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added to N=4 D=4 super Yang-Mills theory a section
Properties – Closed expressions for physical observables
with a brief mentioning of integrability of the dilatation operator, MHV amplitudes and planar limit scattering amplitudes.
For some reason, I was having a look at Arkani-Hamed’s page. In the last of those popular talks, he alludes to the work in Scattering Amplitudes and the Positive Grassmannian where surprising areas of maths play a part, e.g., cluster algebras.
But if
…because understanding observables in QCD/Yang-Mills theory in general is difficult, going to special points in the space of all such theories – such as the point of N=4, D=4 SYM – may be hoped to yield a tractable approximation,
this sounds like they’re benefiting from a very particular situation, which won’t allow the tools to be suitably ’abstract general’.
And yet there’s the idea that they’re doing something more directly, avoiding
…the introduction of a large amount of unphysical redundancy in our description of physics,
including gauge redundancies.
It is therefore surprising to see that even here [in QFT], by committing so strongly to particular, gauge-redundant descriptions of the physics, the usual formalism is completely blind to astonishingly simple and beautiful properties of the gauge-invariant physical observables of the theory.
and
…the Grassmannian picture makes no mention of locality or unitarity, and does not commit to any gauge-redundant description of the physics, allowing it to manifest all the symmetries of the theory.
That sounds like they’re after something intrinsic. Is it only the intrinsic of the very special?
Is it only the intrinsic of the very special?
Yes.
The discussion about “gauge redundancies” just refers to the fact that they consider gauge invariant observables (the diagrams of on-shell amplitudes, which is what all the colorful diagrams in the article depict). This is being emphasized since in contrast a single Feynman diagram in gauge theory is not a gauge invariant quantity.
So a way to summarize what is happening here is that in traditional Feynman perturbation theory one expresses a single gauge invariant physical observable (some scattering amplitude) as a large sum over terms each of which is not gauge invariant and generally has no invariant physical meaning (hence “virtual particles”). Only the end result will have all the right properties.
The point here is that in some special situations this desired end result can be obtained by different means than Feynman diagram technology, more directly. It’s an improved algorithm for computation of observables that applies in special situations.
And one axiom of this special situation is maximal supersymmetry of the theory. You can see this in the article appear right at the beginning, below (2.3). Another characteristic of the special situation considered here is the “planar limit”, which appears first on the bottom of page 10. And then of course there is the restriction to MHV amplitudes.
There are some arguments that some of the insights gained in this special case can be turned into insights that apply also in less specialized situations, in particular in realistic field theories that are not supersymmetric and far from their planar limit. I am not an expert on this, but my impression is that while there are such arguments, the influx to realistic theories is minor at the moment. Elsewhere one sees particle physicists complain about this.
On the other hand, some “on-shell methods” in perturbation theory in recent years, which also originiated in string-theory-motivated setups (such as super Yang-Mills theory) apparently had dramatic impact on LHC physics. See at string theory results applied elsewhere the link to Matthew Strassler’s From string theory to the large hadron collider.
Thanks! Following up on things, I see there’s interest in applying string theoretic calculations to condensed matter physics.
Nice soundbite:
“Maybe string theory is not a unique theory of reality, but something deeper — a set of mathematical principles that can be used to relate all physical theories,” says Green. “Maybe string theory is the new calculus.”
You could also say: string theory is just quantum field theory with second quantization taken seriously.
Ordinary perturbative QFT is
in string theory one considers
and M-theory is at least a faint hint of something related to
In this sense this is just about continuing to do what standard QFT already does. And it turns out that the space of further second quantized (effective) QFTs obtained this way is large and interesting. It contains loads of theories that are unrealistic, but since it populates the space of theories much more densely, it gives new connections between old theories that are realistic. One sees how everything in QFT hangs together.
One can clearly see this for instance in the annual Strings meetings of the last years. To a large extent the talks there are about various QFTs and their relations. These were originally found as second quantized/effective target space theories of strings, but once they have been found this way, these relations can be studied as just QFT.
So you have a “space of theories” (models of physical-theory-in-the sense-of-logic), the points of which stand in various kinds of relation to one another? And there’s a process taking you from other spaces of “$n$ d worldvolume theories” for different $n$ to the original “space of theories”? As you increase $n$, the image grows larger?
Presumably that’s what’s being said at M-theory: dimension reduction being one of the relations between effective QFTs.
What kind of “spaces” are we talking about? Are they really categories?
Yes, I think that’s a good summary!
These “spaces” should eventually be certain higher stacks of course, moduli stacks.
To date there are only baby attempts to say much about what the spaces of all QFTs are.
When string theorists in recent years tried to talk about some subspace of string theories which they called the “landscape of string theory vacua” the result was a huge mess. Nobody had any formal idea of what they were talking about.
There are some hard results on spaces of 2d QFTs. But there also only for the simplest cases. Notably the space of “rational” conformal field theory has been fully captured mathematically (by FFRS formalism). It’s a discrete space of certain algebra objects in certain monoidal categories, in this case. But this is just a tiny subspace of the full answer.
In our book collection Mathematical Foundations of Quantum Field and Perturbative String Theory (schreiber) Soibelman’s contribution is about understanding more of the space of 2d CFTs by identifying it with 2-spectral triples.
Of course what has been fully understood are topological QFTs. Lurie’s theorem says that the spaces of extended TQFTs are equivalently $\infty$-groupoids of fully finite-dimensional objects in symmetric monoidal $(\infty,n)$-categories.
For the issue of second quantization you should beware that the second quantization of a higher dimensional QFT (hence a string theory etc.) is not necessarily a QFT. This is actually unclear. Once it was hoped that it is a string field therory, but the general feeling today is that string field theories capture a lot, but not everything of what the actual ful second quantized theory should be. On the other hand, all work on string field theory to date is just on their action functionals. There is really nothing out there on their quantization.
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