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Computer Science

Quantum Orbit Method in the Presence of Symmetries

Profile image of Nicola CiccoliNicola Ciccoli

2021, Symmetry

https://doi.org/10.3390/SYM13040724
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17 pages

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Abstract

We review some of the main achievements of the orbit method, when applied to Poisson– Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider C∗-algebra quantization obtained through groupoid techniques, and we try to put the results obtained in algebraic or representation theoretical contexts in relation with groupoid quantization.

FAQs

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What are the practical benefits of applying the quantum orbit method?add

The quantum orbit method provides a systematic framework for constructing unitary representations from Poisson manifolds, particularly in nilpotent cases. This was highlighted in the quantization of the Heisenberg group, where all representations can be obtained through a simple induction process.

How does geometric quantization interact with symplectic groupoid structures?add

Geometric quantization applied to symplectic groupoids results in C* algebras that encapsulate the underlying Poisson geometry. Specific examples demonstrate how the quantization of T* M results in trivial C* algebras when the underlying Poisson structure is zero.

What role do Bohr-Sommerfeld conditions play in quantization?add

Bohr-Sommerfeld conditions ensure the existence of polarized sections by requiring trivial holonomy along Lagrangian leaves. This leads to a topological subgroupoid formation, allowing the definition of a convolution product that completes the quantization process.

What distinguishes Poisson-Lie groups from coisotropic subgroups?add

Poisson-Lie groups have Poisson structures characterized by their non-constant rank, while coisotropic subgroups relax this strictness, allowing more flexibility by being invariant under group actions. The distinction impacts their integration and the nature of their associated symplectic structures.

How does the presence of symmetries affect quantization in Poisson manifolds?add

Symmetries in Poisson manifolds often necessitate modified quantization techniques, as seen in cases where the integrability conditions for symplectic groupoids are influenced by these symmetries. The presence of coisotropic subgroups exemplifies this, requiring careful consideration in the quantization process.

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