Deterministic Scalar Field Theory: The Origin of Fundamental Physical Constants

Abstract We present a novel deterministic approach to deriving fundamental physical constants from a nonlinear scalar field theory. Unlike standard quantum‐mechanical interpretations that rely on probability, our model treats matter as stable, vortex‐like excitations (solitons) emerging in a real scalar field governed by a modified Klein–Gordon equation. By applying both variational and numerical methods, the key parameters – including the scalar field amplitude ϕ0, the dimensionless scaling factor λ, and the emergent gravitational constant G – are derived without any adjustable parameters. The resulting values closely match experimental data, thereby offering new insights into the unification of fundamental constants. Introduction Determining the values of fundamental physical constants has long been a central challenge in theoretical physics. Traditionally, constants such as the gravitational constant G, Planck length ℓp, and electron charge e are measured experimentally and then inserted into theoretical models. In contrast, our approach derives these constants from first principles by examining the dynamics of a nonlinear scalar field. In our model, matter is interpreted as a localized vortex-like excitation (soliton) of a real scalar field ϕ(x,t) governed by a modified Klein–Gordon equation: ∂2ϕ∂t2−c2∇2ϕ+m2ϕ+αϕ3=0 The parameters ϕ0 (field amplitude), λ (scaling factor), and α (self-interaction constant) emerge naturally from stability considerations of the vortex solution. Importantly, no parameter is arbitrarily adjusted – the theory is fully deterministic. In what follows, we describe several methods to extract these parameters and demonstrate the consistency of the derived constants with experimental values. Theoretical Basis and Parameter Extraction Our investigation begins with a nonlinear Klein–Gordon-type equation for a real scalar field: ∂2ϕ∂t2−c2∇2ϕ+m2ϕ+αϕ3=0 Here, c is the speed of light, m is a parameter linked to the spatial extent of the vortex, and α characterizes the field’s self-interaction. We explore four independent methods to determine the characteristic amplitude ϕ0 and the scaling factor λ: Method 1: Variational Approach – Uses an analytic Gaussian ansatz to minimize the energy functional. Method 2: Numerical Integration – Involves solving the discretized field equation until a stable vortex emerges. Method 3: Electron Compton Wavelength – Relates ϕ0 to the electron’s reduced Compton wavelength. Method 4: Classical Electron Radius – Derives ϕ0 using the classical electron radius as a characteristic scale. The convergence of these methods reinforces the self-consistency of our deterministic framework. If you find this theory intriguing and share a passion for exploring new ideas, please do not hesitate to contact me. I welcome any collaboration that can help further develop and disseminate these concepts.