The Quantum Structure of Space and Time (293 pages)

(^26) The Quantum Structure of Space and Time
highly non-classical theory of particle dynamics whose statistical predictions largely
coincide with quantum theory [21]. Fundamental noise [22] or spontaneous dynam-
ical collapse of the wave function [23, 241 are the underlying ideas of another class
of model theories whose predictions are distinguishable from those of quantum the-
ory, in principle. Steve Adler has proposed a statistical mechanics of deterministic
matrix models from which quantum mechanics is emergent [25]. Gerard 't Hooft
has a different set of ideas for a determinism beneath quantum mechanism that are
explained in his article in this volume [26]. Roger Penrose has championed a role

for gravity in state vector reduction [27, 281. This has not yet developed into a

detailed alternative theory, but has suggested experimental situations in which the
decay of quantum superpositions could be observed [28, 171.
In the face of an increasing domain of confirmed predictions of quantum the-

ory and the absence as yet of compelling alternatives, it seems natural to extend

quantum theory as far as it will go - to the largest scales of the universe and the

smallest of quantum gravity. That is the course we shall follow in this paper. But

as mentioned in the introduction, usual quantum theory must be generalized to

apply to cosmology and quantum spacetime. We amplify on the reasons in the next

section.

2.1.4 Spacetime and Quantum Theory

Usual, textbook quantum theory incorporates definite assumptions about the nature
of space and time. These assumptions are readily evident in the two laws of evolution
for the quantum state 9. The Schrodinger equation describes its unitary evolution
between measurements.
89
ih-=H9 at

At the time of an ideal measurement, the state is projected on the outcome and

renormalized

The Schrodinger equation (2) assumes a fixed notion of time. In the non-

relativistic theory, t is the absolute time of Newtonian mechanics. In the flat space-

time of special relativity, it is the time of any Lorentz frame. Thus, there are many
times but results obtained in different Lorentz frames, are unitarily equivalent.
The projection in the second law of evolution (3) is in Hilbert space. But in field
theory or particle mechanics, the Hilbert space is constructed from configurations
of fields or position in physical space. In that sense it is the state on a spacelike
surface that is projected (3).
Because quantum theory incorporates notions of space and time, it has changed
as our ideas of space and time have evolved. The accompanying table briefly summa-
rizes this co-evolution. It is possible to view this evolution as a process of increasing