Baggott J. The Quantum Story: A History in 40 Moments

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the quantum story

370

Schrödinger had faced a similar challenge when confronted with the

wavefunction of the electron in a hydrogen atom. In that case, the wave-

function, and speciﬁ cally Born’s probabilistic interpretation, had been

subsumed into the Copenhagen interpretation of quantum theory. But,

as DeWitt now argued, in a quantum theory of gravity the Copenhagen

interpretation was of little help:

The Copenhagen view depends on the assumed a priori existence of a clas-

sical level to which all questions of observation may ultimately be referred.

Here, however, the whole universe is the object of inspection; there is no

classical vantage point, and hence the interpretation question must be re-

argued from the beginning.

If it was assumed that everything there is, is inside the universe, then

there could be no classical ‘measuring device’ sitting outside whose

purpose was to collapse the wavefunction of the universe and make

it ‘real’.

DeWitt had in mind another, altogether more radical interpretation,

devised by American physicist Hugh Everett III in 1957 and submitted

as his Princeton PhD thesis under Wheeler. In Everett’s ‘relative state’

formulation the pure Schrödinger wave mechanics is all that is needed

to make a complete theory. The wavefunction obeys the deterministic,

time-symmetric equations of motion at all times in all circumstances.

There are no discontinuities, the wavefunction does not collapse and this

process occupies no special place in the theory.

However, the apparent restoration of reality in Everett’s formalism

comes with a fairly large metaphysical trade-off. If there is no collapse,

each term in a superposition is taken to be real, and this means that all

experimental outcomes are therefore realized. But, of course, an individ-

ual observer only ever experiences one outcome: the electron is ‘here’, or

‘there’, spin-up, or spin-down. Everett wrote:

Thus with each succeeding observation (or interaction), the observer state

‘branches’ into a number of different states. Each branch represents a dif-

ferent outcome of the measurement and the corresponding eigenstate for

the [superposition]. All branches exist simultaneously in the superposi-

tion after any given sequence of observations.

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The act of measurement causes the world to split into separate worlds,

with different outcomes realized in different worlds. Looking back at

the paradox of Schrödinger’s cat, we can see that the difﬁ culty is now

resolved. The cat is not simultaneously alive and dead in one and the

same world, it is alive in one world and dead in another.

DeWitt would later champion Everett’s formulation as the ‘many

worlds’ interpretation of quantum theory. In his 1967 paper on canonical

quantum gravity he wrote:

Everett’s view of the world is a very natural one to adopt in the quantum

theory of gravity, where one is accustomed to speak without embarrass-

ment of the ‘wave function of the universe’. It is possible that Everett’s

view is not only natural but essential.

Because his equation had been derived by combining Einstein’s gravi-

tational ﬁ eld equations with Schrödinger’s wave mechanics, DeWitt

called it the Einstein–Schrödinger equation. Everybody else called it the

Wheeler–DeWitt equation.

For sure, it was an equation of limited validity and applicability and did

not seem to have any immediate solutions. It was far from a fully ﬂ edged

theory of quantum gravity, but it did serve to highlight the tremendous

problems associated with the quantization of general relativity.

It also represented the beginning of formal attempts to construct a

quantum cosmology.

372

Whilst the Wheeler–DeWitt equation was to be a milestone in the search for a quan-

tum theory of gravity, it quickly proved to be unsatisfactory. DeWitt came to call it ‘that

damned equation’.

The covariant approach didn’t fare much better. In the early 1970s ‘t Hooft and Velt-

man studied the renormalizability of a quantum ﬁ eld theory of gravity. As a warm-up

exercise, they turned their attention ﬁ rst to the renormalizability of Yang–Mills ﬁ eld

theories. Early in 1971 Veltman told ‘t Hooft that they had to have at least one renor-

malizable ﬁ eld theory with massive charged vector bosons. ‘t Hooft told him ‘I can do

that.’ Despite their subsequent success (they won the 1999 Nobel Prize for physics), the

theorists concluded that covariant quantum gravity is plagued with unrenmoralizable

divergences.

With the search for quantum gravity temporarily stalled, a new generation of cos-

mologists were instead exploring the various exotic objects predicted by general relativity.

Gravity might be the weakest of nature’s forces, but it is ultimately irresistible. Grav-

ity binds together and compresses clouds of gas drifting in space. Compression of clouds

with sufﬁ cient mass leads to the ignition of the nuclear fuel at their cores. Stars are born.

Radiation pressure released by burning the nuclear fuel holds back further compression,

and the star enters a period of relative stability.

Hawking Radiation

Oxford, February 1974

hawking radiation

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However, as the fuel is expended, the force of gravity grips tighter. For any mass

greater than about 1.4 times the mass of the sun,

the force of gravity is ultimately

overwhelming. It crushes the body of matter into the obscurity of a black hole.

Black holes were compelling theoretical objects because they pushed physics to extraor-

dinary limits and because, despite their exotic nature, it was possible that they played

important roles in our own universe. In the late 1960s, a young Cambridge University

physicist called Stephen Hawking produced a series of papers on black-hole physics in

collaboration with mathematician Roger Penrose, then at Birkbeck College in London.

General relativity, they claimed, predicted that at the heart of a black hole there beats

a singularity, a region of inﬁ nite density and space–time curvature where the laws of

physics break down. Of course, what goes on in the region of a singularity is completely

hidden from observation by the black hole’s event horizon, a fact that Penrose elevated to

the status of a principle, called the cosmic censorship hypothesis.

Together with Canadian Werner Israel, Australian Brandon Carter, and British physi-

cist David Robinson, Hawking demonstrated that black holes are also one of the sim-

plest objects in the universe. Their properties and behaviour depend only on their mass,

angular momentum, and electric charge, a conjecture called the ‘no hair’ theorem.

All

other information about the matter inside the black hole is lost.

In a moment of inspiration one night in November 1970, Hawking had realized

that the properties of the event horizon meant that this could never shrink—the area

of a black hole could never decrease. He later wrote:

If the rays of light that form the event horizon, the boundary of the black hole, can

never approach each other, the area of the event horizon might stay the same or

increase with time but it could never decrease—because that would mean that at least

some of the rays of light in the boundary would have to be approaching each other.

He called Penrose in a state of excitement the next morning. Penrose agreed. There is one

other well-known physical property that can never decrease: entropy. The second law of

thermodynamics demands that in a spontaneous change entropy always increases.

Could there be a connection between the area of a black hole and its entropy?

This is known as the Chandrasekar limit, named for Indian physicist Subrahmanyan Chan-

drasekhar.

Here ‘hair’ represents all other kinds of information other than mass, angular momentum,

and electric charge, which is lost behind the black hole’s event horizon.

the quantum story

374

Hawking began to show symptoms of amyotrophic lateral sclerosis (ALS),

a type of motor neurone disease, in 1963, shortly after moving to Cam-

bridge. He began to lose neuromuscular control. The initial prognosis

was not good. He was given only a few years to live, and there seemed

little point in completing his PhD.

But it turned out that he had a rare type of ALS which, though no

less debilitating, would take much longer to rob him of control over his

body. He married Jane Wilde in 1965 and, with her help, he recovered his

sense of purpose and enthusiasm for physics. He secured his doctorate

and became ﬁ rst a research fellow, then professorial fellow at Gonville

and Caius College, Cambridge.

Though slow, his disease progressed inexorably. In 1965 he could walk

with the help of a cane. By 1970 he needed a walking frame. By 1972 he

was conﬁ ned to a wheelchair. Hawking and his wife took on local gov-

ernment bureaucracy, occasionally winning small but important battles

against a public infrastructure that did not look kindly on the disabled.

It was one of Wheeler’s students, Jacob Bekenstein, who in his Princeton

PhD thesis made the connection between the non-decreasing area of a

black hole and its entropy. This appeared to resolve a problem with the

status of black holes in the context of the second law of thermodynam-

ics. If a black hole consumed material of high entropy—such as a quan-

tity of gas, for example—then the entropy of the universe outside the

black hole would decrease.

The entropy of the black hole must therefore increase to ensure that

the overall change in entropy associated with the process was either zero

or positive. But, according to the ‘no hair’ theorem, all information is lost

inside the black hole. There appeared to be no way of knowing what the

entropy of the black hole was, and therefore no way of knowing if the

second law was obeyed or violated.

Hawking had argued in 1970 that in such a change the area of the black

hole would increase. If, as Bekenstein was now suggesting, the area is directly

related to the entropy of the black hole, then the second law could be saved.

But Hawking was irritated. Whilst this was a neat solution it carried

with it a number of implications which Bekenstein hadn’t addressed

and which made the idea much less credible. For one thing, a body with

hawking radiation

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entropy also has to have a temperature. And a body with a temperature

has to emit radiation. For a perfect ‘black’ body this would be precisely

the spectrum of radiation that Planck had studied in 1900. But how could

a black hole, with properties and behaviour determined only by its mass,

angular momentum, and electric charge, possess a temperature and emit

radiation?

In August 1972 Hawking confronted Bekenstein at a month-long

summer school on black-hole physics at Les Houches in the French Alps.

During the summer school, Hawking worked together with Brandon

Carter and Yale physicist Jim Bardeen and derived four deﬁ nitive new

laws that governed black-hole physics. They looked remarkably like the

laws of thermodynamics. ‘Each black-hole law, in fact, turned out to

be identical to a thermodynamical law,’ wrote American physicist Kip

Thorne, ‘if one only replaced the phrase “horizon area” by “entropy”, and

the phrase “horizon surface gravity” by “temperature”.’

Bekenstein was more convinced than ever that he was on the right

track, but when challenged on the question of temperature and black-

hole radiation his response involved a certain amount of arm-waving.

The rest of the black-hole physics community was convinced that this

was all a coincidence. Hawking, Carter, and Bardeen subsequently wrote

a paper pointing out the apparently fatal ﬂ aws in Bekenstein’s arguments.

‘I must admit,’ Hawking wrote, ‘that in writing this paper I was motivated

partly by irritation with Bekenstein, who, I felt, had misused my discov-

ery of the increase of the area of the event horizon.’

But, despite its apparent absurdity, the idea of black-hole radiation had

been the subject of some earlier discussions. In June 1971 Thorne had vis-

ited Soviet physicist Yakov Zeldovich in Moscow. Zeldovich had called

Thorne to his apartment early one morning and confronted him with the

proposal that a spinning black hole must radiate. ‘That’s one of the cra-

ziest things I’ve ever heard,’ Thorne declared. ‘How can you make such

a crazy claim? Everyone knows that radiation can ﬂ ow into a hole, but

nothing, not even radiation, can come out.’

Zeldovich argued that a spinning metal sphere emits electromagnetic

radiation and, by analogy, a spinning black hole should emit gravita-

tional energy in the form of waves. Thorne was not aware that a spinning

metal sphere should behave this way, and asked for the explanation. ‘The

the quantum story

376

metal sphere will radiate when electromagnetic vacuum ﬂ uctuations tickle

it,’ Zeldovich said. ‘Similarly, a black hole will radiate when gravitational

vacuum ﬂ uctuations graze its horizon.’

The basis for his argument was quantum electrodynamics. The vac-

uum ﬂ uctuations would produce virtual waves which would be acceler-

ated as they skimmed around the event horizon, gaining energy from the

black hole’s rotation. This transfer of energy would promote the virtual

wave to the status of a real wave, which would appear as radiation.

As the black hole radiated its rate of spin would slow. With its rota-

tional energy drained the black hole would eventually stop spinning.

And it would then stop radiating.

Thorne was not convinced, and they agreed to a bet. If subsequent

theoretical developments showed that Zeldovich was right and spinning

black holes could radiate, then Thorne would buy him a bottle of White

Horse scotch whisky.

Zeldovich published his proposal, but the idea was largely forgotten. It

was not discussed at the Les Houches summer school.

A year later, in August 1973, Hawking and his wife Jane attended a con-

ference in Warsaw to celebrate the ﬁ ve-hundredth anniversary of the

birth of Nicolaus Copernicus. In September they travelled on to Mos-

cow to meet with Zeldovich and his research student, Alexei Starobinsky.

Thorne joined them as a translator and guide.

Hawking now learned about Zeldovich’s crazy proposal from Starob-

insky. The Soviet physicists had proved to their own satisfaction that a

spinning black hole should radiate. ‘They convinced me that, according

to the quantum mechanical uncertainty principle [responsible for the

vacuum ﬂ uctuations], rotating black holes should create and emit parti-

cles,’ Hawking later wrote. He was nevertheless sceptical about Zeldov-

ich and Starobinsky’s proof. He returned to Cambridge to think about it

some more, determined to devise a better theoretical treatment.

In the meantime, Canadian William Unruh, another student of Wheel-

er’s, and Don Page, a student of Thorne’s, provided independent, though

tentative, conﬁ rmation of Zeldovich’s claim. It looked as though Thorne

was going to have to hand over that bottle of whisky.

hawking radiation

377

This was all about quantum effects in the microphysics of a large gravita-

tional body whose properties are predicted by the macrophysics of general

relativity. Obviously, what Hawking needed was a fully ﬂ edged quantum

theory of gravity. In the absence of such a theory, alternative approximate

approaches had to be tried. Hawking chose to retain an essentially classi-

cal, general relativistic description of the black hole itself and apply quan-

tum ﬁ eld theory to the curved space–time around the event horizon.

He found some curious differences in the application of quantum ﬁ eld

theory in curved compared to ﬂ at space–times. However, these paled into

insigniﬁ cance when Hawking realized just what he had discovered. Zel-

dovich had indeed been right: spinning black holes do radiate. Although

Hawking felt that he now had a better mathematical basis for this conclu-

sion, this was relatively old news. What was new was that when the black

hole stopped spinning, it didn’t stop radiating.

‘I didn’t want particles coming out . . . ,’ Hawking explained. ‘I wasn’t

looking for them; I merely tripped over them. I was very sorry because

it destroyed my framework, and I did my best to get rid of them. I was

rather annoyed.’

‘When I did the calculation, I found . . . that even nonrotating black

holes should apparently create and emit particles at a steady rate. At ﬁ rst

I thought that this emission indicated that one of the approximations I

had used was not valid. I was afraid that if Bekenstein found out about

it, he would use it as a further argument to support his ideas about the

entropy of black holes.’

Hawking had found the phenomenon, which would come to be known

as Hawking radiation, but he did not yet have an explanation. He pre-

sented some of his results at an informal seminar in Oxford in November

1973. He then went on to calculate the spectrum of this radiation, and dis-

covered to his astonishment that it is identical to the spectrum of a black

body. When combined in this way, two of the most successful theories of

twentieth-century physics, whose predictions were bizarre and the sub-

jects of endless debate, were now predicting behaviour no different from

nineteenth-century physics.

Where did this radiation come from?

It comes from virtual particle–anti-particle pairs produced in the curved

space–time near the black hole’s event horizon. The particles are produced

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378

within the constraints of Heisenberg’s energy–time uncertainty relation,

with zero net energy. This means that one particle in the pair will possess

positive energy and the other negative energy. Under normal circumstances,

the pair would quickly annihilate. But if the negative-energy particle is

drawn into the black hole before it can be annihilated, it can acquire energy

and become a real particle. Another way of looking at this is to think of the

negative-energy particle as having negative mass. As the negative-mass par-

ticle falls into the back hole, it gains mass and becomes a real particle.

The positive-energy particle created in the erstwhile pair may escape,

to all intents and purposes appearing as though it had been emitted by

the black hole.

The particles or anti-particles so emitted can be of any and all types. As

negative-energy particles spill through the event horizon, the black hole

loses mass and its area decreases. This apparent reduction in entropy is

more than compensated for by the entropy of the emitted radiation, so

there is no violation of the second law of thermodynamics. As the area

decreases, the ‘temperature’ of the black hole increases, as does the rate

of emission. The black hole ‘evaporates’, eventually disappearing alto-

gether in an explosion small by astronomical standards but, as Hawking

estimated, still equivalent to a million 1 megaton hydrogen bombs.

For a black hole with a mass close to that of the sun the evaporation

process would take longer than the present age of the universe, but there

would have been plenty of time for much smaller black holes formed in

the big bang to have evaporated by now.

Hawking fretted over his calculations through Christmas 1973. He sub-

mitted a short paper to the journal Nature in January. He wrote:

. . . Bardeen, Carter and I considered that the thermodynamical similarity

between [the surface gravity] and temperature was only an analogy. The

present result seems to indicate, however, that there may be more to it

than this.

Hawking presented his results ofﬁ cially at the second conference on

quantum gravity held at the Rutherford–Appleton Laboratory near

Oxford in February 1974 (the Nature paper was published in March).

Whilst his conversion was complete, the rest of the physics community

hawking radiation

379

was still very sceptical. The chairman of the session at which Hawking

presented his paper, British physicist John Taylor from King’s College,

London, declared that it was all nonsense. Curiously, Zeldovich was also

reluctant to accept the idea. Having set Hawking on his journey, he could

not accept that a black hole would radiate after it had stopped spinning.

It took several years for the rest of the community to catch up. Hawking’s

welding of quantum ﬁ eld theory and general relativity was novel but it

was not arbitrary. Others began to understand how to apply the same

methods, and what they found conﬁ rmed Hawking’s discovery. Hawk-

ing, in the meantime, had learned to throw caution to the winds. Hav-

ing satisﬁ ed himself that Bekenstein’s instincts were right all along, he

declared without further proof that the four laws of black hole physics

he had worked out with Carter and Bardeen were indeed the laws of ther-

modynamics. ‘I would rather be right than rigorous,’ he told Thorne.

Zeldovich also capitulated. Thorne handed over a bottle of White

Horse scotch whisky in September 1975.

Although Hawking’s discovery had not been based on a fully ﬂ edged

quantum theory of gravity, his successful application of quantum ﬁ eld

theory in curved space–time—and the revelations that followed—

sparked renewed interest in the ﬁ eld. It was a beacon of hope in a sea

of frustration. A few years later it would lead Hawking to resurrect the

covariant approach based on Feynman quantization of general relativity,

this time in a four-dimensional space, or four-space.

Hawking’s achievements were recognized by his election to a Royal

Society Fellowship later in 1974. At 32, he was one of the youngest Fellows

in the Society’s history. His condition continued to deteriorate, however.

His speech had now become so slurred that few could understand him.

Routine physical activities required major effort. Desperate for help, Jane

persuaded him to allow one or more of his graduate students into his

close circle of carers.

He was inducted into the Royal Society in May 1974. He had to be car-

ried up the wheelchair-unfriendly steps into the building. Unable to walk

to the podium to sign his name in the Society’s roll book, he was handed

the book, which he signed with great difﬁ culty.