Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Orbits of the rotation group on spin states

H. Bacry

Published Oct 1, 1974• 46 citations
Publisher

Abstract

A simple theorem on projective spaces generalizes the concept of the Riemann sphere. This leads us to a generalized interpretation of the ray space associated with a finite-dimensional Hilbert space. An application is given about the way the rotation group acts on states of given spin j.
DOI
10.1063/1.1666525

Related papers

Decomposition of Hilbert space in sets of coherent states

Nuno Barros e Sá

Within the generalized definition of coherent states as group orbits we study the orbit spaces and the orbit manifolds in the projective spaces constructed from linear representations. Invariant functions are suggested for arbitrary groups. The group SU (2) is studied in particular and the orbit spaces of its j = 1/2 and 1 representations completely determined. The orbits of SU (2) in CP N can be either two- or three-dimensional, the former being either isomorphic to S 2 or to RP 2 and the latter being isomorphic to quotient spaces of RP 3 . We end with a look from the same perspective to the quantum mechanical space of states in particle mechanics.

arXiv:quant-ph/0009022•Journal of Physics A Mathematical and General•May 31, 2001•3 citations

SOLUTIONS IN SPHERICAL COORDINATES OF GENERAL EQUATIONS OF PARTICLES WITH SPIN

André Jolivet

The wave functions in spherical coordinates were determined in the case of particles with any spin by utilizing a representntion allowing these functions to be expressed by simple formulas. (tr-auth)

•Oct 22, 1962•0 citations

Spin Coherent States

Didier Robert, Monique Combescure

In this chapter we consider the unit sphere \(\mathbb S^2\) of the Euclidean space \(\mathbb R^3\) with its canonical symplectic structure is a phase space. Then coherent states are labelled by points on \(\mathbb S^2\) and allow us to build a quantization of the two-sphere \(\mathbb S^2\). They are defined in each finite dimensional space of an irreducible unitary representation of the symmetry rotation group SO(3) (or its covering SU(2)) of \(\mathbb S^2\) and give a semiclassical interpretation for the spin. As an application we state the Berezin-Lieb inequalities and compute the thermodynamic limit for large spin systems.

Theoretical and mathematical physics•Jan 1, 2021•0 citations

A $C^*$-Algebraic Approach to Parametrized Quantum Spin Systems and Their Phases in One Spatial Dimension

Daniel D. Spiegel

This thesis investigates parametrized quantum spin systems in the thermodynamic limit from a $C^*$-algebraic point of view. Our main physical result is the construction of a phase invariant for one-dimensional quantum spin chains parametrized by a topological space $X$. This invariant is constructed using $C^*$-algebraic techniques and takes values in degree one Čech cohomology $H^1(X;\mathbb{P}\mathrm{U}(\mathscr{H}))$, where $\mathbb{P}\mathrm{U}(\mathscr{H})$ is the projective unitary group of an infinite-dimensional Hilbert space $\mathscr{H}$, endowed with the strong operator topology. Using Dixmier-Douady theory [Bry93, DD63, HJJS08] one may equivalently view this as an element of $H^3(X;\mathbb{Z})$. An exactly solvable model of a one-dimensional spin system parametrized by the 3-sphere $X = \mathbb{S}^3$ is presented and it is shown that its invariant is nontrivial [WQB+22]. We also prove several mathematical results on topological aspects of the pure state space $\mathscr{P}(\mathfrak{A})$ of a $C^*$-algebra $\mathfrak{A}$. We prove that $\mathscr{P}(\mathfrak{A})$, endowed with the weak* topology, has trivial fundamental group for every UHF algebra $\mathfrak{A}$ [BHM+23]. We review the results of [SMQ+22] that show how the outputs of the GNS representation and Kadison transitivity theorem can be understood to depend continuously on their inputs. These results are expanded upon with the construction of a distinguished Čech class in $H^1(\mathscr{P}(\mathfrak{A});\mathrm{U}(1))$ that generalizes the principal $\mathrm{U}(1)$-bundle $\mathbb{S} \mathscr{H} \rightarrow \mathbb{P} \mathscr{H}$ in a representation independent way, where $\mathbb{S} \mathscr{H}$ and $\mathbb{P} \mathscr{H}$ are the unit sphere and projective Hilbert space of $\mathscr{H}$. Finally, we prove a selection theorem in the vein of [SMQ+22] for the weak* topology on $\mathscr{P}(\mathfrak{A})$.

arXiv:2305.07951•May 13, 2023•0 citations

Phase-space spinor amplitudes for spin- 1/2 systems

P. Watson, Anthony J. Bracken

The concept of phase-space amplitudes for systems with continuous degrees of freedom is generalized to finite-dimensional spin systems. Complex amplitudes are obtained on both a sphere and a finite lattice, in each case enabling a more fundamental description of pure spin states than that previously given by Wigner functions. In each case the Wigner function can be expressed as the star product of the amplitude and its conjugate, so providing a generalized Born interpretation of amplitudes that emphasizes their more fundamental status. The ordinary product of the amplitude and its conjugate produces a (generalized) spin Husimi function. The case of spin-$\frac{1}{2}$ is treated in detail, and it is shown that phase-space amplitudes on the sphere transform correctly as spinors under rotations, despite their expression in terms of spherical harmonics. Spin amplitudes on a lattice are also found to transform as spinors. Applications are given to the phase space description of state superposition, and to the evolution in phase space of the state of a spin-$\frac{1}{2}$ magnetic dipole in a time-dependent magnetic field.

arXiv:1010.3522•Physical Review A•Apr 12, 2011•4 citations

A Phase Space Approach to the Projective Representation of the Canonical Commutation Relations

Loucas C. Papaloucas

The projective nature of the representations of canonical commutation relations on the space of states is probed via cohomological considerations. Some interesting geometrical pictures arise, which associate themselves with the phase descriptions of quantum systems.

Progress of Theoretical Physics•Jan 1, 1996•0 citations
Within the generalized definition of coherent states as group orbits we study the orbit spaces and the orbit manifolds in the projective spaces constructed from linear representations. Invariant functions are suggested for arbitrary groups. The group SU (2) is studied in particular and the orbit spaces of its j = 1/2 and 1 representations completely determined. The orbits of SU (2) in CP N can be either two- or three-dimensional, the former being either isomorphic to S 2 or to RP 2 and the latter being isomorphic to quotient spaces of RP 3 . We end with a look from the same perspective to the quantum mechanical space of states in particle mechanics.
In this chapter we consider the unit sphere S2\mathbb S^2S2 of the Euclidean space R3\mathbb R^3R3 with its canonical symplectic structure is a phase space. Then coherent states are labelled by points on S2\mathbb S^2S2 and allow us to build a quantization of the two-sphere S2\mathbb S^2S2. They are defined in each finite dimensional space of an irreducible unitary representation of the symmetry rotation group SO(3) (or its covering SU(2)) of S2\mathbb S^2S2 and give a semiclassical interpretation for the spin. As an application we state the Berezin-Lieb inequalities and compute the thermodynamic limit for large spin systems.
A C∗C^*C∗-Algebraic Approach to Parametrized Quantum Spin Systems and Their Phases in One Spatial Dimension
This thesis investigates parametrized quantum spin systems in the thermodynamic limit from a C∗C^*C∗-algebraic point of view. Our main physical result is the construction of a phase invariant for one-dimensional quantum spin chains parametrized by a topological space XXX. This invariant is constructed using C∗C^*C∗-algebraic techniques and takes values in degree one Čech cohomology H1(X;PU(H))H^1(X;\mathbb{P}\mathrm{U}(\mathscr{H}))H1(X;PU(H)), where PU(H)\mathbb{P}\mathrm{U}(\mathscr{H})PU(H) is the projective unitary group of an infinite-dimensional Hilbert space H\mathscr{H}H, endowed with the strong operator topology. Using Dixmier-Douady theory [Bry93, DD63, HJJS08] one may equivalently view this as an element of H3(X;Z)H^3(X;\mathbb{Z})H3(X;Z). An exactly solvable model of a one-dimensional spin system parametrized by the 3-sphere X=S3X = \mathbb{S}^3X=S3 is presented and it is shown that its invariant is nontrivial [WQB+22]. We also prove several mathematical results on topological aspects of the pure state space P(A)\mathscr{P}(\mathfrak{A})P(A) of a C∗C^*C∗-algebra A\mathfrak{A}A. We prove that P(A)\mathscr{P}(\mathfrak{A})P(A), endowed with the weak* topology, has trivial fundamental group for every UHF algebra A\mathfrak{A}A [BHM+23]. We review the results of [SMQ+22] that show how the outputs of the GNS representation and Kadison transitivity theorem can be understood to depend continuously on their inputs. These results are expanded upon with the construction of a distinguished Čech class in H1(P(A);U(1))H^1(\mathscr{P}(\mathfrak{A});\mathrm{U}(1))H1(P(A);U(1)) that generalizes the principal U(1)\mathrm{U}(1)U(1)-bundle SH→PH\mathbb{S} \mathscr{H} \rightarrow \mathbb{P} \mathscr{H}SH→PH in a representation independent way, where SH\mathbb{S} \mathscr{H}SH and PH\mathbb{P} \mathscr{H}PH are the unit sphere and projective Hilbert space of H\mathscr{H}H. Finally, we prove a selection theorem in the vein of [SMQ+22] for the weak* topology on P(A)\mathscr{P}(\mathfrak{A})P(A).
The concept of phase-space amplitudes for systems with continuous degrees of freedom is generalized to finite-dimensional spin systems. Complex amplitudes are obtained on both a sphere and a finite lattice, in each case enabling a more fundamental description of pure spin states than that previously given by Wigner functions. In each case the Wigner function can be expressed as the star product of the amplitude and its conjugate, so providing a generalized Born interpretation of amplitudes that emphasizes their more fundamental status. The ordinary product of the amplitude and its conjugate produces a (generalized) spin Husimi function. The case of spin-12\frac{1}{2}21​ is treated in detail, and it is shown that phase-space amplitudes on the sphere transform correctly as spinors under rotations, despite their expression in terms of spherical harmonics. Spin amplitudes on a lattice are also found to transform as spinors. Applications are given to the phase space description of state superposition, and to the evolution in phase space of the state of a spin-12\frac{1}{2}21​ magnetic dipole in a time-dependent magnetic field.