From phase space to multivector matrix models

Abstract

Combining elements of twistor-space, phase space and Clifford algebras, we propose a framework for the construction and quantization of certain (quadric) varieties described by Lorentz-covariant multivector coordiantes. The correspondent multivectors can be parametrized by second order polynomials in the phase space. Thus the multivectors play a double role, as covariant objects in D " 2, 3, 4 Mod 8 space-time dimensions, and as mechanical observables of a non-relativistic system in 2 rD{2s´1 euclidean dimensions. The latter attribute permits a dual interpretation of concepts of non-relativistic mechanics as applying to relativistic space-time geometry. Introducing the Groenewold-Moyal ›-product and Wigner distributions in phase space induces Lorentz-covariant non-commutativity and it provides the spectra of geometrical observables. We propose also new (multivector) matrix models, interpreted as descending from the interaction term of a Yang-Mills theory with minimally coupled massive fermions, in the large-N limit, which serves as a physical model containing the constructed multivector (fuzzy) geometries. We also include a section on speculative aspects on a possible cosmological effect and the origin of space-time entropy.

Higher Spin Matrix Models

“…We constructed and provided solutions for a particular model related to type IIB strings, therefore called type IIB higher spin gravity. The matrix models constructed here extend those found in [33] in that now spacetime dimensions are added. An interesting aspect of these matrix models is that they incorporate coordinates of extended objects, as multivector coordinates of rank k are related to k-dimensional objects.…”

“…An interesting aspect of these matrix models is that they incorporate coordinates of extended objects, as multivector coordinates of rank k are related to k-dimensional objects. For instance, part of the rigid equations of motion for (multivector) matrix model in 3`1 dimensions (without fermions and for a constant scalar field) can be reduced to the form Some solutions of this system contain Plucker coordinates of planes through 3`1 dimensions as shown in [33]. An interesting problem to address is whether the classical limit of the respective matrix models (24)- (27), or Label (24), Label (25), and Labels (39)- (41), reproduce Polyakov's string theory in 3`1 dimensions with fine structure [35].…”

“…The system above is formally integrable, but the choices of the algebras involved and of the dimension of the space-time is an engineering problem, it depends of what we want to describe. In what follows we shall solve the system (24)- (27) employing similar techniques than in reference [33].…”

“…The rigid Equations (26) and (27) are written in a form that reminds us of the type IIB matrix model field Equations (7)- (23), but, as we shall justify, the fields involved admit more general labels, i.e., with I labeling target space Lorentz multivectors instead of just vectors. As it was shown in [33], the phase space monomials y αp2q (α " 1, ..., 2 rD{2s ), in the classical level, parametrize an algebraic variety (say M) whose coordinates are conveniently labeled by Lorentz multivectors in D dimensions. M admits a covariant non-commutative deformation (quantization), i.e., introducing the ›-product (22) in the algebra of functions in the space M, which is Lorentz covariant.…”

“…[33]. For that purpose, we can use the identity where νrns pxq are functions of the point x of the spacetime, so that W " g›dg´1, Y µrns " g›X µrns ›g´1, Ψ " 0, Φ "˘θ 2 2…”