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Given an infinite-dimensional Hilbert space $H$, the set $U(H)$ of all unitary operators on $H$ forms a group, known as the unitary group. Now several subgroups of this group play an important role in quantum mechanics, for instance the group of time-translation operators and the group of spatial translation operators. And these subgroups are treated as Lie groups, which I assume means that $U(H)$ is also a Lie group. So my question is, what is the differentiable manifold structure on $U(H)$ used in quantum mechanics?

And what is the topology used in quantum mechanics for this group? Is it the uniform (norm) topology, or the strong operator topology, or what?

EDIT: Assuming the strong operator topology is the right one for quantum mechanics, the second page of this paper mentions "the Frechet Lie group $U(H)$ consisting of all unitary operators on H, equipped with the strong operator topology". That means $U(H)$ has a Frechet manifold structure, i.e. it’s locally isomorphic to an infinite-dimensional Frechet space. (As opposed to an ordinary manifold which is locally isomorphic to a finite-dimensional Euclidean space.) But what is that Frechet manifold structure?

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$U(H)$ is a connected and continuous-path connected topological group with respect to the strong operator topology, which is the most natural one for physical applications. This topology permits to extend some features of Lie groups to the case of an "infinite-dimensional" group where the differentiable structure is not easy to define (there are different points of view and not all properties of standard finite-dimensional smooth manifolds can be extended).

Lie groups have the property that they are almost completely defined by their Lie algebra (it is true for simply connected Lie groups, otherwise this property is valid in a neighborhood of the neutral element only).

In particular, if the group is connected, every element is obtained as a product of elements belonging to one-parameter subgroup generated by vectors in the Lie algebra.

This also happens for $U(H)$ in view of Stone theorem (which also uses the strong operator topology) even if no differentiable structure is chosen over $U(H)$. From an abstract point of view, we can think of selfadjoint operators, as the elements of the Lie algebra of $U(H)$. From the spectral theory of normal operators and the functional calculus, every unitary operator $U$ can be written as $e^{itA}$ for some selfadjoint operator $A$ and some real $t$.

A global definition of a general Lie algebra for $U(H)$ is difficult since selfadjoint operators are unbounded and have different domains so that $[A,B]$ is generally undefined.

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  • $\begingroup$ First of all, can you explain what the physical significance of using the strong operator topology? Second of all, The second page of this paper mentions "the Frechet Lie group $U(H)$ consisting of all unitary operators on H, equipped with the strong operator topology". That means $U(H)$ has a Frechet manifold structure, i.e. it’s locally isomorphic to an infinite-dimensional Frechet space. I want to know what that Frechet manifold structure is. $\endgroup$ – Keshav Srinivasan Apr 16 at 12:58
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    $\begingroup$ The physical significance of the strong operator topology relies on the fact that the spectral theory and the functional calculus used in quantum physics use that topology. The uniform topology is too "strong". For instance a one-parameter group of unitary operators is uniformly continuous if and only if the selfadjoint operator generating that group is bounded and, in infinite dimensional Hilbert spaces, there are very few physically meaningful observables (selfadjoint operators) which are bounded, i.e. they attain a bounded set of values. $\endgroup$ – Valter Moretti Apr 16 at 14:54
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    $\begingroup$ Regarding Frechet topology, that is a complete seminormed topology taking place in several theoretical descriptions of physics...However, the fundamental theorems on strongly continuous unitary representation s of Lie groups used in physics (e.g., Bargmann, Nelson etc...) are proved without referring to that structure of $U(H)$...But focusing on finite dimensional Lie groups. $\endgroup$ – Valter Moretti Apr 16 at 14:59
  • $\begingroup$ A problem with infinite dimensional Lie groups is that there is no Haar measure in general. In principle one can construct quantum theory n the projective space of $H$ viewed as a quotient of $U(H)$ and referring to a measure over that space invariant under the action of every unitary operator. $\endgroup$ – Valter Moretti Apr 16 at 15:07
  • $\begingroup$ In infinite dimensional Hilbert spaces there is no such measure (whereas it exists in finite dimensional spaces and is related to Fubini-Study metric on the projecitve space). In the finite dimensional case QM can be seen as a peculiar type of Hamiltonian mechanics over the projective space. This is not possible (at least with a straightforward generalization) in the infinite-dimensional case. Some of the obstructions arise from the nature of $U(H)$ in the infinite dimensional case. $\endgroup$ – Valter Moretti Apr 16 at 15:07

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