algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
The unitary group of unitary operators on an infinite-dimensional separable complex Hilbert space is traditionally denoted . It does not quite have an established prose name, but is often referred to by some combination of the words “unitary group” and “strong topology”.
Its quotient by the circle subgroup U(1) is the corresponding projective unitary group PU(ℋ).
Throughout, let be any separably infinite-dimensional complex Hilbert space.
The set of bounded operators carries the following topologies, characterized by the conditions under which a sequence converges to a fixed operator :
The limit of this sequence is
…in the norm topology or topology of uniform convergence if
…in the strong operator topology or topology of pointwise convergence if
…in the weak operator topology if
…in the compact-open topology if
uniform convergence above holds for the restrictions and to any compact subset .
The norm topology makes a topological group, in fact a Banach Lie group.
On the weak operator topology and strong operator topology agree and make it a topological group.
The compact-open topology coincides with the strong operator topology
The norm topology on (from Prop. ) is strictly finer than the (weak or strong) operator topology (from Prop. ).
In summary:
(also Espinoza & Uribe 2014, Thm. 1.2)
Equipped with the strong topology (Prop. ), is completely metrizable.
Equipped with the strong topology (Prop. ), is not locally compact.
(Kuiper’s theorem)
The topological unitary group in either the
or the
is contractible in that there is a left homotopy between the identity and the constant function .
The U(1)-quotient space coprojection of U(ℋ) over PU(ℋ) – both in their strong operator topology – is a circle-principal bundle:
Prop. means in particular that is locally trivial, hence that the coset space coprojection admits local sections. See also at coset space coprojection admitting local sections.
Early discussion in relation to PU(ℋ):
Proof that the the weak- and strong- operator topologies agree on :
Proof that weak-, strong- as well as the compact-open topology all agree on :
Martin Schottenloher, The Unitary Group In Its Strong Topology (arXiv:1309.5891), Advances in Pure Mathematics 08 05 (2018) (doi:10.4236/apm.2018.85029)
Jesus Espinoza, Bernardo Uribe, Topological properties of the unitary group, JP Journal of Geometry and Topology 16 1 (2014) 45-55 (arXiv:1407.1869, journal)
Previous influential but wrong claim that they do not:
Michael Atiyah, Graeme Segal, Appendix of: Twisted K-theory, Ukrainian Math. Bull. 1, 3 (2004) (arXiv:math/0407054, journal page, published pdf)
This paper made the mistake of assuming that because various topologies were distinct on , they remain distinct on .
Further properties:
Proof of complete metrizability:
Proof of failure of local compactness:
Proof of Kazhdan's property (T) for :
Last revised on June 20, 2022 at 15:59:12. See the history of this page for a list of all contributions to it.