Note on the Quantum Mechanics of M Theory

TL;DR

The paper argues that M Theory in asymptotically Minkowski space cannot be fully captured by ordinary quantum mechanics because black hole entropy enforces a Planck-scale cutoff on time localization and disrupts the standard spectrum of Heisenberg operators, with a sharp exception in AdS where conventional QM remains viable. It shows that in $d$ noncompact dimensions the high-energy density of states is governed by Schwarzschild black holes, leading to a divergent Fourier transform unless operator matrix elements vanish rapidly, and that in exactly four noncompact dimensions this yields a Hagedorn spectrum undermining a conventional light-cone Hamiltonian. Little string theories, though nonlocal and non-gravitational, also display a Hagedorn spectrum but admit a consistent light-cone DLCQ description, with holographic and DLCQ analyses yielding the same equation of state $E = {M_s \over \sqrt{6k}} S$ and a Hagedorn temperature $T_H = {M_s \over \sqrt{6k}}$. The convergence of holographic and DLCQ results, together with the AdS/CFT correspondence’s positive specific-heat behavior, suggests that flat-space M Theory may require nonstandard quantum descriptions (e.g., DLCQ or little string theory-like frameworks) and invites further exploration of how flat-space physics emerges from AdS/CFT and possible cosmological contexts.

Abstract

We observe that the existence of black holes limits the extent to which M Theory (or indeed any quantum theory of gravity) can be described by conventional quantum mechanics. Although there is no contradiction with the fundamental properties of quantum mechanics, one can prove that expectation values of Heisenberg operators at fixed times cannot exist in an ordinary asymptotic Lorentz frame. Only operators whose matrix elements between the vacuum and energy eigenstates with energy greater than the Planck scale are artificially cut off, can have conventional Green's functions. This implies a Planck scale cutoff on the possible localization of measurements in time. A similar behavior arises also in ``little string theories''. We argue that conventional quantum mechanics in light cone time is compatible with the properties of black holes if there are more than four non-compact flat dimensions, and also with the properties of ``little string theories''. We contrast these observations with what is known about M Theory in asymptotically Anti-de Sitter spacetimes.