything has long been a central goal of
theoretical physics. This section presents Stergios Pellis’s
dimensionless theory, which demonstrates how all fundamental
interactions can be unified through dimensionless relationships between
physical constants (Pellis, 2023). Pellis’s work builds upon the
dimensionless framework established in the preceding sections, extending
it to show how the coupling constants of the fundamental forces can be
connected through mathematical relationships. This theory represents a
significant advancement, revealing deep connections between quantum
mechanics, general relativity, and cosmology that have previously
remained obscured by the use of anthropocentric units.
6.2
Dimensionless Unification of Fundamental Interactions
Pellis’s dimensionless theory demonstrates that the coupling
constants of the fundamental forces can be connected through simple
mathematical relationships. The most fundamental of these is the
dimensionless unification of the strong nuclear and weak nuclear
interactions: e·αs = 10⁷·αw and eπ·αs² = 10⁷·αw, where
αs is the strong coupling constant and αw is the weak coupling constant.
Similarly, the dimensionless unification of the strong nuclear and
electromagnetic interactions is given by eπ·αs·cos(α⁻¹) = 1 and
αs·cos(α⁻¹) = i²ⁱ, where α is the fine-structure constant. For
the weak nuclear and electromagnetic interactions, the unification takes
the form: 10⁷·eπ·αw·cos(α⁻¹) = e. These relationships
demonstrate that the coupling constants are not independent parameters
but are interconnected through mathematical relationships that emerge
naturally when expressed in dimensionless form.
6.3
Dimensionless Unification of Gravitational and Electromagnetic
Interactions
One of the most profound results of Pellis’s dimensionless theory is
the unification of gravitational and electromagnetic interactions. This
is expressed through the relationship 4·e²·α²·αG·NA² =
1, where αG is the gravitational coupling constant and NA is
Avogadro’s number. This equation reveals a deep connection between
gravity and electromagnetism that has previously remained hidden in
dimensional formulations. It connects the microscopic world of atomic
physics with the macroscopic world of thermodynamics and statistical
mechanics.
6.4
Dimensionless Unification of All Four Fundamental Interactions
Pellis’s dimensionless theory culminates in the unification of all
four fundamental interactions through the relationship αs² =
4·10¹⁴·αw²·α²·αG·NA². This equation represents the complete
unification of the strong, weak, electromagnetic, and gravitational
forces. It demonstrates that all fundamental interactions can be
described through pure dimensionless relationships, with no need for
anthropocentric units or dimensional constants.
6.5
Gravitational Constant and Cosmological Constant
Pellis’s dimensionless theory provides exact formulas for the
gravitational constant G and the cosmological constant Λ:
\[G = \frac{c^3}{\hbar} \cdot
\left(\frac{1}{4 \cdot e^2 \cdot \alpha^2 \cdot
N_A^2}\right)\]
\[\Lambda = \frac{1}{\ell_P^2} \cdot
\left(\frac{1}{(2 \cdot e \cdot \alpha^2 \cdot
N_A)^6}\right)\]
These formulas reveal that the gravitational constant and
cosmological constant are not independent parameters but are determined
by fundamental dimensionless relationships.
6.6
Poincaré Dodecahedral Space as the Shape of the Universe
Pellis’s dimensionless theory also provides evidence that the shape
of the universe is Poincaré dodecahedral space (Pellis, 2022). This is
derived from the dimensionless relationships between the density
parameters of baryonic matter, dark matter, and dark energy: ΩB =
e⁻π ≈ 4.32%, ΩD = 2·e¹⁻π ≈ 23.49%, and ΩΛ =
2·e⁻¹ ≈ 73.57%. These dimensionless density parameters sum to
approximately 1.0139, indicating a slightly positively curved universe
consistent with the Poincaré dodecahedral space topology. This suggests
that the universe has a finite, multiply connected topology rather than
being infinite and simply connected (Luminet et al., 2003).
7.0
Temporal Quantization in the Unified Fractal Theory of Time
7.1 Introduction to Fractal
Time
Time has traditionally been conceived as a linear and homogeneous
parameter. However, emerging evidence suggests that temporal dynamics
may instead be governed by fractal, self-similar, and nonlinear
structures (Bassingthwaighte, 1994). This section presents Stergios
Pellis’s Temporal Quantization in the Unified Fractal Theory of Time,
which introduces a ϕ-based fractal formalism as a unifying framework for
the quantization and scaling of time (Pellis, 2025). This framework
integrates the golden ratio ϕ, Fibonacci structures, and operatorial
formulations of fractal calculus, leading to a recursive conception of
temporal flow.
7.2 The Pellis Function
The Pellis Function is defined as f(x) =
360·x⁻² - 2·x⁻³ + (3·x)⁻⁵. When x =
ϕ, the golden ratio, the function approximates with great accuracy the
inverse fine-structure constant α⁻¹. This relationship connects the
constant to the golden angle, θg ≈ 137.5°, which derives from the golden
ratio (Heyrovska, 2005). The equation accurately expresses this
relationship, where the term 360·ϕ⁻² corresponds exactly to the golden
angle, while the corrective terms fine-tune the value to match the
precise experimental constant.
7.3 Fractal Time Flow
Pellis defines Fractal Time as a hierarchy of repeating time scales:
Tn = T₀·ϕⁿ, where Tn is the time scale of
level n, T₀ is the basic unit (e.g., Planck time), and
ϕ is the golden ratio. Each Tn represents a fractal temporal
layer corresponding to quantum, biological, or cosmological phenomena.
This expresses quantized, hierarchical time, where each moment nests
within larger spiraling structures, analogous to musical scales and
biological oscillations (Bailly et al., 2010; Sacco, 2018).
7.4 The Pellis Golden Clock
The Pellis Golden Clock is a theoretical model that unifies
cosmological, biological, and quantum time frequencies through scaling
with the golden ratio. It unifies natural frequencies across domains via
ϕ-scaling: fn = f₀·ϕⁿ, where fn is the
natural frequency of the n-th plane, n ∈ ℤ, and
f₀ is a basis constant. The fractal frequency spectrum is
represented via a Pellis-Fourier transform, F(ω) = ∑n An
eⁱωⁿᵗ, which encodes multi-scale temporal harmonics. This model
captures multi-scale harmonics in ϕ-time and can be applied to heart
rate variability, electroencephalography, DNA oscillations, and cosmic
cycles (Kramer, 2022).
7.5 Applications of Fractal
Time
The fractal time framework has numerous applications. First, in
quantum systems, ϕ-scaled repeating temporal patterns appear in quantum
beats and micro-scale oscillations, and Pellis fractal clocks can model
these patterns (Golmankhaneh et al., 2024). Second, in cosmology, cosmic
time emerges from ϕⁿ-folding of Planck time, connecting molecular,
physiological, and cosmic temporal scales (Khalili-Golmankhaneh, 2018).
Third, in biology, analysis of signals such as heart rate variability
using ϕ-scaled temporal layers enhances detection of nested rhythms and
multi-scale harmonics (Ivanov et al., 1999). Fourth, in genomics, DNA
double helix geometries and protein folding patterns can be described as
ϕ-spirals within the fractal time framework (Marples, 2022). Fifth, in
geophysics, Pellis-type seismic eigenfunctions can describe the temporal
structure of seismic phenomena (Guettari et al., 2025). These
applications demonstrate that the fractal time framework is not merely
theoretical but has practical implications for modeling diverse
physical, biological, and cosmological phenomena.
8.0
Synthesis: Dimensionless Physics as a Unified Framework
8.1 The
Dimensionless Perspective on Physical Law
The analysis presented in this work converges on a singular
conclusion: the fundamental laws of physics are expressions of
relationships between pure, dimensionless ratios. This perspective
reframes our understanding of the cosmos, moving from a picture defined
by arbitrary units to one governed by an intrinsic and universal
mathematical structure. The expression of all fundamental laws in a form
devoid of dimensional constants, as achieved in Section 4.0, reveals
that the universe is governed by a set of pure, dimensionless numbers.
As established by the Buckingham π theorem and the application of Planck
units, constants such as G, c, and ħ are not
fundamental but emerge from a deeper causal network of wave
correlations.
8.2 The Dimensionless
Theory of Everything
Pellis’s dimensionless theory demonstrates that all fundamental
interactions can be unified through dimensionless relationships between
physical constants (Pellis, 2023). The coupling constants of the four
fundamental forces are not independent parameters but are interconnected
through mathematical relationships that emerge naturally when expressed
in dimensionless form. As detailed in Sections 6.2 through 6.5, this
theory provides exact formulas for the gravitational constant G
and the cosmological constant Λ, and provides evidence that the shape of
the universe is Poincaré dodecahedral space (Pellis, 2022).
8.3 Temporal
Quantization and Fractal Time
The Temporal Quantization in the Unified Fractal Theory of Time
demonstrates that time itself is not linear but fractal, governed by the
golden ratio ϕ and Fibonacci structures (Pellis, 2025). As explained in
Section 7.0, the Pellis Function and Pellis Golden Clock provide a
framework for understanding temporal phenomena across all scales as
manifestations of a single, unified fractal structure. This suggests
that biological rhythms, quantum phenomena, and cosmic cycles are all
manifestations of the same underlying fractal structure.
8.4 Implications for
Fundamental Physics
The dimensionless perspective has several profound implications for
fundamental physics. First, the dimensionless relationships between
coupling constants indicate that a unified theory of all fundamental
forces is possible through pure mathematical relationships. Second, the
dimensionless constants of nature, such as α and μ, are not arbitrary
but are determined by deeper mathematical structures. Third, the
dimensionless density parameters indicate a slightly positively curved
universe consistent with the Poincaré dodecahedral space topology.
Fourth, the structure of time itself is not linear but fractal, governed
by the golden ratio. Finally, the dimensionless density ratio ρ′ = ρ/ρP
provides a universal criterion for determining whether a system’s
gravitational behavior is classical or requires a quantum
description.
8.5 Future Directions
The dimensionless framework presented in this work opens several
avenues for future research. One direction is the development of
mathematical frameworks to derive the values of fundamental
dimensionless constants from first principles. Another is the
experimental verification of the dimensionless relationships between
coupling constants and the predictions of the dimensionless theory.
Further research should also involve applying the fractal time
framework, described in Section 7.0, to practical problems in biology,
medicine, and engineering. The dimensionless framework can also be used
to develop a complete theory of quantum gravity. Lastly, cosmological
models can be developed based on the dimensionless density parameters
and Poincaré dodecahedral space topology.
9.0 Conclusion
The comprehensive analysis presented in this work demonstrates that
the fundamental laws of physics are not statements about dimensional
quantities tied to human scales, but are instead expressions of
relationships between pure, dimensionless ratios. This perspective
reframes our understanding of the cosmos, moving from a picture defined
by arbitrary units to one governed by an intrinsic and universal
mathematical structure.
The Buckingham π theorem provides the mathematical foundation for
this dimensionless perspective, proving that any physically meaningful
relationship can be expressed entirely in terms of dimensionless ratios
(Buckingham, 1914). The Planck scale provides a convenient metric for
this dimensionless description, allowing all physical quantities to be
expressed as ratios relative to the emergent scales of nature.
Pellis’s dimensionless theory demonstrates that all fundamental
interactions can be unified through dimensionless relationships between
physical constants (Pellis, 2023). The coupling constants of the
fundamental forces are interconnected through elegant mathematical
relationships that emerge naturally when expressed in dimensionless
form. This theory also provides evidence that the shape of the universe
is Poincaré dodecahedral space (Pellis, 2022).
The Temporal Quantization in the Unified Fractal Theory of Time
demonstrates that time itself is not linear but fractal, governed by the
golden ratio ϕ and Fibonacci structures (Pellis, 2025). This framework
provides a unified structure for understanding temporal phenomena across
all scales.
This dimensionless perspective has profound implications for
fundamental physics and our understanding of the universe. It suggests
that the universe is fundamentally mathematical in nature, with physical
laws expressed as pure numerical relationships. The challenge for future
physics is not just to measure the dimensionless constants of nature,
but to derive them from first principles, revealing the deeper
mathematical structure of the cosmos. As Max Planck envisioned, this
dimensionless framework provides a physics for all times and all
civilizations, built upon the timeless and universal language of
dimensionless numbers (Planck, 1899).
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