TY - JOUR

T1 - Spin 1/2 one- and two-particle systems in physical space without eigen-algebra or tensor product

AU - Andoni, Sokol

N1 - Publisher Copyright:
© 2022 The Author. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd.

PY - 2024

Y1 - 2024

N2 - 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 (Formula presented.), (4) formalizes irreversibility in measurement, and (5) permits adaptive imbedding 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 (triplets) and inversion (singlet), (3) yield the standard total angular momentum, and (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.

AB - 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 (Formula presented.), (4) formalizes irreversibility in measurement, and (5) permits adaptive imbedding 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 (triplets) and inversion (singlet), (3) yield the standard total angular momentum, and (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.

KW - Clifford algebras

KW - Spinor

KW - Vector and spinor representations

U2 - 10.1002/mma.8925

DO - 10.1002/mma.8925

M3 - Journal article

AN - SCOPUS:85143832232

SN - 0170-4214

VL - 47

SP - 1457

EP - 1470

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

ER -