Tuesday, November 20, 2007

The Lie-algebra of symmetries of M-matrix forms an infinite-dimensional Jordan algebra

The notion of Jordan algebra was born as a mathematization of algebra of observables. Also Lie algebra can be seen as this kind of mathematization of the notion of observable. The linear combinations of Hermitian operators with real coefficients are Hermitian and define an algebra under the product A*B= (AB+BA)/2, which is commutative and non-associative but satisfies the weaker associativity condition (xy)(xx)= x(y(xx)).

There exists four infinite families of Jordan algebras plus one exceptional Jordan algebra. The finite-dimensional real, complex, and quaternionic matrix algebras with product defined as above are Jordan algebras. Also the Euclidian gamma matrix algebra defined by Euclidian inner product and with real coefficients is Jordan algebra and known as so called spin factor: now the commutativity is not put in by hand. The exceptional Jordan algebra consists of a real linear space of Hermitian 3× 3 matrices with octonionic coefficients and with symmetrized product.

1. The notion of finite measurement resolution leads to infinite-dimensional Jordan algebra in TGD framework

Hermitian operators of N subset M, where N and M are hyperfinite factors of type II1 and N specifies the measurement resolution, act as maximal symmetries of M-matrix so that finite measurement resolution corresponds to an infinite-D symmetry group and Jordan algebra corresponds now to operators whose action has no detectable physical effect rather than algebra of observables. A so called Hermitian Jordan algebra is in question. Of course, also Lie-algebra commutator i(AB-BA) defines a Hermitian operator in N.

The maximal symmetries of M-matrix mean that the Hermitian generators of the algebra define a generalization of finite-dimensional Jordan algebra. The condition that all Hermitian operators involved are finite-dimensional brings in mind the definition of the permutation group S as consisting of finite permutations only and also the definition of infinite-dimensional Clifford algebra. Thus the natural interpretation of the algebra in question would be as maximal possible dynamical gauge symmetry implied by the finite measurement resolution. The active symmetries would be analogous to global gauge transformations and act non-trivially on all tensor factors in tensor product representation as a tensor product of 2× 2 Clifford algebras.

Quaternionic Jordan algebra is natural in TGD framework since 2× 2 Clifford algebra reduces to complexified quaternions and contains as sub-algebras real and complex Jordan algebras. Also Clifford algebra of world of classical worlds is a generalized Jordan algebra.

2. Octonions and TGD

There are intriguing hints that octonions might be important for TGD.

  1. U(1), SU(2), and SU(3) are the factors of standard model gauge group and also the natural symmetries of minimal Jordan algebras relying on complex numbers, quaternions, and octonions. These symmetry groups relate also naturally to the geometry of CP2.

  2. 8-D Clifford algebra allows also octonionic representation.

  3. The idea that one could make HFF of type II1 a genuine local algebra analogous to gauge algebra can be realized only if the coordinate is non-associative since otherwise the coordinate can be represented as a tensor factor represented by a matrix algebra. Octonionic coordinate means an exception and would make 8-D imbedding space unique in that it would allow local version of HFF of type II1.

  4. These observations partially motivate a nebulous concept that I have christened HO-H duality (see this) - admittedly a rather speculative idea -stating that TGD can be formulated alternatively using hyper-octonions (subspace of complexified octonions with Minkowskian signature of metric) as the imbedding space and assuming that the dynamics is determined by the condition that space-time surfaces are hyper-quaternionic or co-hyper-quaternionic (and thus associative or co-associative). Associativity condition would determine the dynamics.

3. What about the octonionic Jordan algebra?

The question is therefore whether also 3×3 octonionic Jordan algebra might have some role in TGD framework.

  1. Suppose for a moment that the above interpretation for the Hermitian operators as elements of a sub-factor N defining the measurement resolution generalizes also to the case of octonionic state space and operators represented as octonionic matrices. Also the direct sums of octonion valued matrices belonging to the octonionic Jordan algebra define a Jordan algebra and included algebras would now correspond to direct sums for copies of this Jordan algebra. One could perhaps say that the gauge symmetries associated with octonionic N would reduce to the power SU(3)on= SU(3)o× SU(3)o×... of the octonionic SU(3) acting on the fundamental triplet representation.

  2. Triplet character is obviously problematic and one way out could be projectivization leading to the octonionic counterpart of CP2. Octonionic scalings should not affect the physical state so that physical states as octonionic rays would correspond to octonionic CPn. It is not however possible to realize the linear superposition of quantum states in CPn. The octonionic (quaternionic) counterpart of CP2 would be 2× 8-dimensional and U(2)o would act as a matrix multiplication in this space. Realizing associativity (commutativity) condition for 2× 8 spinors defined by octonionic CP2 by replacing octonions with quaternions (complex numbers) would give 2× 4-dimensional (2× 2-dimensional) space.

  3. The first question is whether CP2 as a factor of imbedding space could somehow relate to the octonionic Jordan algebra. Could one think that this factor relates to the configuration space degrees of freedom assignable to CP2 rather than Clifford algebra degrees of freedom? That color does not define spin like quantum numbers in TGD would conform with this. Note that the partial waves associated S2 associated with light-cone boundary would correspond naturally to SU(2) and quaternionic algebra.

  4. Second question is whether the HFF of type II1 could result from its possibly existing octonionic generalization by these two steps and whether the reduction of the octonionic symmetries to complex situation would give SU(3)× SU(3)... reducing to U(2)× U(2)× .... The Lie-algebra of symmetries of M-matrix forms a Jordan algebra.

For a background see the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix".


CarlBrannen said...

Matti; This is, unfortunately for me, over my head. Does this have anything to do with the fact that E8 is the algebra of its own symmetry group?

Matti Pitkänen said...

Well, I am not quite sure which part of posting went over head.

The definition of Hermitian Jordan algebra is simple. You have observables represented by Hermitian operators. Take symmetrized product of observables so that the outcome is unique despite non-commutativity. Associativity is lost and replaced with a weaker property.

The construction of M-matrix as Jordan product implies that Hermitian operators of subalgebra defining resolution (action creates states not distinguishable from original) acts as symmetries and thus commute with M-matrix. This is just standard stuff.

Also the symmetrized product is hermitian element of algebra of resolution so that resolution symmetries form both Lie- and Jordan algebra, which is infinite-dimensional in appropriate sense.

Whether a Jordan algebra of 3x3 octonionic matrices is really involved remains an open question and I am not at all convinced by my arguments;-)! I cannot answer your question about E_8. Tony could certainly tell more about the realtion of E_8 to Jordan algebra.

The Octonions of John Baez contains also material about this. E_8 acts as isometries of octo-octonionic projective plane which could be seen as counterpart of CP_2 with complex numbres replaced with complexified octonions. Complexified octonions appear in the OH duality: M^8 is identified as 8-D plane of complexified octonions consisting of numbers x+ i*y, y imaginary octonion, i commuting imaginary unit. Hence E_8 symmetry would necessarily break down.