Unfinished revolution
C . ROVELLI
One hundred and forty-four years elapsed between the publication of Copernicus’s
De Revolutionibus, which opened the great scientific revolution of the seventeenth
century, and the publication of Newton’s Principia, the final synthesis that brought
that revolution to a spectacularly successful end. During those 144 years, the basic
grammar for understanding the physical world changed and the old picture of
reality was reshaped in depth.
At the beginning of the twentieth century, General Relativity (GR) and Quantum
Mechanics (QM) once again began reshaping our basic understanding of space and
time and, respectively, matter, energy and causality – arguably to a no lesser extent.
But we have not been able to combine these new insights into a novel coherent
synthesis, yet. The twentieth-century scientific revolution opened by GR and QM
is therefore still wide open. We are in the middle of an unfinished scientific revolution.
Quantum Gravity is the tentative name we give to the “synthesis to be
found”.
In fact, our present understanding of the physical world at the fundamental level
is in a state of great confusion. The present knowledge of the elementary dynamical
laws of physics is given by the application of QM to fields, namely Quantum
Field Theory (QFT), by the particle-physics Standard Model (SM), and by GR.
This set of fundamental theories has obtained an empirical success nearly unique
in the history of science: so far there isn’t any clear evidence of observed phenomena
that clearly escape or contradict this set of theories – or a minor modification of
the same, such as a neutrino mass or a cosmological constant.1 But, the theories in
this set are based on badly self-contradictory assumptions. In GR the gravitational
field is assumed to be a classical deterministic dynamical field, identified with the
(pseudo) Riemannian metric of spacetime: but with QM we have understood that
all dynamical fields have quantum properties. The other way around, conventional
1 Dark matter (not dark energy) might perhaps be contrary evidence.
Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. Daniele Oriti.
Published by Cambridge University Press. c Cambridge University Press 2009.
4 C. Rovelli
QFT relies heavily on global Poincaré invariance and on the existence of a
non-dynamical background spacetime metric: but with GR we have understood
that there is no such non-dynamical background spacetime metric in nature.
In spite of their empirical success, GR and QM offer a schizophrenic and confused
understanding of the physical world. The conceptual foundations of classical
GR are contradicted by QM and the conceptual foundation of conventional QFT
are contradicted by GR. Fundamental physics is today in a peculiar phase of deep
conceptual confusion.
Some deny that such a major internal contradiction in our picture of nature exists.
On the one hand, some refuse to take QM seriously. They insist that QM makes no
sense, after all, and therefore the fundamental world must be essentially classical.
This doesn’t put us in a better shape, as far as our understanding of the world is
concerned.
Others, on the other hand, and in particular some hard-core particle physicists, do
not accept the lesson of GR. They read GR as a field theory that can be consistently
formulated in full on a fixed metric background, and treated within conventional
QFT methods. They motivate this refusal by insisting than GR’s insight should not
be taken too seriously, because GR is just a low-energy limit of a more fundamental
theory. In doing so, they confuse the details of the Einstein’s equations (which
might well be modified at high energy), with the new understanding of space and
time brought by GR. This is coded in the background independence of the fundamental
theory and expresses Einstein’s discovery that spacetime is not a fixed background,
as was assumed in special relativistic physics, but rather a dynamical field.
Nowadays this fact is finally being recognized even by those who have long
refused to admit that GR forces a revolution in the way to think about space and
time, such as some of the leading voices in string theory. In a recent interview
[1], for instance, Nobel laureate David Gross says: “ [...] this revolution will likely
change the way we think about space and time, maybe even eliminate them completely
as a basis for our description of reality”. This is of course something that
has been known since the 1930s [2] by anybody who has taken seriously the problem
of the implications of GR and QM. The problem of the conceptual novelty of
GR, which the string approach has tried to throw out of the door, comes back by
the window.
These and others remind me of Tycho Brahe, who tried hard to conciliate Copernicus’s
advances with the “irrefutable evidence” that the Earth is immovable at the
center of the universe. To let the background spacetime go is perhaps as difficult
as letting go the unmovable background Earth. The world may not be the way it
appears in the tiny garden of our daily experience.
Today, many scientists do not hesitate to take seriously speculations such as
extra dimensions, new symmetries or multiple universes, for which there isn’t a
Unfinished revolution 5
wit of empirical evidence; but refuse to take seriously the conceptual implications
of the physics of the twentieth century with the enormous body of empirical evidence
supporting them. Extra dimensions, new symmetries, multiple universes and
the like, still make perfectly sense in a pre-GR, pre-QM, Newtonian world,
while to take GR and QM seriously together requires a genuine reshaping of our
world view.
After a century of empirical successes that have equals only in Newton’s and
Maxwell’s theories, it is time to take seriously GR and QM, with their full conceptual
implications; to find a way of thinking the world in which what we have
learned with QM and what we have learned with GR make sense together – finally
bringing the twentieth-century scientific revolution to its end. This is the problem
of Quantum Gravity.
1.1 Quantum spacetime
Roughly speaking, we learn from GR that spacetime is a dynamical field and we
learn from QM that all dynamical field are quantized. A quantum field has a granular
structure, and a probabilistic dynamics, that allows quantum superposition of
different states. Therefore at small scales we might expect a “quantum spacetime”
formed by “quanta of space” evolving probabilistically, and allowing “quantum
superposition of spaces”. The problem of Quantum Gravity is to give a precise
mathematical and physical meaning to this vague notion of “quantum spacetime”.
Some general indications about the nature of quantum spacetime, and on
the problems this notion raises, can be obtained from elementary considerations.
The size of quantum mechanical effects is determined by Planck’s constant . The
strength of the gravitational force is determined by Newton’s constant G, and the
relativistic domain is determined by the speed of light c. By combining these three
fundamental constants we obtain the Planck length lP =
G/c3 ∼ 10−33 cm.
Quantum-gravitational effects are likely to be negligible at distances much larger
than lP, because at these scales we can neglect quantities of the order of G, or 1/c.
Therefore we expect the classical GR description of spacetime as a pseudo-
Riemannian space to hold at scales larger than lP, but to break down approaching
this scale, where the full structure of quantum spacetime becomes relevant. Quantum
Gravity is therefore the study of the structure of spacetime at the Planck
scale.
1.1.1 Space
Many simple arguments indicate that lP may play the role of a minimal length, in
the same sense in which c is the maximal velocity and the minimal exchanged
action.
6 C. Rovelli
For instance, the Heisenberg principle requires that the position of an object of
mass m can be determined only with uncertainty x satisfying mvx > , where v
is the uncertainty in the velocity; special relativity requires v < c; and according
to GR there is a limit to the amount of mass we can concentrate in a region of size
x, given by x > Gm/c2, after which the region itself collapses into a black hole,
subtracting itself from our observation. Combining these inequalities we obtain
x > lP. That is, gravity, relativity and quantum theory, taken together, appear to
prevent position from being determined more precisely than the Planck scale.
A number of considerations of this kind have suggested that space might not be
infinitely divisible. It may have a quantum granularity at the Planck scale, analogous
to the granularity of the energy in a quantum oscillator. This granularity of
space is fully realized in certain Quantum Gravity theories, such as loop Quantum
Gravity, and there are hints of it also in string theory. Since this is a quantum
granularity, it escapes the traditional objections to the atomic nature of space.
1.1.2 Time
Time is affected even more radically by the quantization of gravity. In conventional
QM, time is treated as an external parameter and transition probabilities change
in time. In GR there is no external time parameter. Coordinate time is a gauge
variable which is not observable, and the physical variable measured by a clock is
a nontrivial function of the gravitational field. Fundamental equations of Quantum
Gravity might therefore not be written as evolution equations in an observable time
variable. And in fact, in the quantum-gravity equation par excellence, the Wheeler–
deWitt equation, there is no time variable t at all.
Much has been written on the fact that the equations of nonperturbative Quantum
Gravity do not contain the time variable t. This presentation of the “problem of
time in Quantum Gravity”, however, is a bit misleading, since it mixes a problem
of classical GR with a specific Quantum Gravity issue. Indeed, classical GR as
well can be entirely formulated in the Hamilton–Jacobi formalism, where no time
variable appears either.
In classical GR, indeed, the notion of time differs strongly from the one used in
the special-relativistic context. Before special relativity, one assumed that there is
a universal physical variable t, measured by clocks, such that all physical phenomena
can be described in terms of evolution equations in the independent variable t.
In special relativity, this notion of time is weakened. Clocks do not measure a universal
time variable, but only the proper time elapsed along inertial trajectories. If
we fix a Lorentz frame, nevertheless, we can still describe all physical phenomena
in terms of evolution equations in the independent variable x0, even though this
description hides the covariance of the system.
Unfinished revolution 7
In general relativity, when we describe the dynamics of the gravitational field
(not to be confused with the dynamics of matter in a given gravitational field),
there is no external time variable that can play the role of observable independent
evolution variable. The field equations are written in terms of an evolution parameter,
which is the time coordinate x0; but this coordinate does not correspond to
anything directly observable. The proper time τ along spacetime trajectories cannot
be used as an independent variable either, as τ is a complicated non-local
function of the gravitational field itself. Therefore, properly speaking, GR does
not admit a description as a system evolving in terms of an observable time variable.
This does not mean that GR lacks predictivity. Simply put, what GR predicts
are relations between (partial) observables, which in general cannot be represented
as the evolution of dependent variables on a preferred independent time
variable.
This weakening of the notion of time in classical GR is rarely emphasized: after
all, in classical GR we may disregard the full dynamical structure of the theory and
consider only individual solutions of its equations of motion. A single solution of
the GR equations of motion determines “a spacetime”, where a notion of proper
time is associated to each timelike worldline.
But in the quantum context a single solution of the dynamical equation is like a
single “trajectory” of a quantum particle: in quantum theory there are no physical
individual trajectories: there are only transition probabilities between observable
eigenvalues. Therefore in Quantum Gravity it is likely to be impossible to describe
the world in terms of a spacetime, in the same sense in which the motion of a
quantum electron cannot be described in terms of a single trajectory.
To make sense of the world at the Planck scale, and to find a consistent conceptual
framework for GR and QM, we might have to give up the notion of time
altogether, and learn ways to describe the world in atemporal terms. Time might be
a useful concept only within an approximate description of the physical reality.
1.1.3 Conceptual issues
The key difficulty of Quantum Gravity may therefore be to find a way to understand
the physical world in the absence of the familiar stage of space and time. What
might be needed is to free ourselves from the prejudices associated with the habit
of thinking of the world as “inhabiting space” and “evolving in time”.
Technically, this means that the quantum states of the gravitational field cannot
be interpreted like the n-particle states of conventional QFT as living on a given
spacetime. Rather, these quantum states must themselves determine and define a
spacetime – in the manner in which the classical solutions of GR do.
8 C. Rovelli
Conceptually, the key question is whether or not it is logically possible to understand
the world in the absence of fundamental notions of time and time evolution,
and whether or not this is consistent with our experience of the world.
The difficulties of Quantum Gravity are indeed largely conceptual. Progress in
Quantum Gravity cannot be just technical. The search for a quantum theory of gravity
raises once more old questions such as: What is space? What is time? What is
the meaning of “moving”? Is motion to be defined with respect to objects or with
respect to space? And also: What is causality? What is the role of the observer
in physics? Questions of this kind have played a central role in periods of major
advances in physics. For instance, they played a central role for Einstein, Heisenberg,
and Bohr; but also for Descartes, Galileo, Newton and their contemporaries,
as well as for Faraday and Maxwell.
Today some physicists view this manner of posing problems as “too philosophical”.
Many physicists of the second half of the twentieth century, indeed, have
viewed questions of this nature as irrelevant. This view was appropriate for the
problems they were facing. When the basics are clear and the issue is problemsolving
within a given conceptual scheme, there is no reason to worry about
foundations: a pragmatic approach is the most effective one. Today the kind of
difficulties that fundamental physics faces has changed. To understand quantum
spacetime, physics has to return, once more, to those foundational questions.
