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Journal of Mathematical Physics

A note on families of spin observables and a generalization of Wigner’s theorem in C2

Norbert Poschadel

Published Nov 1, 2000• 2 citations
Publisher

Abstract

We present an elementary geometric method to derive a full quantum mechanical description of the spin of a spin-1/2 particle from the only requirement that no two different orthonormal bases in R3 could be distinguished from each other by any spin measurement. Further, this method is used to obtain a generalization of Wigner’s theorem on symmetry operations for the special Hilbert space C2.

DOI
10.1063/1.1290057

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How robust is the persistence obstruction for Wigner's friend's perception? A closed-form visibility threshold

Manuel Flores Gordillo

What this note does. Guérin, Baumann, Del Santo and Brukner (Commun. Phys. 4, 93 (2021)) proved that, except for two special choices of Wigner's measurement basis, no joint probability distribution consistent with unitary quantum mechanics describes Wigner's friend's perceived outcomes before and after Wigner's measurement. Their theorem is exact and binary: it asks whether two effective perception observables commute. This note asks how much the friend's records must degrade before the obstruction disappears, and answers in closed form. Writing Wigner's basis parameter as c = |a² − b²| and smearing both perception observables to visibility η, a persistent-reality description exists if and only if η ≤ 2/(√(1−c²) + √(1+3c²)). The curve equals 1 exactly at the two compatible cases of the qualitative theorem and is minimized at √3/2 ≈ 0.866 when c² = 2/3. What is new, and what is not. Stated plainly, because each ingredient is due to someone else. The scenario, the assumptions P1–P3 and the two effective perception POVMs are from Guérin et al. The equivalence between a linear two-time distribution and a joint POVM is established in the proof of their Theorem III.1 and is used as stated there, not re-proved here. The joint-measurability criterion for unbiased qubit observables is Busch's (Phys. Rev. D 33, 2253 (1986)) and is standard. Given the Bloch vectors, the threshold follows from that criterion in two lines; this note does not present the computation as difficult. The contribution is the robustness question, the closed-form answer, the worst-case basis, and an elementary observation about the single-time claim. A clarification the note also records. The frequently repeated claim that no single global density operator reproduces both agents' descriptions is false in its single-time operational form: the two states agree on the friend's entire record algebra, differing only in coherences she cannot access, so Wigner's own state satisfies both agents' operational constraints. The claim becomes a theorem only through the two-time persistence requirement. This is elementary and expected to be folklore among specialists; it is recorded because the inference is common and because it locates where the real obstruction lives. Status and scope. This is a preprint, not a peer-reviewed article. Two honest caveats are stated in the text rather than buried. The noise model is a stipulation on the effective observables, not a decoherence channel derived from a modified unitary model; Del Santo, Manzano and Brukner (Phys. Rev. Research 7, 033279 (2025)) have a physically motivated degradation parameter for the same scenario, and the map between the two is not known here. And smearing the first, sharp observable means degrading the friend's perception of her own outcome, which requires an interpretive commitment some readers will not grant; smearing only the later readout provably achieves nothing, which is why the symmetric family is used. Verification. All algebra was recomputed symbolically and independently: reconstruction of the perception POVMs from the unitary model, reproducing Eqs. (12)–(15) of Guérin et al.; derivation of the threshold and of the minimum at c² = 2/3; and a semidefinite-programming feasibility test of the joint-POVM constraints at five values of c on both sides of the threshold, agreeing with the curve in all ten cases. The three scripts are deposited with the note. Use of AI assistance. This note was prepared with substantial assistance from AI language models, used for symbolic computation, drafting, literature search, and adversarial review of the arguments. Every mathematical claim was independently recomputed symbolically and cross-checked numerically. No specialist in quantum foundations has refereed it. The author is responsible for the content. This disclosure is made unconditionally.

•Jul 29, 2026•0 citations

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

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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 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.

•Sep 3, 2021•0 citations
We present an elementary geometric method to derive a full quantum mechanical description of the spin of a spin-1/2 particle from the only requirement that no two different orthonormal bases in R3 could be distinguished from each other by any spin measurement. Further, this method is used to obtain a generalization of Wigner’s theorem on symmetry operations for the special Hilbert space C2.
Spin-s systems Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often distinguish between two types of angular momentum: orbital angular momentum , which a system of particles possesses due to particle motion through space; and spin angular momentum , which is an intrinsic property of a particle. The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description. We have already dealt with spin systems, particularly the example of a spin-½ particle. Our approach began with the empirical observation that a measurement of any spin component of a spin-½ particle could yield only the results +ħ/2 or −ħ/2. We introduced the basis states | z ±〉 for the two-dimensional Hilbert space ℋ. We also gave other basis states such as {| x ±〉} and {| y ±〉} in terms of the | z ±〉 states. From basis states and measurement values we constructed operators for the spin components S x , S y , and S z . With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56). Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic.
Abstract In this chapter we explain how we can describe quantum theory within the general framework given in Chapter 1. We do not discuss various theories concerning the question of how quantum theory can (or cannot) be deduced from experimental facts or from a verifiable axiomatic system. Instead we start by introducing the assumptions of quantum theory. The main content is the algebraic viewpoint of quantum theory based on the theory of operator algebras, the GNS construction which supplies the fundamental relation between the description of the states by Hilbert space vectors and the description of the states in this general framework, the Wigner theorem which is basic for the mathematical description of quantum mechanical symmetries, and its generalizations. The basic definitions and theorems about Hilbert spaces and operator algebras are collected in the appendix.
What this note does. Guérin, Baumann, Del Santo and Brukner (Commun. Phys. 4, 93 (2021)) proved that, except for two special choices of Wigner's measurement basis, no joint probability distribution consistent with unitary quantum mechanics describes Wigner's friend's perceived outcomes before and after Wigner's measurement. Their theorem is exact and binary: it asks whether two effective perception observables commute. This note asks how much the friend's records must degrade before the obstruction disappears, and answers in closed form. Writing Wigner's basis parameter as c = |a² − b²| and smearing both perception observables to visibility η, a persistent-reality description exists if and only if η ≤ 2/(√(1−c²) + √(1+3c²)). The curve equals 1 exactly at the two compatible cases of the qualitative theorem and is minimized at √3/2 ≈ 0.866 when c² = 2/3. What is new, and what is not. Stated plainly, because each ingredient is due to someone else. The scenario, the assumptions P1–P3 and the two effective perception POVMs are from Guérin et al. The equivalence between a linear two-time distribution and a joint POVM is established in the proof of their Theorem III.1 and is used as stated there, not re-proved here. The joint-measurability criterion for unbiased qubit observables is Busch's (Phys. Rev. D 33, 2253 (1986)) and is standard. Given the Bloch vectors, the threshold follows from that criterion in two lines; this note does not present the computation as difficult. The contribution is the robustness question, the closed-form answer, the worst-case basis, and an elementary observation about the single-time claim. A clarification the note also records. The frequently repeated claim that no single global density operator reproduces both agents' descriptions is false in its single-time operational form: the two states agree on the friend's entire record algebra, differing only in coherences she cannot access, so Wigner's own state satisfies both agents' operational constraints. The claim becomes a theorem only through the two-time persistence requirement. This is elementary and expected to be folklore among specialists; it is recorded because the inference is common and because it locates where the real obstruction lives. Status and scope. This is a preprint, not a peer-reviewed article. Two honest caveats are stated in the text rather than buried. The noise model is a stipulation on the effective observables, not a decoherence channel derived from a modified unitary model; Del Santo, Manzano and Brukner (Phys. Rev. Research 7, 033279 (2025)) have a physically motivated degradation parameter for the same scenario, and the map between the two is not known here. And smearing the first, sharp observable means degrading the friend's perception of her own outcome, which requires an interpretive commitment some readers will not grant; smearing only the later readout provably achieves nothing, which is why the symmetric family is used. Verification. All algebra was recomputed symbolically and independently: reconstruction of the perception POVMs from the unitary model, reproducing Eqs. (12)–(15) of Guérin et al.; derivation of the threshold and of the minimum at c² = 2/3; and a semidefinite-programming feasibility test of the joint-POVM constraints at five values of c on both sides of the threshold, agreeing with the curve in all ten cases. The three scripts are deposited with the note. Use of AI assistance. This note was prepared with substantial assistance from AI language models, used for symbolic computation, drafting, literature search, and adversarial review of the arguments. Every mathematical claim was independently recomputed symbolically and cross-checked numerically. No specialist in quantum foundations has refereed it. The author is responsible for the content. This disclosure is made unconditionally.
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.