Octonions, trace dynamics and noncommutative geometry—A case for unification in spontaneous quantum gravity

Abstract

Abstract We have recently proposed a new matrix dynamics at the Planck scale, building on the theory of trace dynamics and Connes noncommutative geometry program. This is a Lagrangian dynamics in which the matrix degrees of freedom are made from Grassmann numbers, and the Lagrangian is trace of a matrix polynomial. Matrices made from even grade elements of the Grassmann algebra are called bosonic, and those made from odd grade elements are called fermionic—together they describe an ‘aikyon’. The Lagrangian of the theory is invariant under global unitary transformations and describes gravity and Yang–Mills fields coupled to fermions. In the present article, we provide a basic definition of spin angular momentum in this matrix dynamics and introduce a bosonic(fermionic) configuration variable conjugate to the spin of a boson(fermion). We then show that at energies below Planck scale, where the matrix dynamics reduces to quantum theory, fermions have half-integer spin (in multiples of Planck’s constant), and bosons have integral spin. We also show that this definition of spin agrees with the conventional understanding of spin in relativistic quantum mechanics. Consequently, we obtain an elementary proof for the spin-statistics connection. We then motivate why an octonionic space is the natural space in which an aikyon evolves. The group of automorphisms in this space is the exceptional Lie group G 2 which has 14 generators [could they stand for the 12 vector bosons and two degrees of freedom of the graviton?]. The aikyon also resembles a closed string, and it has been suggested in the literature that 10-D string theory can be represented as a 2-D string in the 8-D octonionic space. From the work of Cohl Furey and others it is known that the Dixon algebra made from the four division algebras [real numbers, complex numbers, quaternions and octonions] can possibly describe the symmetries of the standard model. In the present paper we outline how in our work the Dixon algebra arises naturally and could lead to a unification of gravity with the standard model. From this matrix dynamics, local quantum field theory arises as a low energy limit of this Planck scale dynamics of aikyons, and classical general relativity arises as a consequence of spontaneous localisation of a large number of entangled aikyons. We propose that classical curved space–time and Yang–Mills fields arise from an effective gauging which results from the collection of symmetry groups of the spontaneously localised fermions. Our work suggests that we live in an eight-dimensional octonionic universe, four of these dimensions constitute space–time and the other four constitute the octonionic internal directions on which the standard model forces live.