Mass in motivic quantum gravity

From the proceedings of Group32, the 32nd International Colloquium on Group Theoretical Methods in Physics (9–13 July 2018, Prague):

M D Sheppeard

Abstract: A physical approach to a category of motives must account for the emergent nature of spacetime, where real and complex numbers play a secondary role to discrete operations in quantum computation. In quantum logic, the cardinality of a set is initially replaced by a dimension of a linear space, making contact with the increasing dimensions in an operad. The operad of associahedra governs tree level scattering, and is closely related to the permutohedra and cube tiles, where cube vertices can encode components of a spinor in higher dimensional octonionic approaches. A study of rest mass generation begins with the cosmological infrared scale, set by the neutrino masses, and its related see-saw mechanism. We employ the anyonic ribbon spectrum for Standard Model states, and consider its relation to magic star algebras, giving a context for the Koide rest mass phenomenology of charged leptons and quarks.



There are at least two things to discuss here - how the emergence of space-time works in this kind of model, and how specific mass matrices are obtained - because this author is not just aiming to construct an emergent physics, but to connect with phenomenology.

Regarding emergence, one can see just from the abstract that this is supposed to involve scattering polytopes (i.e. like the amplituhedron). That kind of framework is like the S-matrix in ordinary quantum field theory, but with a step up in abstraction.

The "S-matrix" (S for scattering) is a matrix whose rows and columns correspond to asymptotic states, and whose elements are transition amplitudes. If we begin with m particles of specified kinds coming in from "infinity" (i.e. so far separated that they are initially experiencing zero interactions), and after their interaction, we have n particles of specified kinds moving out to infinity (i.e. once again, an asymptotic state in which the particles are too far apart to interact)... then the S-matrix gives us an amplitude for that incoming->outgoing transition.

This is a ubiquitous and completely standard perspective in particle physics. Normally, the amplitudes in the S-matrix can be obtained by some kind of path integral, i.e. a Feynman sum over space-time histories of interaction. However, these polytope methods obtain the scattering amplitudes through constructions that don't involve space-time. Rather, there's some other kind of object - such as a geometric object with one dimension for each incoming and outgoing particle, which you then integrate over - from which the amplitudes can be calculated.

Meanwhile, what are "motives"? It is an algebraic concept that I won't try to define at this point. But I will mention category theory as more people will know about that. Category theory is a highly abstract approach to mathematics, which begins with "objects" and "arrows" between them. The utility of category theory comes about because innumerable aspects of real mathematics, like the representation theory of groups, can fit into this framework. I understand a motive to be a kind of category of algebraic objects, so the "arrows" will be tracking associations between algebraic objects, like the mapping between elliptic curves and modular forms that proved Fermat's last theorem.

A categorical or motivic approach to scattering amplitudes, is going to express the amplitudes for going from 'initial condition' to 'final condition', in terms of arrows between objects. For example, the calculations that arise with polytope methods, should belong to particular algebraic categories. That's all I will say about the motivic aspect for now, except to remark that there is an existing literature on motives in scattering theory.

Regarding the other topic, mass matrices, the abstract tells us that a kind of seesaw mechanism is proposed. That's a standard thing although I don't know how it will fit into the framework here. But I will mention one aspect of the paper that I do recognize. This is an association of braids and matrices.

A braid (in the topological sense) is a generalized permutation. The threads of the braid are permuted, but the braid carries extra information in the form of path dependence, just as spinor transformations are rotations with some extra path information. Braids actually form a group, and groups can be represented by matrices, so it seems part of the idea here is to bridge algebra and geometry through a categorical study of matrix representations of braids. I don't yet understand how that works in the scattering context, though I suppose a scattering formalism should be able to represent single-particle states as a kind of trivial case of scattering (one particle in, same particle out).
 

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