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gave a bare minimum to Schur orthogonality relation (for the moment just because I want to be able to link to it).
Is there a ’behind the scenes’ story for this relation, perhaps fitting in with
1-dimensional Dijkgraaf-Witten theory as a prequantum field theory comes down to be essentially a geometric interpretation of what group characters are and do. One may regard this as a simple example of geometric representation theory.
from prequantum field theory. How does physics view the irreducibility of the characters?
Hmm, what does this mean there
It follows that in a discussion of quantization the path integral for the partition function of 1d DW theory is given by the Schur integral over the group character c?
What is the Schur integral?
In DW physics the character $c$ is the exponentiated action functional
$\mathbf{B}c : \mathbf{B} G \to \mathbf{B} U(1)$As such it is given by the trace of a 1-dimensional representation and is hence irreducible.
By the Schur integral of a character $c$ I mean
$\frac{1}{\vert G \vert} \underset{g \in G}{\sum} c(g) = \langle c,1\rangle \,.$I have added that remark here.
Thanks. So back to my question, is there a way to see the orthogonality relation between two exponentiated action functionals?
Oh, I see, sorry.
So given two action functionals $c_1$ and $c_2$ there is their tensor product and so the path integral for $c_1 \otimes (-c_2)$ is now the Schur inner product $\langle c_1, c_2 \rangle$.
Orthogonality hence translates in the 1d DW theory to the tensor product pairing followed by the fact that the partition function on the circle is either 1 for the trivial action functional or 0 otherwise.
Re #4, is there a way to keep entries which replicate common parts of each other up to date when modifying one of them? I mean, now you added #3 to Remark 12 of prequantum field theory, we need to do the same for Remark 6 of FQFT. So I’ll do that now. Maybe it doesn’t matter too much.
True, generally I shouldn’t copy long passages like this but instead “!include” them. Here I needed the material quickly…
Re #3, do the characters as traces of irreps there have anything to do with the particles as irreps idea, as in unitary representation of the Poincaré group?
That’s an interesting question. Let me think about that…
Then beyond $\mathbf{B} U(1)$ as a target, elsewhere there is $\mathbf{B}^n U(1)$ and then Fiorenza and Valentino’s $n$-characters into $\mathbf{B} Pic(n Vect)$.
Are these all traces (literally or figuratively) of representations?
Generally,there is
the $\infty$-group of of phases $\mathbf{B}^{n-1}U(1)$;
a linear representation (“the superposition principle”) $\mathbf{B}^n U(1) \to \mathbf{B}(GL_1(E))$;
a local Lagrangian $\mathbf{Fields} \to \mathbf{B}^n U(1) \to \mathbf{B}GL_1(E)$;
for each homotopy type of a closed manifold $\Sigma$ the $\Sigma$-shaped higher dimensional trace $[\Pi(\Sigma),\mathbf{Fields}] \to \Omega^{dim \Sigma} \mathbf{B}GL_1(E)$.
The traces discussed above are more a way to invoke the classicla result of Schur, not so much intrinsic to the QFT.
Sorry to be dim, but where does unitary representation of the Poincaré group fit in?
It talks there about unitary representations, i.e., continuous homomorphisms from the Poincaré group (or universal cover of the connected component of the isometry group of Minkowski spacetime) $G = SL_2(\mathbb{C}) \ltimes \mathbb{R}^4$ to the unitary group of a Hilbert space $H$.
Back in the blog discussion behind that entry, an instance of $H$ is given (for spin 0 particles) as the space of sufficiently nice functions $\mathbb{R}^4 \to \mathbb{R}$ satisfying the Klein-Gordon equation. Later on it mentions that we’re working with the strong operator topology on $U(H)$. (I see you and Todd were wondering about an nPOV on this once.)
So what’s going on there? $G$ is something like $Diff(\Sigma)$ of space-time $\Sigma$, and it’s looking to act on scalar fields $[\Sigma, \mathbf{Fields}]$? I see this is about single noninteracting relativistic quantum particles.
So consider two field species $\mathbf{Fields} = \mathbf{Fields}_1 \times \mathbf{Fields}_2$ where the first is “field of gravity” and the second is “scalar field”. Then one may pick a fixed gravitational field configuration and ask for the remaining stabilizer subgroup of $Diff(\Sigma)$ fixing that. That fixed gravitational field configuration would be called a “background metric” and that stabilizer subgroup its “isometry group”. For $\Sigma = \mathbb{R}^{d-1,1}$ and that background metric the Minkowski metric, the stabilizer subgroup is the Poincaré group. It now acts on the remaining scalar fields.
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