Abstract
Temporal and Configurational Relationalism having each produced constraints, we now need to ascertain whether these form a consistent whole. Thereby the third aspect of Background Independence is Constraint Closure; any impasses with this are the Constraint Closure Problem: the third facet of the Problem of Time. At the classical level, the Dirac algorithm is a powerful and general technique for handling Constraint Closure. In this chapter, we give many examples of its use, as well as recasting it in Temporal Relationalism compatible form. The outcome is a consistent set of machinery for handling all of these first three Problem of Time facets together.
The outcome of Constraint Closure is a closed constraint ‘algebraic structure’ (i.e. algebra or algebroid). We also consider the corresponding lattice of constraint subalgebraic structures. Having two sources of constraints, we furthermore classify blockwise splits of constraints into direct products, semidirect products, Thomas-type integrabilities and two-way integrabilities. General relativity’s constraints’ own Dirac algebroid is of Thomas type. On the other hand, supergravity’s constraint algebroid is of two-way type, by which its linear constraints do not close by themselves. Due to this, a number of notions used in general relativity itself, such as Wheeler’s superspace, and the possibility of treating the quadratic Hamiltonian constraint differently, evaporate.
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Notes
- 1.
While this remains a linear ansatz in \(\boldsymbol{\mathsf{O}}\), this book does not exceed this mandate. Also the \(\boldsymbol{\mathsf{b}}\) are ‘base objects’, which in this book are usually the \(\boldsymbol{\mathsf{Q}}\), whether or not accompanied by \(\boldsymbol{\mathsf{P}}\), and the \(c\) are constants.
- 2.
Whereas this statement may look innocuous, in Part III we shall see that its quantum counterpart is not. Moreover, classical-level considerations themselves need to justify why Poisson brackets are in use rather than e.g. i) Lagrange, ii) Peierls, iii) Schouten–Nijenhuis, iv) Nambu and associator brackets. Brief answers are as follows [32]. Poisson brackets are more convenient than i) and ii). We do not consider iii) since this corresponds to multisymplectic formulations, in which time and space are treated on an even more common footing than spacetime co-geometrization. I.e. here the usual time derivative based momenta are accompanied by spatial derivative based analogues. This is motivated by the introduction of the notion of spacetime being taken to imply necessity of joint treatment of further spatial and temporal notions. However, this motivation runs contrary to Broad’s point that a formulation breaking isolation between space and time does not imply an end to the distinction between these notions. Once this is understood, the advent of spacetime clearly does not imply any necessity to replace the standard temporally-distinguished symplectic formulation with a time-and-space covariant multisymplectic one. Finally, not using iv) follows from second-order theory alongside noncommutative but associative Quantum Theory sufficing for most purposes.
- 3.
Rigged phase space \(\boldsymbol{\mathfrak{R}}\mbox{ig-}\boldsymbol{\mathfrak{P}}\text{hase}\) is a more minimalistic alternative at this stage [37]. This corresponds to specifying that the physical \(\boldsymbol{\mathsf{Q}}\) are distinguishable from the corresponding \(\boldsymbol{\mathsf{P}}\). Moreover, the nontrivial (\(\boldsymbol{\mathsf{P}}\) and \(\boldsymbol{\mathsf{Q}}\) mixing) canonical transformations do not preserve this additional structure. The corresponding morphisms are now the group \(\mathit{Point}\) of \(\mathfrak{q}\)-morphisms, which is much smaller than \(\mathit{Can}\). This does not affect the type of brackets in use.
- 4.
One might augment this to a quadruple by considering varying the type of group action of \(\mathfrak{g}\) on \(\mathfrak{T}(\mathfrak{q})\).
- 5.
Again, we drop the spinorial index in this book’s schematic presentation. This constraint is also accompanied by a conjugate constraint.
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Anderson, E. (2017). Brackets, Constraints and Closure. In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_24
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