## Friday, September 3, 2010

### M Theory Lesson 350

A general hyperdeterminant begins with an $n$ dimensional matrix of shape

$(k_1 + 1) \times (k_2 + 1) \times \cdots \times (k_n + 1)$

and the special set $A$ is now the full set of entries $a_{i_{1} i_{2} \cdots i_{n}}$. The first main theorem (page 446) says that the hyperdeterminant of this matrix is non trivial if and only if

$k_{j} \leq \sum_{i \neq j} k_{i}$

for all possible $j$. For a hypercubic matrix, where all the $k_{i}$ are the same, this is certainly true. That is, the hypercube for $n$ qudits always gives us something. The next proposition says that this hyperdeterminant is invariant (in a suitable sense) under what could be called a SLOCC group: $SL(V_1) \times \cdots \times SL(V_{n})$. Another cool result (page 454) tells us that the degree of our hyperdeterminant is given by a sum over partitions $\lambda$

$N(k_1, \cdots, k_{n}) = \sum_{\lambda} (d_{\lambda k}) (1 + m_2 + \cdots + m_p)! \prod_{i = 2}^{p} \frac{(i - 1)^{m_i}}{(m_i)!}$

where the $m_i$ are the indices of the partition $\lambda$ (so that $m_1 = 0$) and $d_{\lambda k}$ counts binary matrices with row sums in the ordered partition $k$ (of the $k_i$) and column sums in $\lambda$. The cubic case $(k + 1)^{3}$ is given by

$N(k,k,k) = \sum 2^{k - 2j} \frac{(j + k + 1)!}{(j!)^{3} (k - 2j)!}$

for a sum over $0 \leq j \leq k/2$. We see that for three qubits, where $k = 1$, there is only one term and it gives the degree $4$ of the classical polynomial.

1. The stuff about the entropy and numbers like 27 is really mind blowing, especially when you consider that it was written long before physicists started to connect hyperdeterminants to entropy. How can anyone not speculate wildly about what is going on when they read such things?

2. Phil, yeah this is bloody amazing stuff. In another (but not unrelated) context, Connes would call it the Tip of The Iceberg ... a sign that everything we know is but a drop in an ocean whose shore is beside us.

I figure the only reason that few physicists have read it (going by Google anyway) is because the mathematicians don't bother telling mere mortals what they are really doing.

3. That is quite right, ea :)

4. It seems that for $V^n$ the hyper-determinant allows recursive definition in terms of the n-linear formula for ordinary determinants gradually reducing $V^k$ to $V^{k-1}$ in given step. I wonder whether anything analogous holds true when dimensions on product are different.

By definition hyper-determinant is a polynomial in its arguments which vanishes only if there exists a non-trivial point at which all partial derivatives of the function defining it vanish.

What is interesting is a possible connection to highly symmetric classical variational principles. For instance, Higgs potentials which are hyper-determinants would by definition allow extrema which are non-trivial and therefore break the symmetry. This kind of model would be very predictive. Since homogenous functions are in question, the extrema allow scale invariance and form a linear sub-space rather than consisting of single point. In SUSYs this happens.

For extremals of classical action all functional derivatives of the action vanish for extremal. Could one formulate interesting variational principles in terms of some kind of infinite-D generalization of hyper-determinant in which one has formally tensor product of the finite-D field space over points of the space-time? Could one consider integrable theories as theories for which action reduces to infinite-D hyper-determinant for some preferred choices of field variables? What could one say about the symmetries of this kind of action?

5. Infinite-D generalizations of hyperdeterminants might be assigned with the vanishing of higher variations of action at criticality. The equations are homogenous in the deformation of the critical extremal and formally multilinear in the deformations at different points of space-time so that one has at least formally an infinite tensor power of field space. Actually the equations reduce to single point in local case and only the appearance of partial derivatives of field variables remains a remnant of non-locality becoming explicit when one discretizes the system.

For vanishing second variation equations are linear and the ordinary determinant of the infinite-D matrix defining the variation vanishes.
For a vanishing third variation equations would be quadratic and one might think that here the vanishing of a generalized hyper-determinant could code for the vanishing of variation for sufficiently symmetric actions.

This kind of hierarchy of generalized hyper-determinants could relate to a critical system in which one has a hierarchy of criticalities. In TGD framework Kahler action with infinite-dimensional vacuum degeneracy defines this kind of hierarchy.

6. I think even matematicians in different areas have completely different languages and it is hard to cross over. In alegbraic geometry and elliptic curve is an abelian variety and a special case of all kinds of other stuff. You are doing well to decipher the book.

7. I found the generating formula they give to be useful for calculating the degrees to higher order. It has a sum over symmetric polynomials in the denominator.

By the way, I was having trouble posting comments here. I think the solution is don't use IE.

8. Sorry about comment moderation, but I get spam and I need to use the spam filters.

9. I added to my blog a little article about possible application of hyper-determinants to the field equations characterizing n:th order criticality which is very relevant notion in TGD Universe due to the infinite-D vacuum degeneracy of Kahler action. These equations are formally multilinear equations in the variation and under special conditions also genuinely multilinear equations. For local action principles genuine multi-linearity generically fails and one obtains non-linear terms.

In TGD framework effective 2-dimensionality however implies genuine multilinearity since the deformations of space-time surface are expressible as non-local functionals of those for partonic 2-surfaces and their tangent space data.

This relates also closely to integrability and absence of local divergences in the functional integral. Hence infinite-D hyper-determinants defined as generalizations of Gaussian determinants might be very relevant for characterizing the occurrence of higher order phase transitions in TGD Universe.

See this.