Nambu variant of Local Resolution of Problem of Time and Background Independence
arXiv:1908.03168
Abstract
A Local Resolution of the Problem of Time has recently been given, alongside reformulation as A Local Theory of Background Independence. The classical part of this can be viewed as requiring just Lie's Mathematics, albeit entrenched in subsequent topological and differential-geometric developments and extended to contemporary Physics' state spaces. We now widen this approach by mild recategorization to one based on Nambu's generalization of Lie's Mathematics, as follows. i) In this approach, the Lie derivative still suffices to encode Relationalism. ii) Closure is now assessed using the Nambu bracket - with slots rather than 2, so the first nontrivially Lie case has 3 slots - and a `Nambu Algorithm' analogue of the Dirac and Lie Algorithms. This produces a class of Nambu algebraic structures of generators or of first-class constraints. iii) Nambu observables are defined by Nambu brackets zero-commutation with generators or with first-class constraints; we use the Nambu analogue of the Jacobi identity to simplify this discussion relative to a previous treatment. These Nambu brackets relations can moreover be recast as explicit PDEs to be solved using the Flow Method. Nambu observables themselves form Nambu algebras. Lattices of Nambu constraint or generator algebraic substructures furthermore induce dual lattices of Nambu observables subalgebras. iv) Deformation of Nambu algebraic structures encountering Rigidity gives a means of Constructing more structure from less. v) Reallocation of Intermediary-Object Invariance gives the general Nambu algebraic structure's analogue of posing Refoliation Invariance for GR. We also draw some motivation from M-Theory's use of Nambu Mathematics along the lines of Bagger, Lambert and Gustavsson, finding some qualitative distinctions between this, GR and Supergravity as regards how Background Independence is realized.
arXiv comment30 pages, including 3 figures
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