The n-Category Café

eric said

Nietzsche must have forgotten that Life itself is a journey without any goal except dying ofcourse.

Well, I think it’s pushing it a bit to deduce that much from a single aphorism. For instance, the following aphorism appears, in The Wanderer and his Shadow, a little before the one John quoted.

End and goal: Not every end is the goal. The end of a melody is not its goal; and yet: if a melody has not reached its end, it has not reached its goal. A parable.

Actually, I’m only writing this post so that I can get my favouritist Nietzsche aphorism in, and this also appears in the same place in The Wanderer and his Shadow.

Pleasure tourists.- They climb the hill like animals, stupid and perspiring: no one has told them there are beautiful views on the way.

I can’t climb a hill without it coming to mind.

Death is the end of life, not the goal. The goal, you get to choose.

Thanks! Fixed.

For b) I guess you’re talking about 2d QFTs?

Constructive quantum field theory is vastly easier in 2 spacetime dimensions than in higher dimensions — for example, scalar fields with polynomial interactions were rigorously constructed by Segal, Glimm and Jaffe in the 1970s. But general nonlinear $\sigma$-models have long seemed out of reach. This has got to be fixed.

What really excites me, though I don’t understand the details yet, are Summers’ remarks on constructing higher-dimensional quantum field theories — starting not from field Lagrangians but from particle interactions!

I wrote #

I know that people have thought about [AQFT nets from continuum limits of lattice models], but no real discussion in the literature seems to exist.

I just found something:

there is

Dirk Schlingemann, Constructive Aspects of Algebraic Euclidean Field Theory.

This builds on

Dirk Schligemann, From Euclidean Field theory to Quantum Field Theory

which defies Euclidean AQFT nets (algebraic statistical field theory, that is) and shows how these may be Wick rotated to Haag-Kastler nets under certain conditions.

In Constructive Aspects of Algebraic Euclidean Field Theory the idea is to construct Euclidean nets from lattice models, in the hope that some of them would then admit a Wick rotation.

That last step is not discussed here, but Schlingemann sets up plenty of machinery to talk about Euclidean lattice models , block spin transformations, continuum limits etc. in the AQFT context, relating them to inductive limits over the relative $C^*$-alegrbas, etc.

I haven’t yet looked at this in detail, but this seems to be the kind of discussion I was hoping would exist somewhere.

I am being told that the following thesis is a considerable achievement for construction of 2-d AQFTs:

Gandalf Lechner, On the Construction of Quantum Field Theories with Factorizing S-Matrices.

A summary is in hep-th/0502184.

This crucially uses wedge algebras (algebras localized in spacelike wedges left or right of a point in 2d Minkowski space) to construct from them the usual double-cone algebras. This makes use of an old insight by Bert Schroer on “polarization free generators”.

I still have to absorb this. But it was indicated to me that this might help also in constructing from a local net the corresponding parallel transport 2-functor. As indicated here.

Cool!

Those at the Minnesota $n$-categories workshop in 2004 may remember that Tony was there. Great to have an artist interested in $n$-categories, just as, as Tony showed, Picasso made a close study of 4-dimensional geometry around the turn of the nineteenth century. You can read about all this in Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought.

Any idea why the blue lines are there?

Any idea why the blue lines are there?

The short answer is: they’re decorative. They’re known as strapwork or girih, and are part of the aesthetic tradition.

Exactly how these kinds of lines appear as markings on a certain set of tiles was figured out by Lu and Steinhardt in Decagonal and quasi-crystalline tilings in medieval Islamic architecture, but there’s a nice summary of their work on Wikipedia.

You can also see those lines on the Darb-i Imam shrine in Isfahan:

Doesn’t it have perfect 5-fold symmetry if you choose the right centre of rotation, apart from the fact that it’s cut off at the edges?

Jamie wrote:

Doesn’t it have perfect 5-fold symmetry if you choose the right centre of rotation […]?

It probably does. Many Penrose tilings formed by inflation have this property, too.

What a 2d pattern can’t have is perfect 5-fold symmetry about one center and also two linearly independent translation symmetries.

(In fact, right now it seems to me that even perfect 5-fold symmetry about one center and one translation symmetry is impossible.)

Doesn’t it have perfect 5-fold symmetry if you choose the right centre of rotation, apart from the fact that it’s cut off at the edges?

I think there are subsets with perfect 5-fold symmetry, but the whole thing is not a subset of any finite pattern with perfect 5-fold symmetry.

If you look at Lu and Steinhardt’s Figure S7A, the Darb-i Imam portal John shows above is constructed by subdividing a figure with four large decagons and three bowties. Each decagon is subdivided in a way that retains its perfect rotational symmetry, but the whole assembly doesn’t have that symmetry (and can’t be extended in a way that does).

Oops, there was a glitch in this image where I omitted some tiles because I wasn’t taking account of a certain degeneracy.

That’s fixed now, and this cluster looks much more “icosahedrally symmetric” to the naked eye.

Of course real quasicrystals in nature will almost never be centred on the single point of perfect icosahedral symmetry like this, but it’s still nice to see what’s theoretically possible.

Eric V. wrote:

This post has got absolutely nothing to do with the workshop…

Yeah, it doesn’t — so I moved it here, where it fits in a bit better. Please try to post comments in vaguely appropriate places.

I’ve heard about Marinov’s claims, but I also heard they’ve been heavily criticized, so I’ll wait until something new happens before getting very interested:

• Richard van Noorden, Heaviest element claim criticised, Chemistry World, May 2, 2008.
• Mitch André Garcia, Addressing Marinov’s Element 122 Claim, ChemistryBlog, April 29, 2008.

It would be cool to see the legendary island of stability — I’ve been waiting for it ever since high school! However, I sort of doubt Marinov has found it.

Beautiful! I see a lot of lines that extend across the whole picture you’ve shown us — a clear sign of long-range order. But I also see some lines that go on for quite a while but then fizzle out. Do some lines go on forever?

(Just an idle question.)

If the plane you project onto passes through a point of the lattice (and is also oriented in a nice symmetrical way, which is taken for granted if you want to produce tilings with edges that belong to a nice symmetrical set of directions) then there are reflection symmetries of the whole setup that require there to be infinite lines passing through the lattice point that lies on the plane.

If you give the plane a generic offset, as in the picture above, then I’m not sure if there are any truly infinite lines. I suspect not, but I don’t know an easy proof of that.

Of course, the actual edges in the lattice that project onto these long lines are really zigzagging wildly in orthogonal dimensions. Unlike the rhombic tilings, where all the possible edge directions have distinct projections, the triangular tilings are degenerate in that respect.

I’ve written an applet now that draws quasiperiodic tilings by projecting triangles from $A_n$.

At the risk of stating the obvious, I wanted to spell out the precise connection between the rhombic tilings based on hypercubic lattices, and the rhombic tilings based on $A_n$.

In the hypercubic lattice $I_{n+1}$, the Voronoi cells are unit hypercubes centred on the lattice points, and the Delaunay cells are unit hypercubes whose vertices are lattice points. Dual to each $m$-face of a Voronoi cell, there is an $(n+1-m)$-face of a Delaunay cell, whose vertices consists of the set of lattice points whose Voronoi cells share our chosen $m$-face. For example, in $I_3$, dual to any $2$-face F of a Voronoi cell there is a Delaunay $1$-face, or edge, joining the two lattice points whose Voronoi cells share the $2$-face F.

The original deBruijn method involved projecting onto a suitable plane P every Voronoi $2$-face in $I_{n+1}$ that was dual to a Delaunay $(n-1)$-face that intersected P. So how can we get exactly the same result using $A_n$, rather than $I_{n+1}$?

If S is the subspace of $\mathbb{R}^{n+1}$ satisfying $x_1 + x_2 + ... + x_{n+1} = 0$, the Voronoi cells of $A_n$ turn out to be the projections onto S of unit hypercubes in $\mathbb{R}^{n+1}$ centred on the $A_n$ lattice points: in other words, projections of those Voronoi cells in $I_{n+1}$ whose centres lie in $A_n$. Each $A_n$ Voronoi cell has two less vertices than the original hypercube, because those two vertices project into the interior. What’s more, each $m$-boundary of a Voronoi cell in $A_n$ is the projection onto S of a particular $m$-face of the hypercube in $\mathbb{R}^{n+1}$.

But the Delaunay cells of $A_n$ turn out to be intersections of S with the Delaunay cells in $I_{n+1}$, not projections.

Given any $m$-boundary, M, of a Voronoi cell in $A_n$, the Delaunay $(n-m)$-boundary D dual to M is the convex hull of all the lattice points in $A_n$ whose Voronoi cells share M.

But if you want, you can also find D via $I_{n+1}$, as follows:

1. find the $m$-face M’ of the $I_{n+1}$ Voronoi cell that projects to M
2. find the dual Delaunay $(n+1-m)$-face D’ of M’ in $I_{n+1}$
3. intersect D’ with S, to get the Delaunay $(n-m)$-boundary D back down in $A_n$.

So the requirement in the original deBruijn method of selecting Delaunay $(n-1)$-faces in $I_{n+1}$ that intersect P is really the same as selecting Delaunay $(n-2)$-faces in $A_n$ that intersect P, so long as P lies in S. And of course projecting a Voronoi $2$-face first from $\mathbb{R}^{n+1}$ into S and then down to P is exactly the same as projecting straight from $\mathbb{R}^{n+1}$ to P.