Abstract:
This paper examines the mathematical foundations of the Yang-Mills mass-gap problem, challenging the presumption that physical invariants can be derived through pure formal logic. When the identity principle is employed, the traditional commutative infrastructure collapses into a trivial state where Planck's reduced constant could equal zero ($\hbar = 0$) and/or a particle's lifespan is infinite ($t = \infty$). Consequently, this yields a vanishing minimal mass bound (\(\Delta = 0\)), and serves as a counter-example to any proposed proof of a non-zero mass gap. We conclude that mathematical formalism acts as a malleable descriptive tool rather than an absolute arbiter of physical existence, demonstrating that non-zero invariants like the mass gap require an empirical anchor established through experimental observation and measurement.
Proving the Mass Gap?
\begin{equation} \Delta = \frac{h}{tc^2} \end{equation} The mass $\Delta$ is the least mass above zero possible and $\hbar$ is Planck's reduced constant. The speed of light $c$ was mathematically derived (thanks to empirical measurements) by James Clerk Maxwell in the following manner: \begin{equation} c = \sqrt{\frac{1}{\epsilon_0\mu_0}} \end{equation} where $\epsilon_0$ is the permittivity of free space and $\mu_0$ is the permeability of free space, and $t$ is the particle's lifespan. The particle's observed energy $E$ is as follows: \begin{equation} (3)\hspace{.3 cm} E = \frac{\hbar}{t} \Longrightarrow E\psi = i\hbar\frac{d}{dt} exp(-i\frac{E}{\hbar} t) \end{equation} The commutator of $E$ and $t$ is as follows: \begin{equation} [E,t] = Et - tE \end{equation} Multiply both sides by $\psi$ and make substitutions (see (3)): \begin{equation} [E,t]\psi = i\hbar\frac{d}{dt}t\psi - ti\hbar\frac{d}{dt}\psi \end{equation} The derivative product rule: \begin{equation} [E,t]\psi = i\hbar\psi + ti\hbar\frac{d}{dt}\psi- ti\hbar\frac{d}{dt}\psi \end{equation} Complete the operations: \begin{equation} [E,t]\psi = i\hbar\psi + tE\psi- tE\psi \end{equation} Cancel the $\psi$'s: \begin{equation} [E,t] = i\hbar + tE- tE \end{equation} More substitutions (see (3)): \begin{equation} [E,t] = i\hbar + t\frac{\hbar}{t}- t\frac{\hbar}{t} \end{equation} Simplify: \begin{equation} [E,t] = (i\hbar + \hbar)- \hbar \end{equation}Premise 1: $E = E$.
Premise 2: $t = t$.
Premise 3: $\hbar = \hbar$.
The above premises indicate that a particle's observed energy $E$ or lifespan $t$ maintain their respective identities, and their measured values don't change if $E$ and $t$ merely switch positions on paper: $Et \to tE$. If we assume the premises are false, that the identity principle does not apply, then we cannot prove that a thing is what it is. We cannot prove that a mass gap is what it is. It could be zero (no mass gap) rather than some non-zero value. Alternatively, if we adhere to the above premises, the following inferences are inevitable: \begin{equation} Et = i\hbar + \hbar \end{equation} \begin{equation} tE = \hbar \end{equation} \begin{equation} E = \frac{i\hbar + \hbar}{t} = \frac{\hbar}{t} \end{equation} \begin{equation} E - E = 0 \end{equation} \begin{equation} \frac{i\hbar + \hbar}{t} - \frac{\hbar}{t} = 0 \end{equation} \begin{equation} (16) \hspace{.3 cm}\frac{i\hbar}{t} = 0 \end{equation} \begin{equation} i = \sqrt{-1} \neq 0 \end{equation} Multiply both sides of (16) by $\frac{-i}{c^2}$. Paradoxically, the least mass above zero is \begin{equation} \Delta = \frac{h}{tc^2} = 0 \end{equation} However, we know there is a least mass above zero greater than zero, that there is in fact a mass gap; we know that $\hbar > 0$, but we only know these things via experimental observations. The above demonstration provides a counter-example that contradicts any proposed mathematical proof of a mass gap. A proof is invalid if a contradiction exists. Therefore, mathematics cannot establish whether or not there is a mass gap. Math is a blind-logic machine that will take any premise(s) and logically output the inevitable right-or-wrong conclusion. Only experimental observations determine whether a conclusion is correct.
References
1. E. Witten, A. Jaffe. "Quantum Yang Mills Theory." Clay Mathematics Institute.
Maki, M. Nakagawa, and S. Sakata, "Remarks on the Unified Model of Elementary Particles," Progress of Theoretical Physics 28(5), 870 (1962).
Muta, Taize. "Foundations of Quantum Chromodynamics," World Scientific Publishing Co. (1987).
D. Fleisch. "A Student's Guide to Maxwell's Equations." Cambridge University Press (2008).
© 2026 G.M. Jackson
