M Theory Lesson 228

Observe that in any dimension, the diagonal Fourier operator $D$ that generalises $\sigma_{Z}$ will have the elementary basis vectors as eigenvectors. Similarly, the basic 1-circulant $(23 \cdots n 1)$ provides a basis set that is complementary to the basis set for $D$. An example of a set of vectors complementary to the $D$ basis in three dimensions is the set $(\omega^{2}, \omega^{2}, 1)$, $(1, \omega^{2}, \omega^{2})$ and $(1, \omega, 1)$, which are the eigenvectors for the operator As with the truncated braid algebras constructed from permutations and diagonals, this equals $D \cdot (231)$. So we see that the quantum Fourier transform is very closely related to complementary basis sets.

M Theory Lesson 227

Recall that the two dimensional Fourier operator diagonalises a circulant via conjugation. For the Pauli matrices, again ignoring factors of 2, we have so the Fourier operator cycles between the mutually unbiased bases for the two dimensional space. This shows how the non-zero entries of $\sigma_{Z}$ really are identified with the spin eigenvalues of the Fourier generator $\sigma_{X}$. Observe also how these Fourier maps

$\sigma_{X} \rightarrow \sigma_{Z} \rightarrow \sigma_{Y} \rightarrow \sigma_{X}$

cycle through the three directions of the trefoil knot quandle. In other words, this quantum action acts on the embedding space for a trefoil by interchanging the knot crossings. If these crossings are drawn on the squares of an associahedron Riemann sphere, then the action cycles the points $0$, $1$ and $\infty$.

The Dark Side IV

If you thought the Dark Force had thrown enough mysteries into standard cosmology, think again! Now there is a Dark Flow, seen as a remarkable motion of 700 galaxy clusters over a scale of 6 billion light years, as indicated by WMAP data. It appears that $\Lambda$CDM, as valiantly as it has tried to keep up with developments, may finally bite the dust. The motion is being touted as a gravitational pull from matter beyond our horizon. Enough of this rubbish.

Hat tip: Mottle. See the NASA news. And here is Louise's post.

M Theory Lesson 226

Returning to the regular theme of constructing n-categories out of prime building blocks, note that the first three finite fields to appear as truth values for Fourier type spaces of dimension $p$ (prime) are the primes $2p + 1$, namely 5, 7 and 11. These primes have already turned up in a number of places, most notably in the buckyball trinity.

What primes are of the form $2p + 1$ for $p$ prime? This is not an arithmetic progression, because the sequence of primes $p$ is not the ordinals. For the next prime, namely 7, we find that $2p + 1$ is not prime. Similarly, $2p + 1$ is not prime for $p = 13$, 17 or 19. The first five primes that work give values for $2p + 2$ of 6, 8, 12, 24 and 48.

Since all primes $p > 3$ are of the form $6n \pm 1$, for $2p + 1$ to be prime we require at least that $p = 3n$, which cannot be true for a prime, or $p = 3n - 1$ for some $n$. For example, $23 = 3 \times 8 - 1$ and $11 = 3 \times 4 - 1$. So now we are actually interested in primes in the arithmetic progression $3m + 2$. In this case, Dirichlet's theorem tells us that there are infinitely many such primes.

Note that $m$ must always be odd, but when do these larger $m$ correspond to primes $2p + 1 = 6m + 5$? Again, by Dirichlet's theorem, there are infinitely many primes in this sequence. With the handy table, one can quickly find more $p$ such that the truth values in dimension $p$ form a finite field of $2p + 1$ elements.

M Theory Lesson 225

In lesson 173 we came across the ternary quandle rules:

$X \circ Y = Z$
$Y \circ Z = X$
$Z \circ X = Y$

In the world of MUBs, a useful representation of these rules is with the Pauli matrices: Ignoring factors of 2, the operators $i \sigma_X$, $i \sigma_Y$ and $i \sigma_Z$ obey the quandle rules when multiplication is the Lie bracket. Up to phase, the eigenvectors form a set of three MUB bases for a two dimensional space. One usually uses complex numbers, because the fourth root of unity is essential in defining the full set. However, ignoring normalisation, all matrix entries (truth values) belong to the finite set $\{ 0, \pm 1, \pm i \}$. Moreover, the unnormalised eigenvectors all have eigenvalues $\pm 1$, just like the basis Fourier operator $\sigma_X$.

Observe that quandles naturally associate the braid group $B_3$ with two dimensional MUBs, rather than three. It was also more natural in the zeta value algebras to let $B_n$ correspond to $d = n - 1$. This is another way of seeing why mass is not naturally described by only three stranded diagrams.

Now let us view the Jacobi rule on a Lie algebra as a little computer program. The program initialises a variable $\alpha$ to zero. It then takes an input $v = (X,Y,Z)$ and performs the following operations: 1. take the (right bracketed) triple product $m(X,Y,Z)$ (Lie bracket), 2. add $m$ to $\alpha$, 3. shift $v$ so that $(X,Y,Z) \mapsto (Y,Z,X)$, 4. output $v$ and $\alpha$. Now three iterations of this program returns $\alpha = 0$ again.

Did we really need complex numbers or Lie algebras here? We have already seen how sixth roots of unity, in three dimensions, are enough to see modular structure, since six equals two times three. In that case, the addition of 0 gives seven possible matrix entries, and there are four MUBs.

M Theory Lesson 224

Without worrying about drawing all paths through the graph defining a small category, we can start to discuss the Yoneda lemma, the first fundamental theorem of category theory.

A diagram (such as the cube) internal to a category C (such as Set) may be thought of as a functor from the small category formed from the diagram, which embeds the diagram in the category C. We will call this functor $F$. That is, instead of a vertex labelled $C_i$, we imagine that $C_i = F(A_i)$ where $A_i$ labels an object on the diagram.

Now consider a second functor $H$ from the diagram into the category C, defined as follows for the cube in Set. The object $H(C_i)$ is the set of all paths from $C_i$ into the target of the cube. The number of elements is defined by the numbers on the diagram where 0 includes the empty path from the target to itself. For $t$ the target, a category theorist would call such a (actually contravariant) functor Hom$(-,t)$, where the dash stands for the argument and Hom is short for homomorphism.

Observe that the target object on the new cube is the one element set. The correct way to introduce two dimensional structures to categories is to describe natural transformations between functors. Such a 2-arrow $\eta: H \Rightarrow F$ is described by a whole family of commuting squares made using arrows $\eta_{X}: H(X) \rightarrow F(X)$ for any object $X$. So what are these $\eta_{X}$ for the functors in question?

Since $H(t)$ is the one element set, $\eta_{t}$ picks out an element of the original target set $F(t)$, such as the set $C_0 \cup C_1 \cup C_2$ of the last lesson. In fact, the Yoneda lemma tells us that the set of all possible natural transformations $\eta$ is isomorphic to the set $F(t)$. Here $\eta$ sends a basic projection (of a square onto a 1-arrow) onto a representative of the projection in the 1-arrow of the cube. The message is that the higher dimensional arrows can deconstruct higher dimensional spaces into simple one dimensional paths, much as in the case of space filling curves for complicated geometries based on the real or complex numbers.

Holiday

After two days relaxing in Wanaka (with hot pools, venison, beer, cake and pizza) I ended my little holiday by finishing all 30 levels of this very entertaining cube game, for which I was awarded, via electronic certificate, the degree of Master of Spatial Logics. It gets tricky around level 23.

M Theory Lesson 223

Recall that internal precategories in Set are formed from three sets, say $C_0$, $C_1$ and $C_2$. Now imagine these are any distinct (finite) sets. Since elements of sets are sets, one can form a three element set $\{ C_0, C_1, C_2 \}$, which is isomorphic to any three element set. This set is obtained by taking unions of the component sets, and the six ways of doing this are the paths on the parity cube. In categorical terms, this cube is canonically given by colimits (pushouts) on the three initial directions in space. In other words, the permutations of three letters exist for free in the topos Set. In M Theory, it is convenient to view all groups and groupoids as derived structures. In order to specify the maps (group operations) between paths on the cube, one must fill in the squares with higher dimensional cells, but these don't live in the one dimensional Set.

As a Chu space, a permutation on three letters should be a map $3 \times 3 \rightarrow \Omega$ into the two point set of Boolean truth values. For example, the circulant $(231)$ sends the Cartesian product elements $(C_0,C_1)$, $(C_1,C_2)$ and $(C_2,C_0)$ to the value 1 and all other pairs to the value 0. Similarly, the identity $(123)$ labels $(C_0,C_0)$, $(C_1,C_1)$ and $(C_2,C_2)$ as true. In a topos, one automatically pulls back such arrows along the arrow true $1 \rightarrow \Omega$. Since the pullback arrow into 1 is unique, this square is described by the arrow into $3 \times 3$. For the identity $(123)$, this is naturally the diagonal map $\Delta: x \mapsto (x,x)$ into $3 \times 3$. Thus the permutations generalise the diagonal map by selecting different arrows.

Working

I have been busy working. Some customers today saw one of these naughty critters (nightmare pests in NZ) run by the cafe. They are extremely common, but one doesn't often see them this far from the forest. Plenty of rabbits. That is, another pest. In fact all mammals are pests in NZ, with the exception of a small native bat. No, I'm not exempting humans.

Operads in 2d

Schreiber, well known for saying he doesn't like operads, has now found considerable enthusiasm for them (in 2d). What will happen to string theory now?

Of course, great mathematicians like Manin have been thinking about such two dimensional structures for many years. Now Manin may not care much about physics, but some of his collaborators are most definitely in the Connes camp, far from endorsing the naive predictions of string theory. So what gives? We live in interesting times.

M Theory Lesson 222

A useful notion in categorical Galois theory is the idea of precategory. This is a collection of objects $O$ and morphisms $A$ (ie. a graph) together with source and target maps ($s$ and $t$) and suitable maps $m$ and $e$ expressing composition and identities. But in a category we can define an internal precategory using the pullback diagram where certain pieces through $O$ commute. That is, the lower and right hand triangles express the fact that taking a source or target of an identity arrow defines the identity on objects. The equation $s m = s p_1$ says that taking the source of a composition is the same as taking the source of the first arrow in the pair. Similarly, taking the target of the second arrow is the same as taking the target of the pair. The pullback square expresses the fact that one can only compose arrows when the target of the first matches the source of the second.

Observe that not all pieces of the diagram commute. For example, it is not true that $sm = tm$, unless the composition forms a loop. But this is always true for a one object category, such as a group, in which case one is permitted to draw in an arrow $1_{A}$ diagonally across the pullback square, and then basically everything commutes.

Time at the LHC

As Tommaso Dorigo and many others have already pointed out, the official start date for the LHC is tomorrow! Unfortunately, it is therefore obligatory for theorists to make predictions today. So far these have mostly been along the lines of

there will definitely be supersymmetry

which isn't really a prediction at all, since it doesn't specify measurable parameters which have meaning for experimentalists. According to Woit, the predictions of Veltman include no fairy field and the realisation that string theory is mumbo jumbo.

Can we do any better? Tony Smith's composite Higgs is a possibility, but many approaches would lay claim to it. Is there something distinctive that might arise from a more Galoisian gravity? What truly new $p = 5$ process might we observe? The first possible experiment along these lines that came to my mind has nothing to do with the LHC. The difficulty is in imagining a material that could create a field quite unlike the usual suspects. A high temperature ceramic superconductor is one possibility. Having eliminated magnetic fields, one could look for Stern-Gerlach type pentuplet splittings of monoenergetic electron (or muon) beams.

I predict that the surprises (besides a lack of fairy field and sparticles) will mostly come in the analysis of multijet processes, where QCD predictions fail spectacularly. If I wasn't such a scatter brain, I would still be working on operad combinatorics for QCD.

Problem of Time II

As indicated by the outside of an astrolabe disc, whether in hours or months, time divisions are defined in terms of periodic motions in the heavens. Time lies on a discrete circle before a continuous one, since measurement never resolves the intervals indefinitely. Yet even before astrolabes, and modern science, Zeno proposed the riddle of Achilles and the tortoise, relying not on a continuous time, but on the assumption of an infinitely divisible space.

Why should space appear infinitely divisible when time does not? Of course, for the purposes of 19th century physics, the continuum is a useful construction, but now one expects this classical geometry to emerge (ie. be derived) from a collective of the large scale observations of like observers, such as ourselves. If so, where does the notion of scale originate? To begin with, we are now used to the correspondence between cosmic time and cosmic scale in an expanding cosmology, so we can ask instead: from where does our notion of cosmic time originate? We immediately observe that an estimate of cosmic time is something that, as sensible observers, we are largely in agreement upon.

What does it mean to view this kind of time on a discrete circle? We are not talking about cyclic universes, or any construction that proposes structure outside what we can possibly observe. So discrete cosmic time is a concept of quantization on cosmic scales. This sounds a bit like the old style Bohr correspondence, which after all had a good phenomenological basis. Now let us not assume that this cosmic time is universal for all observers. Then its range of values is analogous to the quantized energy levels of a hydrogen atom.

Banks on Holography

Carl, Louise ... and everybody else ... you simply must see T. Banks' outstanding talk at the current PI Multiverse conference. He outlines a relatively mainstream, but completely original, analysis of holography using basic quantum mechanics and general relativity.

An FRW type cosmology is obtained from a dense black hole fluid which satisfies causality (causal diamond) constraints arising from finite dimensional holographic pixel Hamiltonians on a lattice, which in turn are analysed in terms of noncommutative function algebras inspired by matrix models and M theory. Their main model considers a physical region of a dilute black hole gas, arising from a large fluctuation of the maximal entropy stable fluid.

He is inclined (but reluctant in the end) to give up inflation altogether, because the model generates homogeneity and isotropy without it. Banks also points out that many field theory prejudices regarding the nature of an emergent geometry from fluctuating classical spacetimes simply cannot be correct from this point of view, which advocates a large $N$ (matrix size) limit construction from towers of causal diamonds of increasing cosmic time, for each observer.

Of course Banks prefers Dark Energy to a varying speed of light, and this leads to some criticism of the landscape on the grounds that it cannot reproduce universes of this kind, with suitable values for $\Lambda$. On the other hand, he recognises the need for a richer mathematical discussion of the quantum operators (his Hamiltonians are only supposed to be a simple solution to the constraints).

M Theory Lesson 221

Recall that a two dimensional kind of 1-operad is the little squares operad, which is like trees made of squares instead of points. Since matrices may be Chu spaces, we can also talk about the substitution of matrices into matrices as an operad of spaces. Note that the trivial case of the one dimensional number 1 acts like an operad of lattice squares, since matrices look like a square array of points. Restricting attention to matrices with entries 0 and 1 means assuming that all spaces belong to the category Chu(Set,2). In other words, a Chu space operad is like a topological space operad. However, the Chu space operations involve two different compositions, $+$ and $*$. For initial Chu spaces with $(p_1, s_1)$ and $(p_2, s_2)$ points and states, the new Chu space will have

$(p,s) = (p_1,s_1) + (p_2,s_2) = (p_1 + p_2, s_1 * s_2)$

and indeed the calculator tells us that the 0 for addition is the space $(0,1)$. For trees, these operations can be interpreted as in the graftings That is, in representing the initial spaces by 1-ordinal trees $p_i$ and $s_i$, the new Chu space is easily represented by a pair of 2-ordinal trees, where the number of leaves gives the final number of points and states. Compound additions of Chu spaces then result in higher dimensional ordinal trees. This is very different to the creation of associahedra, which are all labelled by 1-ordinal trees. However, selecting a sequence of Chu spaces with $0,1,2, \cdots$ points, say $\{ C_{p} \}$, results in well defined spaces

$C_{\sum n_i} \equiv C_{n_1} + C_{n_2} + \cdots + C_{n_k}$

What is an interesting such sequence? We could choose the quantum Fourier basis 1-circulant sequence, beginning with the Pauli swap matrix and then the $3 \times 3$ matrix $(231)$, interpreted as Set like Chu spaces! Then as the number of states is chosen to equal $p$, the space $C_{\sum n_i}$ will have $s = \prod n_i$ states. Restriction to a prime number of points, as in the MUB problem, results only in spaces with states given by prime factorisations for the integers.

Problem of Time I

It was possible to tell the precise hour of the day with the aid of a pocket sized device a very long time ago. Hipparchus, who recorded star positions from 147 to 127 BC, is said to have invented the astrolabe, a small plate with fitted dials that uses the position of the sun and stars to tell the time. This required knowledge of trigonometry, which Hipparchus also used to calculate the eccentricity of the orbits of the moon and sun, thereby forcing the Earth to be shifted from the true centre of a circular orbit for the sun.

Astrolabes were used extensively in the Islamic world from the 8th century, and Abd al-Rahman al-Sufi listed many uses for it, from astrology to navigation and surveying. A typical astrolabe has a stereographic projection of the night sky on the base plate, and a dial marked with the main constellations, which can be turned to the setting or rising position. The outer edge of the base is marked with the hours of the day. On the reverse side of the plate is usually a rod with sights, which may be used to determine, for instance, declinations of stars (albeit not very accurately).

M Theory Lesson 220

A simple example of an adjunction between two 1-categories is a Galois connection. We are interested in the case of sets with a partial order, so there exists an arrow from $a$ to $b$ in a set $S$ whenever $a \leq b$. The natural functions (in this case functors) between such sets (here considered as categories) are monotone functions, so a Galois connection consists of two monotone functions $f: S \rightarrow T$ and $g: T \rightarrow S$ such that $f(a) \leq x$ iff $a \leq g(x)$. Observe that, at least when $g$ is onto, $fg$ is idempotent as a map from $T$ to $T$. This follows from $gfg (y) \geq g(y)$ (and $gfg (y) \leq g(y)$), which is true because the application of the one rule to $f(x) \leq f(x)$ gives $x \leq gf (x)$, and we can find a $y$ such that $x = g(y)$.

Usually the sets $S$ and $T$ are quite different. Consider the original example for finite number fields. The finite field with $p^{n}$ elements, for $p$ prime, is an extension of the field with $p$ elements. Take the finite set of all fields in this large field which contain the field with $p$ elements. For any such field $K$, there is a map $K \mapsto \textrm{Gal}(K)$ which sends $K$ to its Galois group in the large field. The dual connection map takes any subgroup $H$ of the main Galois group to all elements of the large field that are fixed by $H$.

Now let $p = 3$ and $n = 3$. There are fields of 3, 9 and 27 elements that extend the field $F_{3}$. The main Galois group is all isomorphisms of the 27 element field that fix the field $F_{3}$. The dual connection map sends the trivial group inside this group to the full 27 element field, the whole Galois group to $F_{3}$ and an intermediary subgroup (bit from trit) to the 9 element field. All this is encoded in an elementary pair of arrows between the 2 kinds of three element set.