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

Abstract

A novel representation of spin 1/2 combines in a single geometric object the roles of the standard Pauli spin vector and spin state. Under the spin-position decoupling approximation it consists of the ordered sum of three orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) is Hermitian; (2) is oriented due to ordering; (3) reproduces all standard expectation values, including the total one-particle spin modulus A; (4) constrains basis opposite spins to have same orientation; (5) allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled spin pairs have opposite orientation and precisely related gauge phases; (2) the dimensionality of the spin space doubles due to variation of orientation; (3) the four maximally entangled states are naturally defined by the four improper rotations in 3D: reflections onto the three orthogonal frame planes (triplets) and inversion (singlet). The cross-product terms in the expression for the squared total spin of two particles relates to experiment and they yield all standard expectation values after measurement. Here spin is directly defined and transformed in 3D orientation space, without use of eigen algebra and tensor product as in the standard formulation. The formalism allows working with whole geometric objects instead of only components, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘on lab’ (uncontrolled gauge phase) spin transformations.