[Paper Review] From Quantum Optics to Non-Commutative Geometry : A Non-Commutative Version of the Hopf Bundle, Veronese Mapping and Spin Representation

Kazuyuki Fujii|ArXiv.org|Feb 26, 2005

Noncommutative and Quantum Gravity Theories13 references3 citations

TL;DR

This paper constructs a non-commutative version of the Hopf bundle using the Jaynes-Cummings model and Quantum Diagonalization Method, showing that Dirac strings—classical singularities—only appear in ground-state configurations, not in excited states. It further develops non-commutative analogues of the Veronese mapping and spin representations of SU(2), revealing that classical geometric and algebraic structures do not universally survive non-commutative deformation.

ABSTRACT

In this paper we construct a non-commutative version of the Hopf bundle by making use of Jaynes-Commings model and so-called Quantum Diagonalization Method. The bundle has a kind of Dirac strings. However, they appear in only states containing the ground one (${\cal F} imes \{\ket{0}\} \cup \{\ket{0}\} imes {\cal F} \subset {\cal F} imes {\cal F}$) and don't appear in remaining excited states. This means that classical singularities are not universal in the process of non-commutativization. Based on this construction we moreover give a non-commutative version of both the Veronese mapping which is the mapping from $\fukuso P^{1}$ to $\fukuso P^{n}$ with mapping degree $n$ and the spin representation of the group SU(2). We also present some challenging problems concerning how classical (beautiful) properties can be extended to the non-commutative case.

Motivation & Objective

• To generalize classical fiber bundles, particularly the Hopf bundle, into a non-commutative geometric framework using quantum optical models.
• To investigate whether classical singularities like Dirac strings persist under non-commutative deformation, challenging the universality of such singularities.
• To extend classical constructions such as the Veronese mapping and spin representation of SU(2) to non-commutative settings.
• To explore the feasibility and challenges of defining non-commutative Chern classes and tensor decompositions in quantum groups.
• To provide a quantum field-theoretic foundation for non-commutative geometry via cavity QED models, particularly the Jaynes-Cummings model.

Proposed method

• Utilizes the Jaynes-Cummings model as a non-commutative deformation of the Berry Hamiltonian, where x and y coordinates are quantized while z remains classical.
• Applies the Quantum Diagonalization Method (QDM) to diagonalize the non-commutative Hamiltonian, enabling construction of a non-commutative Hopf bundle structure.
• Defines a non-commutative complex coordinate via stereographic projection using the operator Z = (1/R(N)+θ)a†, with R(N) = √(N + θ²), generalizing the classical CP¹ coordinate.
• Constructs a non-commutative Veronese mapping from a non-commutative CP¹ to CPⁿ by lifting the classical degree-n map using operator-valued coordinates.
• Attempts to generalize the spin representation of SU(2) by lifting the standard Clebsch-Gordan decomposition to non-commutative operators, but encounters obstructions due to non-commutativity.
• Uses unitary transformations T analogous to Clebsch-Gordan matrices to attempt block-diagonalization of tensor products, but finds that T†(V⊗V)T ≠ diag(1, Φ₁(V)) due to non-commutativity.

Experimental results

Research questions

• RQ1 Can the classical Hopf bundle be consistently deformed into a non-commutative fiber bundle using quantum optical models?
• RQ2 Do classical singularities such as Dirac strings persist universally in non-commutative geometric constructions, or are they state-dependent?
• RQ3 How can the Veronese mapping from CP¹ to CPⁿ be generalized to a non-commutative setting, and what role does the mapping degree play?
• RQ4 Is it possible to construct a non-commutative version of the spin representation of SU(2), and what are the obstructions in higher spin cases?
• RQ5 Can non-commutative Chern classes be defined via such constructions, and what are the implications for non-commutative characteristic classes?

Key findings

• Dirac strings appear only in states containing the vacuum state (F×{|0⟩} ∪ {|0⟩}×F), not in excited states, indicating that classical singularities are not universal under non-commutative deformation.
• The non-commutative Hopf bundle is constructed via the Quantum Diagonalization Method applied to the Jaynes-Cummings model, yielding a fiber bundle structure over a non-commutative base space.
• A non-commutative Veronese mapping is defined by lifting the classical degree-n map using operator-valued coordinates, with the mapping degree potentially linked to a future non-commutative Chern class.
• The non-commutative spin representation is successfully constructed only for j=1 and j=3/2, indicating that generalization to arbitrary spin is obstructed by non-commutativity.
• Tensor product decomposition fails to preserve block-diagonal form under non-commutative unitary transformations, as T†(V⊗V)T ≠ diag(1, Φ₁(V)), revealing a fundamental obstruction in non-commutative representation theory.
• In the classical limit, the non-commutative coordinate Z reduces to the standard CP¹ coordinate Z_c = (x+iy)/(r+z), confirming consistency with the classical theory.