M is for Magic

As we have seen, Carl Brannen's QFT uses circulant matrices. By resetting a mass scale, one may renormalise a 1-circulant

XYZ
ZXY
YZX

by a constant $\lambda = \frac{1}{Y + Z - 2X}$ so that the resulting circulant obeys the condition $2X = Y + Z$. This turns the circulant into a magic square. For 2-circulants the condition is instead $2Z = X + Y$. Although perhaps not a very useful observation, it is certainly entertaining! The total number of $5 \times 5$ normal (ie. matrices built from the first few ordinals) magic squares was only computed in 1973, and the number of $6 \times 6$ ones is still unknown. There is only one $3 \times 3$ normal square, up to rotation and reflection.

A paper by A. Adler uses circulants to find an algorithm for generating higher order normal magic n-cubes, by playing with p-adic L functions. For $p = 3$, Adler constructs two cute normal magic cubes: a $3 \times 3 \times 3$ cube and a $27 \times 27 \times 27$ cube. I was further intrigued by this paper of Adler's, containing the conjecture that magic n-cubes always form a free monoid. It shows first that sets of magic squares contain prime squares, out of which all others are constructed, and then that generating functions built from cardinalities for magic cubes have the remarkable property of being everywhere divergent!

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.

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.

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.

Mermin Magic

This week's PIRSA lectures include an enjoyable talk by M. Skotiniotis on his 2007 paper about epistemic models for hidden variable versions of Spekkens' toy quantum mechanics. In particular, the Mermin-Peres magic square is introduced. This is a $3 \times 3$ square of tensor products of Pauli operators of the form

$X^1$, $X^2$, $X^1 X^2$
$Y^2$, $Y^1$, $Y^1 Y^2$
$X^1 Y^2$, $X^2 Y^1$, $Z^1 Z^2$

corresponding to two qubits in three directions, which is related to the 2-direction three qubit Mermin pentagram of the form The number theoretic nature of these objects is discussed in the arxiv link. M theorists will notice the likeness of the magic square to certain mixing matrices in HEP phenomenology.

Quantum Computation

A well funded industry these days is research into the real world implementation of small sets of carefully chosen quantum operations, combinations of which are able to closely simulate any quantum computation. This is known as universal quantum computation, or UQC.

For example, in this paper by Kitaev and Bravyi, it is shown that the Clifford operations (which stabilise Pauli operators) along with certain magic states are sufficient for UQC. They assume that Clifford operations may be implemented ideally, and that the preparation of magic states is faulty. However, they first consider ideal magic states, such as
$|H \rangle = \textrm{cos} \frac{\pi}{8} |0 \rangle + \textrm{sin} \frac{\pi}{8} |1 \rangle$
$|T \rangle = \textrm{cos} \beta |0 \rangle + \omega \textrm{sin} \beta |1 \rangle$
where $\omega$ is a primitive eighth root of unity and $\textrm{cos} 2 \beta = \sqrt{3}^{-1}$. In particular, $| T \rangle$ may be used to implement a one qubit phase gate for the $12$-th root of unity. With the Clifford operations, this gate gives UQC.

The only Clifford generator that is not obviously a fun (field with one element) operation is the one qubit Hadamard gate (Fourier transform), but recall that (in MUB maths) this basis behaves like the zero of a finite field, or the standard choice of marked point for a fancy set! Now let's do UQC without complex numbers.

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 $M^{\infty} = D$.

M Theory Lesson 256

Larger products of MUB operators also lead to nice magic matrices. For example, consider this stunning exact magic matrix:
which has numerical values roughly equal to Observe that this matrix is similar in form to, although not the same as, the CKM quark mixing matrix.

Magic Motives

Speaking of orbifold Euler characteristics, let's put the magic formula

$f(n) = \prod_{m=0}^{n - 1} \frac{m!}{(m + n)!}$

in terms of Euler characteristics. First, let $m = 2g - 2$ be the Euler characteristic of a closed surface of genus $g$. This already suggests allowing non-orientable surfaces to account for odd values of $m$. Then consider moduli spaces $M_{m,n}$ for $(n + 1)$ punctured surfaces. The orbifold Euler characteristic of such a space will be denoted by $E_{m,n}$. Using Mulase's expression for $E_{m,n}$ and assuming it may be extended to the non-orientable case, one finds a natural definition for the moment coefficients of the form

$f(n) = \frac{1}{(n + 1)!} \prod_{m=0}^{n - 1} b_{m + 2} E_{m,n}^{-1}$

which is a product over surfaces of genus $g$ limited by $n$, and where $b_{i}$ is a Bernoulli number (for even $m$ these are defined in terms of zeta values for odd negative reals). One should take more care with the non-orientable factors, but this simple exercise shows that the zeta moments are naturally dependent on categorical invariants associated to complex moduli.

Neutrinos Again II

Recall that a renormalised circulant matrix is a kind of magic square, where we don't worry about summing along diagonals. In neutrino physics, the unitarity of mixing forces the (squared) mixing matrix to be a magic square with rows and columns summing to 1. The tribimaximal case was first discussed by Harrison et al, where the whole matrix follows from the entries $U_{13}$, $U_{23}$ and $U_{12}$. Labelling columns by $\nu_{1}$, $\nu_{2}$, $\nu_{3}$ and rows by $e$, $\mu$, $\tau$ the matrix $U^{2}$ is

$\frac{2}{3}$ $\frac{1}{3}$ $0$
$\frac{1}{6}$ $\frac{1}{3}$ $\frac{1}{2}$
$\frac{1}{6}$ $\frac{1}{3}$ $\frac{1}{2}$

In terms of the standard mixing angles this corresponds to $\theta_{13} = 0$, $\theta_{23} = \frac{\pi}{4}$ and $\textrm{sin} \theta_{12} = \frac{1}{\sqrt{3}}$ with no additional (Dirac) CP violating phase. Given the excellent experimental agreement with this case, the question is, what is the justification for choosing $U_{13} = 0$, $U_{23} = \frac{1}{\sqrt{2}}$ and $U_{12} = \frac{1}{\sqrt{3}}$? Most physicists expect some deviation from tribimaximal mixing, but perhaps there is a good reason for things being so simple. For instance, observe that we can reorder the columns arbitrarily so that $U^{2}$ is derived (assuming one democratic column) from a diagonal

$\frac{2}{3}$, $\frac{1}{2}$, $\frac{1}{3}$

which is the length 3 Farey sequence. That is, it has the modular group property that for consecutive fractions $\frac{a}{b}$ and $\frac{c}{d}$, one has $bc - ad = 1$.

On the other hand, what mixing do we get if we substitute Carl's neutrino Koide rule for the one assumed by Harrison et al? Note that Harrison et al use the $3 \times 3$ circulant mass matrix for the charged leptons. On using the same quantum Fourier diagonalisation operator for both the charged leptons and neutrinos (see page 7 in Harrison et al) one would find that $U^{\dagger} U = 1$, so the tribimaximal mixing matrix would be replaced by the identity! It is the interplay of $3 \times 3$ circulants and $2 \times 2$ circulants that gives rise to the observed tribimaximal mixing.

M Theory Lesson 203

As usual Carl has jumped ahead with a post on mixing matrices as magic squares. For reference, let us collect here some actual figures for the CKM matrix, given in this article from the Particle Data Group. Absolute value signs are omitted.

$M_{ud} = 0.97377 \pm 0.00027$
$M_{us} = 0.2257 \pm 0.0021$
$M_{ub} = 4.31 \pm 0.30 \times 10^{-3}$
$M_{cd} = 0.230 \pm 0.011$
$M_{cs} = 0.957 \pm 0.017 \pm 0.093$
$M_{cb} = 41.6 \pm 0.6 \times 10^{-3}$
$M_{td} = 7.4 \pm 0.8 \times 10^{-3}$
$M_{ts} = 40.6 \pm 2.7 \times 10^{-3}$
$M_{tb} > 0.78$

This is a little different to the values given in the wikipedia article. Standard Model analyses of these quantities can be quite complicated. Following the notation from before, in a very simple ideal double circulant the magic square property demands that $a + b = c + d$. For the CKM values (squared) we see that rows and columns do indeed sum to 1, and $c + d \simeq 1$ because $b$ is so small.

M Theory Lesson 276

Thanks to Phil for commenting about magic matrices and MUB operators. The interaction of the matrix $R_3$ with the Fourier operator $F_3$ is expressed in relations such as where all entries have the same norm, and under normalisation are just phases. Note that $2 + \overline{\omega} = 1 - \omega$. The odd phase differs from the other phase by a special angle. The moduli of these matrices are all permutation matrices, which are also trivially magic with row sum $1$. The special angle, in radians, is given by $\theta = 0.2928428$, which corresponds to a sin squared of $1/12$. Actually, the phase difference is $\pi - 2 \theta$. By cubing $1 - \omega$, we see that the basic phase here is just $\pi/6$, a $12$th root. Unsurprisingly, the value of $\textrm{sin} \theta$ also turns up in the Fourier transform of the neutrino tribimaximal mixing matrix in circulant form.

Magic Matrix

Philip Gibbs has now provided a webpage with his solution to the problem of showing that any $3 \times 3$ unitary matrix can be turned into a magic matrix by multiplication of its rows and columns by phase factors.

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.

M Theory Lesson 128

In Geometric Representation Theory lecture 13 you can hear James Dolan talking about braid diagrams and Hecke operators.

First, we think of Hecke operators as magic matrices with sets as elements. Composition of operators is sort of like a renormalised matrix multiplication. These can be redrawn as braid diagrams where we don't worry too much about the crossings. Matrix multiplication will be replaced by braid compositions. Dolan gave an example like this one: The dots on the top line come from the cardinalities $R$ and $B$, whereas the dots on the bottom line come from $R'$ and $B'$. The left hand strands represent the set in the top left box. Note that the total number of dots on the top and bottom is always the same. Braids with real crossings supposedly come in handy when considering not sets but vector spaces, or rather representations of groups like $GL(n, F_{q})$ over a finite field with $q$ elements (Langlands, anyone?). This links a simple set cardinality with a knot parameter $q$. But the knotty $q$ can take many complex values, most notably a complex root of unity. Fortunately, we already know that cardinalities can also take such values.

Another example, this time for a $3 \times 3$ matrix, shows how to associate an element of $B_3$ with a matrix whose entries sum to $3$. Sticking to the set interpretation, a zero is given by an empty square. M theorists will find such diagrams familiar by now. If you enjoyed lecture 13, in lecture 14 you can see John Baez write up the three matrices $\mathbf{1}$, $(231)$ and $(312)$ which underlie the mass Fourier transform.

Neutrinos Again VII

Recall that Carl's magic form for the experimentally verified tribimaximal mixing involves a sum of a 1-circulant and a 2-circulant. Such objects naturally live in a group algebra for the permutation group $S_{3}$. That is, let the 6 elements of $S_{3}$ (three 1-circulants and three 2-circulants) represent unit basis vectors for a six dimensional vector space, nominally $\mathbb{C}^{6}$. The tribimaximal mixing matrix for neutrinos is then expressed in the form

$\frac{1}{\sqrt{3}} (e^{i \theta_{1}}) + \frac{\sqrt{2}}{\sqrt{3}} (e^{i \theta_{2}})$

where $\theta_{1} = - \theta_{2} = \frac{\pi}{4}$ are phases in two complex directions, $((231),(213))$ and $((123),(321))$. Given the simplicity of the coefficients, a restriction of the number field would be feasible here. M theorists will also recognise the dimension of twistor space.

Quotes of The Week

There have been attempts to observe time lags in gamma flares and in gamma-ray bursts, but we have never seen something like this....

said Daniel Ferenc of U.C. Davis, discussing the new MAGIC result.

The observation of this group of galaxies that is almost devoid of dark matter flies in the face of our current understanding of the cosmos

said Arif Babul of the University of Victoria, discussing the Abell 520 cluster.

Not only has no one ever found a void this big, but we never even expected to find one this size

said Lawrence Rudnick of U. Minnesota regarding the void that corresponds to a cold spot on the WMAP map.

The Concorde cosmology is ready to crash

said Louise Riofrio on her blog.

Dear Santa

I know it's a bit late for this year, but I found the perfect cheap present for a budding M theorist: the Sudokube! Of course, some basic knowledge of magic squares makes it too easy to solve, but it would look good on the shelf. And if you don't mind me saying so, Santa, I was a bit disappointed with The Golden Compass. Why were all the physicists male? And that extended arm double ice axe arrest was just plain ridiculous.

Magic Motives

The most fashionable of all phrases amongst real mathematicians studying QFT, occasionally mentioned here, are the words motivic cohomology. This is supposed to be the mother of all cohomology theories. But what is cohomology anyway?

As a basic concept, cohomology relies on the logic of ordinary sets, particularly the difference between unions and disjoint unions. The latter type of union keeps track of which set each element actually came from, so it tends to double count intersections. In the category Set, disjoint union is a coproduct. Ordinary unions are a little less natural, but intersections are pullbacks of subset monics.

Universality is a basic categorical concept. A cohomology theory usually assumes a standard set of axioms for a cohomology functor $H^{*}$ from some category of (commutative) spaces (which we want invariants for) to a category of groups (where invariants live). The existence of group inverses is related to the arbitrariness of path directions in a commutative space. A universal cohomology theory is supposed be about an elusive category of motives.

But thinking of the field with one element, what happens if we have sets with funny actions, instead of just sets? Are our spaces necessarily commutative? Perhaps the underlying logic of intersection and union should be modified to properly account for the geometry of the absolute point. This might require drawing higher dimensional limit diagrams, or looking at higher dimensional target categories, not necessarily groupoids. Fortunately, some really smart people are now thinking about noncommutative motives, but it may take a while for us poor physicists to see what is going on.