Thanks very much, Urs – I really appreciate all this hunting down of online articles.

But while looking around, I found this webpage here which gives a rather detailed – albeit informal – survey of what is known for which types of groups. On the following pages it them seems to restrict attention to reductive algebraic groups.

This looks interesting, but I am slightly dismayed by the fact that this apparently won’t apply directly to the Poincaré group because (I have it on reasonable authority) the Poincaré group is not reductive. Oh well.

The definition of “topological direct integral” is on the second page.

Okay, thanks. That sure looks like the usual direct integral, although if I’m missing some subtlety here, I’m hoping it won’t go unnoticed for long. One slight curious thing is the role played by $\mathcal{S}$: the notation for the direct integral has this $\mathcal{S}$ attached to it, although $\mathcal{S}$ isn’t used directly in the construction of the direct integral Hilbert space. As best I can tell, this $\mathcal{S}$ plays a role analogous to the rigging of a Hilbert space, as in “rigged Hilbert space”.

Meanwhile, I’m hoping someone can help answer a question I tried asking in comment 14:

Is the following a standard maneuver in the business…?

Some further reading around seems to point encouragingly in the direction of “yes”, but expert confirmation would be wonderful.

]]>Ah, I have found it. Here.

The definition of “topological direct integral” is on the second page.

]]>A definition of “topological direct integrals” is apparently given here, or at least concretely referenced there. But I cannot currently access that article.

]]>I am under the impression that “topological direct integral” is equivalent terminology for “direct integral”, but let me further check.

Meanwhile, just for completeness: the statements in the eom entry are also in the paper-published “Encyclopedia of Mathematics”-article here (also referring to “topological direct integral”s, though)

]]>Hi Todd,

I am not yet sure that I have found a genuinely good reference.

But while looking around, I found this webpage here which gives a rather detailed – albeit informal – survey of what is known for which types of groups. On the following pages it them seems to restrict attention to reductive algebraic groups.

]]>Thanks for your help, Urs.

That should be true. It is stated here for instance.

I guess you mean this statement:

“A unitary representation of a locally compact group $G$ in a Hilbert space $H$ admits a decomposition as a topological direct integral of irreducible unitary representations of $G$, if either $G$ or $H$ is separable (for non-separable groups and spaces this is not generally true).”

That seems helpful, but I’m not sure what a “topological direct integral” is (googling wasn’t too helpful here). Is it just a translation from a Russian expression that means the same thing as what we call a “direct integral” in mathematical English?

An MO question here seems to suggest Folland as a reference for this.

Okay, for the abelian case here. I *guess* that statement is the same thing as what I was proposing. Do you happen to know whether Folland employs the language of direct integrals of Hilbert spaces?

What I’m really after here is more interpretive. Let me put it this way. Is the following a standard maneuver in the business: starting with a strongly continuous unitary representation $\rho: G \to U(H)$ ($G$ abelian), consider the von Neumann algebra in $B(H)$ generated by the image of $\rho$? (Followed by applying the direct-integral form of the spectral theorem for abelian von Neumann algebras, since the von Neumann algebra so generated is abelian.)

These are probably really basic questions [see, Neel? ;-)], but my command of this material is very faltering and beginner-like. I’m asking to test to what extent I’d be able to mathematically justify the long comment I made to the Café, and I’m just at the beginning where I’m considering the role of mass.

]]>An MO question here seems to suggest Folland as a reference for this.

]]>Is there some nice theorem to the effect that for a certain class of locally compact groups $G$ (reductive Lie groups, perhaps?), every unitary $G$-representation […] can be expressed as a direct integral of irreducible unitary G-representations?

That should be true. It is stated here for instace.

]]>I might have a go at this, having posted that long comment at the Café. But I might could use a little help.

Here is a general question. Is there some nice theorem to the effect that for a certain class of locally compact groups $G$ (reductive Lie groups, perhaps?), every unitary $G$-representation [a strongly continuous homomorphism $G \to U(H)$, where $H$ is a separable Hilbert space] can be expressed as a direct integral of irreducible unitary $G$-representations? If so, is there some online reference where I can read about it?

Here is a somewhat more restricted question. Let’s say I am interested in unitary representations of $\mathbb{R}^n$. Is it possible to develop a niche for the type of theorem above along the following lines? The idea is to consider the image of $\mathbb{R}^n \to U(H)$ as generating an abelian von Neumann algebra on $H$, and then quote the spectral theorem for abelian von Neumann algebras $A$, which realizes $H$ as unitarily equivalent to a direct integral

$U: H \to \int_{x \in X} H_x$over some Borel measure space $X$, and $A$ as equivalent to bounded functions on $X$,

$\Phi: A \stackrel{\sim}{\to} L^\infty(X),$so that the action

$A \hookrightarrow B(H) \stackrel{conj_U}{\to} B(\int_{x \in X} H_x)$is identified with the action

$A \stackrel{\Phi}{\to} L^\infty(X) \stackrel{diag}{\to} B(\int_{x \in X} H_x).$(?) This is only my guess, just reading around and guessing how to patch stuff together.

]]>The page seemed confused about whether it was still only about the irreps, so I tried to clarify that a bit.

]]>I have taken the liberty to change the page name to

unitary representation of the Poincaré group

which seems to be a more canonical choice of title.

Then I have added a lead-in paragraph in the Idea-section, and added references: Wigner’s original article and a pointer to a survey in Haag’s book.

It would be nice if this entry would be further expanded. Todd has posted more material over to the $n$Café here, maybe somebody feels like working this into the entry?

]]>Okay, I thouhgt of another way to state the informal idea and then quickly tried to produce a nice-looking entry, even though still very stubby, at irreducible representation, by taking the material out of the query box and arranged into a decent entry.

]]>What would be a better way to say it in informal words?

The only other thing I can think of is "cannot be further reduced" :-)

Anyway. Zoran, you are clearly right about what an irrep is.

]]>While you're right, Zoran (and I think Urs agrees), let's recognize that Urs was linking from an article on *unitary representations*, where the notions coincide. So I think the slip, if you want to call it that, was understandable after all.

Zoran,

there was no definition there, just an idea sentence, even in quotation marks. Just go ahead and put in the definition.

]]>I created irreducible representation with nothing much in it yet, but just so I could make irrep redirect to it

]]>I made a stubby start at unitary irreps of the Poincare group, titled this way to save space. Very eager to get to the bottom of things; this subject *can't* be that hard.