The Causal Quaternionic Field Theory: A Non-Markovian Approach to the Metric Tensor and Quantum Entanglement

This paper introduces the Causal Quaternionic Field Theory (CQFT), a theoretical framework that redefines the foundational axioms of quantum mechanics and general relativity by rejecting the fungibility of numerical values. We propose that physical space possesses a non-Markovian topological memory encoded within the imaginary axes (i,j,k) of a quaternionic metric tensor. By introducing the Causal Tensor Product (⊠) governed by a strict chronological index (τ), we resolve the tensor product problem that historically hindered quaternionic quantum mechanics (e.g., S. Adler). In this framework, causal asymmetry is strictly preserved, logically prohibiting retrocausality at the algebraic level. Furthermore, we reinterpret quantum entanglement not as non-local statistical collapse, but as a deterministic topological tunnel shared within the hidden axes of the field. The paper outlines the mathematical framework, proposes the architecture for Photonic Causal Tensor Chips (PCTC), and presents a falsifiable experimental setup—the Causal Hysteresis Test of Entangled Photons—to empirically distinguish our model from the standard Copenhagen interpretation.