Holographic inflation revised

Introduction The formalism of holographic space-time (HST) is an attempt to write down a theory of quantum gravity which can treat space-times more general than those accessible to traditional string theory. String theory, roughly speaking treats space-times which are asymptotically flat or anti-de Sitter (AdS). As classical space-times, these contain infinite area causal diamonds on which strict boundary conditions are imposed. The corresponding quantum theory has a unique ground state, and the existing formalism describes the evolution of small fluctuations around that ground state in terms of evolution operators involving the infinite set of possible small fluctuations at the boundary. Local physics is obscure in the fundamental formulation of the theory. It emerges only by matching the fundamental amplitudes to those of an effective quantum field theory, in a restricted kinematic regime. In the AdS case, one must also work in a regime where the AdS radius is much larger than the length scale defined by the string tension. That string length scale is bounded below by the Planck length. In regimes where the two scales are close, there are no elementary stringy excitations. The HST formalism works directly with local quantities. Its important properties are summarized as follows: • The fundamental geometrical object, a time-like trajectory in space-time, is described by a quantum system with a time dependent Hamiltonian. Four times the logarithm of the dimension (= entropy) of the Hilbert space of the system is viewed as the quantum avatar of the area of the holographic screen of the maximal causal diamond along the trajectory. The causal diamond associated with a segment of a time-like trajectory is the intersection of the interior of the backward light cone of the future endpoint of the segment, with that of the future light cone of the past point. The holographic screen is the maximal area surface on the boundary of the diamond. When the entropy of Hilbert spaces is large, space-time geometry is emergent. The case of infinite dimension must be treated by taking a careful limit. The Hilbert space comes with a built-in nested tensor factorization: t is a discrete parameter, which labels the length of a proper time interval along the trajectory. is the Hilbert space describing the causal diamond of that interval. describes all operators, which commute with those in the causal diamond.