The Quantum Structure of Space and Time (293 pages)

Quantum Mechanics 27

Table 2.1 P

Newtonian Physics

Special Relativity

General Relativity

Quantum Gravity

M-theory, Loop

quantum gravity,

Posets, etc.

ihort History of Sp

Fixed 3-d space and

a single universal

time t.

Fixed flat, 4-d

spacetime with

many different

timelike directions.

Fixed, but curved

spacetime geometry.

Geometry is not

fixed, but rather a

quantum variable

Spacetime is not

even a fundamental

variable

etime and Quantum Theory

Non-relativistic Quantum

Theory:

The Schrodinger equation

ifi(a’P/at) = H’P

holds between measurements in the

Newtonian time t.

Relativistic Quantum Field

Theory:

Choose a Lorentz frame with time t.

Then (between measurements)

ifi(C”/at) = H’P.

The results are unitarily equivalent to

those from any other choice of Lorentz

frame.

Quantum Field Theory in Curved

Spacetime:

Choose a foliating family of spacelike

surfaces labeled by t. Then (between

measurements)

ifi(a’P/at) = H’P.

But the results are not generally

unitarily equivalent to other choices.

The Problem of Time:

What replaces the Schrodinger

equation when there is no fixed

notion of time(s)?

?

generalization of the concepts in the usual theory. Certainly the two laws of evolu-
tion (2) and (3) have to be generalized somehow if spacetime geometry is not fixed.
One such generalization is offered in this paper, but there have been many other
ideas [29]. And if spacetime geometry is emergent from some yet more fundamental
description, we can certainly expect that a further generalization - free of any
reference to spacetime ~ will be needed to describe that emergence. The rest of
this article is concerned with these generalizations.

2.1.5

This section reviews, very briefly, the elements of the modern quantum mechanics of

closed systems4 aimed at a quantum mechanics for cosmology. To keep the present

The Quantum Mechanics of Closed Systems

4See, e.g. [5-7] for by now classic expositions at length or [30] for a shorter summary.