I live in the US, and I had never heard the word "dilatation" (as opposed to "dilation") before. However, it seems it's in use in, for example, the Scottish Qualifications Authority's specification for "Advanced Higher Mathematics", so it's definitely in use in some dialects/texts. As more of a math person rather than a physics person, I was surprised (though not unpleasantly) that there was no mention of the complex numbers in the video. I'm excited to see how this connects to physics as the series continues.

@OnePlusOneSpace

Жыл бұрын

I'd thought dilatation was the more common in the field (and I haven't heard it outside of this context), but actually dilation is the term used in the most famous textbook of CFT. I'm also British, I wonder if that has anything to do with it. The lack of complex numbers was intentional and I was wondering if it would be noticed! It's important for generalising to higher dimensions that the complex structure is not used as a crutch. Thanks for the comment!

@plazmi1

Жыл бұрын

@@OnePlusOneSpace in english definitly most common is dilation, but most languages translates this word based on dilatation (for example polish, german, spanish or french). Complex numbers appear within the topic of conformal geometry quite naturally, because all holomorphic and injective complex automorphisms are exactly conformal automorphisms, and every other geometry have only the trivial conformal mappings, so the only interesting conformal structure is the structure of holomorphic functions. The reason why it is true is crucial for deep understanding of conformal geometry or complex analysis, so i would not dismiss complex numbers while talking about conformality, however its more advanced topic than what you discussed in the video :)

@OnePlusOneSpace

Жыл бұрын

For sure conformal symmetry in two dimensions is a very rich and beautiful topic, and I hope to get onto this eventually!

@chriscopeman8820

Жыл бұрын

Are complex number conformal maps the same as your conformal transformations?

@diribigal

Жыл бұрын

@@chriscopeman8820 A conformal transformation of the plane is indeed the same thing as a "conformal map" in a complex analysis class.

Please continue introducing us to field theory! QFT is so hard to understand - please consider covering as well

What a great surprise, thank you so much for this wonderful video, I wanted something more about conformal symmetry on KZread !

Thank you, thank you for this. You explained this complex and important concept quite well!

Amazing content! I don't think there are many people tackling this type of stuff, I really hope ur channel grows faster haha

Thank you very much for this great video! It has cleared up several questions that I have had about conformal field theory. Most resources I have found on this subject have been hard to understand. This one is a lot easier.

Thank you for this video. It might be interesting for some to mention that while it is true that the described transformations are all transformations of the plane (2d flat space) and of the 2-sphere, there is a very interesting detour hiding here. As explained in the video, a conformal map is one which preserves angles. This is a local condition, for to find out what happens to any angle, you can (and have to) zoom in as far as you like into the angle. On the other hand, most (or, depending who you ask, all) maps we consider can be seen as being very nicely approximated by linear maps when you zoom in far enough. This is a generalisation of the derivative for functions \R -> \R, which also says that if you zoom in close enough, the function is roughly a line. For conformal transformations, this linear approximation then also has to preserve angles. In all dimensions, we understand well which linear transformations are conformal, i.e. preserve angles. In 2 dimensions, these turn out to be most maps (a bit more precisely: a linear transformation in 2 dimensions is given by 4 parameters, and a general conformal linear transformation is given by 2 parameters). However, in 3 and more dimensions dimensions, a smaller and smaller „portion“ of linear transformations is conformal (again more precisely: in 3 dimensions, a general linear transformation is given by 9 parameters, but a conformal linear transformation is given by only 4 parameters, and in 4 dimensions, the figures are 16 and 7 respectively - the number of parameters for conformal linear transformations is roughly half that for general linear transformations, but what counts is the absolute difference of the number of parameters). So in high dimensions, we might expect that requiring that the function should behave like a conformal linear map is a very strong restriction. This is indeed the case. Liouville‘s theorem states that a („nice“) conformal map defined on some subset of \R^n for n > 2 must in fact come from linear orthogonal transformations, translations, and „inversion“ (which is just one special map). But what about n = 2? Maybe it‘s not so bad in 2 dimensions? In fact, it is not. While conformal linear maps in 2 dimensions are much more restricted than general linear maps, there is a nice way to restate that restriction. It turns out (and this is in fact very easy to see when you look into the formulas) that you can view these linear approximations as complex numbers - so conformal maps defined on subsets of the plane may be viewed as maps from complex numbers to complex numbers which are locally approximated by complex numbers. These maps are called „complex differentiable“ maps. There’s quite a lot of them, and the study of them, complex analysis, is rich and beautiful. But how come then that there still only are few conformal transformations of the plane or the sphere? This is because while there are a lot of complex differentiable maps, their behaviour is very restricted in a lot of ways. For instance, a conformal transformation of the plane must be a complex differentiable map which is in addition injective (one-to-one). One can show that there are only few injective complex differentiable maps. That way, one recovers a version of the above Liouville theorem for dimension 2 (and although this requires one to develop some chunk of complex analysis, I‘d argue it‘s simpler than the Liouville theorem for high dimensions).

4:50 I suppose that Cartan-Dieudonné may be worth mentioning, as translations, rotations, and dilations may be represented as compositions of two reflections, hence one would prove insufficient.

Very good video!!! I would like to put forward the idea that the sphere exists as in Huygens’ Principle of 1670 that says, “Every point on a light wave front has the potential for a new spherical 4πr² light wave". We can think of the point as a photon ∆E=hf. The spherical surface forms a boundary condition or manifold with light photon energy continuously transforming potential energy PE into kinetic Eₖ=½mv² energy of matter in the form of electrons. This form our ever-changing world with the movement of positive and negative charges. Because of the spherical geometry, we have to square the radius r² as in Ψ², t², e² and c².

One of the best best video I have seen in #some3

Liked, subbed, and notifications set. Looking forward to part 2!

Absolutely incredible video ❤

Great stuff. It'd be nice to have a visualisation of what the SCT looks like on the plane, since the dual animations of the other 3 are really informative. I'm totally new to this topic, but intuitively it seems like there are other conformal transformations on the sphere that aren't captured here. For example, rotating the plane rotates the sphere about the vertical axis, but what about rotating the sphere about other axes? Is that conformal, and what does that look like on the plane? Can you construct that transformation from the 4 core ones you talked about?

@OnePlusOneSpace

Жыл бұрын

Thanks for the feedback, and great questions! Your intuition is correct, a rotation about any other axes also gives a conformal transformation. The reason I didn't show those (and the special conformal transformation) on the plane is that they don't fix the north pole of the sphere, and so on the plane the transformations could potentially move points 'to infinity', which breaks the animation. Nevertheless, if you're careful and restrict to a set of points which won't get sent to infinity, the transformations can be visualized on the restricted domain, which is still useful. I have been thinking about doing a video which has a catalogue of these transformations on the sphere and plane just to have them all collected in one place. Actually, I have a video where rotations about the other axes is visualized, which perhaps I should have signposted more clearly, in my 'Möbius maps' video, and which leverages the use of complex numbers more. And yes, rotations about the other axes can be constructed from the 4 core ones. Secretly, translations and SCTs are actually made of 2 independent transformations each, due to the two different directions you can translate (/SCT) in. Then heuristically, there are 6 transformations in total, which is also what you get by counting a boost and rotation for each axis. Either set of 6 is a complete set of generators for the conformal transformations, and it's possible to switch from one set to the other.

It is fantastic. Thank you

This is awesome

My dad once used to say Science is simpled for not complexed.❤ I cannot go surfacing without my dad's assertation

very good!

awesomeeee content!

I learned something from this; appreciate it! I'm not sure how far toward understanding Ads/CFT I might go though... This is just the baby step. There are earlier and simpler uses of conformal mapping, that solve useful problems in classical physics. One of the most famous ones arose very early in aerodynamics, where conformal mapping can transform a circle into an airfoil shape, and then the known solutions for a dipole plus a rotation can be mapped by this, to solve the airflow around the wing in the Euler flow approximation. This isn't physical because there is no drag. As a result an extra "trick" must be assumed: the rotation (and hence the lift) is specified so that the rear stagnation line does leave the cusp of the trailing edge. This approximation is about as far as analytic methods can take you in aerodynamics. . . The Navier Stokes equation being a bitch. It cannot get the drag at all, and cannot predict behavior near stall. But it does get the pressures over the wing surface pretty well. This plus Prandtl boundary layer scaling and a large body of empirical knowledge basically was the state of aerodynamics up through the 1930s

If I understand correctly then the special conformal transformation is a sort of translation "centered" on 0? Can it be said that the SCT is linear? And conversely that translations have a corresponding linear operator on some dual space?

@OnePlusOneSpace

Жыл бұрын

Good questions! The answer is quite subtle: it depends on which space is under consideration. On the sphere, it is true that the SCT is translation centred at zero rather than infinity. But then on the plane, the SCT doesn't look much like a translation (try to visualize how the points move). Re linearity, the SCT is neither linear on the sphere (which has no notion of linearity as it is a curved surface) nor on the plane, but it can be realized as a linear map (as can translations) on Minkowski space. But this is a long story, which I developed in my first video!

Gaussian curvature preserves isometries but missed out on dilations. Maybe there is a curvature that preserves all conformal mappings?

The thing that I don't really get is the appearance of spatial translations on the celestial sphere. I understand aberration upon a boost and rotations, but shouldn't the distant stars look the same around me, regardless what coordinates I am on? Is it related to parallax?

@OnePlusOneSpace

Жыл бұрын

The key subtlety here is that the translations of the plane are not the same thing as translations in spacetime, so yeah moving in spacetime doesn't change what you see, but this doesn't correspond to translation on the plane. The corresponding transformation on spacetime would be a mix of a boost in one direction, say x, and rotation about an axis at pi/2 or 90 degrees from the original, say y.

@user-hq3dq3vx5x

Жыл бұрын

​@@OnePlusOneSpace I assumed (don't know why) that we are trying to describe the Poincaré group here. But PSL(2;C) is isomorphic to SO^+(1;3), not to Poincaré. Thanks for the quick reply!

Worth noting that dilatations and rotations are not enough to produce all linear maps, and (in general) must center around the origin to be linear maps.

It is interesting to note that conformal transformations are also called conformal maps or conformal mappings, and trace their history back to map making from ancient times through the late Middle Ages to early Renaissance because astronomers and navigators needed flat maps that preserved the angles when projected from the celestial sphere or from the spherical surface of the earth. By the 5th century B.C., it was widely accepted that the Earth is a sphere. There is a widespread misconception that ancient peoples thought the Earth was flat. This was simply not the case. In the 5th century BCE, the ancient Greeks Empedocles and Anaxagoras offered arguments for the spherical nature of the Earth. In about 240 BCE, Eratostenes of Cyrene was the first known person to calculate the earth's circumference and it's axial tilt. By about 77 CE when the Roman writer Pliny the Elder was writing the first part of his Natural History, the fact that the Earth is a sphere was treated as common knowledge: "We all agree on the earth’s shape. For surely we always speak of the round ball of the Earth" (Pliny, Natural History, II.64). These views continued into the medieval period, since even the changing hours of daylight throughout the year made it evident that the Earth was round. Around 723 CE to 725 CE, the monk Bede explained to his students: "The reason why the same days are of unequal length is the roundness of the Earth, for not without reason is it called ‘‘the orb of the world’’ on the pages of Holy Scripture and of ordinary literature. It is, in fact, a sphere set in the middle of the whole universe. It is not merely circular like a shield [or] spread out like a wheel, but resembles more a ball, being equally round in all directions ..." (Bede, The Reckoning of Time, translated by Faith Wallis (Liverpool University Press, 1999), p. 91). This belief was also reflected in many medieval maps. The idea that the Earth was round was not limited to tracts on science and natural history. Much medieval art also depicted the Earth as a sphere. Many writers also assumed the Earth was a sphere. Dante’s Divine Comedy even discussed how the shape of the world created different time zones, and how different stars were visible in the southern and northern hemispheres. It is often stated incorrectly that in the epoch of Columbus that the earth was thought to be flat. This is simply untrue. It may be that at that time some uneducated people thought the earth was flat, but this was not the prevailing view at the time. It had also been known since ancient times that there can be no isometry (that is, a distance-preserving map) from a spherical region to a planar one. This is one of the reasons why the oldest known maps of the celestial sphere were not on flat surfaces such as paper, but rather on tortoise shells. The ancients also realized that if distances cannot be preserved on planar maps of spherical surfaces, then at least it would be useful to conserve angles. When the angles between curves on the celestial sphere or on the spherical surface of the earth were equal to the corresponding angles on a map, the map was said to be conformal. The stereographic projection is a conformal map projection whose use dates back to antiquity. A stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the projection plane) perpendicular to the diameter through the point. It maps circles on the sphere to circles or lines on the plane. The stereographic projection was likely known in its polar aspect to the ancient Egyptians, though its invention is often credited to Hipparchus, who was the first Greek to use it. It was originally known as the planisphere projection. "Planisphaerium" by Ptolemy is the oldest surviving document that describes it. One of its most important uses was the representation of celestial charts. The term planisphere is still used to refer to such charts. Ship navigators of the late Middle Ages to early Renaissance realized the usefulness of having angle preserving maps for navigation. A navigator on the high seas, orienting himself by the stars, has to go through some rather involved calculations to determine the distance he has just traveled. However, the stars tell him quickly the direction in which he is traveling, and hence the angle which his course is making with lines of latitude and longitude. It is believed that the map created in 1507 by Gualterius Lud used stereographic projection, as did the later maps of Jean Roze (1542) and many others, to preserve angles from the spherical surface of the earth to the plane of flat maps. The famous Mercator world map of 1569 uses a conformal map projection. Directions, angles, and shapes are maintained in the Mercator projection. Any straight line drawn on this projection represents an actual compass bearing. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator but accelerates with increasing latitude to become infinite at the poles. It is also worthwhile to point out that conformal transformations (also called conformal mappings or biholomorphic maps) have a long history of application to solving problems in classical physics in addition to the more modern application to Conformal Field Theory. The idea behind the use of conformal mapping in classical physics is to take the solution to a very simple boundary condition problem and to fold, stretch, or otherwise deform the boundary by a conformal transformation to match the boundary for a more complicated problem of interest. The same transformation that changes the boundary will also deform the field lines and constant potential lines of the simple problem to those of the more complicated problem. As an undergraduate physics major in the late '70s to early '80s, I learned conformal mapping to solve Laplace's equation in two dimensions with given boundary conditions (aka boundary-value problems) applied to a variety of phenomena in classical physics including the steady-state temperature distribution in solids, electric potential or field distribution in electrostatics, and inviscid and irrotational flow (potential flow). As a double major in physics and mathematics, I also learned more about the mathematics of conformal transformations in a course on complex analysis.

@OnePlusOneSpace

Жыл бұрын

Thank you so much for the lovely detailed comment, I actually didn't know the Earth was known to be spherical as far back as the Greeks. It was nice to hear about the history of conformal maps rooted in mapmaking, as well as your own experience with conformal maps in maths and physics :)

The most well known nonconformal transformation, while technically also a shearing, would be uneven scaling, aka stretching/squishing

Shouldn’t rotations and dilation not give all invertible linear maps, on account of not including skew things? So, I guess, uh, it would be like SO(R^2) times R_+ ?

@OnePlusOneSpace

Жыл бұрын

Yes, they don't give all invertible linear maps. And that's correct, they're given by the direct product of SO(2) and R_+. If you like, you can also think of this as the multiplicative complex group of non-zero complex numbers under multiplication.

@drdca8263

Жыл бұрын

@@OnePlusOneSpace Ah! Yes, that’s a better description than the product I said! Not sure why I didn’t think to describe it that way. I should have thought to describe it that way, as, based on my past experiences, I should be familiar with the [Möbius transformations? Is that the right name? I mean the complex differentiable bijections from CP^1 to CP^1.] .

@OnePlusOneSpace

Жыл бұрын

Exactly, these are Möbius transformations, and in fact the group of conformal symmetries of 2d space is precisely the group of Möbius transformations

At 4:28 you say that compositions of translations, rotations and dilatations are the only (smooth) conformal mappings of the plane. This is not true, in 2D the conformal maps are the holomorphic (complex differentiable) functions with nonzero derivative. However, for dimensions greater than 2 a similar statement is correct! (Liouville’s theorem for conformal maps) I guess n>2 is the case we ultimately care about for physics, but unfortunately it’s much harder to visualize because of the 3-sphere required.

2:15 Isn't it proportional to mass?

@OnePlusOneSpace

Жыл бұрын

You're right, weight is proportional to mass. Then mass is in turn proportional to volume, with density as the constant of proportionality

[[ Obligatory geometric algebra comment (conformal geometric algebra)]]

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Anggap aja Allah yang lewat..❤

Upvoted just because you explained how the "physics" and "math" meanings are actually related. More critically, you lost me at the end. "do it upside down"? I don't see what the transformation was in the animation. A couple other things could be clearer.

Just say inversion instead of special conformal transformation.

@OnePlusOneSpace

Жыл бұрын

Special conformal transformations are distinct from inversion, though they are related. A nice way to view SCTs is as translations conjugated by inversion. But SCTs give a continuous transformation, while inversion is a discrete transformation (of order 2).

@bentoomey15

Жыл бұрын

@@OnePlusOneSpace What do you mean by "discrete transformation" here? As a function from the Riemann sphere to itself, inversion is continuous (but reverses angles, so isn't an SCT). One can compose with a reflection (also continuous) about a line through the circle's center (which re-reverses the angles) to obtain an SCT, "inversion-and-reflection" --- perhaps the OP meant this SCT when they said "inversion." If one topologizes the group of all conformal symmetries of the sphere, then the set containing just this transformation is one point (a discrete set). This one point is contained in the one-parameter (continuous) family of SCTs. As a more familiar example of the same kind of relationship, consider "rotations" (a one parameter family of transformations with an infinitesimal generator, just like SCTs) and "the rotation by 180 degrees" (one of those transformations, which is a continuous map, and has order 2).

Hehe Jacobians.

completely unclear what the special transformation does

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Dilation?