Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

TL;DR

This paper introduces a geometric algebra-based representation of spin-1/2 systems that simplifies understanding spin correlations and transformations without relying on eigen-algebra or tensor products.

Contribution

It presents a novel Hermitian spin-1/2 formalism using vectors in 3D space, replacing traditional matrix-based methods, and provides a geometric interpretation of entanglement and measurement.

Findings

The new formalism reproduces standard quantum results.

Maximally entangled pairs are in phase with opposite handedness.

The approach offers a clear geometric picture of spin correlations.

Abstract

Under the spin-position decoupling approximation, a vector with a phase in 3D orientation space endowed with geometric algebra, substitutes the vector-matrix spin model built on the Pauli spin operator. The standard quantum operator-state spin formalism is replaced with vectors transforming by proper and improper rotations in the same 3D space -- isomorphic to the space of Pauli matrices. In the single spin case the novel spin 1/2 representation: (1) is Hermitian; (2) shows handedness; (3) yields all the standard results and its modulus equals the total spin angular momentum S_tot; (4) formalizes irreversibility in measurement; (5) permits adaptive embedding of the 2D spin space in 3D. Maximally entangled spin pairs: (1) are in phase and have opposite handedness; (2) relate by one of the four basic improper rotations in 3D: plane-reflections for triplets and inversion for singlet; (3) yield the standard total angular momentum; (4) all standard expectation values for bipartite and partial observations follow. Depending on whether proper and improper rotors act one or two sided, the formalism appears in two complementary forms, the spinor or the vector form, respectively. The proposed scheme provides a clear geometric picture of spin correlations and transformations entirely in the 3D physical orientation space.