Mycelial Signaling Fields and Low-Energy Quantum Scalar Fields:
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\title{Mathematical Equivalence Between Mycelial Signaling Fields and Low-Energy Quantum Scalar Fields}
\author{C.L. Vaillant}
\date{\today}
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\maketitle
\begin{abstract}
We demonstrate a formal mathematical equivalence between the dynamics of electrical signal propagation in biological mycelial networks and the low-energy regime of real quantum scalar field theory. By establishing a structural isomorphism between the damped diffusion equation for mycelial potentials and the non-relativistic limit of the Klein--Gordon equation, we identify shared operator forms and spectral behaviors. The correspondence extends to nonlinear corrections, stochastic forcing, and higher-order correlations. We discuss the continuum approximation for biological networks, highlight potential biological analogs of field interactions, and propose empirical validation strategies. This framework suggests a universality of field-like computation underlying both biological and physical systems.
\end{abstract}
\section{Introduction}
Electrical signaling in fungal mycelial networks has been observed to encode environmental and physiological information \cite{adamatzky2021fungal,fricker2017mycelial}. These signals propagate as spatially distributed voltage variations that can exhibit cooperative, adaptive, and self-organizing behavior \cite{boddy2021fungal}.
In physics, scalar quantum fields describe excitations of continuous media governed by wave and diffusion-like operators. In the low-energy limit, their equations of motion reduce to forms mathematically similar to those that govern mycelial electrical propagation.
This paper formalizes that analogy by demonstrating a direct mapping between the mycelial diffusion equation and the non-relativistic Klein--Gordon equation. We show that both systems share identical spectral kernels and nonlinear extensions, suggesting a shared universality class. Importantly, this is a \textit{mathematical analogy}—not a claim of quantum effects in biology—but it provides a powerful unifying framework for interdisciplinary analysis.
\section{Mycelial Signal Propagation Model}
Let $\phi_m(x,t)$ represent the electrical potential field across a mycelial network:
\[
\phi_m(x,t): \mathbb{R}^3 \times \mathbb{R} \rightarrow \mathbb{R}.
\]
Signal propagation is modeled as:
\begin{equation}
\frac{\partial \phi_m}{\partial t} = D \nabla^2 \phi_m - \gamma \phi_m + S(x,t),
\label{eq:mycelial}
\end{equation}
where $D$ is the effective ionic diffusion coefficient, $\gamma$ is a decay constant representing dissipation, and $S(x,t)$ is an external stimulus term.
Although real mycelial networks are discrete and anisotropic, this continuum model approximates their large-scale behavior. At spatial scales larger than typical hyphal separations, the net propagation of ionic or electrical activity can be represented as a diffusive process with an effective coefficient $D$, analogous to mean-field treatments in porous media or reaction–diffusion systems.
\section{Scalar Field in Quantum Field Theory}
A real scalar field $\phi_q(x,t)$ in quantum field theory satisfies the Klein--Gordon equation:
\begin{equation}
\left(\Box + m^2\right)\phi_q(x,t) = 0,
\qquad
\Box = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2.
\label{eq:KG}
\end{equation}
Here $m$ is the mass parameter, which sets the inverse correlation length of field fluctuations.
To derive the low-energy limit explicitly, let us introduce the ansatz
\begin{equation}
\phi_q(\mathbf{x},t) = e^{-imc^2t/\hbar}\psi(\mathbf{x},t),
\end{equation}
where $\psi$ varies slowly in time. Substituting this into Eq.~\eqref{eq:KG} and neglecting the second-order time derivative term $\partial_t^2 \psi \ll (mc^2/\hbar)\partial_t\psi$, we obtain:
\begin{equation}
i\hbar\,\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2 \psi + \frac{m c^2}{2}\psi.
\end{equation}
The constant $\tfrac{1}{2}mc^2$ term can be absorbed into the energy reference or neglected for real-valued fields representing low-energy amplitudes.
In units where $\hbar=c=1$ and focusing on the real part of $\psi$, this simplifies to the diffusion–decay form:
\begin{equation}
\frac{\partial \phi_q}{\partial t} = \nabla^2 \phi_q - m^2 \phi_q.
\label{eq:lowenergy}
\end{equation}
Equation~\eqref{eq:lowenergy} thus emerges as the low-energy, real-valued reduction of the Klein--Gordon equation, showing that scalar quantum fields relax according to diffusion-like dynamics.
\section{Structural Equivalence}
Comparing Eq.~\eqref{eq:mycelial} (for $S=0$) and Eq.~\eqref{eq:lowenergy}, the mapping
\[
D \leftrightarrow 1, \qquad \gamma \leftrightarrow m^2
\]
establishes a one-to-one correspondence between parameters. Both systems are scalar fields obeying diffusion–decay dynamics governed by identical operator structures.
This equivalence implies that information or perturbations in a mycelial network evolve according to the same mathematics that describe relaxation processes in low-energy quantum fields.
\section{Spectral Density and Stochastic Forcing}
To analyze spectral behavior, we include a stochastic noise term $\eta(x,t)$ representing spontaneous fluctuations:
\begin{equation}
\frac{\partial \phi_m}{\partial t} = D \nabla^2 \phi_m - \gamma \phi_m + S(x,t) + \eta(x,t),
\label{eq:stochastic}
\end{equation}
where $\langle \eta(x,t)\rangle = 0$ and
\[
\langle \eta(x,t)\eta(x',t') \rangle = 2A\,\delta(x-x')\delta(t-t').
\]
Taking the Fourier transform of Eq.~\eqref{eq:stochastic} for $S=0$, we obtain
\[
\phi_m(k,\omega) = \frac{\eta(k,\omega)}{-i\omega + Dk^2 + \gamma},
\]
leading to a power spectrum
\begin{equation}
\langle |\phi_m(k,\omega)|^2 \rangle \propto \frac{1}{\omega^2 + (Dk^2 + \gamma)^2}.
\end{equation}
Integrating over frequency yields the steady-state spatial spectrum:
\begin{equation}
\langle \phi_m(k)\phi_m(-k)\rangle \propto \frac{1}{k^2 + \gamma/D}.
\label{eq:mycelial-spectrum}
\end{equation}
In QFT, the analogous vacuum two-point function is
\begin{equation}
\langle \phi_q(k)\phi_q(-k)\rangle \propto \frac{1}{k^2 + m^2}.
\label{eq:qft-spectrum}
\end{equation}
Thus, the equivalence $m^2 \leftrightarrow \gamma/D$ confirms a shared spectral kernel between mycelial and quantum fields.
\section{Source Term Analogy}
Including a driving term $S(x,t)$ gives:
\begin{equation}
\frac{\partial \phi_m}{\partial t} = D\nabla^2\phi_m - \gamma\phi_m + S(x,t),
\end{equation}
analogous to a scalar field coupled to an external current:
\begin{equation}
(\Box + m^2)\phi_q = J(x,t).
\end{equation}
In both cases, the system’s evolution reflects the convolution of the source with a Green’s function of identical mathematical form. Hence, environmental or nutrient stimuli act as external field perturbations in the biological domain.
\section{Nonlinear Corrections and Biological Interpretation}
Empirical data show that fungal electrical activity exhibits nonlinear thresholding, signal amplification, and saturation. These can be represented as:
\begin{equation}
\frac{\partial \phi_m}{\partial t} = D\nabla^2\phi_m - \gamma\phi_m + \lambda \phi_m^n.
\end{equation}
The analogous interacting field equation in QFT is:
\begin{equation}
(\Box + m^2)\phi_q + \lambda \phi_q^n = 0.
\end{equation}
Here, $\lambda$ sets the interaction strength.
For example:
\begin{itemize}
\item A $\phi^3$ term models cooperative or excitatory behavior, where signals reinforce each other once a threshold is reached.
\item A $\phi^4$ term represents saturation or self-limiting dynamics, stabilizing field growth and maintaining bounded activity.
\end{itemize}
These correspondences link measurable biological nonlinearities to formal field-theoretic interactions.
\section{Higher-Order Correlations}
Beyond two-point correlations, higher-order functions such as
\begin{equation}
\langle \phi(k_1)\phi(k_2)\phi(k_3)\rangle
\end{equation}
capture non-Gaussian dependencies. In mycelial systems, detecting such three- or four-point correlations in multichannel voltage recordings would demonstrate nontrivial coupling structures analogous to interaction vertices in quantum field theory.
\section{Emergence and Universality}
Both systems display emergent, collective organization from local rules:
\begin{itemize}
\item \textbf{Mycelial networks:} adaptive routing, spatial optimization, and distributed memory formation \cite{fricker2017mycelial,boddy2021fungal}.
\item \textbf{Quantum fields:} vacuum polarization, renormalization flow, and particle-like excitations \cite{peskin1995intro,zee2010qft}.
\end{itemize}
These phenomena suggest a shared universality class—distinct systems manifesting the same low-level equations and hence similar emergent behaviors.
\section{Experimental Validation}
Potential experimental tests include:
\begin{enumerate}
\item Measuring spatiotemporal voltage fields in live mycelial networks using microelectrode arrays.
\item Extracting spectral densities and higher-order correlation functions.
\item Fitting observed spectra to theoretical kernels $(k^2 + m^2)^{-1}$.
\item Applying controlled stimuli $S(x,t)$ to characterize nonlinear response and saturation.
\end{enumerate}
Experimental confirmation of these spectral and dynamic parallels would support the universality hypothesis and open pathways for using field-theoretic tools to describe biological computation.
\section{Analogy Versus Physical Correspondence}
The equivalence derived herein is a mathematical analogy, not a claim that mycelial systems operate quantum mechanically. Nevertheless, the shared equations, correlation structures, and emergent organization imply that similar field-theoretic formalisms may govern both biological and physical processes. The analogy therefore provides a conceptual bridge between information propagation in living systems and the statistical field frameworks of physics.
\section{Conclusion}
We have demonstrated that the damped diffusion equation describing mycelial electrical signaling is mathematically equivalent to the low-energy limit of the Klein--Gordon equation in scalar quantum field theory. This equivalence encompasses linear, spectral, and nonlinear regimes. By interpreting biological computation in field-theoretic terms, one may apply analytical tools such as renormalization and spectral analysis to study emergent intelligence in living systems.
This work thus opens an avenue for interdisciplinary exploration of universal dynamical principles shared across physics and biology.
\section*{Acknowledgments}
The author thanks collaborators and interdisciplinary colleagues for insightful discussions connecting theoretical field formalisms with experimental fungal electrophysiology.
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\bibitem{adamatzky2021fungal}
A.~Adamatzky, ``Fungal Computing: Information Processing in Mycelial Networks,'' \textit{Biosystems}, vol.~199, 104305, 2021.
\bibitem{fricker2017mycelial}
M.~D.~Fricker, ``Mycelial Networks: Structure and Function,'' \textit{Nature Reviews Microbiology}, vol.~15, pp.~626--635, 2017.
\bibitem{boddy2021fungal}
L.~Boddy, M.~D.~Fricker, and N.~Tlalka, ``Fungal Networks: Ecological Intelligence and Resilience,'' \textit{Fungal Biology Reviews}, vol.~35, pp.~30--40, 2021.
\bibitem{peskin1995intro}
M.~E.~Peskin and D.~V.~Schroeder, \textit{An Introduction to Quantum Field Theory}, Addison-Wesley, 1995.
\bibitem{zee2010qft}
A.~Zee, \textit{Quantum Field Theory in a Nutshell}, Princeton University Press, 2010.
\end{thebibliography}
\end{document}
