Duality – The World Has No Dimensionality At All
By Sascha Vongehr | September 28th 2010 09:51 PM | 15 comments | Print | E-mail | Track Comments

Dr. Sascha Vongehr [风洒沙] studied phil/math/chem/phys in Germany, obtained a BSc in theoretical physics (electro-mag) & MSc (stringtheory)...

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Duality is one of the most important insights in physics and philosophy of physics. What does duality mean? It means that there are dual descriptions of one and the same observed physics. Duality implies that you cannot possibly decide whether the one description T or the other, to T dual description T is right or wrong, because both descriptions lead to the exact same observations for the observers who live in a universe that can be described by the theories T and T’. Both are equally right or wrong, as they are dual descriptions. This is not about interpretation or the often abused Ocham’s razor, since dual descriptions are transformable into each other. They are the same although they look incompatible.

Why is duality relevant? In philosophy of physics, they are still publishing about the “hole argument” in general relativity, unaware of what is going on at the cutting edge. In blogs and pop-science, people ask about how many dimensions the universe really has: two, four, 10? Grasping duality elevates you above these discussions.

Duality had its heyday in string theory, where there are several types of dualities. Wise people may have known the gist of duality thousands of years ago, but now we have mathematical models that show the validity and consistency of such. One such duality is for example the Maldacena (or “AdS-CFT”) duality, proposed by Juan Maldacena.

I met Juan before he became really famous, and I asked him a really stupid question about this duality; I hope he forgot all about it. Maldacena’s duality is about a universe that is described by general relativity inside but by a non-relativistic theory on the boundary around the universe on “the outside”:

You either describe the stuff inside with general relativity, or you can equally well describe just corresponding “degrees of freedom” on the surface around the universe. The observer does not know whether she lives inside the universe or whether she is made up from stuff on a surface “around the universe”. The universe inside may not "exist" in any sense more than just as implied by what is happening on the boundary surface.

I wrote the previous sentence to make you for a moment think about that possibility, as if it makes sense, which it does not, because it comes better: there is no difference between the descriptions. Being on the surface or being inside, these are the exact same situation! I will describe this in a post on the black-hole instantiation of this kind of “holographic duality”.

Note well: a sphere’s surface is only two dimensional while the interior has three dimensions. The boundary around something has one dimension less than whatever is inside!

So let us put it yet again in another, more pointed way: There is a surface of N dimensions without general relativity, the stuff of which obeys some rules that allow for evolution and all that, only to end up with conscious systems that argue in all earnest that the world fundamentally must have N+1 dimensions and that anybody who does not pledge full allegiance to general relativity as the fundamental last answer is a total quack.

This is basically the state today, and the sad part is: we already know this for quite a number of years. As always, progress goes on funeral by funeral, established philosophers are mostly windbags, and pop-science sells via time-travel and worm-holes, but fails to communicate insight.

The holographic description is two dimensional, as it has only two space dimensions. That does not mean that the world is actually two dimensional. The world does not have two, four or 10 dimensions. The world does not have any specific dimensionality at all. Our descriptions have.
This holds also for the description that has evolved as being the model of our world used by our brains. It looks three dimensional because this is the most useful way to model it in order to eat, have children, and so on. That does not mean that the world is three dimensional.

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I see where the "holographic" prefix comes from; you are embedding a three dimensional image on a two dimensional surface.

If you have dual theories of dimension N and N+1, how many measurements do you need to describe the state of the system? N or N+1?

What you are saying sounds in part like stating a theory can be formulated in different ways. Classical mechanics has a Newtonian or Hamiltonian formulation, for example.

Your duality reminds me of this: The model that the earth orbits the sun was a rival to an epicycle earth-centered theory. The epicycle theory in a sense provided a coordinate system with many parameters that could be related to the heliocentric coordinates. Your duality must mean more then this because the epicycle coordinates could only assume values from a two dimensional subspace of the epicycle space.

"Your duality reminds me of this: The model that the earth orbits the sun was a rival to an epicycle earth-centered theory."

Such rivals are mostly well known to be mutually incompatible. They lead to different predictions. Qualitatively new is how one now has theories that are known to be completely equivalent and yet do not agree on such naively fundamental aspects like the number of spatial dimensions.

Transformations are always possible. I can transform everything inside-out and then claim, look, the described observer is now inside-out like my socks but she does not realize it. The new dualities now discussed are equally exact (as transformations) but much more equivalent also in naturalness. The physics on the boundary surface is equally natural. Nothing says, hey actually we should not describe it here and in 2D but in the inside and in 3D. There are at most advantages of a certain description relative to a given task.

On the other hand, yes the main insight that fundamental dualities are not impossible could and should have been an old one since long before the advent of epicycles.
I didn't write my comment very clearly. You are of course correct that the Copernican and epicycle theories lead to different predictions because of the way the epicycle coordinates change with time. However, the epicycle coordinates could be made to change in such a way that the motion of the Copernican theory is duplicated.

Anyway, I am still a bit confused about this duality. Does the higher dimensional theory have a constraint that reduces it to the lower dimensional one? How many initial conditions are needed to specify a state?

About the degrees of freedom (DoF). Sure, this is an important point, because in a classical theory, a dimension more also implies more freedom. In fact, the holographic idea was discovered by thinking about the information bound in black holes. The DoF of a black hole increase proportional to the area of the event horizon (as far as we know - there could also be more restriction still). This was a hint for that a 2D rather than 3D description might be sufficient.

Arguing with DoF you may defend some sort of principle of parsimony: the number of dimensions is "truly true" if it suffices to give you all the DoF. However, are the internal DoF due to dimensions or not? x,y,z, quark colors, electric charge, weak charge, maybe Higgs on top, and there you have 10D (plus time), so there is super string theory if you count like that. These problems add to the fact that as long as there is quantum mechanical entanglement, the DoF are always less than they could be classically (also the theory described on the outside boundary is still quantum mechanical), so maybe even less than 2D (fractal dimension) is still enough.
No Sascha. Internal degrees of freedom don't matter. The absolute number of degrees of freedom is totally irrelevant for the argument what constitutes the true number of dimensions. This number is dictated by the * scaling * behavior of the degrees of freedom, not by the number of degrees of freedom. Internal degrees of freedom (and degrees of freedom associated with compact dimensions) disappear from the scaling behavior.

We agree on the scaling, that is what I meant by writing "The DoF of a black hole increase proportional to the area". So, if you are still far away from the resolution that could possibly resolve a truly fundamental scale (say the cells of a Game of Life world), the scaling is an important part of what one argues dimensions with. However, if plainly due to entanglement the degrees of freedom grow much slower than expected from the geometry one experiences, then what is the "true" dimension? It is the dimension of the model, one has maybe two, the other three, both models describe important aspects of the phenomenal world, which as such does not necessarily have any particular one.
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Meaningless sidebar:

This duality of which Sascha speaks has nothing to do with the dualities that mathematicians associate with a vector (or tensor) space.

Nor is it directly related to the duality between the Heisenberg and Schrödinger methodologies of quantum mechanics, albeit, it has some common ground in that it will be a warning to not mistake a map for the territory. I don’t know who famously made such an error — somebody somewhere … or its one of those urban myths.
Mathematical embeddings differing in dimensionality can yield the same testable implications. That sounds quite plausible, but does it do justice to Sascha's narrative? I bet a full appreciation of his point builds on Poincare, topology, and the rich recent mathematics underlying string theory.

I tell my brightest students that the most important mathematical abstraction to get comfortable with is a vector space. I gather that powerful new properties of certain physically useful vector spaces are still being discovered.

"Don't mistake the map for the territory" is always good advice!

Sascha, I like your English, even "biting the grass"!

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As Sascha says, “The observer does not know whether she lives inside the universe or whether she is made up from stuff on a surface “around the universe”. The universe inside may not be there at all, if you like, at least not in any sense more than just as implied by what is happening on the boundary surface.”

Maybe its one of those complementarity things, as in how a description of matter in-falling to a black hole is going to be very different depending on whether one chooses a viewpoint far away from the horizon where space is “almost” flat, or one chooses to give a description from a frame comoving with the in-falling matter.

For cosmology, here is what I have in mind. For an earth-bound observer, the edge of the observable universe is a holographic boundary, somewhat akin to the main illustration for this blog. Everything on that boundary is delocalized and yet it contains the same amount of information as what resides “inside”.

For an observer asymptotically close to this edge and peering all the way back to earth, the earth itself is right by the outer limits of his or her observable universe.

What does the universe look like at its outer limits? From the perspective of someone by those limits, it looks as we see everything right here and now on Earth!

One’s choice of section of the boundary to make this observation does not matter because of the delocalization of everything in the hologram. The older way of phrasing this was to say that there is no preferred location. It is called the Copernican Principle, yet today it is called holography … and it shows up in many places, including the non-local aspects of entanglement.

OK. My bad. I think I’ll take a catnap.
"Maybe its one of those complementarity things ... Copernican Principle"

You are talking about incompatible observers kept apart by horizons. Duality is not that. I was only describing one observer. There is only one observer but possibly several mutually dual theories T ~ T' ~ T''.
I am trying to grasp the insight here;

"dual descriptions are transformable into each other. They are the same although they look incompatible."

it really sounds like observer dependant views, is this really the reason behind duality?

I liked the comment the Copernician Principle and entanglement... never thought of that.

"sounds like observer dependant views ... liked the comment the Copernician Principle ..."

It is not the same and that is why I do not really "like" that "Copernican" comment (just using big names again). The observer is only one, a single state of consciousness in some sort of situation that she is conscious off and sees in some way. The (self-) description of that observer can be one where she is 3D inside or one where she is described as some 2D system on a boundary surface. The observer does exactly not change, but the theory (language) used is totally different yet also equivalent.

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All I am doing is working from the non-local aspects of holography … which are no stranger than the fact that when Alice enters a horizon, her loveliness is spread out and distributed throughout the entire area of the horizon. Her dual description is plunging to the “singularity”.

One way of understanding how there can be a faithful description T’ with one less dimension than T is in the way that phase is not an observable. Yes, it has to do with gauge theories and how I can land on my feet when in a range of height differences. There are subtleties of course, a la Yakir Aharanov et. al, and yes, fields can be defined by how a vector and other objects are changed when taken (parallel transported) around a loop.

No catnip involved here Sir.

A constructive thing to do is to give examples of these mappings from T to T’.

What is needed is a simple and intelligible rendition of the essential mathematics. Can it be done here? I never implied that it was a simple coordinate transformation that preserves the number of dimensions. I am not your straw man.

I wish I had quoted Susskind’s principle of complementarity for black holes so that you can claim that Susskind is on drugs (cookies actually). The better thing to do is for us to put things in our own words and perfect our writing skills.
Below is a repost of a comment I made earlier on a semi-retired blog.

The one thing that people remember from the Bell inequality violation with states like |ud>-|du>
is that it is telling us something about how quantum mechanics is non-local in ways that classical physics is not. Exactly how it pulls this off is subject to debate. Be that as it may, a look at Erik Verlinde’s website shows that he feels that this non-locality is also at the heart of Holography and essential to understanding how gravity and space can be emergent from entropic gradients. Below are a few quotes from his site: (my apologies for using quotes ...)

“[T]he microscopic degrees of freedom on the holographic screens should not be seen as being associated with local degrees of freedom in actual space. They are very non local states. This is what holography tells us. In fact, they can also not be only related to the part of space contained in the screen, because this would mean we can count micro states independently for every part of space, and in this way we would violate the holographic
principle. There is non locality in the microstates.”

“There is action at a distance hidden in gravity, even relativistically. The ADM and Komar definitions of mass make this non-local aspect of gravity very clear. This  non local aspect of gravity is precisely what the holographic principle is about.”

“The logic here is: thermodynamics + holographic principle -> gravity.

... gravity follows in a very simply fashion from holography [the derivation] the other way [from gravity to holography] is much more complicated.”

[end quote]

Keep it Simple.

purr

"All I am doing is ..."