Edward Witten 2012 黑洞的量子力学
主要内容:
发在Science上。是个系列review：
全文introduction：
INTRODUCTION
Inescapable Pull
BLACK HOLES, ONCE THE PRESERVE OF THEORY AND SCIENCE FICTION, ARE WELL-
established inhabitants of the universe. Observations of the motions of stars
orbiting the center of the Milky Way have proved beyond doubt that a black
hole 4 million times as massive as the Sun resides there. Many other galaxies
are thought to host similarly heavy or even heavier black holes at their cen-
ters. Scattered out beyond the center, there are thought to be millions of lighter,
stellar-mass black holes, produced when the most massive stars collapse in on
themselves at the end of their lives. This week, Science explores the current state
of understanding of black holes with a series of Perspectives and Reviews.
Thorne (p. 536) describes what happens when black holes collide: Distur-
bances in the curvature of space-time produced in these collisions are the target
of a number of international gravitational wave observatories. In another Per-
spective, Witten (p. 538) explains how black holes can be understood in terms
of quantum mechanics and how this understanding, developed over the past four
decades, has been applied to more down-to-earth problems such as high-temper-
ature superconductors and heavy-ion collisions.
Fender and Belloni (p. 540) review the phenomenology of stellar black holes
accreting mass from binary star companions. These systems are luminous in
the x-ray regime, and many of them undergo transient bright outbursts during
which the mass accretion rate onto the black hole can change dramatically over
a period of days. The detailed study of these outbursts has led to progress in the
understanding of black hole accretion.
Volonteri (p. 544) reviews the formation of the massive black holes that
reside in the centers of galaxies and how they affect, and are affected by, gal-
axy evolution. This area of study has grown in importance over the last decade
because of the strong observed correlation between black-hole and galaxy prop-
erties and the systematic detection of very distant active galaxies powered by
massive black holes.
Finally, in the Reports section, Webb et al. (p. 554) present radio observations
of a hyperluminous x-ray source in an external galaxy, which are consistent with
the presence of a black hole with a mass between that of stellar-mass black holes
and massive black holes. Intermediate-mass black holes are still mysterious, and
their existence, contrary to that of their light and massive cousins, is still a matter
of much debate.
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文章信息:
Quantum Mechanics of Black Holes
Abstract
The popular conception of black holes
reflects the behavior of the massive black holes found by astronomers
and described
by classical general relativity. These objects
swallow up whatever comes near and emit nothing. Physicists who have
tried
to understand the behavior of black holes from a
quantum mechanical point of view, however, have arrived at quite a
different
picture. The difference is analogous to the
difference between thermodynamics and statistical mechanics. The
thermodynamic
description is a good approximation for a
macroscopic system, but statistical mechanics describes what one will
see if one
looks more closely.
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In quantum mechanics, if a time-dependent transition is possible from an initial state |i> to a final state |f>, then it is also possible to have a transition in the opposite direction from |f> to |i>.
The most basic reason for this is that the sum of quantum mechanical
probabilities must always equal 1. Starting from this
fact, one can show that, on an atomic time scale,
there are equal probabilities for a transition in one direction or the
other
(1).
This seems, at first, to contradict the whole idea of a black hole. Let B denote a macroscopic black hole—perhaps the one in the center of the Milky Way—and let A be some other macroscopic body, perhaps a rock or an astronaut. Finally, let B* be a heavier black hole that can be made by combining A and B. General relativity tells us that the reaction A + B → B* will occur whenever A and B get close enough. Quantum mechanics tells us, then, that the reverse reaction B* → A + B can also happen, with an equivalent amplitude.
The reverse reaction, though, is one in which the heavier black hole B* spontaneously emits the body A, leaving behind a lighter black hole B.
That reverse reaction is exactly what does not happen, according to
classical general relativity. Indeed, the nonoccurrence
of the reverse reaction, in which a black hole
re-emits whatever it has absorbed, is often stated as the defining
property
of a black hole.
It seems, then, that black holes are
impossible in light of quantum mechanics. To learn more, let us consider
another physical
principle that is also seemingly violated by the
existence of a black hole. This is time-reversal symmetry, which says
that
if a physical process is possible, then the
time-reversed process is also possible. Clearly, black holes seem to
violate this
as well.
Time reversal is a subtle concept, and elementary particle physicists have made some unexpected discoveries about it (2).
However, for applications to black holes, the important problem with
time reversal is that in everyday life, it simply
does not appear to be valid. We can spill a cup of
water onto the ground, but the water never spontaneously jumps up into
the cup.
The explanation has to do with randomness
at the atomic level, usually called entropy. Spilling the cup of water
is an irreversible
operation in practice, because it greatly increases
the number of states available to the system at the atomic level, even
after one specifies all of the variables—such as
temperature, pressure, the amount of water, the height of the water
above
the ground, and so on—that are visible
macroscopically. The water could jump back up into the cup if the
initial conditions
are just right at the atomic level, but this is
prohibitively unlikely.
The second law of thermodynamics says that
in a macroscopic system, like a cup of water, a process that reduces
the randomness
or entropy in the universe can never happen. Now
suppose that we consider what is sometimes called a mesoscopic
system—much
larger than an atom, but not really macroscopic.
For example, we could consider 100 water molecules instead of a whole
cupful.
Then we should use statistical mechanics, which
tells us that a rare fluctuation in which randomness appears to diminish
can
happen, but very rarely. Finally, at the level of a
single particle or a handful of particles, we should focus on the
fundamental
dynamical equations: Newton’s laws and their modern
refinements. These fundamental laws are completely reversible.
Since the late 19th century, physicists have understood that thermodynamic irreversibility arises spontaneously by applying
reversible equations to a macroscopic system, but it has always been vexingly hard to make this concrete.
Black Hole Entropy and Hawking Radiation
Now let us go back to the conflict between black holes and quantum mechanics. What is really wrong with the reverse reaction
B* → A + B, wherein a heavier black hole B* decays to a lighter black hole B plus some other system A?
A great insight of the 1970s [originating from a suggestion by Bekenstein (3)]
is that what is wrong with the reverse reaction involving black holes
is just like what is wrong with a time-reversed movie
in which a puddle of water flies off the wet
ground and into the cup. A black hole should be understood as a complex
system
with an entropy that increases as it grows.
In a sense, this entropy measures the ignorance of an outside observer about what there is inside the black hole. When a black
hole B absorbs some other system A in the process A + B → B*, its entropy increases, along with its mass, in keeping with the second law of thermodynamics. The reverse reaction B* → A + B diminishes the entropy of the black hole as well as its mass, so it violates the second law.
How can one test this idea? If the irreversibility found in black hole physics is really the sort of irreversibility found
in thermodynamics, then it should break down if A is not a macroscopic system but a single elementary particle. Although a whole cupful of water never jumps off the floor
and into the cup, a single water molecule certainly might do this as a result of a lucky fluctuation.
This is what Hawking found in a celebrated calculation (4).
A black hole spontaneously emits elementary particles. The typical
energy of these particles is proportional to Planck’s
constant, so the effect is purely quantum
mechanical in nature, and the rate of particle emission by a black hole
of astronomical
size is extraordinarily small, far too small to
be detected. Still, Hawking's insight means that a black hole is
potentially
no different from any other quantum system, with
reactions A + B → B* and B* → A + B occurring in both
directions at the microscopic level. The irreversibility of classical
black hole physics is just like the
familiar irreversibility of thermodynamics,
valid when what is absorbed by the black hole is a macroscopic system.
Although it was a shock at the time,
perhaps in hindsight we should not be surprised that classical general
relativity does
not describe properly the emission of an atom or
elementary particle from a black hole. After all, classical general
relativity
is not a useful theory of atoms and individual
elementary particles, or quantum mechanics would never have been needed.
However, general relativity is a good
theory of macroscopic bodies, and when it tells us that a black hole can
absorb a macroscopic
body but cannot emit one, we should listen.
Black Holes and the Rest of Physics
Is the quantum theory of black holes
just a theoretical construct, or can we test it? Unfortunately, the
usual astrophysical
black holes, formed from stellar collapse or in
the centers of galaxies, are much too big and too far away for their
microscopic
details to be relevant. However, one of the
cornerstones of modern cosmology is the study of the cosmic microwave
radiation
that was created in the Big Bang. It has
slightly different temperatures in different directions. The theory of
how this came
about is in close parallel with the theory of
Hawking radiation from black holes, and its success adds to our
confidence that
the Hawking theory is correct.
Surprisingly, in the past 15 years,
the theory of Hawking radiation and related ideas about quantum black
holes have turned
out to be useful for theoretical physicists
working on a variety of more down-to-earth problems. To understand how
this happened,
we need one more idea from the early period.
The Membrane Paradigm for Black Holes
The entropy of an ordinary body like Earth or the Sun is basically a volume integral; to compute the entropy, one computes
the entropy density and integrates it over the interior of Earth or the Sun.
Black holes seem to be different.
According to Bekenstein and Hawking, the entropy of a black hole is
proportional to the
surface area of the black hole, not to its
volume. This observation led in the early days to the membrane paradigm
for black
holes (5).
The idea of the membrane paradigm is that the interactions of a black
hole with particles and fields outside the hole can
be modeled by treating the surface or horizon of
the black hole as a macroscopic membrane. The membrane associated with a
black hole horizon is characterized by
macroscopic properties rather similar to those that one would use to
characterize any
ordinary membrane. For example, the black hole
membrane has temperature, entropy density, viscosity, and electrical
conductivity.
In short, there was a satisfactory
thermodynamic theory of black hole membranes, but can one go farther and
make a microscopic
theory that describes these membranes? An
optimist, given the ideas of the 1980s, might hope that some sort of
quantum field
theory would describe the horizon of a black
hole. What sort of theory would this be, and how could one possibly find
it?
Gauge-Gravity Duality
The known forces in nature other than
gravity are all well described in the standard model of particle physics
in terms of
quantum field theories that are known as gauge
theories. The prototype is Maxwell’s theory of electromagnetism,
interpreted
in modern times as a gauge theory.
Quantum gauge theory is a subtle yet
relatively well-understood and well-established subject. The principles
are known, but
the equations are hard to solve. On the other
hand, quantum gravity is much more mysterious. String theory has given
some
insight, but the foundations are still largely
unknown.
In the 1990s, string theorists began to discover that aspects of black hole physics can be modeled by gauge theory (6, 7). Such insights led to a remarkable new way to use gauge theory to study black holes and other problems of quantum gravity
(8).
This relied on the fact that string theory has extended objects known as branes (9), which are rather like membranes except that in general they are not two-dimensional. In fact, the word “brane” is a riff
on “membrane.”
Branes can be described by gauge
theory; on the other hand, because black holes can be made out of
anything at all, they can
be made out of branes. When one does this, one
finds that the membrane that describes the horizon of the black hole is
the
string theory brane.
Of course, we are cutting corners with
this very simple explanation. One has to construct a string theory
model with a relatively
large negative cosmological constant (in
contrast with the very small positive one of the real world), and then,
under appropriate
conditions, one gets a gauge theory description
of the black hole horizon.
Solving the Equations of Gauge Theory
Gauge-gravity duality was discovered
with the aim of learning about quantum gravity and black holes. One can
turn the relationship
between these two subjects around and ask
whether it can help us better understand gauge theory.
Even though gauge theory is the
well-established framework for our understanding of much of physics,
this does not mean that
it is always well understood. Often, even if one
asks a relatively simple question, the equations turn out to be
intractable.
In the past decade, the gauge theory
description of black holes has been useful in at least two areas of
theoretical physics.
One involves heavy ion collisions, studied at
the Relativistic Heavy Ion Collider at Brookhaven. The expanding
fireball created
in a collision of two heavy nuclei turns out to
be a droplet of nearly ideal fluid. In principle, this should all be
described
by known equations of gauge theory—quantum
chromodynamics, to be precise—but the equations are intractable. It
turns out that
by interpreting the gauge theory as a
description of a black hole horizon, and using the Einstein equations to
describe the
black hole, one can get striking insight about a
quantum almost-ideal fluid (10). This has become an important technique in modeling heavy ion collisions.
Condensed matter physics is described
in principle by the Schrödinger equation of electrons and nuclei, but
for most systems,
a full understanding based on the Schrödinger
equation is way out of reach. Nowadays, there is great interest in
understanding
quantum critical behavior in
quasi–two-dimensional systems such as high temperature superconductors.
These systems are studied
by a wide variety of methods, and no one
approach is likely to be a panacea. Still, it has turned out to be very
interesting
to study two-dimensional quantum critical
systems by mapping them to the horizon of a black hole (11). With this approach, one can perform calculations that are usually out of reach.
Among other things, this method has
been used to analyze the crossover from quantum to dissipative behavior
in model systems
with a degree of detail that is not usually
possible. In a sense, this brings our story full circle. The story began
nearly
40 years ago with the initial insight that the
irreversibility of black hole physics is analogous to the
irreversibility described
by the second law of thermodynamics. In general,
to reconcile this irreversibility with the reversible nature of the
fundamental
equations is tricky, and explicit calculations
are not easy to come by. The link between ordinary physics and black
hole physics
that is given by gauge-gravity duality has given
physicists a powerful way to do precisely this. This gives us
confidence
that we are on the right track in understanding
quantum black holes, and it also exhibits the unity of physics in a most
pleasing
way.
References and Notes
- ↵ The precise mathematical argument uses the fact that the Hamiltonian operator H is hermitian, so that the transition amplitude <f|H|i> is the complex conjugate of the transition amplitude <i|H|f> in the opposite direction.
- ↵ The precise time-reversal symmetry of nature also includes reflection symmetry and charge conjugation.
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Acknowledgments: This research was supported in part by NSF grant PHY-096944.
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