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# Conformal Field Theory - Science topic

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Conformal mapping is a powerful tool to solve 2D boundary value problems that would would otherwise not be able to be solved analytically
Have these ideas been extended to 3D applications? Can someone point me to a suitable reference?
A conformal map is a function in mathematics that preserves angles but not necessarily lengths locally. In more technical terms, let and be open subsets of At a location, a function is said to be conformal (or angle-preserving) if it retains angles between directed curves while also retaining orientation.
the conformal map, A transformation of one graph into another in which the angle of intersection of any two lines or curves remains constant. The most prominent example is the Mercator map, which is a two-dimensional depiction of the earth's surface that includes compass directions.
In contrast, if the Cauchy-Riemann equations are met and the derivative at the point is not zero, one may demonstrate that there is an a > 0 and such that the above is true. As a result, the map retains angles. As a result, a map is a conformal map if and only if it is a one-to-one, onto analytic function of D to D.
Kind Regards
Qamar Ul Islam
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Conformal Mapping is used to solve 2D electrostatic phenomena, but what are the steps to follow, and how one get to know about which mapping function should use to map?
Is there any web reference or Book?
I want to solve Poisson's Equation in 2D geometry, please provide some good reference or Book for this.
The real or the imaginary part of any complex function is harmonic. The mapping is to transform a question in complex plane domain A (not easy to solve) to B (easy to solve or even known). Then you transform the solution back to your original domain.
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It turns out that, in systems exhibiting the quantum Hall effect, chiral excitations belonging to different edge channels have different propagation speeds, as people, including myself, have discussed in numerous papers. There really usually is a non-trivial "spectrum" of different propagation speeds. This implies that a theory describing such a system is NOT conformal in the strict sense of the word!
The problem is that many incompressible Hall fluids have several different "c's", depending on which chiral edge channel you consider; i.e., the propagation speeds of different chiral edge modes are usually different from each other!
(Quite apart from that, there tends to be a small amount of dispersion in the propagation of edge modes - but you may want to neglect that.)
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I would like to design a circular arc array antenna in CST Mwstdio, i was tried with far field array (by mentioning positions of antennas in array) which is in post processing tab and i also tried without using far field array (by creating each antenna in array). But both results are not matched. Those are supposed to be equal.
what is the best way to design circular arc array in CST with accurate results?
I agree with Ali Shahi, may be due to mutual coupling between antenna elements in the array. So, you expect to keep distance between elements at least Lamda/2 of you resonant frequency.
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In the paper "Conformal field theories associated to regular chiral vertex operator algebras I: theories over the projective line" by Kiyokazu Nagatomo and Akihiro Tsuchiya, there was the announcement of a forthcoming paper about the higher genus case. Has this second paper been published?
It is probably best to ask the authors but at least Google Scholar does not seem to find the reference you ask for among the citations for the first part:
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In an FRW cosmology expressed in conformal coordinates the classical 'mechanical' action is as shown in the first image. The Dirac equation in conformal FRW spacetime is shown in the second image. In both cases the effect of the scale factor is to make the mass appear as if dynamic. (The 'spin connection' term results in a pure phase adjustment to the wavefunction and is of no import. )
In this coordinate system the EM fields are completely unaffected by the expansion; the EM action for the fields and the field-current interaction can be written as if in Minkowski spacetime.
Of course there is no new physics in switching from the traditional RW coordinate system to a conformal system. (In case you are concerned, though in the conformal system there is no Cosmological red-shift of radiation, there is instead a progressive blue shift of matter. Observationally these are indistinguishable. To make the same point in RW coordinates is messier, since both the matter and EM actions depend on the scale-factor.)
Mass renormalization offsets an infinite mass due to EM self-energy with an appropriately chosen mechanical mass so as to leave a finite positive 'observed' mass. Presumably the latter is the subject of the mechanical action, and scales accordingly with cosmological expansion.
But the electromagnetic part is unaffected by expansion. So it seems that renormalization in conformal FRW spacetime must cancel a constant infinity, and add a finite part that scales with expansion
Note this would be a problem even without renormalization. I.E. even if the electromagnetic contribution to the mass was finite and non-zero, because the total mass would not then scale proportionally with expansion.
I presume that when the Higgs mechanism is adopted it is understood that the mechanical part of the mass is set to zero. Since the EM self-action will still be present the question above changes to one about the relative scaling with expansion of the Higgs mass and the electromagnetic self-energy of a charge. Specifically: the two contributions to the mass will not scale proportionally unless the Higgs field is conformally invariant.
Are the above inferences correct?
Do those that think about renormalization in curved spacetime have a work-around?
... well, there is, but I think that this discomfort is not something specific to such a problem, rather it is a type of unease that more generally appears when we work with the methods of QFT in curved space-times.
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It seems everything propagates in waves (eg.,EM field propagation). Can one suggest other theoretical modes of any other physical field propagation?
I think that wave propagation is at the foundation of the material world, which is ruled by the very hard constraint of stability: If it does not oscillate, then it would go to infinity, impossible for material 'things'.
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We know that a free scalar field on a diff-invariant 1+1 dimensional background (i.e. bosonic string theory on the worldsheet) contributes to the central charge of the Virasoro algebra with a constant term.
Is there any examples of a 1+1d QFT that has instead a central charge contribution that diverges when the regularization cut off is removed in the canonical quantization approach?
Such an energy momentum tensor would presumably lead to an infinite 2-pointfunction i.e. the smeared field (testfunctiond on the lightray) would not represent operators (i.e. not even unbounded). Please check (the computation is very simple) and send your answer back
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