S-matrix Reloaded

This simple and beautiful PI talk by Freddy Cachazo is a must see! He begins with some history on the S-matrix and proceeds to whiz through twistors and MHV diagrams.

Operadification III

The multicategorical analogue of the natural number diagram

$1 \rightarrow N \rightarrow N$

looks like

$\Delta^{\cap} \rightarrow \textrm{Tree} \rightarrow \textrm{Tree}$

where the category $\Delta^{\cap}$ (dimension not specified) has objects $n$ represented by single level trees, the associahedra trees. That is, since we are allowed any number of input identity arrows, the simplest one object category has an arrow for each $n$. The category Tree, by definition, extends these single level trees to $k$-ordinal trees of $k$ levels. In other words, the ordinals $N$ in Set are replaced by the levels of the $k$-ordinal trees. This is how we wanted to represent $n$ in the quantum world, in association with dimension.

Now recall that the $k$-ordinal trees can represent Batanin's polytopes, which are topological spaces. The successor map simply adds a leaf to every top branch. For example, the sequence of $k$ dimensional spheres arises as a version of the ordinals in this sense.

Multicategorical arithmetic therefore compares an ordinary cosimplicial object in $C$ with a weakened kind of
cohomological object Tree $\rightarrow C$. By truncating the categories at level $k$, one obtains a multicategorical analogue of modular number objects. There are many motives for studying this kind of arithmetic.

Still Down South VI

Well, there goes the opera ticket. Actually, I finally received one short email from a representative of the British government, but since it came from Wellington there is ample room to doubt the vague promises and excuses therein.

Operadification II

Underlying the concept of natural number object is the basic recursion theorem. The composition of the arrows

$f \circ q: 1 \rightarrow A \rightarrow A$

in Set is just the evaluation $f(q)$. This arrow $f(q): 1 \rightarrow A$ can itself be used as input for the same diagram, by appending another copy of $f$ to the right, to obtain the arrow $f(f(q))$. That is, the natural number object commuting diagram extends indefinitely to the right by appending extra copies of the successor and the function $f$. Once the comparison arrow $u: N \rightarrow A$ assigns zero to $q$, it follows that it must assign $f(q)$ to 1, $f(f(q))$ to 2, and so on.

Thus the definition of recursion, as a possibly infinite process, demands the full set $N$ rather than some finite ordinal set. But for periodic recursive functions, satisfying $f(f(f(\cdots(q)))) = f(q)$ for $n + 1$ brackets on the left hand side, modular arithmetic using the set n acts as a universal diagram. For example, if $f$ represents rotation by an $n$-th root of unity, then it is periodic in this sense.

Operadification

An important characteristic of the topos Set is the existence of a natural number object, namely the object $N$ of counting numbers along with a diagram

$1 \rightarrow N \rightarrow N$

where the second arrow is a successor function, plus one. This diagram is universal in the sense that it is initial in the category of all such diagrams. A general diagram in this category replaces the object $N$ by another set $A$.

In the quantum world, however, $N$ is better described by the dimensions of simple vector spaces. Including the ordinal maps, we can think of $N$ as a whole category, usually called $\Delta$. But $\Delta$ lives in a category of categories, Cat, rather than Set. So instead of maps $u: N \rightarrow A$ characterising the universality of arithmetic, we end up looking at functors $U: \Delta \rightarrow C$, which are basic mathematical gadgets known as cosimplicial objects.

The commuting square in Set that compares the successor function with a map $f: A \rightarrow A$ is replaced by a (weakened) commuting square that compares an increment in dimension to a functor $F: C \rightarrow C$ via the cosimplicial functor $U$. In other words, quantum arithmetic really is about cohomological invariants after all.

And let's not forget that in this higher dimensional operadic world, $1$-ordinals are merely the simplest kind of trees. The category $\Delta$ should really be replaced by a category whose objects are trees.

M Theory Lesson 264

Recall that circulants are always magic as well as square magic in the sense that the sum of squares along a row or column is a fixed constant. In particular, any Koide mass matrix $M$ has this property.

But MUB operators such as $F_3$ are not magic. For example, the action of $F_3 F_2$ (the neutrino mixing matrix) on $M$ results in a $1 \times 2$ block matrix in terms of the square roots of the masses, because $F_3$ diagonalises, and $F_2$ then acts on a pair of mass eigenvalues. The fact that this matrix is not magic is the same as the statement that $m_1 \neq m_2$. The fact that it is not square magic follows from the statement that $s < 0$ in

$\textrm{cos} \delta = \frac{s - 6v}{s}$,

where $\delta$ is the angle shared by all three masses. This property is shared by the hadron fits.

Good, Bad and Ugly

People who hear that I am interested in physics often say to me that they would like to know more about String Theory. Although not articulated, the implication is usually that there must be Only One String Theory. In reality, string theorists come in several flavours: good, bad and ugly. For example, our friend kneemo is a good string theorist, who recognises that the so called physical predictions of main stream String Theory are probably just plain wrong.

Then there is mottle, who recently said:

String theory is one theory, it predicts many possible vacua (Lorentz-invariant or dS-invariant solutions to its equations of motion), and we live in one of them.

Well, at least this statement settles any doubt one might have had that the bad string theorists might be willing to alter their physics a little to fit the facts. Clearly, the facts don't matter to them. Ugly string theorists include people who boast loudly that they are not string theorists at all, but are not very convincing.

Still Down South V

That was a very enjoyable late summer trip, with perfect weather. Thanks to Kerie and Tasman (the dog). One cannot have such beautiful valleys to oneself in England. Nonetheless, tomorrow I phone Australia, yet again, in an attempt to find out how my visa application is progressing. It appears to be no longer possible to speak to Wellington.

Still Down South IV

After two months of waiting for what should be a straightforward visa, I have now settled into life in Wanaka. Now I'm off into the hills for 2 days.

M Theory Lesson 263

In summary, this approximate solution for the CKM matrix uses the four parameters $A^2 = 676$, $B^2 = 697$, $C^2 = 29$ and $X^2 = 1$. With an appropriate choice of signs, the magic circulant parameters become

$3I = 26 + 2 \sqrt{697}$
$3J = 26 - \sqrt{697} - \sqrt{3}$
$3K = 26 - \sqrt{697} + \sqrt{3}$
$3R = \sqrt{53} - 2 \sqrt{2}$
$3G = \sqrt{53} + \sqrt{2} + \sqrt{87}$
$3B = \sqrt{53} + \sqrt{2} - \sqrt{87}$

and the resulting (square) matrix values are given by

0.9495, 0.0502, 0.0003
0.0498, 0.9465, 0.0037
0.0008, 0.0033, 0.9960

Observe that the largest disagreement with experiment is in the very small $V_{td}$ and $V_{ub}$ values.

M Theory Lesson 262

Using only integer values for the squares of entries in 27 $F^{\dagger} V F$, it follows that the small (squares of) entries in the imaginary part must sum to 3. On solving for the variables $I$, $J$, $K$, $R$, $G$ and $B$ one has the freedom of signs in sorting out the CKM values.

Let us just look at the up-down entry, $I + iR$. The solution is given by

$3I = 26 + 2 \sqrt{697}$
$3R = \sqrt{53} - 2 \sqrt{2}$

and so the up-down entry $V_{ud}$ must equal $0.9744 = \sqrt{I^2 + R^2}$. Fortunately, according to wikipedia, this value is $0.9742 \pm 0.0002$. It should not be difficult for the reader to find expressions for the other CKM entries, based on the Fourier transform.

Aside: Tommaso Dorigo continues with his excellent series of posts on fairy fields.

M Theory Lesson 261

In general, the Fourier transform of a magic circulant sum takes the form The condition that it be in $U(1) \times SU(2)$ reduces the number of parameters to four, as in where it is assumed that $B^2 + C^2 = 726$. For the CKM matrix $V$, we have seen that the parameters $A$, $B$ and $C$ are very simple numbers satisfying such rules. $X$ is a small parameter ($X < \sqrt{3}$) that we might adjust to fit the experimental values.

CKM Recipe II

In the approximation of a cubed root cosine by a square root, if one starts with the exact number $26/27$ (for the CKM row sum), then equality implies a numerator not of 723, but of 722.9792412. On the other hand, assuming an exact numerator of 723 results in the number $26.003436/27$. Anyway, $723 = 697 + 26$ and these integers appear in the decomposition of the symmetric magic matrix $U$ into 1-circulant and 2-circulant integer matrices: Observe that the small value of 3 (off the $U(2) \times U(1)$ block) limits the number of positive integer decompositions to four. The 2-circulant piece always takes the form $(26,0,0) + kD$, where $D$ is the democratic matrix and $k \in 0,1,2,3$.

CKM Recipe

Take the following real unitary magic matrix. Take the square root of each entry to form another real matrix. The bottom right $2 \times 2$ corner is the real part of the Fourier transform of the CKM matrix. The top left corner is very close to the real part of the row sum for the cubed root of the CKM matrix, which itself has a row sum with real part $26/27$. That is, the following approximate relation holds:

$\textrm{cos}(\frac{1}{3} \textrm{cos}^{-1}(\frac{26}{27})) \simeq \frac{\sqrt{723}}{\sqrt{729}}$

The norms of the Fourier transform blocks were previously observed to be 1. This fixes the imaginary part of the $1 \times 1$ piece. We will then consider another unitary magic matrix for the imaginary component.

Pioneering C Change

For years, Louise Riofrio has patiently explained her varying speed of light cosmology. Now, thanks to DIY QG, I would like to bring attention to this paper, by Antonio F. Ranada and Alfredo Tiemblo, which explains the Pioneer anomaly in terms of a varying speed of light cosmology, or rather the mismatch between atomic and astronomical time.

Quote of the Month

I think there is a widespread misconception that there are many ignored alternatives. I think that most good string theorists would be eager to look into alternatives IF it had even *some* of the good features of a fundamental theory. But most people don’t seem to realize how difficult it is to come up with even a mildly promising alternative. It is not like there are tons of alternatives out there and theorists are ignoring them.

Somebody at Not Even Wrong

M Theory Lesson 260

This CKM business is getting a bit messy, so let us recall that the Fourier transform of a complex circulant sum takes the form Carl Brannen's values for the CKM matrix (perhaps slightly inaccurate) are

$I = 0.973313$
$J = -0.008577$
$K = 0.000466$
$R = 0.040013$
$G = 0.225762$
$B = -0.004273$.

In particular, $(I + J + K) + i(R + G + B) = 0.965202 + 0.261502 i$, which has norm 1. The point is that, focusing on the real parts, it is better to think of these numbers in the form $a/27$. No doubt Carl will fix the numerical fit and blog about it shortly. The number 27 (or rather its square, $729 = 3^6$) is a natural normalisation factor for products of MUB type matrices.

M Theory Lesson 259

Complex magic matrices also multiply to yield new magic matrices. If the row and column sums of two magic matrices are $e^{i \theta_1}$ and $e^{i \theta_2}$, the row and column sum of their product will be $e^{i (\theta_1 + \theta_2)}$.

Carl's parameterization of the CKM matrix $V$ results in a row sum phase with $\theta = -0.27308859$, close to a 23rd root of unity. In other words, the row sum is the complex number $0.96294248 - 0.26970686 i$. The $n$th power of such a complex matrix will have a row sum with $n$ times the angle, $n \theta$.

Observe that the number 0.96294248 is very close to $26/27$. This corresponds to the fact that $2 - 2 \times 0.9629 = 8/9$, which is the real part of a factor in a product form for the CKM matrix. That is, let $V = AB$. Now assume that the row sums for $A = A_1 + i A_2$ and $B = B_1 + i B_2$ are such that $A_1 + A_2 = B_1 + B_2 = 1$, where these numbers may be complex. Then it follows that the real part of $A_1$ equals $8/9$. We should probably check to see how an exact figure of $8/9$ compares with experiment.

Abtruse Goose

The brilliance of Abtruse Goose is apparent in the variety of reactions to the Arguing with String Theorists episode.

Clifford Johnson, a string theorist, casually laughs at the accuracy of the cartoon without appearing to understand it at all. As a result, Peter Woit puts up with idiotic comments about his sex life and of course Lubos Motl also weighs in, patting himself on the back for his PC approval of the girl string theorist (the bitch slapper). Woit laments that nobody else is doing their blogging duty and criticising string theory.

Yes, this is all rather tiresome, isn't it? Since nobody has actually bothered to define String Theory, it clearly encompasses any successful theoretical predictions that might appear in the next few decades. Why are people complaining? If they were serious about changing the culture of theoretical physics, they would do something about it. Oh, but hang on a minute ... That might require actually taking risks and displaying courage. Well, we won't be holding our breath then. Obama can't be expected to sort out this mess on his own.

Still Down South III

According to a local newspaper, from February 20 all UK visas for New Zealand citizens will be processed in Australia. When I phoned Wellington this morning, I was quite amazed by the incredible transformation in their helpfulness. I sent an email requesting a progress report on my visa application (this was certainly not possible previously) and received a friendly, automated reply! Unfortunately, tomorrow is a public holiday and one suspects that nothing more will happen until next week. I wonder what the record time is for visa processing?

M Theory Lesson 258

A general unitary magic 1-circulant may be written as the sum of two magic 1-circulants, as in $(a,b,b) + (0,c,0)$.

The $n$-th power of this sum has a binomial expansion for which at least one matrix factor in each product has a power greater than or equal to $n/2$. Since $DM = D$ (where $D$ is the unitary democratic matrix), for any such 1-circulant $M$ it follows that the limit of the power as $n \rightarrow \infty$ must also be $D$. These arguments apply to matrices over restricted domains for the rationals or reals. Similar arguments apply to 2-circulants.

Now general magic unitary matrices that are written as sums of two circulants, such as the approximate norm square of the CKM matrix, may also be expanded binomially to a sum of products that converges to $D$.

M Theory Lesson 257

Unitary magic matrices with non-negative rational entries, such as the norm square of the neutrino mixing matrix, form a semigroup because the product of two such matrices results in another matrix of the same kind. Restricting to 1-circulant unitary magic matrices results in a smaller semigroup, since products of 1-circulants are again 1-circulants. Observe that in a product of the form the difference between the two entries in the resulting circulant is $(a - b)(d - c)$, namely the product of the differences in the components. In particular, the power $M^{n}$ of a single such 1-circulant $M$ results in a difference of $(a - b)^{n}$, which cannot be zero for finite $n$ if $a \neq b$. So the only way such a power can result in the democratic unitary magic matrix $D = (1/3,1/3,1/3)$ is if it is an infinite power. Moreover, since $a, b < 1$, it is always the case that an infinite power will converge to $D$, that is $M^{\infty} = D$.

Matrix Power I

A nonassociative array product is naturally defined by replacing multiplication with power and addition with multiplication, as in the $2 \times 2$ case Observe that matrices which are magic under normal matrix multiplication have analogues which are magic under these power products, in the sense that the multiplication of entries along each row and column is equal to some constant. For example, the analogue of the usual democratic matrix is the matrix with entries the cubed root of unity. Note that the identity matrix still acts as an identity, but we will not bother to define a zero element, because we don't much care if things turn out to be like ordinary fields or not.

Demanding a magic sum of 1, and a magic product of 1 in the new nonassociative algebra, results in a mapping of the positive real interval $[0,1]$ to the complex unit circle. Scalar multiples do not exist in the new array product. If we replace zero by the number 1, then all permutation matrices must be mapped to the (power) democratic matrix with unit entries, namely three times the original democratic matrix.