The Planck-Unified Field: Physics as a Projection from a Unitless Substrate

J. Rogers, SE Ohio, USA, 19 July 2025, 1837

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Abstract

We argue that physics, at its foundation, is always and everywhere Planck-scale physics. The act of setting $c = h = G = 1$ does not describe a remote, high-energy regime but rather reveals the universal substrate of reality itself. Within this substrate, all physical quantities—time, energy, mass, frequency, momentum, and inverse length—are unified as the same numerical value for the same state of existence. Observed distinctions arise only from our projection of these unitless, harmonized values into conventional unit systems.

We propose a simple, nonlinear, universal field theory for this direct-unity view: a single, dimensionless scalar field, $S_u$, that encodes the state of the universe as shared time experiences. Projection from this field to SI units recovers all familiar laws and constants. Thus, the apparent complexity of physics is a translational artifact of a deeper, coordinate-free and unit-independent reality—the absolute physics sought by Einstein.

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I. Physics Is Planck-Unified

In conventional physics, we habitually distinguish between time, space, mass, energy, frequency, momentum, and wavelength. This distinction, we argue, is not fundamental—it is an artifact of our measurement conventions. By setting the three physical conversion constants $c = h = G = 1$, we reveal the true, underlying unity of all measurable quantities:

$$\text{Time} = \text{Frequency}^{-1} = \text{Mass} = \text{Energy} = \text{Momentum} = \text{Length}^{-1}$$

We can represent this unified physical state by a single, dimensionless, universal state variable, $S_u$, defined at every spacetime point. This is not a notational trick or a high-energy limit—this is the nature of the physical universe at all scales. We are always at the Planck scale. The labels “high energy” and “low energy” merely describe different configurations of the $S_u$ field as measured by our human-scale rulers.

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II. A Universal, Nonlinear Field Dynamics

We postulate that the universe is described by a single, dimensionless, real scalar field, $S_u$, governed by the Lagrangian:

$$L = S_u (\partial_\mu S_u)^2 - V(S_u) \tag{1}$$

Here, $S_u$ is the universal Planck-state variable, representing the singular "substance" from which all physics arises. The nonlinear kinetic term $S_u (\partial_\mu S_u)^2$ reflects the self-referential nature of a truly universal field. The potential, for which we may use the canonical form

$$V(S_u) = \lambda(S_u^2 - v^2)^2,$$

introduces structure and allows for spontaneous symmetry breaking. This formalism is inherently coordinate-free and unit-free; all physical meaning is intrinsic to the state of the field itself.

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III. Measurement as Projection

The act of measurement is the act of projecting the unitless reality of the $S_u$ field onto a chosen coordinate chart. When we measure a physical quantity (e.g., energy), we:

1. Evaluate the relevant quantity within the $S_u$ theory, yielding a dimensionless number.

2. Project this number into our chosen unit system by multiplying by the corresponding Planck unit conversion factor (e.g., $E_P = \sqrt{hc^5/G}$), which is itself a function of the constants $h$, $c$, and $G$ as defined in our coordinate system.

Every prediction of the Standard Model and General Relativity is thus a translation from this absolute, Planck-unified structure to our SI ruler.

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IV. Recovery of Standard Physics from the $S_u$ Field

To demonstrate that this framework is a computationally exact rewriting of familiar physics, we explicitly recover the Standard Model and General Relativity as projections of the harmonized Planck-scale dynamics.

A. Electrodynamics as a Phase Excitation

Let us model a photon-like excitation as a small phase oscillation around the vacuum state:

$$S_u(x) = v e^{i\theta(x)}.$$

For small oscillations, the kinetic term becomes:

$$L_{\text{kin}} \approx v^3 (\partial_\mu \theta)^2.$$

To connect this to electrodynamics, we project into SI units. The vector potential $A_\mu$ relates to the phase gradient via:

$$A_\mu = \frac{h}{2\pi e} \, \partial_\mu \theta.$$

Substituting this into the kinetic term and matching it to the Maxwell Lagrangian $L_{EM} = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu}$ uniquely determines the vacuum structure parameter $v$ in terms of the constants of electromagnetism.

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B. Higgs Mechanism from the Universal Potential

The potential $V(S_u)$ already contains the Higgs mechanism. Letting $S_u = v + h(x)$, small fluctuations $h(x)$ yield a Lagrangian with a mass term:

$$m_H^2 = 2\lambda v^2.$$

In SI units, the Higgs mass becomes:

$$m_{H, \text{SI}} = \sqrt{2\lambda} \, v \sqrt{\frac{hc}{G}}.$$

Thus, the Higgs mass is entirely determined by the universal field parameters and the projection into the SI system.

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C. Gravitation as $S_u$ Curvature

Large-scale modulations of $S_u$ are equivalent to metric dynamics. The kinetic term $S_u (\partial_\mu S_u)^2$ recovers the Einstein-Hilbert action in the low-energy limit. The gravitational constant $G$ is not fundamental—it is derived:

$$G = \frac{hc}{m_P^2},$$

where $m_P$ is defined via the vacuum expectation value $v$. Gravity thus emerges from the structure of the $S_u$ vacuum.

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D. The Standard Model and GR as Projections

The established theories of physics are not approximations. They are exact translations of the Planck-unified, coordinate-free $S_u$ dynamics onto specific unit conventions. Every constant, field, and interaction is a manifestation of measurement projection.

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V. Cosmological and Quantum Interpretations

• Cosmology: The Big Bang corresponds to the condensation of the $S_u$ field into its vacuum state. Structure formation and cosmic evolution are histories of this field.

• Quantum Mechanics: “Collapse” is not stochastic but a nonlinear dynamical reconfiguration of the $S_u$ field. Particles and forces are stable topological excitations—solitons—of the field.

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VI. Conclusion

Physics is already, and always, Planck-unified. There is no physical distinction between “Planck physics” and “everyday physics” beyond what our arbitrary rulers impose. All physical structures—particles, forces, spacetime, and constants—are present in, and projected from, the configuration and dynamics of the unitless, universal field $S_u$.

This is not a theory of emergence from a hidden substratum; it is the recognition that all of physics is the direct, coordinate-free, unit-invariant reality of the Planck-unified field. The complexity and multiplicity of observed phenomena are illusions born of our measurement conventions.