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.
$X \circ Y = Z$
$Y \circ Z = X$
$Z \circ X = Y$
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.
posted by Kea | 6:26 PM
8 Comments:
Congratulations on reaching Post 3^2*5^2! Thanks for the comment on Lumo's blog. Talk of a certain theory has reached so many threads that one has trouble keeping track of them all.
Hi Louise. Of course, Mottle assures me that we're completely wrong and stupid to disagree with the string theory viewpoint. But I agree that more and more people do seem interested in better theories.
The sqrt(2) in the denominators of the sigma_x and sigma_y operators is an obvious typo. These operators have eigenvalues of +-1.
On the other hand, the sqrt(2) does need to divide the eigenvector, if the eigenvector is to be displayed in normalized form.
And when you fix this, go ahead and delete this comment.
Hi Carl. Yes, I thought of putting the 1/sqrt(2) with the vectors, but there seems to be something nice about defining the operators this way, and it all still works, unless I've made some silly errors.
Hmmmmmmm.
But if you put the sqrt(2)s on the operators, then the multiplication rules you're advertising don't work any more. You end up wrong by factors of sqrt(2) or 2.
Yeah, OK, I changed it.
Work is progressing on the meson paper. A new result:
Let X, Y, and Z be three bases that are mutually unbiased with respect to each other, for instance from the Pauli algebra.
Consider the algebra generated by complex multiples of products of the projection operators (density matrix states) of these three bases.
A basis for the algebra consists of all the products uv where u is from one of the bases and v is from another, PLUS all the raising and lowering operators.
For example, if z and /z are the projection operators for spin in the z direction, and x and y are the projection operators for spin in the x and y directions, then
z x /z and
z y /z
are two raising operators for Z.
Yes, eventually we should have a nice picture of a modern bootstrap mechanism for QFT.
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