Ninguno

DOES TIME EXIST IN QUANTUM GRAVITY?

Claus Kiefer
Institute for Theoretical Physics,
University of Cologne,
Z¨ lpicher Strasse 77, 50937 K¨ ln, Germany.
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http://www.thp.uni-koeln.de/gravitation/

Abstract
Time is absolute in standard quantum theory and dynamical in
general relativity. The combination of both theories into a theory
of quantum gravity leads therefore to a “problemof time”. In my
essay I shall investigate those consequences for the concept of time
that may be drawn without a detailed knowledge of quantum gravity.
The only assumptions are the experimentally supported universality
of the linear structure of quantum theory and the recovery of general
relativity in the classical limit. Among the consequences are the fundamental timelessness of quantumgravity, the approximate nature
of a semiclassical time, and the correlation of entropy with the size
of the Universe.

1 Time in Physics
On December 14, 1922, Albert Einstein delivered a speech to students and
faculty members of Kyoto University in which he summarized how he created his theories of relativity [1]. As for the key idea in finding special relativity in 1905, he emphasized: “Ananalysis of the concept of time was my
solution.” He was then able to complete his theory within five weeks.
An analysis of the concept of time may also be the key for the construction of a quantum theory of gravity. Such a hope is supported by the fact
that a change of the fundamental equations in physics is often accompanied by a change in the notion of time. Let me briefly review the history
oftime in physics.
Before Newton, and thus before the advent of modern science, time
was associated with periodic motion, notably the motion of the ‘Heavens’.
It was therefore a countable time, each tick corresponding to one period;
there was no idea of a continuum.
1

It was Newton’s great achievement to invent the notion of an absolute
and continuous time. Such a concept was needed for theformulation
of his laws of mechanics and universal gravitation. Although Newton’s
concepts of absolute space and absolute time were heavily criticized by
some contemporaries as being unobservable, alternative relational formulations were only constructed after the advent of general relativity in
the 20th century [2].
In Einstein’s theory of special relativity, time was unified with space toform a four-dimensional spacetime. But this “Minkowski spacetime” still
constitutes an absolute background in the sense that there is no reactio
of fields and matter – Minkowski spacetime provides only the rigid stage
for their dynamics. Einstein considered this lack of back reaction as very
unnatural.
Minkowski spacetime provides the background for relativistic quantum field theory and theStandard Model of particle physics. In the nonrelativistic limit, it yields quantum mechanics with its absolute, Newtonian time t. This is clearly seen in the Schr¨ dinger equation,
o
i

∂ψ
ˆ
= Hψ .
∂t

(1)

It must also be noted that the presence of t occurs on the left-hand side
of this equation together with the imaginary unit, i; this fact will become
important below. In relativisticquantum field theory, (1) is replaced by its
functional version.
The Schr¨ dinger equation (1) is, with respect to t, deterministic and
o
time-reversal invariant. As was already emphasized by Wolfgang Pauli,
the presence of both t and i are crucial for the probability interpretation
of quantum mechanics, in particular for the conservation of probability
in time.
But the story is not yetcomplete. It was Einstein’s great insight to see
that gravity is a manifestation of the geometry of spacetime; in fact, gravity is geometry. This led him to his general theory of relativity, which he
completed in 1915. Because of this identification, spacetime is no longer
absolute, but dynamical. There is now a reactio of all matter and fields
onto spacetime and even an interaction of spacetime...