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

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The time had ﬁ nally come to turn attention away from weapons of war and return to the

problems of quantum theory. Theoretical physics had been in the doldrums for nearly

two decades. ‘The last eighteen years have been the most sterile of the century,’ remarked

Columbia University physicist Isidor Rabi in 1947.

The debate about interpretation was over, or so most physicists thought. But there

were plenty of other things to be worrying about. Dirac’s breakthrough had created a

relativistic equation for the electron which predicted the property of electron spin and the

existence of the anti-electron. This was indeed a triumph but it also quickly became a

dead-end. Quantum mechanics could successfully predict the properties and behaviour

of ‘stationary’ states in which the integrity of individual quantum particles such as elec-

trons is preserved, but it could not deal with transitional situations in which particles are

destroyed and new particles created.

For example, it had become apparent that when an electron collides with a positron,

the two particles could be expected to ‘annihilate’ to form high-energy (gamma-ray) pho-

tons. Conversely, at high enough energies, gamma-ray photons could spontaneously cre-

ate an electron–positron pair. If quantum mechanics couldn’t handle this, what could?

Some physicists had acknowledged that a quantum theory of ﬁ elds was required, starting

with a quantum version of Maxwell’s electromagnetic ﬁ eld.

Shelter Island

Long Island, June 1947

In a ﬁ eld theory, the strength of the ﬁ eld is speciﬁ ed at each point in space–time. Fields may

be scalar (magnitude only), vector (magnitude and direction), or tensor (a generalization of a

vector ﬁ eld in which the vector varies from one point to the next).

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172

These physicists recognised that ﬁ elds are more fundamental than particles. It was

believed that a proper quantum ﬁ eld description should yield particles as the quanta of

the ﬁ elds themselves, carrying the force from one interacting particle to another. It seemed

clear that the photon was the quantum of the electromagnetic ﬁ eld, created and destroyed

when charged particles interacted. To account for the creation and annihilation of elec-

trons, a quantum theory of an ‘electron ﬁ eld’ was needed.

Heisenberg and Pauli had developed a quantum ﬁ eld theory in 1929. When applied

to the electron ﬁ eld it became a form of quantum electrodynamics (QED). It was found

that the resulting ﬁ eld equations did not yield exact solutions, and had had to resort to

an approach based on a perturbation expansion.

They then ran into a major problem.

Higher-order terms in the expansion should have provided smaller and smaller correc-

tions to the result, but instead they found that some terms in the expansion yielded inﬁ -

nite corrections.

This was a result that made no physical sense.

The inﬁ nities were identiﬁ ed to be due to the ‘self-energy’ of the particles. They

resulted from particles interacting with their own ﬁ elds, a consequence of treating

them theoretically as point particles, with no volume or shape. There was no obvious

solution.

And there was yet more trouble brewing. The problem with the inﬁ nities was a theo-

retical or mathematical problem. In 1947, new experimental data pointed to problems of

a more practical kind. Closer inspection had showed that the predictions of Dirac’s theory

did not, after all, agree with experiment.

Quantum theory was in crisis once again.

In 1945 Duncan MacInnes, a physical chemist at the Rockefeller Institute

in New York, conceived the idea of a series of small, intimate conferences

at which both distinguished scientists and young, up-and-coming schol-

ars could address the problems facing their discipline. He had grown frus-

trated and wary of large-scale conferences at which academics appeared

to talk past each other without resolving the issues. ‘The programs are

In a perturbation expansion, the equation is cast in terms of a zeroth-order (or zero-interaction)

expression which can be solved exactly. To this is added additional (perturbation) terms which form

a power series (ﬁ rst-order, second-order, etc.), each in principle providing a smaller and smaller cor-

rection to the zeroth-order result. The accuracy of the ﬁ nal result then depends on the number of

perturbation terms included in the calculation.

In these early versions of QED, the ﬁ rst-order terms are solvable and produce physically

realistic results. It is the second-order terms that mushroom to inﬁ nity.

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173

so crowded that presentation of the papers takes most of the time and

discussion is slighted if it takes place at all,’ he wrote.

Much of Europe was still recovering from the aftermath of a devastat-

ing war, and European physicists were slowly picking up the pieces of

their academic careers. The atom bomb was seen as a triumph of Ameri-

can science and ingenuity (supported, of course, by many European

émigré physicists and British physicists who had contributed to the

Manhattan Project). Perhaps it was time to put American physics ﬁ rmly

on the world stage, to show that it had come of age, by establishing an

American equivalent of the Solvay conferences.

MacInnes secured ﬁ nancial support from the US National Academy

of Sciences. The shortlist of proposed conference topics included quan-

tum mechanics. Both Pauli, who had moved to Princeton University in

1940, and American physicist John Wheeler were asked to advise on

attendance and discussion topics. After many exchanges, the ﬁ rst of the

National Academy-sponsored conferences was held on 2–4 June 1947 at

the Ram’s Head Inn, a small clapboard hotel and inn on thinly populated

Shelter Island, at the eastern end of New York’s Long Island.

When the physicists arrived at Greenport, Long Island, they were

treated like celebrities. The prestigious group was given a motorcycle

escort which ran through red lights. The conference attracted the atten-

tions of the press: ‘Twenty three of the country’s best-known theoretical

physicists—the men who made the atomic bomb—gathered today in

a rural inn to begin three days of discussion and study, during which

they hope to straighten out a few of the difﬁ culties that beset modern

physics.’

Among them were Oppenheimer, Hans Bethe, who had led the Theo-

retical Division at Los Alamos and had returned to academia at Cornell

University, Victor Weisskopf, Isidor Rabi, Edward Teller, John Van Vleck,

John von Neumann, and Hendrik Kramers. The new generation was rep-

resented by Wheeler, Abraham Pais, Richard Feynman, Julian Schwinger,

and former Oppenheimer students Robert Serber and David Bohm.

Wil-

lis Lamb, also a former Oppenheimer student and now an expert in the

application of microwave techniques at Columbia University’s Radiation

Einstein was invited to attend but declined for reasons of ill health.

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174

Laboratory, joined the group to report on some disturbing new results

on the spectrum of the hydrogen atom.

The emphasis of the conference was on informal discussion rather than

the formal presentation of prepared papers. The physicists were the only

occupants of the Inn, which had opened early in the season to accom-

modate them. When Lamb opened the conference on the morning of 2

June, what he had to say dominated subsequent discussion.

A few months previously Lamb had announced the results of new,

more detailed spectroscopic studies of the hydrogen atom. Trained as a

theoretician, Lamb had realized that if he wanted to put some of his ideas

to the test, then he was going to have to do the experiments himself. He

focused his attention on the behaviour of two atomic states of hydrogen

which share the same principal quantum number but possess different

values of the orbital angular momentum quantum number.

According to Dirac’s theory these states were predicted to have the

same energy (such states are said to be degenerate). As a consequence of

this degeneracy, the microwave spectrum involving transitions to these

states was expected to show a single line.

Not so. Lamb had discovered with graduate student Robert Retherford

that there are in fact two lines. One of the hydrogen-atom states is shifted

in energy relative to the other, a phenomenon that quickly became

known as the Lamb shift. The physicists gathered on Shelter Island now

heard the full details of Lamb’s latest experimental results. Oppenheimer

and Weisskopf led the discussion that followed. Oppenheimer speculated

that a purely quantum electrodynamic effect was the ‘least improbable

explanation’ of the shift.

Rabi then stood to deliver a second experimental challenge. He

reported the results of experiments conducted by his Columbia Univer-

sity students John Nafe and Edward Nelson and his Columbia colleagues

Polykarp Kusch and H.M. Foley. They had found that the g-factor of the

electron is not exactly 2, as required by Dirac’s relativistic theory. It is

slightly larger, more like 2.00244.

To be sure, this is a small difference; a little over 0.1 per cent. But such

differences were now well outside the bounds of experimental error. For

the theoreticians, these experimental ﬁ ndings were profoundly interest-

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175

ing and stimulating. It seemed that the problems with QED were nothing

to do with zeros or inﬁ nities, after all. The problems were rather to do

with the very ﬁ nite differences between two energy states of the hydrogen

atom and the very ﬁ nite g-factor for the electron.

The discussion continued long into the night, over hastily eaten meals.

The delegation split into groups of two and three, the corridors echoing

their arguments, as the physicists regained their passion. ‘It was the ﬁ rst

time people who had all this physics pent up in them for ﬁ ve years could

talk to each other without somebody peering over their shoulders and

saying “Is this cleared?”,’ remarked Schwinger.

The second day of the conference was concerned with theory. Kramers

opened the session with a short lecture summarizing his recent work on

a classical theory of the electron. He outlined a new approach to treat-

ing the mass of an electron in an electromagnetic ﬁ eld. Suppose the self-

energy of the electron appears as an additional contribution to the mass

of the electron, he explained. The mass that we observe in experiments

would then include this contribution. In other words, the observed

(or ‘dressed’) mass of the electron is equal to an intrinsic or ‘bare’ mass

plus an ‘electromagnetic mass’ arising from the electron’s interaction

with its own electromagnetic ﬁ eld.

The ‘bare’ mass is a purely theoretical quantity. It is the mass that the

electron would possess in the absence of an electromagnetic ﬁ eld. Clearly,

the mass that the physicists had to deal with is the observed or dressed

mass. This meant that the equations had to be rewritten in terms of the

observed mass. In other words, they had to be ‘renormalized’.

Here was more food for thought. In subsequent discussions, Weisskopf

and Schwinger suggested that a calculation of the Lamb shift based on

Kramer’s ideas might actually yield a ﬁ nite answer.

The young New Yorkers Feynman and Schwinger were both preco-

cious physicists and mathematical prodigies. Schwinger had spent much

of his time at the City College of New York ignoring his classes, sitting

in the library reading advanced mathematics and physics texts. He had

written his ﬁ rst (unpublished) paper in 1934, at the age of sixteen. When

he ﬂ abbergasted Rabi by explaining a subtle mathematical point in con-

nection with the Einstein, Podolsky, Rosen paper, Rabi arranged for him

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176

to transfer to Columbia. He graduated in 1936 and secured his doctor-

ate under Rabi three years later. He then worked for two years with

Oppenheimer at the University of California, Berkeley, before moving to

Purdue University in Indiana. He became an associate professor at Har-

vard University in February 1946, and was promoted to full professor a

year later. He was just 29.

Where Schwinger was a quiet introvert, Feynman was a loud extravert.

He was blessed with an intuitive ability to solve complex mathematical

problems in his head, an ability to ‘visualize’ the physical interpretation

of the equations and deduce the solutions. Feynman had graduated from

MIT in 1938 and moved to Princeton to work with Wheeler shortly after-

wards. In 1943 he was recruited to Los Alamos, where his bongo-playing

and safe-cracking antics would be the subject of endless anecdotes told

at the many parties hosted by the Oppenheimers, fuelled by Robert’s

legendary vodka martinis.

In June 1946 Feynman had borrowed Klaus Fuchs’ car to drive to a hos-

pital in Albuquerque where his wife Arline was seriously ill with tuber-

culosis. After suffering delays caused by three ﬂ at tyres, he arrived in

time to be with her as she died.

From Los Alamos Feynman went to Cornell University, but struggled

to regain his enthusiasm for physics. Eventually, his attention was cap-

tured by a seemingly trivial problem with a spinning, wobbling plate he

had seen in the cafeteria. He decided to play:

I went on to work out equations of wobbles. Then I thought about how

electron orbits start to move in relativity. Then there’s the Dirac equation

in electrodynamics. And then quantum electrodynamics. And before I

knew it (it was a very short time) I was ‘playing’—working, really—with

the same old problem that I loved so much, that I had stopped work-

ing on when I went to Los Alamos: my thesis-type problems; all those

old-fashioned, wonderful things.

The contrasts between Feynman and Schwinger extended to the way they

did physics. Schwinger was conservative. His was a thoughtful, consid-

ered, deductive approach. He was committed to doing it right, no matter

how elaborate or unwieldy the mathematics. Feynman was a radical. He

was more of a visionary, his approach no less considered but much more

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intuitive. He was happy to push through the mathematics in order to get

quickly to a satisfying solution.

Schwinger had been quiet for much of the conference. On the third

and ﬁ nal day Oppenheimer asked Feynman to present his most recent

work. Feynman now stood to describe his new take on non-relativistic

quantum mechanics, ‘ . . . with a clear voice, great rush of words and

illustrative gestures sometimes ebullient.’

The basis of Feynman’s approach can be found in some of the simplest

observations of classical physics. Classical light rays travel in straight lines

because straight lines represent the least amount of time required for light

to travel from its source to its destination. This was a principle ﬁ rst enun-

ciated by Pierre de Fermat (of ‘last theorem’ fame) in 1657. But how is the

light supposed to ‘know’ in advance what the path of least time is?

It is the wave nature of light that provides a solution to this particular

mystery. Light does not need to know the path of least time in advance

because it takes all paths from its source to its destination.

Imagine a wave oscillating up and down as it moves forward through

space towards its destination. Clearly, a second wave with the same wave-

length, starting out from the same place but heading for a place just slightly

displaced from the destination of the ﬁ rst wave will look very similar in

terms of the time and space dependences of the wave amplitude. However,

as we increase the displacement of the destination of the wave further and

further, the time and space dependences of the waves become increas-

ingly ‘misaligned’: at speciﬁ c points in time and space peak no longer lines

up with peak and trough no longer lines up with trough. The result is

destructive interference and a loss of coherence of the light.

We get constructive interference and maximum coherence for light

paths that do not differ signiﬁ cantly in terms of distance and therefore

time. The mystery is now resolved. When light travels through a single

medium (such as air), the light paths that do not differ signiﬁ cantly in

terms of distance and time are all clustered around the shortest, straight-

line path from source to destination, which is also the path of least time.

The principle is one of least time and not least distance. If the light

passes from one medium into another at an oblique angle (for example,

from air into water), then the path of least distance remains the straight

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178

line drawn from source to destination. But light slows down in water, and

this path of least distance does not minimize the amount of water that

light has to travel through to get to the destination. The more water the

light has to travel through, the longer it takes. We can identify a ‘path of

least water’ which enters the water vertically and goes straight down to

the destination, but this would involve a very round-about path, length-

ening the distance, and hence the time travelled, through the air. The pre-

ferred path—the one of least time—represents a compromise. It is not

the path of least distance (which is why a stick half-immersed in water

appears bent) nor is it the path of least water.

Feynman elevated these relatively simple physical principles into an

alternative formulation of non-relativistic quantum theory equivalent

to wave and matrix mechanics. He represented the passage of a quan-

tum particle from one place to another as a sum

over all the possible

paths the particle could take or, alternatively, as the sum over all pos-

sible ‘histories’ of the particle’s motion. The probability of ﬁ nding the

particle at a speciﬁ c location can be determined from the amplitudes of

all the various paths it can take to get there.

Here is one particularly illuminating way to think about Feynman’s

approach. Imagine inserting a screen between a point source of light and

a photographic plate. If we drill a hole through the screen then the ampli-

tude for the photons to propagate from the source to the photographic

plate is simply the amplitude of the path through the hole. If we now

drill a second hole, we need to sum the amplitudes of the paths through

both holes. We can keep repeating this for three, four, ﬁ ve holes, each

time summing the amplitudes of all the possible paths. By the time we

have drilled an inﬁ nite number of holes in the screen, the screen is no

longer there. We conclude that we must still sum the amplitudes of all

the possible paths from the source to the photographic plate.

This is a determinedly particle representation—the wave aspect of

quantum particles is captured in terms of their phases, whatever their ori-

gin, and the way these phases combine. A certain kind of combination

results in effects which appear much like wave interference.

Strictly speaking, an integral.

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179

It was as though the Copenhagen interpretation had never been

conceived.

However, apart from providing an interesting, more visualizable

reinterpretation of quantum mechanics, Feynman’s approach yielded

no new results and, arguably, no new insights. He had tried, and so far

failed, to develop a path-integral formulation for the Dirac equation. His

approach was simply the result of playing with some new ideas. Abraham

Pais later described the effect Feynman’s lecture had on the assembled

audience: ‘At that time no one could follow what he was talking about.’

Sufﬁ ce to say, they were not overwhelmed.

Feynman was nevertheless awed by his experiences at the Shelter Island

conference. ‘There have been many conferences in the world since,’ he

explained many years later, ‘but I’ve never felt any to be as important as

this . . . The Shelter Island conference was my ﬁ rst conference with the big

men . . . I had never gone to one like this in peacetime.’

After the conference, Bethe returned to New York and picked up a train

to Schenectady, where he was engaged as a part-time consultant to Gen-

eral Electric. Like many of his contemporaries, he had been thinking

deeply about the Lamb shift and the discussions during the conference

now prompted him to attempt a calculation. As he sat on the train he

played around with the equations of QED.

The existing theories of QED predicted an inﬁ nite shift, a consequence

of the self-interaction of the electron with the electromagnetic ﬁ eld. Bethe

now followed Kramers’ suggestion and identiﬁ ed the divergent term in

the perturbation expansion with an electromagnetic mass effect.

So, he reasoned, what if we now subtract the expression for a free

electron (including the divergent self-energy term) from the expression

for the electron bound in a hydrogen atom (also including the divergent

self-energy term). It sounds as though subtracting inﬁ nity from inﬁ n-

ity should yield a nonsensical answer, but Bethe now found that in a

non-relativistic version of QED this subtraction produced a result that,

though it still diverged, did so much more slowly. He ﬁ gured that in a

fully relativistic QED this normalization procedure would eliminate the

divergence completely and give a physically realistic answer.