Composite Complex Quantum Systems

Eric Needham

This restricted record contains the fourth paper in the ENSO series Foundations of Quantum Physics. The paper examines the conditional step from a local imaginary structure to a global complex scalar field in quantum mechanics. Building on the preceding papers in the series, it starts from the result that bounded return forces a local complex structure J² = -I on a single closure plane. It then asks whether composite systems must share one global imaginary unit, or whether each subsystem may retain its own local complex structure. The central result is that two apparently separate problems are the same hinge. In the composition language, the question is whether the local imaginary structures of two systems satisfy J_A ⊗ 1 = 1 ⊗ J_B, selecting one shared global imaginary and the complex tensor product. In the transport language, the question is whether parallel transport preserves the local complex structure, HJH⁻¹ = +J, or conjugates it, HJH⁻¹ = -J. The paper shows that these are equivalent conditions. The result is a sharp conditional: closure forces the local imaginary, but the global imaginary follows only if composite phase is globally coherent. Closure alone reaches the fork but does not select the branch. The remaining question — whether nature enforces global phase coherence across composite systems — is connected to the ongoing debate over real versus complex quantum theory. The upload includes a Python validation script that explicitly constructs the two-system real tensor product, verifies the aligned and anti-aligned sectors, and demonstrates the equivalence between the composition condition and the transport/no-Möbius condition. For further information about the ENSO Framework, please contact Eric Needham:ensotheory1@gmail.com

Jeff Murugan, Victor Van Aswegen, Hendrik J R Van Zyl

We investigate the behaviour of Krylov complexity as the number of degrees of freedom of a quantum system increases. Specifically, we show that for tensor products of commuting subsystems, the leading excess complexity is controlled by a weighted variance of the subsystem Lanczos coefficients. This provides a quantitative measure of dynamical heterogeneity and a precise characterization of synchronous evolution. We also show that Krylov entropy is able to disambiguate the information associated with subsystems composition from that generated by dynamical mismatch, and that the same Lanczos variance controls the leading mismatch contribution to both complexity and the entropy. Our results establish excess complexity and Krylov entropy as complementary probes of the emergence of collective dynamics in large quantum systems.

Heinz K. H. Siedentop

Gerd Niestegge

An axiomatic approach to the mathematical formalism of quantum mechanics, based upon a certain concept of conditional probability, has been proposed in two recent papers by the author. It leads to Jordan operator algebras and thus comes rather close to the standard Hilbert space model of quantum mechanics, but still includes the so-called exceptional Jordan algebras, for which a Hilbert space representation does not exist. This approach is now extended by defining a mathematical model of composite systems. Such a model is required for the study of the joint distribution of two quantum observables. A very general type of observables (not only the real-valued observables corresponding to the self-adjoint operators) is considered. The joint distribution is defined, using the concept of conditional probability, and exhibits a certain dependence on the succession of the observations which is different from the classical case and unknown so far in quantum mechanics. Finally, it turns out that, at least in the finite-dimensional case, a really satisfying model of the composite system exists only if each single system is modeled by a complex Jordan matrix algebra (or a direct sum), and the model then becomes the tensor product. This provides some reasoning why the exceptional Jordan algebras can be ruled out, why quantum mechanics needs the complex numbers and the complex Hilbert space, and why the tensor product is the right choice for the model of a composite system.