I read your paper with considerable interest. I am always looking for connections between what I am thinking with others, or between what others are thinking. I posted this on Ray's blog site. Mine essay is at
http://www.fqxi.org/community/forum/topic/494
where I indicate aspects of how exception algebras may work with respect to the cosmological constant.
I have been giving the matter of quantions some study. I am not entirely decided about their status as yet. My sense is they are an interlinking between two complex number or quaternions in a way which defines norms differently. This might have something to do with S-matrix. So I will outline some aspects of S-matrix theory and black hole complementarity, and then try to make possible links to quantions.
The holographic principle and black hole complementarity are generalizations of the S-matrix. Susskind's treatment of strings falling onto a black hole according to a distant observer treats the S-matrix on a domain which is causally defined on an infinite domain of support according to the tortoise version of the radial Schwarzschild coordinate
r* = r - 2m ln|r - 2m|
The S-matrix requires an infinitely extended domain by which fields are causally related, which is "manufactured" by this coordinate. In these coordinates the string exhibits a range of strange behavior, which I am not going to review again in great detail. Yet the string ends up covering the black hole horizon and is frozen their according to this distant observer. To an infalling observer on a commoving frame with the string none of this is the case, but rather the string enters the black holes with no apparent change and then exhibits tidal forces of an extreme nature near the interior of singularity. The string is a form of S-matrix theory, and the two cases reflect the existence of two S-matrices, each according to state space elements which are incommensurate with each other, or according to noncommutative operators. This is one way of looking at the so called black hole complementarity principle. There is then a superposition of the string in these two bases of states, and for this reason the distant observer may see the string frozen above the event horizon and also "burned up" by Hawking radiation made of quanta scattered from the string according to the infalling observer's frame.
The ordered S-matrix defines each vertex, or particle, and its neighbor. In a linear chain a general state is an S-matrix channel of the form
|φ> = |p_1, . . . , p_i, . . . , p_j , . . . , p_n>
This state or S-matrix channel is related to but distinction from the channel
|φ'> = |p_1, . . . , p_j, . . . , p_i, . . . , p_n>
The particles or vertices p_i and p_j have exchanged their neighbors, which means some "relationship" structure to the amplitude has been fundamentally changed. The S-matrix is written according to S = 1 - 2πiT, so two states or channels |p_1, . . ., p_n> and |q_1, . . . , q_n> are related to each other by the S-matrix as
(p_1, . . . , p_n|Sjq_1, . . . , q_n> = (p1, . . . , p_n|(1 - 2πiT)|q_1, . . . , q_n>
= (p_1, . . . , p_n|q_1, . . . , q_n> - 2πi(p_1, . . . , p_n|T|q_1, . . . , q_n>:
For the < | the in channel and | > as the out channel p_n and q_1 are neighbors, and neighbors through the T-matrix. This eliminates an open vertex in the chain. The vertices or particles p_1 and q_n are the open elements in the chain and defines an "anchor" for the chain, and are thus defined as neighbors in this manner.
A four point function and the transition matrix defined by vertex operators T = V(p_1)ΔV(p_3) will contruct the Euler-beta function for coherent states of the S-matrix. This is the connection of course between string theory and the old bootstrap or S-matrix theory. Now for two S-matrices, which pertain to the different domains of causality on a black hole this theory is made more difficult. The S-matrix is a braiding operation of sorts between elements of a quantum group G. So we might model this as a commutator structure (braiding) between two elements a and b \in G. So we might denote this as ab --- ba. Now let us assume the states we observe are super-positions of incommensurate states involving two quantum groups G and G'. We will then have a structure of the sorts (ab)c --- a(bc), that exist in an associahedron I_2(5) with a homotopy structure. This homotopy then connects to a K-theoretic field theory, which I discuss in my paper
http://www.fqxi.org/community/forum/topic/494.
I will not belabor this part of the things, until later or somebody takes an active interest in what I am suggesting here.
The black hole complementarity principle. The complementarity is an odd structure, for Hawking radiation is due to a Bogoliubov transformation between basis elements. In this setting the theory of spacetime is classical and the fields scatter off the black hole or spacetime with an event horizon. The response of the black hole or spacetime is a metric back reaction, which is a classical response to a quantum scattering. Yet black hole complementarity has demonstrated that quantum information is preserved for the case of a BZT black hole in an anti-de Sitter spacetime. So a connection between the quantum principles of unitarity (or maybe more generally modularity) and a classical field theory which exhibits thermal physics (black hole entropy and Bekenstein bounds etc) exists within this AdS/CFT setting. Yet we do not as yet understand how quantum information is preserved. We just know that it is.
So the quantumal rules of Grgin seem to segue into the picture here. The permitted multiplication rules
(fαg)αh (gαh)αf (hαf)αg = 0
gα(fσh) = (gαf)σh fσ(gαh)
(fσg)σh − fσ(gσh) = agα(hαf)
Connects the Jordan exceptional algebra to a quantum algebraic system. The associator is then by the homotopy equivalence mapped to a quantum group as a system of permutations (related permutahedra) with one set of norms determined by the underlying permutative rules or associahedra and the other by standard rules of complex conjugation in quantum mechanics. So the associator is [f, g, h]σ = agα(hαf) which induces the map between the octonions and the quantion group. This seems like an interesting problem to develop.
Cheers LC