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NY Review, JANUARY 19, 2017 ISSUE
The Trouble with Quantum Mechanics
Steven Weinberg
The development of quantum mechanics in the first decades of the
twentieth century came as a shock to many physicists. Today, despite the
great successes of quantum mechanics, arguments continue about its
meaning, and its future.
1.
The first shock came as a challenge to the clear categories to which
physicists by 1900 had become accustomed. There were particles—atoms,
and then electrons and atomic nuclei—and there were fields—conditions of
space that pervade regions in which electric, magnetic, and
gravitational forces are exerted. Light waves were clearly recognized as
self-sustaining oscillations of electric and magnetic fields. But in
order to understand the light emitted by heated bodies, Albert Einstein
in 1905 found it necessary to describe light waves as streams of
massless particles, later called photons.
Then in the 1920s, according to theories of Louis de Broglie and Erwin
Schrödinger, it appeared that electrons, which had always been
recognized as particles, under some circumstances behaved as waves. In
order to account for the energies of the stable states of atoms,
physicists had to give up the notion that electrons in atoms are little
Newtonian planets in orbit around the atomic nucleus. Electrons in atoms
are better described as waves, fitting around the nucleus like sound
waves fitting into an organ pipe.1 The world’s categories had become all
muddled.
Worse yet, the electron waves are not waves of electronic matter, in the
way that ocean waves are waves of water. Rather, as Max Born came to
realize, the electron waves are waves of probability. That is, when a
free electron collides with an atom, we cannot in principle say in what
direction it will bounce off. The electron wave, after encountering the
atom, spreads out in all directions, like an ocean wave after striking a
reef. As Born recognized, this does not mean that the electron itself
spreads out. Instead, the undivided electron goes in some one direction,
but not a precisely predictable direction. It is more likely to go in a
direction where the wave is more intense, but any direction is possible.
Probability was not unfamiliar to the physicists of the 1920s, but it
had generally been thought to reflect an imperfect knowledge of whatever
was under study, not an indeterminism in the underlying physical laws.
Newton’s theories of motion and gravitation had set the standard of
deterministic laws. When we have reasonably precise knowledge of the
location and velocity of each body in the solar system at a given
moment, Newton’s laws tell us with good accuracy where they will all be
for a long time in the future. Probability enters Newtonian physics only
when our knowledge is imperfect, as for example when we do not have
precise knowledge of how a pair of dice is thrown. But with the new
quantum mechanics, the moment-to-moment determinism of the laws of
physics themselves seemed to be lost.
All very strange. In a 1926 letter to Born, Einstein complained:
Quantum mechanics is very impressive. But an inner voice tells me that
it is not yet the real thing. The theory produces a good deal but hardly
brings us closer to the secret of the Old One. I am at all events
convinced that He does not play dice.2
As late as 1964, in his Messenger lectures at Cornell, Richard Feynman
lamented, “I think I can safely say that no one understands quantum
mechanics.”3 With quantum mechanics, the break with the past was so
sharp that all earlier physical theories became known as “classical.”
The weirdness of quantum mechanics did not matter for most purposes.
Physicists learned how to use it to do increasingly precise calculations
of the energy levels of atoms, and of the probabilities that particles
will scatter in one direction or another when they collide. Lawrence
Krauss has labeled the quantum mechanical calculation of one effect in
the spectrum of hydrogen “the best, most accurate prediction in all of
science.”4 Beyond atomic physics, early applications of quantum
mechanics listed by the physicist Gino Segrè included the binding of
atoms in molecules, the radioactive decay of atomic nuclei, electrical
conduction, magnetism, and electromagnetic radiation.5 Later
applications spanned theories of semiconductivity and superconductivity,
white dwarf stars and neutron stars, nuclear forces, and elementary
particles. Even the most adventurous modern speculations, such as string
theory, are based on the principles of quantum mechanics.
Many physicists came to think that the reaction of Einstein and Feynman
and others to the unfamiliar aspects of quantum mechanics had been
overblown. This used to be my view. After all, Newton’s theories too had
been unpalatable to many of his contemporaries. Newton had introduced
what his critics saw as an occult force, gravity, which was unrelated to
any sort of tangible pushing and pulling, and which could not be
explained on the basis of philosophy or pure mathematics. Also, his
theories had renounced a chief aim of Ptolemy and Kepler, to calculate
the sizes of planetary orbits from first principles. But in the end the
opposition to Newtonianism faded away. Newton and his followers
succeeded in accounting not only for the motions of planets and falling
apples, but also for the movements of comets and moons and the shape of
the earth and the change in direction of its axis of rotation. By the
end of the eighteenth century this success had established Newton’s
theories of motion and gravitation as correct, or at least as a
marvelously accurate approximation. Evidently it is a mistake to demand
too strictly that new physical theories should fit some preconceived
philosophical standard.
In quantum mechanics the state of a system is not described by giving
the position and velocity of every particle and the values and rates of
change of various fields, as in classical physics. Instead, the state of
any system at any moment is described by a wave function, essentially a
list of numbers, one number for every possible configuration of the
system.6 If the system is a single particle, then there is a number for
every possible position in space that the particle may occupy. This is
something like the description of a sound wave in classical physics,
except that for a sound wave a number for each position in space gives
the pressure of the air at that point, while for a particle in quantum
mechanics the wave function’s number for a given position reflects the
probability that the particle is at that position. What is so terrible
about that? Certainly, it was a tragic mistake for Einstein and
Schrödinger to step away from using quantum mechanics, isolating
themselves in their later lives from the exciting progress made by others.
2.
Even so, I’m not as sure as I once was about the future of quantum
mechanics. It is a bad sign that those physicists today who are most
comfortable with quantum mechanics do not agree with one another about
what it all means. The dispute arises chiefly regarding the nature of
measurement in quantum mechanics. This issue can be illustrated by
considering a simple example, measurement of the spin of an electron. (A
particle’s spin in any direction is a measure of the amount of rotation
of matter around a line pointing in that direction.)
All theories agree, and experiment confirms, that when one measures the
amount of spin of an electron in any arbitrarily chosen direction there
are only two possible results. One possible result will be equal to a
positive number, a universal constant of nature. (This is the constant
that Max Planck originally introduced in his 1900 theory of heat
radiation, denoted h, divided by 4π.) The other possible result is its
opposite, the negative of the first. These positive or negative values
of the spin correspond to an electron that is spinning either clockwise
or counter-clockwise in the chosen direction.
But it is only when a measurement is made that these are the sole two
possibilities. An electron spin that has not been measured is like a
musical chord, formed from a superposition of two notes that correspond
to positive or negative spins, each note with its own amplitude. Just as
a chord creates a sound distinct from each of its constituent notes, the
state of an electron spin that has not yet been measured is a
superposition of the two possible states of definite spin, the
superposition differing qualitatively from either state. In this musical
analogy, the act of measuring the spin somehow shifts all the intensity
of the chord to one of the notes, which we then hear on its own.
This can be put in terms of the wave function. If we disregard
everything about an electron but its spin, there is not much that is
wavelike about its wave function. It is just a pair of numbers, one
number for each sign of the spin in some chosen direction, analogous to
the amplitudes of each of the two notes in a chord.7 The wave function
of an electron whose spin has not been measured generally has nonzero
values for spins of both signs.
There is a rule of quantum mechanics, known as the Born rule, that tells
us how to use the wave function to calculate the probabilities of
getting various possible results in experiments. For example, the Born
rule tells us that the probabilities of finding either a positive or a
negative result when the spin in some chosen direction is measured are
proportional to the squares of the numbers in the wave function for
those two states of the spin.8
The introduction of probability into the principles of physics was
disturbing to past physicists, but the trouble with quantum mechanics is
not that it involves probabilities. We can live with that. The trouble
is that in quantum mechanics the way that wave functions change with
time is governed by an equation, the Schrödinger equation, that does not
involve probabilities. It is just as deterministic as Newton’s equations
of motion and gravitation. That is, given the wave function at any
moment, the Schrödinger equation will tell you precisely what the wave
function will be at any future time. There is not even the possibility
of chaos, the extreme sensitivity to initial conditions that is possible
in Newtonian mechanics. So if we regard the whole process of measurement
as being governed by the equations of quantum mechanics, and these
equations are perfectly deterministic, how do probabilities get into
quantum mechanics?
One common answer is that, in a measurement, the spin (or whatever else
is measured) is put in an interaction with a macroscopic environment
that jitters in an unpredictable way. For example, the environment might
be the shower of photons in a beam of light that is used to observe the
system, as unpredictable in practice as a shower of raindrops. Such an
environment causes the superposition of different states in the wave
function to break down, leading to an unpredictable result of the
measurement. (This is called decoherence.) It is as if a noisy
background somehow unpredictably left only one of the notes of a chord
audible. But this begs the question. If the deterministic Schrödinger
equation governs the changes through time not only of the spin but also
of the measuring apparatus and the physicist using it, then the results
of measurement should not in principle be unpredictable. So we still
have to ask, how do probabilities get into quantum mechanics?
One response to this puzzle was given in the 1920s by Niels Bohr, in
what came to be called the Copenhagen interpretation of quantum
mechanics. According to Bohr, in a measurement the state of a system
such as a spin collapses to one result or another in a way that cannot
itself be described by quantum mechanics, and is truly unpredictable.
This answer is now widely felt to be unacceptable. There seems no way to
locate the boundary between the realms in which, according to Bohr,
quantum mechanics does or does not apply. As it happens, I was a
graduate student at Bohr’s institute in Copenhagen, but he was very
great and I was very young, and I never had a chance to ask him about this.
Today there are two widely followed approaches to quantum mechanics, the
“realist” and “instrumentalist” approaches, which view the origin of
probability in measurement in two very different ways.9 For reasons I
will explain, neither approach seems to me quite satisfactory.10
3.
The instrumentalist approach is a descendant of the Copenhagen
interpretation, but instead of imagining a boundary beyond which reality
is not described by quantum mechanics, it rejects quantum mechanics
altogether as a description of reality. There is still a wave function,
but it is not real like a particle or a field. Instead it is merely an
instrument that provides predictions of the probabilities of various
outcomes when measurements are made.
It seems to me that the trouble with this approach is not only that it
gives up on an ancient aim of science: to say what is really going on
out there. It is a surrender of a particularly unfortunate kind. In the
instrumentalist approach, we have to assume, as fundamental laws of
nature, the rules (such as the Born rule I mentioned earlier) for using
the wave function to calculate the probabilities of various results when
humans make measurements. Thus humans are brought into the laws of
nature at the most fundamental level. According to Eugene Wigner, a
pioneer of quantum mechanics, “it was not possible to formulate the laws
of quantum mechanics in a fully consistent way without reference to the
consciousness.”11
Thus the instrumentalist approach turns its back on a vision that became
possible after Darwin, of a world governed by impersonal physical laws
that control human behavior along with everything else. It is not that
we object to thinking about humans. Rather, we want to understand the
relation of humans to nature, not just assuming the character of this
relation by incorporating it in what we suppose are nature’s fundamental
laws, but rather by deduction from laws that make no explicit reference
to humans. We may in the end have to give up this goal, but I think not yet.
Some physicists who adopt an instrumentalist approach argue that the
probabilities we infer from the wave function are objective
probabilities, independent of whether humans are making a measurement. I
don’t find this tenable. In quantum mechanics these probabilities do not
exist until people choose what to measure, such as the spin in one or
another direction. Unlike the case of classical physics, a choice must
be made, because in quantum mechanics not everything can be
simultaneously measured. As Werner Heisenberg realized, a particle
cannot have, at the same time, both a definite position and a definite
velocity. The measuring of one precludes the measuring of the other.
Likewise, if we know the wave function that describes the spin of an
electron we can calculate the probability that the electron would have a
positive spin in the north direction if that were measured, or the
probability that the electron would have a positive spin in the east
direction if that were measured, but we cannot ask about the probability
of the spins being found positive in both directions because there is no
state in which an electron has a definite spin in two different directions.
4.
These problems are partly avoided in the realist—as opposed to the
instrumentalist—approach to quantum mechanics. Here one takes the wave
function and its deterministic evolution seriously as a description of
reality. But this raises other problems.
The realist approach has a very strange implication, first worked out in
the 1957 Princeton Ph.D. thesis of the late Hugh Everett. When a
physicist measures the spin of an electron, say in the north direction,
the wave function of the electron and the measuring apparatus and the
physicist are supposed, in the realist approach, to evolve
deterministically, as dictated by the Schrödinger equation; but in
consequence of their interaction during the measurement, the wave
function becomes a superposition of two terms, in one of which the
electron spin is positive and everyone in the world who looks into it
thinks it is positive, and in the other the spin is negative and
everyone thinks it is negative. Since in each term of the wave function
everyone shares a belief that the spin has one definite sign, the
existence of the superposition is undetectable. In effect the history of
the world has split into two streams, uncorrelated with each other.
This is strange enough, but the fission of history would not only occur
when someone measures a spin. In the realist approach the history of the
world is endlessly splitting; it does so every time a macroscopic body
becomes tied in with a choice of quantum states. This inconceivably huge
variety of histories has provided material for science fiction,12 and it
offers a rationale for a multiverse, in which the particular cosmic
history in which we find ourselves is constrained by the requirement
that it must be one of the histories in which conditions are
sufficiently benign to allow conscious beings to exist. But the vista of
all these parallel histories is deeply unsettling, and like many other
physicists I would prefer a single history.
There is another thing that is unsatisfactory about the realist
approach, beyond our parochial preferences. In this approach the wave
function of the multiverse evolves deterministically. We can still talk
of probabilities as the fractions of the time that various possible
results are found when measurements are performed many times in any one
history; but the rules that govern what probabilities are observed would
have to follow from the deterministic evolution of the whole multiverse.
If this were not the case, to predict probabilities we would need to
make some additional assumption about what happens when humans make
measurements, and we would be back with the shortcomings of the
instrumentalist approach. Several attempts following the realist
approach have come close to deducing rules like the Born rule that we
know work well experimentally, but I think without final success.
The realist approach to quantum mechanics had already run into a
different sort of trouble long before Everett wrote about multiple
histories. It was emphasized in a 1935 paper by Einstein with his
coworkers Boris Podolsky and Nathan Rosen, and arises in connection with
the phenomenon of “entanglement.”13
We naturally tend to think that reality can be described locally. I can
say what is happening in my laboratory, and you can say what is
happening in yours, but we don’t have to talk about both at the same
time. But in quantum mechanics it is possible for a system to be in an
entangled state that involves correlations between parts of the system
that are arbitrarily far apart, like the two ends of a very long rigid
stick.
For instance, suppose we have a pair of electrons whose total spin in
any direction is zero. In such a state, the wave function (ignoring
everything but spin) is a sum of two terms: in one term, electron A has
positive spin and electron B has negative spin in, say, the north
direction, while in the other term in the wave function the positive and
negative signs are reversed. The electron spins are said to be
entangled. If nothing is done to interfere with these spins, this
entangled state will persist even if the electrons fly apart to a great
distance. However far apart they are, we can only talk about the wave
function of the two electrons, not of each separately. Entanglement
contributed to Einstein’s distrust of quantum mechanics as much or more
than the appearance of probabilities.
Strange as it is, the entanglement entailed by quantum mechanics is
actually observed experimentally. But how can something so nonlocal
represent reality?
5.
What then must be done about the shortcomings of quantum mechanics? One
reasonable response is contained in the legendary advice to inquiring
students: “Shut up and calculate!” There is no argument about how to use
quantum mechanics, only how to describe what it means, so perhaps the
problem is merely one of words.
On the other hand, the problems of understanding measurement in the
present form of quantum mechanics may be warning us that the theory
needs modification. Quantum mechanics works so well for atoms that any
new theory would have to be nearly indistinguishable from quantum
mechanics when applied to such small things. But a new theory might be
designed so that the superpositions of states of large things like
physicists and their apparatus even in isolation suffer an actual rapid
spontaneous collapse, in which probabilities evolve to give the results
expected in quantum mechanics. The many histories of Everett would
naturally collapse to a single history. The goal in inventing a new
theory is to make this happen not by giving measurement any special
status in the laws of physics, but as part of what in the post-quantum
theory would be the ordinary processes of physics.
One difficulty in developing such a new theory is that we get no
direction from experiment—all data so far agree with ordinary quantum
mechanics. We do get some help, however, from some general principles,
which turn out to provide surprisingly strict constraints on any new theory.
Obviously, probabilities must all be positive numbers, and add up to 100
percent. There is another requirement, satisfied in ordinary quantum
mechanics, that in entangled states the evolution of probabilities
during measurements cannot be used to send instantaneous signals, which
would violate the theory of relativity. Special relativity requires that
no signal can travel faster than the speed of light. When these
requirements are put together, it turns out that the most general
evolution of probabilities satisfies an equation of a class known as
Lindblad equations.14 The class of Lindblad equations contains the
Schrödinger equation of ordinary quantum mechanics as a special case,
but in general these equations involve a variety of new quantities that
represent a departure from quantum mechanics. These are quantities whose
details of course we now don’t know. Though it has been scarcely noticed
outside the theoretical community, there already is a line of
interesting papers, going back to an influential 1986 article by Gian
Carlo Ghirardi, Alberto Rimini, and Tullio Weber at Trieste, that use
the Lindblad equations to generalize quantum mechanics in various ways.
Lately I have been thinking about a possible experimental search for
signs of departure from ordinary quantum mechanics in atomic clocks. At
the heart of any atomic clock is a device invented by the late Norman
Ramsey for tuning the frequency of microwave or visible radiation to the
known natural frequency at which the wave function of an atom oscillates
when it is in a superposition of two states of different energy. This
natural frequency equals the difference in the energies of the two
atomic states used in the clock, divided by Planck’s constant. It is the
same under all external conditions, and therefore serves as a fixed
reference for frequency, in the way that a platinum-iridium cylinder at
Sèvres serves as a fixed reference for mass.
Tuning the frequency of an electromagnetic wave to this reference
frequency works a little like tuning the frequency of a metronome to
match another metronome. If you start the two metronomes together and
the beats still match after a thousand beats, you know that their
frequencies are equal at least to about one part in a thousand. Quantum
mechanical calculations show that in some atomic clocks the tuning
should be precise to one part in a hundred million billion, and this
precision is indeed realized. But if the corrections to quantum
mechanics represented by the new terms in the Lindblad equations
(expressed as energies) were as large as one part in a hundred million
billion of the energy difference of the atomic states used in the clock,
this precision would have been quite lost. The new terms must therefore
be even smaller than this.
How significant is this limit? Unfortunately, these ideas about
modifications of quantum mechanics are not only speculative but also
vague, and we have no idea how big we should expect the corrections to
quantum mechanics to be. Regarding not only this issue, but more
generally the future of quantum mechanics, I have to echo Viola in
Twelfth Night: “O time, thou must untangle this, not I.”
1
Conditions on sound waves at the closed or open ends of an organ pipe
require that either an odd number of quarter wave lengths or an even or
an odd number of half wave lengths must just fit into the pipe, which
limits the possible notes that can be produced by the pipe. In an atom
the wave function must satisfy conditions of continuity and finiteness
close to and far from the nucleus, which similarly limit the possible
energies of atomic states. ↩
2
Quoted by Abraham Pais in ‘Subtle Is the Lord’: The Science and the Life
of Albert Einstein (Oxford University Press, 1982), p. 443. ↩
3
Richard Feynman, The Character of Physical Law (MIT Press, 1967), p. 129. ↩
4
Lawrence M. Krauss, A Universe from Nothing (Free Press, 2012), p. 138. ↩
5
Gino Segrè, Ordinary Geniuses (Viking, 2011). ↩
6
These are complex numbers, that is, quantities of the general form a+ib,
where a and b are ordinary real numbers and i is the square root of
minus one. ↩
7
Simple as it is, such a wave function incorporates much more information
than just a choice between positive and negative spin. It is this extra
information that makes quantum computers, which store information in
this sort of wave function, so much more powerful than ordinary digital
computers. ↩
8
To be precise, these “squares” are squares of the absolute values of the
complex numbers in the wave function. For a complex number of the form
a+ib, the square of the absolute value is the square of a plus the
square of b. ↩
9
The opposition between these two approaches is nicely described by Sean
Carroll in The Big Picture (Dutton, 2016). ↩
10
I go into this in mathematical detail in Section 3.7 of Lectures on
Quantum Mechanics, second edition (Cambridge University Press, 2015). ↩
11
Quoted by Marcelo Gleiser, The Island of Knowledge (Basic Books, 2014),
p. 222. ↩
12
For instance, Northern Lights by Philip Pullman (Scholastic, 1995), and
the early “Mirror, Mirror” episode of Star Trek. ↩
13
Entanglement was recently discussed by Jim Holt in these pages, November
10, 2016. ↩
14
This equation is named for Göran Lindblad, but it was also independently
discovered by Vittorio Gorini, Andrzej Kossakowski, and George Sudarshan. ↩
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