Nijenhuis-type variants of Local Theory of Background Independence
arXiv:1908.00193
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 Topology and Differential Geometry developments and extended to the setting of contemporary Physics' state spaces. We now generalize this approach by mild recategorization to one based on Nijenhuis' generalization of Lie's Mathematics, as follows. 1) Relationalism is encoded using the Nijenhuis-Lie derivative. 2) Closure is assessed using the Schouten-Nijenhuis bracket, and a `Schouten-Nijenhuis Algorithm' analogue of the Dirac and Lie Algorithms. This produces a class of Gerstenhaber algebraic structures of generators or of constraints. 3) Observables are defined by a Schouten--Nijenhuis brackets relation, reformulating the constrained canonical case as explicit PDEs to be solved using the Flow Method, and forming their own Gerstenhaber algebras of observables. Lattices of Schouten-Nijenhuis-Gerstenhaber constraint or generator algebraic substructures furthermore induce dual lattices of Gerstenhaber observables subalgebras. 4) Deformation of Gerstenhaber algebraic structures of generators or constraints encountering Rigidity gives a means of Constructing more structure from less. 5) Reallocation of Intermediary-Object Invariance gives the general Schouten-Nijenhuis-Gerstenhaber algebraic structure's analogue of posing Refoliation Invariance for GR. We finally point to general Gerstenhaber bracket and Vinogradov bracket generalizations, with the former likely to play a significant role in Backgound-Independent Deformation Quantization and Quantum Operator Algebras.
4 pages
References in corpus (14)
- A Local Resolution of the Problem of Time
- Theory of Transformation Groups, by S. Lie and F. Engel (Vol. I, 1888). Modern Presentation and English Translation
- A Local Resolution of the Problem of Time. II. Configurational Relationalism via a generalization of Group Averaging
- Shape Theory. III. Comparative Theory of Backgound Independence
- A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
- A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism
- A Local Resolution of the Problem of Time. IV. Quantum outline and piecemeal Conclusion
- A Local Resolution of the Problem of Time. V. Combining Temporal and Configurational Relationalism for Finite Theories
- A Local Resolution of the Problem of Time. IX. Spacetime Constructability
- A Local Resolution of the Problem of Time. VII. Constraint Closure
- Shape Theories. I. Their Diversity is Killing-Based and thus Nongeneric
- Shape Theories. II. Compactness Selection Principles
- Problem of Time and Background Independence: classical version's higher Lie Theory
- A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics