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arXiv:1805.03987

Normalized non-redundant vector tomographic portraits of spin states

Ya. A. Korennoy

Published May 10, 2018• 0 citations
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Abstract

Non-redundant and normalized four-component vector tomographic portrait fully describing the states of spin 1/2 quantum particles was introduced. Dequantizer and quantizer for such portrait were found, and generalization to the case of spin (2^N-1)/2 was done (N is a natural number). It was shown that such a portrait is completely defined by a thriple of non-complanar vectors with the lengthes equal or less then unity. A clear geometric interpretation of the choice of parameters for finding normalized dequantizers and quantizers is presented and numerical examples of such dequantizers and quantizers for spin 1/2 are given.

DOI
10.48550/arxiv.1805.03987

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Non-redundant and normalized four-component vector tomographic portrait fully describing the states of spin 1/2 quantum particles was introduced. Dequantizer and quantizer for such portrait were found, and generalization to the case of spin (2^N-1)/2 was done (N is a natural number). It was shown that such a portrait is completely defined by a thriple of non-complanar vectors with the lengthes equal or less then unity. A clear geometric interpretation of the choice of parameters for finding normalized dequantizers and quantizers is presented and numerical examples of such dequantizers and quantizers for spin 1/2 are given.
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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.
Abstract A novel representation of spin 1/2 combines in a single geometric object the roles of the standard Pauli spin vector operator and spin state. Under the spin-position decoupling approximation it consists of three orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) is Hermitian; (2) shows handedness; (3) reproduces all standard expectation values, including the total one-particle spin modulus 𝑆tot = (ℏ/2)√3; (4) constrains basis opposite spins to have same handedness; (5) allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled pairs have precisely related gauge phases; (2) the dimensionality of the spin space doubles due to variation of handedness; (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). Cross-product terms in the expression for the squared total spin of two particles relate 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 done 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.
The positive vector optical tomogram fully describing the quantum state of spin 1/2 particle without any redundancy is introduced. Reciprocally the vector symplectic tomogram and vector quasidistributions W⃗(q,p)\vec W({\mathbf q},{\mathbf p})W(q,p), Q⃗(q,p)\vec Q({\mathbf q},{\mathbf p})Q​(q,p), P⃗(α⃗)\vec P(\vecα)P(α) are introduced. The evolution equations for proposed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are obtained. The quantum system of charged spin 1/2 particle in arbitrary electro-magnetic field is considered in proposed representations and evolution equations which are analogs of Pauli equation are obtained. The propagator of evolution equation in the case of homogeneous and stationary magnetic field in Landau gauge is found and the evolution of initial entangled superposition of lower Landau levels in the vector optical representation is considered. The system of linear quantum oscillator with spin in vector optical tomography representation is considered and the evolution of initial entangled superposition of two lower Fock states and spin-up, spin-down states is studied in this representation.
Abstract A new representation for spin 1/2 in the even 3D subalgebra of the spacetime algebra (STA) combines in a single geometric object the roles of the standard Pauli spin vector and spin state. It is a vector quantity comprising a gauge phase. In the one-particle case the representation (1) is Hermitian; (2) chiral; (3) reproduces all standard expectation values, including the total one-particle spin modulus ; (4) constrains a spinor basis representing opposite spins to preserve handiness (chirality); (5) the gauge phase allows to explicitly formalize irreversibility in spin measurement. In the two-particle case it (1) identifies entangled spin pairs as having opposite handiness and precise gauge phase relations; (2) doubles the dimensionality of the spin space due to variation of handiness; (3) the four maximally entangled states are naturally derived by pairing spins that are reflections (triplets) and inversions (singlet) of each-other. The cross-product terms in the expression for the squared total spin of two particles can be affected by experiment and they yield the standard expectation values after measurement. Here I directly define and transform spin in 3D orientation space, without invoking concepts like abstract Hilbert space and tensor product as in the standard formulation. The STA 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.