Archive for September, 2007

M Theory Lesson 109

In the easy book Special matrices of Mathematical Physics the chapter on circulant matrices begins with the statement: making a Fourier transform is equivalent to diagonalising a circulant matrix, and an inverse Fourier transform takes a diagonal matrix into a circulant matrix. In fact, for a circulant built from the numbers $X_1 , X_2 , \cdots , X_n$ the eigenvalues $\lambda_k$ are related by the following pair of $\mathbb{Z}_n$ (Abelian) transforms

$\lambda_k = \sum_{j} e^{\frac{2 \pi i}{n} jk} X_j$
$X_j = \frac{1}{n} \sum_{k} e^{- \frac{2 \pi i}{n} jk} \lambda_k$

Perhaps non-commutative Fourier transforms should be about circulants of circulants! In fact, examples of quantum groups (Hopf algebras) in the book are built from matrices of matrices. The simple quantum analogue of phase space is viewed as the space underlying the circulant/diagonal matrix space, such as the discrete torus $\mathbb{Z}_n \otimes \mathbb{Z}_n$, where the cyclic group $\mathbb{Z}_n$ belongs to the self-dual schizophrenic object $U(1)$ of Stone (Pontrjagin) duality.

A circulant $C$ is diagonalised via $M^{-1} C M$ with $M_{ij} = \frac{\omega^{- ij}}{\sqrt{n}}$ for $\omega$ the primitive $n$-th root of unity. Given that the eigenspace projectors are circulant, a circulant matrix can be written in the basis of projectors. It can also be written as a degree $n$ polynomial in a nice circulant $S$ representing a basic shift operator. In the $3 \times 3$ case this is the familiar

001
100
010

corresponding to the permutation $(312)$, and a circulant takes the form $X_1 + X_2 (231) + X_3 (312)$. A spectral function $f$ on a circulant is defined to be the circulant

$f(C) = \frac{1}{n} \sum_j (\sum_k f(\lambda_k) \omega^{-jk}) S^j$

The Fourier transform pair is particularly simple in terms of the matrices $S^j$ and the projectors. Thus it is useful to define a matrix product via convolution, $G.H = \sum_j (G*H)_j S^j$ where

$(G*H)_j = \sum_k G_k H_{j – k}$

Now let the diagonal matrix with entries $\delta_{ij} \omega^{i}$ be denoted $D$. Then one has $DS = \omega SD$, a simple non-commutativity condition, which extends to the Weyl rule $D^m S^n = \omega^{mn} S^n D^m$ for a quantum torus. This setup has the feature that the $n \rightarrow \infty$ process is associated with a continuum limit, which is just what we want to do with n-categories. To quote Weyl: The problems of mathematics are not problems in a vacuum …

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M Theory Lesson 108

Well, it is fun playing with circulants, but as Carl points out we might want to worry about idempotency also. For 1-circulants of the form

XYY
YXY
YYX

the only determinant 1 idempotent is the identity matrix. Proof: since $Y = Y^2 + 2XY$ then $X^2 = – 2Y^2 – \frac{1}{2} Y + \frac{1}{2}$, and from the relation $Y = Y^2 + 2XY$ it follows that $X = – \frac{1}{2} Y + \frac{1}{2}$. Thus $- 2 Y^2 = 2 Y^2$ and so $Y = 0$. By symmetry, the circulant matrices given by $(X,Y,Z) = (0,1,0)$ and $(0,0,1)$ also satisfy both the determinant and idempotency constraints. If we assume that $Y = \overline{Z} = r e^{i \theta}$ and that $X$ is real, it follows that the discriminant

$1 – 8X (a – ib)$

must be a positive real so that $b = 0$ and $Y = Z$ is real, and thus once again the identity is the only solution.

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M Theory Lesson 107

First let $Y = r e^{i \theta}$. Continuing our conversation on determinant cubics, for a complex circulant with $Z = \overline{Y}$ we can solve the cubic

$X^3 – 3 r^2 X + (1 + Y^3 + {\overline{Y}}^{3}) = 0$

for $X = X(Y, \overline{Y})$ with Chebyshev radicals, under certain restrictions on $Y$. In terms of the Chebyshev root function $C$ (omitting the subscript) the solutions are

$X_1 = r C(t) – \frac{1}{3}$
$X_2 = -r C(-t) – \frac{1}{3}$
$X_3 = r C(-t) – r C(t) – \frac{1}{3}$

where $-t = r^{-3} (1 + 2r cos (3 \theta))$. Note that for $r = 1$, $t = 0$ when $\theta = \frac{2 \pi}{9}$ and $C(t) = \sqrt{3}$. But for the case $r = 1$ (which might not be interesting since we have used the determinant to renormalise the matrix) this solution set makes sense only provided $cos 3 \theta < 0.5$ and then

$C(t) = 2 cos (\frac{cos^{-1} (0.5 t)}{3})$

Observe that the solution condition states that $\theta > \frac{\pi}{9}$. Now observe that $\frac{2}{9} \frac{\pi}{9}$ so the neutrino type cubic has three real solutions for $X$, but only $X_1 = 1.5644$ is positive. For a general positive real determinant $D$, $t = D + 2 cos (3 \theta))$ and the solution condition says that $cos (3 \theta) < \frac{2 – D}{2}$ which is less restrictive if $D$ is small.

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M Theory Lesson 106

Let $\omega$ be the primitive cube root of unity. The determinant of a $3 \times 3$ complex circulant

XYZ
ZXY
YZX

is given by

$(X + Y + Z)(X + \omega Y + \omega^{2} Z)(X + \omega^{2} Y + \omega Z)$
$= X^3 + Y^3 + Z^3 – 3XYZ$

On setting the determinant to 1 (another choice of normalisation) the inverse of the circulant takes the simple form

(XX – ZY)(ZZ – XY)(YY – XZ)
(YY – XZ)(XX – ZY)(ZZ – XY)
(ZZ – XY)(YY – XZ)(XX – ZY)

which is again a 1-circulant since matrix multiplication is closed for this set. For 2-circulants, on the other hand, inverses can be 2-circulant. An inverse of an idempotent will also be idempotent.

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Search Terms

After telling Tommaso Dorigo that search terms for this blog were boring, it now turns out that AF is at least high on a google search for pizza + theory + cheers (with absolutely no qualification regarding any specific physics or mathematics). Who are all these people? Don’t they know* that I give women a bad name? If you’re looking for a great pizza recipe, folks, I’m afraid you came to the wrong place! Hmmm. How about seared venison, English spinach, Kalamata olives, a fine Kapiti goat’s cheese and roast kumara on a thin crust. * This may have something to do with my lack of faith in the Dark Force ($\Lambda$), Higgs bosons or SUSY partners.

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Mt Ruapehu

This post is just to share a nice photo from the news today. We are being warned to stay away from Mt Ruapehu’s crater after Tuesday night’s small eruption. One climber, who happened to be high on the mountain, almost died when a large rock crushed his legs. Fortunately nobody else was injured, despite the skifields being busy with the school holidays.

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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!

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Blogging Cats

The categorical blogosphere is rapidly expanding, now with Jeff Morton’s new blog, Theoretical Atlas, and the Infinite Seminar. Meanwhile Urs Schreiber reports on being mobbed by reporters in Croatia, as he attends the maths conference on Categories and Geometry. It sounds like a fantastic conference: Batanin, Cartier, Kontsevich, Leinster and many others will be there. And what looks like the first real categorical physics conference has just ended in Texas. Lastly, check out the hot catsters lectures on You Tube.

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Bookworms

If I were a Springer-Verlag Graduate Text in Mathematics, I would be W.B.R. Lickorish’s An Introduction to Knot Theory.

I am an introduction to mathematical Knot Theory; the theory of knots and links of simple closed curves in three-dimensional space. I consist of a selection of topics which graduate students have found to be a successful introduction to the field. Three distinct techniques are employed; Geometric Topology Manoeuvres, Combinatorics, and Algebraic Topology.

Which Springer GTM would you be? The Springer GTM Test

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United We Stand

Today’s quote comes from Tony Smith (who has some noteworthy string theory papers on the CERN server). To summarise an interesting conversation here with Matti Pitkanen, he said

If you map the twistor 5 dimensional hypersurface in CP3 to S5, then you would have a twistor version of my view of string segments as local WorldLines in the Feynman PathIntegral picture. Global WorldLines would follow, as [Michael Rios] said: … the continuous string would be recovered as a long polymer of projective space line segments …

Meanwhile Carl Brannen posts on the snuark mass interaction calculation, based on his version of the measurement algebra. On the cosmological side, Louise Riofrio continues to keep us updated.

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