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For discussion at geometry of physics I need a way to point to the concept of “locality” in QFT, so I gave it a small entry: local quantum field theory.
Here, too, one can consider the analog of extended quantum field theory and ask for extended Lagrangians that are not only local as top-degree forms on spacetime/worldvolume, but which also have an “extended” to lower dimensional subspaces.
Is there a noun missing after “extended”, or should it be “extension”?
Local Lagrangians are expected to yield local quantum field theories under quantization.
Is there a conceptual reason for this expectation?
Yes, it should have been “extension”, have fixed that now. Thanks.
And yes, a local Lagrangian is one that depends only on local data. An action functional induced by a local Lagrangian is manifestly built by incrementally adding up local contributions and satisfies the expected additivity/multiplicativity under decomposition of its domain. The same is then expected for the quantization of this local functional.
This is made rather precise by formulating quantum and pre-quantum field theory both as functors on n-categories of cobordisms, as in the last section of dcct. The locality is all encoded by the n-functoriality on cobordisms.
pre-quantum field theory … as functors on n-categories of cobordisms
So there’s some relation between that and
it “depends only on finitely many derivatives of the fields” at each point, which formally means that it is a horizontal form on the jet bundle of the field bundle?
Plausible, I guess, but I wouldn’t be able to establish the connection.
You are right, I should try to give you a more formal statement. For the moment allow me to give just the following simple argument:
it is clear that if you have a functional on a space of paths which is the integral over a 1-form whose value at any point of the path explicitly depends on what the path is like a finite distance away from this point, then this functional cannot satisfy the decomposition property that its value on paths of some length is the sum of its values of the two halfs of the path.
Now if the 1-form here depends at each point on non-finitely many derivatives, that’s generically the same as depending on the value of the path a finite distance away, by Taylor expansion.
At the other extreme, if the 1-form here depends on no derivative, then the functional manifestly satisfies that decomposition property, in that case it is just the transgression of the 1-form to path space.
Then one needs to convince oneself that dependence of finitely many derivatives does not destroy this property, only the limit of infinitely many derivatives does.
Is there a parallel with how $k$-jet $\infty$-toposes for a cohesive $\infty$-topos remain cohesive?
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