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


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 …

13 Responses so far »

  1. 1

    Anonymous said,

    Kea, you mention “matrices of matrices”.

    This may be off-topic with respect to quantum groups, but with respect to Jordan algebras and their generalizations, Rosenfeld in his book Geometry of Lie Groups says (page 97) that the Freudenthal algebra Fr(3,O) of E6 automorphisms can be defined as Zorn-type matrices some of whose entries are elements in Jordan algebras.
    On my web page at
    I discuss how this might be extended to algebras for E7 and E8 by using generalized “matrices” that have cube (for E7) or tesseract (for E8) structure.

    Maybe you might find such structures useful in other contexts (however, maybe possibly related) than generalized Jordan and Lie algebras.

    Tony Smith

  2. 2

    Anonymous said,

    Why, Tony, this is in fact exactly the sort of application we have in mind! Thanks for the link.

    Remote Kea

  3. 3

    Matti Pitkanen said,

    Dear Kea and others,

    I understood that finite-dimensional matrix representations of matrices with non-commuting elements are constructed by replacing the elements of matrices with multiples of selected matrices so that the elements cease to commute in particular manner. I wonder whether these matrices could correspond to finite-dimensional representations of quantum groups, say SU(2)_q.

    This brings in mind hyperfinite factors of type II_1 and the inclusion N subset M defining quantum measurement theory with finite measurement resolution characterized by N and with complex rays of state space replaced with N rays. What this really means is far from clear!

    a) Naively one expects that matrices whose elements are elements of N give a representation for M. Now however unit operator has Tr equal to one and one cannot visualize the situation in terms of matrices in case of M and N.

    b) The state space with N resolution would be formally M/N consisting of N rays. For M/N one has finite-D matrices with non-commuting elements of N. In this case quantum matrix elements should be multiplets of selected elements of N, not all possible elements of N. One cannot therefore think in terms of the tensor product of N with M/N regarded as a finite-D matrix algebra. ‘

    c) What does this mean? Obviously one must pose a condition implying that N action commutes with matrix action just like C: this poses conditions on the matrices that one can allow. Connes tensor product does just this.

    The starting point is the Jones inclusion sequence

    N \subset M \subset M \otimes_N M …

    M\otimes_N M is Connes tensor product which can be seen as elements of the ordinary tensor product commuting with N action so that N indeed acts like complex numbers in M. M/N is in this picture represented with M in which operators defined by Connes tensor products of elements of M. The replacement M–> M/N corresponds to the replacement of the tensor product of elements of M defining matrices with Connes tensor product.

    One can try to generalize this picture to zero energy ontology.

    a) M\otimes_N M would be generalized by M_+\otimes M_- M_+ would create positive energy states and second M_- negative energy states and N would create zero energy states in some shorter time scale resolution: this would be the precise meaning of finite measurement resolution.

    b) Connes entanglement with respect to N would define a non-trivial(!) and unique(!) generalization of S-matrix, M-matrix a depending only on pair (N,M_+,M_-).

    I proposed the analog of this already for few years ago but without reference to zero energy ontology. Interactions would be an outcome of a finite measurement resolution and at the never-achievable limit of infinite measurement resolution the theory would be free as I indeed proposed! This would be the counterpart of asymptotic freedom!

  4. 4

    Anonymous said,

    Matti said “… hyperfinite factors of type II_1 and the inclusion N subset … What this really means is far from clear! …”.

    As John Baez said in his week 175:
    “… it’s easy to construct a type II1 factor. Start with the algebra of 1 x 1 matrices, and stuff it into the algebra of 2 x 2 matrices as follows:
    ( x 0 )
    x |-> ( )
    ( 0 x )
    This doubles the trace, so define a new trace on the algebra of 2 x 2 matrices which is half the usual one. Now keep doing this, doubling the dimension each time, using the above formula to define a map from the 2^n x 2^n matrices into the 2^(n+1) x 2^(n+1) matrices, and normalizing the trace on each of these matrix algebras so that all the maps are trace-preserving. Then take the union of all these algebras… and finally, with a little work, complete this and get a von Neumann algebra! … This is … the only II1 factor … that contains a sequence of finite-dimensional von Neumann algebras whose union is dense in the weak topology. A von Neumann algebra like that is called “hyperfinite”, so this guy is called “the hyperfinite II1 factor”.
    … the algebra of 2^n x 2^n matrices is a Clifford algebra, so the hyperfinite II1 factor is a kind of infinite-dimensional Clifford algebra. But the Clifford algebra of 2^n x 2^n matrices is secretly just another name for the algebra generated by creation and annihilation operators on the fermionic Fock space over C^(2n). …”.

    In that case, John Baez is working over the complex numbers and getting complex Clifford algebras Cl(2n,C)
    which have a 2-fold periodicity property
    Cl(2n,C) = Cl(2,C) (x) …(n times tensor)… (x) Cl(2,C)
    so that, as Matti mentioned, the building blocks of the usual hyperfinite II1 factor are 2×2 complex matrices (therefore related to SU(2) and tensor products thereof, and their unions and completion.

    If you do the same tensor-union-completion construction for real Clifford algebras, you have to work with real 8-periodicity
    Cl(8n,R) = Cl(8,R) (x) …(n times tensor)… (x) Cl(8,R)
    and you get a generalized hyperfinite II1 factor whose basic building blocks are Cl(8,R) = 16×16 real matrix algebra,
    which gives you Spin(8) and my physics model that allows you to calculate force strengths, particle masses, K-M parameters, etc.

    At lower-than-Planck energies, octonionic symmetry is broken in that a preferred quaternionic substructure of the octonionic 8-dim spacetime freezes out, giving the M4 x CP2 Kaluza-Klein structure that Matti and I both use.
    The quaternionic version of the real Cl(8) Clifford algebra is the real Cl(2,6) Clifford algebra = 8×8 quaternionic matrix algebra.

    Tony Smith

  5. 5

    Matti Pitkanen said,

    Yes Tony,

    you are right. There are also constructions based on tensor products of matrix algebras of arbitrary dimension and you argument would select Clifford algebras in unique role. These factors are extremely many-faceted.

    Cl(8) appears also in TGD in preferred role and for obvious reasons. I even developed an argument for how M^4xCP_2 emerges purely number theoretically from the condition that associativity defines the fundamental dynamics.

    The symmetry breaking of Poincare times color results in given sector of world of classical worlds by a choice of quantization axes for color and Poincare quantum numbers and this leads also to quantum group symmetries also selecting preferred quantization axes. There is also dual description based on M^8 interpreted as “hyper-octonions” which is not closed algebra under octonion multiplication. The world of classical worlds is union over sectors corresponding to different choices so that these symmetries are exact.

    Well, I cannot prevent myself from repeating that the real beauty of these HFFs is that Connes tensor product allows to reduce S- and density-matrices to the notion of measurement resolution. Just finite measurement resolution as a fundamental dynamical principle! I am afraid that if no one prevents me I will go to the local market place to declare this truth;-). M-matrix is indeed a product of unitary S-matrix and square root of density matrix so that zero energy quantum physics include also thermodynamics.

    The most compelling argument for zero energy ontology is the possibility of describing coherent states of Cooper pairs without giving up fermion number, charge, etc. conservation and automatic emerges of length scale dependent notion of quantum numbers (qnumbers identified as those associated with positive energy factor).


  6. 6

    Anonymous said,

    Matti, I hope that you continue “… repeating that the real beauty of these HFFs is that Connes tensor product allows to reduce S- and density-matrices to the notion of measurement resolution. Just finite measurement resolution as a fundamental dynamical principle! …”.
    Maybe if every day you “… go to the local market place to declare this truth;-) …”
    some day some influential physicist will have sense enough to understand it, and to remove the subject (and those who work on it) from the blacklist, and to teach it to graduate students.

    Tony Smith

    PS – Connes is an influential expert on such factors, and Connes has a blog at
    but it seems to deal more with his particular noncommutative geometry than with other models that use such factors in other ways, so I have not ever posted any comment about such factors there.
    I did see where Carl Brannen posted there on 24 September 2007 a comment (to a post about Galois) mentioning his approach, but now a week later nobody has responded to his comment,
    so that further discourages me from attempting to discuss such things with Connes by commenting there.

  7. 7

    Matti Pitkanen said,

    I posted a couple of comments to Connes site few months ago but did not have courage to look their fate;-)! Incredible this fear of becoming publicly insulted by a person whom one really respects.


  8. 8

    Kea said,

    Off topic, but this is sidesplittingly hilarious: today I’m ranked number 3 on the new site MathBloggers, which seems to be drawing some traffic. LOL! Heh, thanks for the interesting remarks Matti and Tony.

  9. 9

    kneemo said,

    today I’m ranked number 3 on the new site MathBloggers

    Even better, I see you’re ranked number 2!

  10. 10

    CarlBrannen said,

    This is an interesting find, that circulant matrices are related to diagonalized ones by Fourier transform.

    Somehow this seems to get back to the fact that when we Fourier transform QFT, we get momentum space, where things are much easier to calculate. And it certainly puts some explanation into why the Koide mass matrices are circulant.

    By the way, while I haven’t thought of the mass matrices as fundamental, but instead have been working with QFT as the fundamental thing, any mass matrix for 3 distinct masses can be put into circulant form. It is very interesting to hear that this is a way of taking the Fourier transform of the masses. This in itself is worth an immediate paper on arXiv I think.

    This also gives an excuse for why the mass matrices of the baryon resonances are also fairly simple when expressed in circulant form.

    You’ve been writing some amazing things the last few days. (Translation: I could actually understand some of it, your other stuff goes over my head.)

  11. 11

    Kea said,

    This in itself is worth an immediate paper on arXiv I think.

    Why yes, I agree of course. To me the Fourier transform aspect is fundamental, and I think Laurent Freidel would agree with me about that. Care to write the paper and find an endorser?

  12. 12

    CarlBrannen said,

    Kea, you should write the paper. I’m not in academia so I have little motivation for running the scientific publishing gauntlet. You, on the other hand, need to get some stuff out, and besides, it’s your idea.

    I think it’s a good idea, career wise, to get things like this in print now. I have plans to make the baryon formulas well known in 2008. It would be nice to have articles to refer to, this would be one.

    For the endorser, I suggest talking to Koide.

  13. 13

    Kea said,

    All right. I will try.

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