[Paper Review] The infinite unitary group, Howe dual pairs, and the quantization of constrained systems

N.P. Landsman|ArXiv.org|Nov 23, 1994

Advanced Operator Algebra Research41 references3 citations

TL;DR

This paper reconstructs irreducible unitary representations of the infinite unitary group $U_0(\mathcal{H})$ via geometric quantization and Rieffel induction, using Howe dual pairs and symplectic quotients. It shows that quantization via Rieffel induction reproduces Kirillov-Olłshanskii's representations, but only when half-form corrections are omitted or modified weight shifts are applied, revealing a potential obstacle in quantizing constrained systems.

ABSTRACT

The irreducible unitary representations of the Banach Lie group $U_0(\H)$ (which is the norm-closure of the inductive limit $\cup_k U(k)$) of unitary operators on a separable Hilbert space $\H$, which were found by Kirillov and Ol'shanskii, are reconstructed from quantization theory. Firstly, the coadjoint orbits of this group are realized as Marsden-Weinstein symplectic quotients in the setting of dual pairs. Secondly, these quotients are quantized on the basis of the author's earlier proposal to quantize a more general symplectic reduction procedure by means of Rieffel induction (a technique in the theory of operator algebras). As a warmup, the simplest such orbit, the projective Hilbert space, is first quantized using geometric quantization, and then again with Rieffel induction. Reduction and induction have to be performed with either $U(M)$ or $U(M,N)$. The former case is straightforward, unless the half-form correction to the (geometric) quantization of the unconstrained system is applied. The latter case, in which one induces from holomorphic discrete series representations, is problematic. For finite-dimensional $\H=\C^k$, the desired result is only obtained if one ignores half-forms, and induces from a representation, `half' of whose highest weight is shifted by $k$ (relative to the naive orbit correspondence). This presumably poses a problem for any theory of quantizing constrained systems.

Motivation & Objective

• To reconstruct irreducible unitary representations of the infinite unitary group $U_0(\mathcal{H})$ using quantization techniques.
• To investigate the role of symplectic reduction and Rieffel induction in realizing coadjoint orbits as quantized spaces.
• To address inconsistencies in quantization when applying half-form corrections in the context of constrained systems.
• To clarify the correspondence between orbits and representations in infinite-dimensional unitary groups.
• To test the robustness of Rieffel induction in the setting of dual pairs for infinite-dimensional Lie groups.

Proposed method

• Realizing coadjoint orbits of $U_0(\mathcal{H})$ as Marsden-Weinstein symplectic quotients within a Howe dual pair framework.
• Applying Rieffel induction to quantize these symplectic quotients, using representations of $U(M)$ or $U(M,N)$ as input.
• Comparing results from geometric quantization of projective Hilbert space with those from Rieffel induction.
• Using holomorphic discrete series representations as induction data, particularly analyzing weight shifts relative to naive orbit correspondence.
• Analyzing the effect of half-form corrections in geometric quantization and their incompatibility with desired representation results.
• Performing reduction and induction with either $U(M)$ or $U(M,N)$, and assessing consistency across both cases.

Experimental results

Research questions

• RQ1 Can the irreducible unitary representations of $U_0(\mathcal{H})$ be reconstructed via symplectic reduction and Rieffel induction?
• RQ2 What is the role of half-form corrections in the quantization of coadjoint orbits of $U_0(\mathcal{H})$?
• RQ3 Why does inducing from holomorphic discrete series representations fail to reproduce the correct representations unless weight shifts are applied?
• RQ4 How do dual pairs facilitate the quantization of constrained systems in infinite-dimensional settings?
• RQ5 What are the implications of these results for a general theory of quantizing constrained systems?

Key findings

• The coadjoint orbits of $U_0(\mathcal{H})$ are realized as symplectic quotients via a Howe dual pair construction.
• Geometric quantization of the projective Hilbert space yields the expected Fock-Bargmann representation.
• Rieffel induction from $U(M)$ reproduces the correct representations without half-form corrections.
• When using $U(M,N)$ and holomorphic discrete series, the correct result is only obtained if the highest weight is shifted by $k$ relative to the naive correspondence.
• The half-form correction leads to inconsistency, suggesting a fundamental obstacle in quantizing constrained systems using standard geometric quantization.
• The failure to reproduce representations without modifying the weight or omitting half-forms indicates a potential flaw in general quantization schemes for constrained systems.