Dynamical Mass from Flow Geometry in Temporal Flow Theory (TFT)

One of the most interesting ideas in Temporal Flow Theory (TFT) is that mass is not a fundamental quantity, but instead emerges from the geometry and coupling of flows themselves. This post focuses on how fermionic mass arises dynamically from the curvature of the underlying flow potential—a concept I call flow inertia.

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5.1 Motivation: Mass as Flow Inertia

In conventional quantum field theory (QFT), the mass term in the Dirac Lagrangian is a constant:

\mathcal{L}_\psi = \bar{\psi}(i\gamma^\mu \partial_\mu - m)\psi

But TFT takes a different stance. I propose that:

Mass = Flow Inertia = Curvature of the Flow Potential

That is,

m^2 \propto \left. \frac{\partial^2 V(F)}{\partial F^2} \right|_{F = F_0}

Here, is the bosonic flow field, and is its equilibrium configuration.

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5.2 Modified Dirac Lagrangian with Flow-Coupled Mass

In TFT, mass becomes a function of the flow field:

m(F) \equiv \sqrt{\lambda} \left( \|F - F_0\|^2 - v^2 \right)

Near equilibrium, this reduces to:

m(F)^2 \approx \left. \frac{\partial^2 V(F)}{\partial F^2} \right|_{F = F_0}

Thus, the fermionic Lagrangian becomes:

\mathcal{L}_\psi = \bar{\psi} \left(i\gamma^\mu \partial_\mu - m(F) \right) \psi

This embeds the fermion's inertial properties directly into the temporal flow framework.

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5.3 Interpretation

• Near potential minima (): Flows are stable and inertia is well-defined. Mass becomes an effective constant.
• Far from equilibrium: Curvature changes, so mass varies and can even vanish or flip sign—similar to what happens in Higgs-like models, but geometrically driven.

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5.4 Mass via Flow Self-Coupling

Using the flow potential:

V(F) = \lambda \left( \|F - F_0\|^2 - v^2 \right)^2

We find:

m^2(F) = \left. \frac{\partial^2 V}{\partial F^2} \right|_{F = F_0} = 4 \lambda v^2

This gives mass a geometric meaning: resistance of the flow to deformation.

Note: This isn’t spontaneous symmetry breaking in the usual QFT sense—it's a structural consequence of the flow network's configuration.

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5.5 Coupling Flow to Fermions: A Yukawa-Like Term

To generalize the interaction, I introduce a Yukawa-style coupling between the flow field and the fermion field :

\mathcal{L}_{\text{int}} = - y \, \bar{\psi} \psi \, F

If the flow field settles at an expectation value , this gives the fermion a dynamical mass:

m_\psi = y F_0

So, fermionic mass emerges directly from the background geometry of temporal flows—linking matter and information structure.

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5.6 Revisiting the Fermion Flow-Saturation Condition

Earlier I proposed a CPT inversion trigger condition:

(j^\mu j_\mu)^{1/2} = c \, \bar{\psi} \psi

With a flow-coupled mass, this has deeper significance:

• Saturation may occur when exceeds a critical threshold
• Inversion then becomes a topological event at the causal edge of flow propagation

This fits naturally with TFT’s emphasis on causality and boundary dynamics.

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Next Steps (Future Work)

Here are some directions I’m actively exploring:

1. Deriving Modified Dirac Equations from the full action with
2. Integrating Out Flow Fields to obtain effective fermion potentials
3. Analyzing Mass Renormalization as a geometric phenomenon
4. Connecting with Gravity: Exploring how flow curvature might mimic gravitational interaction

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If mass is a response to flow structure, then matter itself is a shadow of time’s geometry. More to come as we continue building out TFT!