Abstract
The macroscopic dimensions of space-time should not be input but rather output of a general model for physics. Here, dimensionality arises from a recently discovered mathematical bifurcation: “positive versus indefinite manifold pairings.” It is used to build actions on a “formal chain” of combinatorial space-times of arbitrary dimension. The context for such actions is 2-field theory where Feynman integrals are not over classical, but previously quantized configurations. A topologically enforced singularity of the action can terminate the dimension at four and, in fact, the final fourth dimension is Lorentzian due to light-like vectors in the four dimensional manifold pairing. Our starting point is the action of “causal dynamical triangulations” but in a dimension-agnostic setting. Curiously, some hint of extra compact dimensions emerges from our action.
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Notes
- 1.
In topology it is natural to associate a negative integer as the dimension of the empty set and setting this to be − 1 avoids delaying the nontrivial steps of the construction.
- 2.
Actually we will work with a Euclidean, Wick rotated version of S.
- 3.
See Appendix A for both finite and \({L}^{2}\) sums and pairings.
- 4.
Today a similar situation exists in dimension three. Three-manifolds admit rather disjoint understandings: hyperbolic geometry and Chern-Simons theory linked only weakly by the “volume conjecture.”
- 5.
We use “linearize” not to mean “to approximate by a linear system,” but rather “to replace a set by the complex vector space it spans,” e.g. as in the passage from a category to a linear category.
- 6.
Between Hamiltonian and Lagrangian formalisms.
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Dedicated to the 80th Anniversary of Professor Stephen Smale
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Freedman, M.H. (2012). Quantum Gravity via Manifold Positivity. In: Pardalos, P., Rassias, T. (eds) Essays in Mathematics and its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28821-0_6
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