A "general boundary" formulation for quantum mechanics and quantum gravity

TL;DR

The paper introduces a general boundary formulation that assigns boundary state spaces $\mathcal{H}_S$ and region amplitudes $\rho_M$ to space-time regions, enforcing a holographic, composition-consistent structure akin to a TQFT. Through holographic quantization, it defines $\mathcal{H}_S=C(K_S)$ and $\rho_M$ via a boundary-path integral, showing that both NRQM and scalar QFT arise within this framework and that NRQM inherently includes multi-particle sectors and pair creation tendencies. It further connects the approach to quantum gravity by noting that 3D gravity is a TQFT realizable in this scheme and outlining a path to non-perturbative 4D gravity via spin foam models, all within a local, boundary-centric, covariant formalism. This formulation aims to unify quantum theories under a covariant, boundary-based paradigm with potential progress toward a complete quantum gravity theory.

Abstract

I propose to formalize quantum theories as topological quantum field theories in a generalized sense, associating state spaces with boundaries of arbitrary (and possibly finite) regions of space-time. I further propose to obtain such ``general boundary'' quantum theories through a generalized path integral quantization. I show how both, non-relativistic quantum mechanics and quantum field theory can be given a ``general boundary'' formulation. Surprisingly, even in the non-relativistic case, features normally associated with quantum field theory emerge from consistency conditions. This includes states with arbitrary particle number and pair creation. I also note how three dimensional quantum gravity is an example for a realization of both proposals and suggest to apply them to four dimensional quantum gravity.

Figures (2)

• Figure 1: A contribution to the path integral over $R=A\cup B$ between boundaries $S_1$ and $S_2$. The inner region $B$ is crossed twice, corresponding to a two-particle state on its boundary $S$.
• Figure 2: Allowing to "forget" connectivities introduces pair creation and annihilation. The connections $(a,b)$ and $(c,d)$ on the left are changed to $(a,d)$ and $(c,b)$ on the right.