nLab
PU(ℋ)

Contents

Context

Group Theory

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

The projective unitary group on an infinite-dimensional separable complex Hilbert space \mathcal{H} is traditionally denoted PU()PU(\mathcal{H}), being the quotient of the unitary group U(ℋ) by its circle subgroup S 1S()S^1 \simeq S(\mathbb{C}) \simeq U(1).

Properties

General

Proposition

The U(1)-quotient space coprojection of U(ℋ) over PU(ℋ) – both in their strong operator topology – is a circle-principal bundle:

S 1 U() PU()U()/S 1 \array{ S^1 &\hookrightarrow& \mathrm{U}(\mathcal{H}) \\ && \big\downarrow \\ && PU(\mathcal{H}) \mathrlap{ \; \simeq \; \mathrm{U}(\mathcal{H})/S^1 } }

(Simms 1970, Thm. 1)
Remark

Prop. means in particular that U()PU()\mathrm{U}(\mathcal{H}) \xrightarrow{\;} PU(\mathcal{H}) is locally trivial, hence that the coset space coprojection U()U()/S 1\mathrm{U}(\mathcal{H}) \xrightarrow{\;} \mathrm{U}(\mathcal{H})/S^1 admits local sections. See also at coset space coprojection admitting local sections.

Proposition

In its operator topology (here), PU()PU(\mathcal{H}) is a well-pointed topological group.

(Hebestreit & Sagave 2020, p. 23, using Dardalat & Pennig 2016, Prop. 2.26)

Homomorphisms into PU()PU(\mathcal{H})

Since Γ\Gamma \,\coloneqq\, PU(ℋ) is not a (finite-dimensional) Lie group, it falls outside the applicability of the general theorem that nearby homomorphisms from compact Lie groups are conjugate. Nevertheless, the conclusion still holds, at least for domain GG a discrete, hence finite group:

The PU(ℋ)-space of homomorphisms Hom(G,PU())Hom\big(G,\, PU(\mathcal{H})\big) is a disjoint union (as here) of orbits of the conjugation action.

This is established in Uribe & Lück 2013, Sec. 15, p. 38.

Literature

General discussion:

Concerning well-pointedness:

Last revised on September 19, 2021 at 09:59:13. See the history of this page for a list of all contributions to it.