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started field (physics).
So far there is an Idea-section, a general definition with some remarks, and the beginning of a list of examples, which after the first spelled out (gravity) becomes just a list of keywords for the moment.
More later.
In which entry one is supposed to cover the older range of the notion which includes all fields of quantities in geometrical picture, for example in differential geometry, like vector fields and tensor fields? (The entry field says for fields in differential geometry look at vector field – this restricts the general concept to a subfield and a subexample and it suggests unduly that the concept in physics is not a specialization and outgrow of the same concept).
They are the historical prototype which also specialized to electromagnetic etc. fields which then became prototypes to fields in QFT (a mainstream physicist still thinks of a field in QFT as some field of operators, often in distribution (generalized function) sense). Personally I dislike when Lawvere says quantity for what is more traditionally called and more suggestive field of quantities (I put it in brackets just because I like the terminology I used as high schooler, undergraduate and so on), partly because in past it often confused me (I tend to forget what other specific people assign to common terms). Though of course, he is also right, as the intensive quantities are space-dependent, hence fields. It would if we would be here eventually both systematic and modern and also linguistically and historically comprehensive (I hope my contribution here is not taken as a critics), and, especially, having a big picture (across fields of science) in concept/terminology.
I also point out that there is an entry quantum field (quite an unambiguous topic) with zero content.
In which entry one is supposed to cover the older range of the notion which includes all fields of quantities in geometrical picture, for example in differential geometry, like vector fields and tensor fields?
The entry field (physics) will cover all kinds of physical fields. Brief remarks on tensor fields are already there, I’ll further expand. Of course there is also tensor field.
But notice that few physical fields are really tensor field. The field strength of the electromagnetic field is a $(0,2)$-tensor, but the whole field is not a tensor field, but a 2-cocycle in differential cohomology. Similarly for Yang-Mills fields.
The field of gravity looks like it is a rank $(2,0)$-tensor, but there is a constraint on it (non-degeneracy) which makes fields of gravity be not the same as all $(2,0)$-tensors even disregarding diffeomorphis invariace. The field of gravity is instead a vielbein field, which is not a tensor field.
The more modern perspective is to declare that fields are sections of some bundle (not necessarily a tensor product of tangent and cotangent bundles). The “field bundle”. This solves some of the above problems, but is still not satisfactory. I have collected some remarks on this in
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(a mainstream physicist still thinks of a field in QFT as some field of operators
I’ll come to quantum fields. These are maybe more properly discussed under quantum observable. I’ll get to that. But a little later.
I also point out that there is an entry quantum field (quite an unambiguous topic) with zero content.
So I have removed that entry and made quantum field redirect to field (physics).
Oh it sseems that my post up there is not clear or misunderstood. Urs, in 1 I meant that a vector field does not need to be one related to physics, the same with tensors, spinors, fields of distributions, fields of currents, fields of operators, of Verma modules. These are all similar to physics, but it can be fields of any kind of quantities, nothing to do with physics. So I was aksing where to put this wider perspective and how.
What is this wider concept other than a function (or generalised function) on a manifold (or other sort of space) $M$? (Possibly taken to be a section of a bundle over $M$, maybe not; of course a map $M \to A$ is the same as a section of the trivial bundle $M \times A \to M$.) I mean, what is there to say about these that would not be said at function or manifold or bundle or section (or quite possibly all of them)?
By the way, Urs, you created physical field some time ago (but put essentially nothing there). I would consider that a better name (similarly with physical theory, although you did make that a redirect).
We now have quantum field redirecting to both quantum field theory and field (physics); the former redirect is actually active. I'm not sure that it shouldn't simply be its own page. (In fact, I'm now inclined to move the nearly empty stub physical field to quantum field and then move field (physics) to physical field!)
Thanks, Toby, for noticing this. I have tried to clean up a bit now, making everything that refers to physical fields now point to field (physics). I am agostic about whether to rename that entry to physical field. Please do if you think that’s better.
That gave me occasion to also touch some other entries related to this, many of which deserve much more attention. For instance at quantum field theory I made the very first sentence say something like “quantum field theory is the quantum mechanics of physical fields”, which now no longer sends the reader in circles and also avoids the conflation with quantum field (which however still redirects to field (physics) right now, I’ll try to write something about that soon and then maybe split it off).
Strange, quantum field used to be a separate entry, without content but a separate entry (as it should be I think). In cobordism approach and some other approaches also favoured in $n$Lab quantum field theory is explained as a theory/functor in equivalent but different language from the old familiar language based on familiar quantum fields over a FIXED manifold. I think the entry on quantum fields could wisely have emphasis on things like the fact that quantum fields have some problem with being well defined and measured at a point, distribution aspects etc. freedoming the space in quantum field theory for theory/functor aspects.
What is this wider concept other than a function (or generalised function) on a manifold (or other sort of space) M? (Possibly taken to be a section of a bundle
I am roughly happy with this definition. Though there are more general things like representation spaces, Verma modules etc., like in operator algebras one quite often talks on “continuous field of Hilbert spaces” (cf. http://en.wikipedia.org/wiki/Hilbert_C*-module and http://www.math.harvard.edu/~lurie/261ynotes/lecture18.pdf).
Zoran,
the entry field (physics) already does or will do everything you demand here. Except that there is currently no discussion of quantum field in the sense of “operator-valued distribution” or similar. As I said above, I think the latter really a topic in quantum observables. But I’ll include some comments on this, too. As soon as we have genuine material, we’ll split off quantum field as a separate entry, of course.
I have now considerable expanded the discussion at
I have been further editing field (physics). More tomorrow.
I have now added discussion of a bunch of examples
Now I am finally (this was really what I wanted to do all along…) working on the last Examples-section
where the interesting stuff goes.
I have now filled in more material in the remaining Examples-sections. One could write much more, but for the moment I am running a bit out of steam and time. I hope to be expanding some of this further in the next days, though.
I also expanded the discussion of general properties, this is now in the sections
There will be many typos left for the moment. Will proof-read when I have the energy.
I will now also be transporting this material into the section Fields at geometry of physics.
added a further example: Supergeometric B-field
I have added more details to the section
I have added a paragraph to field (physics) on the discussion of why the standard notion of field bundle is problematic.
Remember the story was that I had pointed out that for gauge theories there cannot in fact be a description by field bundles that is also local and that the only reason that this is never really noticed in the literature is that most of it restricts to physics on Minkowski spacetime and or restricts to perturbation theory.
Now Igor Khavkine noticed (and kindly pointed me to) a recent article that does use tools accurate enough to actually see the problem:
The paragraph that I have added to the entry is this one:
This failure of locality is often not recognized in the literature, since many if not most descriptions of physics restrict to trivial spacetime topology and/or restrict to perturbation theory only. A formulation accurate and encompassing enough to see this issue is AQFT on curved spacetimes. A reference that explicitly runs into this non-locality issue of the field bundle in gauge theory in this context is (Benini-Dappiaggi-Schenkel 13): the authors define a functor from spacetimes equipped with a $G$-principal bundle that assigns the algebras of observables of the corresponding Yang-Mills fields built from the field bundle of connections on the given principal bundles; and they observe that the result fails to be a local net in that the inclusion of observables of a smaller spacetime into a larger patch may fail the isotony axiom (BDS, remark 5.6). The authors then try to circumvent this by restricting to trivial instanton sectors.
But notice that instanton sectors are non-negligible phenomena. …
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