Lorentz Invariance and Renormalization Naturally Emerge in Temporal Flow Physics

 

Lorentz Invariance, Renormalization, and Phase Structure in Temporal Flow Theory

In Temporal Flow Physics (TFP), the fundamental object is not spacetime, but a discrete network of 1D temporal flows $F_i(t)$, each evolving forward in causal order. These are not spatial vectors embedded in a background — space emerges from the correlations between them. This shifts the burden of Lorentz invariance: instead of being imposed as a symmetry of background spacetime, it must emerge from intrinsic properties of the flow dynamics.

Lorentz Invariance as an Emergent Principle

We begin by identifying the only intrinsic, observer-independent scalar associated with a pair of flows $F_i(t)$ and $F_j(t)$: the proper time separation

$$\tau_{ij} = |t_i - t_j| = N_{ij} t_p$$

where $N_{ij} \in \mathbb{Z}$ is the number of Planck-time steps between flows $i$ and $j$. Importantly, any function of $\tau_{ij}$ is manifestly invariant under boosts since it arises from the proper-time ordering intrinsic to the causal flow structure.

The complex interaction kernel between flows is defined as:

$$\mathcal{A}(F_i, F_j) = \exp\left(- \frac{\|F_i - F_j\|}{l_p} \right) e^{i \theta_{ij}}$$

We now seek to understand the structure of the phase term $\theta_{ij}$.

Gauge-Invariant Phase and Proper Time Contribution

Suppose that flow interactions preserve an internal gauge freedom (e.g., global or local phase). Then the physically meaningful phase difference is the gauge-invariant quantity:

$$\phi_{ij} = \theta_i - \theta_j$$

We propose that this phase difference contains a term proportional to the proper time between flows:

$$\phi_{ij} = \frac{\tau_{ij}}{l_p} + \Delta\phi^{\text{topo}}_{ij}$$

where:

• The first term ensures the phase accumulates linearly with proper time separation (a signature of consistent Lorentz-invariant propagation).

• The second term $\Delta\phi^{\text{topo}}_{ij}$ encodes nonlocal or topological effects — for instance, winding around compactified temporal structures or interference from other flows.

Flow Separation Must Be Timelike

Within TFP, since space emerges from comparisons between temporal flows, any meaningful separation $\Delta F_{ij}$ is inherently timelike. This is because there is no pre-existing space in which to define spacelike intervals — causality comes first.

This assumption is also consistent with the dynamics of flow fluctuation fields $\delta F(x)$, which propagate on an emergent lightcone defined by coarse-grained alignment statistics:

$$G_{\mu\nu}(x) = \langle \partial_\mu \delta F(x) \partial_\nu \delta F(x) \rangle$$

At leading order, fluctuations respect causal structure, implying that the dominant propagator support lies on timelike or lightlike separations.

Summary Equations

To summarize:

• Proper-time separation:

$$\tau_{ij} = |t_i - t_j|$$

• Phase structure:

$$\phi_{ij} = \theta_i - \theta_j = \frac{\tau_{ij}}{l_p} + \Delta\phi^{\text{topo}}_{ij}$$

• Interaction kernel (gauge-invariant form):

$$\mathcal{A}(F_i, F_j) = \exp\left(- \frac{\|F_i - F_j\|}{l_p} \right) e^{i \phi_{ij}}$$

This structure satisfies all the constraints:

• Lorentz invariance emerges from $\tau_{ij}$ dependence.

• Gauge invariance from relative phase $\phi_{ij}$.

• Timelike flow separation is built into the causal foundation of the theory.

This provides a concrete realization of how TFP encodes relativistic and quantum properties through temporal flow interactions.