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

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the quantum story

390

These were not the only problems. The low-energy vibrational patterns

correspond to massless particles, yet the three generations of particles of

the Standard Model all have mass. The theory also predicts many more

vibrational patterns than there are particles.

Despite the great rush of enthusiasm, it was clear that superstring the-

ory was quickly losing any sense of uniqueness and hence a capacity to

make unique predictions. It would have some way to go before it could

fulﬁ l its early promise as the theory of everything.

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Superstring theory promised much, and momentum built through the mid-1980s as

more theoreticians began to grapple with its structure. As this was a quantum ﬁ eld theory

of strings, it seemed that the covariant approach to quantum gravity, favoured largely by

the particle physics community, was winning out.

But, for those physicists grounded in general relativity, there was still (at least) one

signiﬁ cant objection to superstring theory. In order for the strings to vibrate, superstring

theory had to assume a background space–time for them to vibrate in. The theory was

not background-independent, and there appeared to be no easy way to free it from this

dependence. Many superstring theorists did not even appear to acknowledge that this was

a problem.

The canonical approach to quantum gravity had stalled with the Wheeler–DeWitt

equation. The dying embers had been stirred in 1983 by Hawking and American physicist

James Hartle at the Enrico Fermi Institute in Chicago. They had derived a wavefunction

of the universe from a speciﬁ c solution of the Wheeler–DeWitt equation and had used

Feynman’s path-integral approach to explore aspects of big bang cosmology.

Their solution was based on a ‘no boundary’ assumption, applied to the ﬁ nite space–time

of a simple model universe.

No boundary to space–time implies that there was no begin-

ning to the universe, as we commonly understand the meaning of the word ‘beginning’.

Quanta of Space and Time

Santa Barbara, February 1986

To begin to understand what it means for space–time to have no boundary, think how you

might answer the question: ‘What lies north of the north pole?’

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When at a conference at the Vatican in 1981 Pope John Paul II suggested that cosmologists

conﬁ ne their speculations to moments after the Creation, Hawking did not have the cour-

age to explain that his no-boundary assumption meant that there could be no moment

of Creation. ‘I had no desire to share the fate of Galileo,’ he explained, ‘with whom I feel a

strong sense of identity, having been born exactly 300 years after his death!’

The ground-state wavefunction was found to correspond to an expanding universe.

Excited states were also found which expanded and then collapsed, but also had a ﬁ nite prob-

ability of tunnelling to states describing continually expanding universes. But the approach

was to prove to be another blind alley, and enthusiasm quickly waned. More gloom gathered

around the canonical approach.

As superstring theorists wrestled with string vibrations in ten dimensions, curled-up

Calabi–Yau spaces, and a general loss of uniqueness, a series of discoveries was being

made that would resurrect the canonical approach and provide a genuine rival to super-

strings as a theory of quantum gravity. A rival that was, moreover, completely independ-

ent of any assumed space–time background.

It would become known as loop quantum gravity.

In August 1982 Amitabha Sen at the University of Maryland submitted a

paper to the journal Physics Letters in which he proposed to recast general

relativity in terms of a three-dimensional ‘spin system’. This is a system

based on ‘spinors’, or spin-vectors ﬁ rst developed by the French math-

ematician Élie Cartan in 1913. Spinors had ﬁ rst been deployed in quantum

theory by Pauli in May 1927, in the form of his spin matrices. This basic

spinor structure had subsequently dropped out of Dirac’s relativistic

wave equation, for both the electron and positron.

In Sen’s development, the spinors had no physical meaning. They were a

mathematical device which allowed him to recast general relativity in a space

of complex vectors more comprehensive in terms of its ability to accom-

modate geometrical information. What he found was a new way to express

the constraints of the ADM Hamiltonian formalism of general relativity.

The formalism was much simpler.

Sen’s work was picked up a few years later by Indian physicist Abhay

Ashtekar at Syracuse University in New York. Ashtekar had studied

gravitational physics and general relativity in Chicago, securing his PhD

in 1974. He recognized that Sen’s use of spinors could be the basis for a

complete reformulation of the constrained Hamiltonian form of general

quanta of space and time

393

relativity, leading to a much simpler version of the Wheeler–DeWitt

equation. By changing from a space–time metric to a space of spin-con-

nections, on which the spinors are propagated, the theory begins to look

remarkably like a gauge ﬁ eld theory. ‘One could now import into general

relativity techniques that have been highly successful in the quantization

of gauge theories,’ he later wrote.

Lee Smolin, recently appointed as an assistant professor at Yale Univer-

sity, heard about Ashtekar’s work and invited him to Yale to give a semi-

nar on the subject. Smolin had graduated in physics and philosophy and

gained his PhD at Harvard in 1979. He had held postdoctoral positions

at the Institute for Advanced Study, the Institute for Theoretical Physics

in Santa Barbara, and the Enrico Fermi Institute in Chicago before join-

ing the faculty at Yale. He had seen Sen’s papers and had pondered their

signiﬁ cance, but he had not followed them up. Now he learned ﬁ rst-hand

from Ashtekar what these ideas promised.

In December 1985 Ashtekar submitted a paper to Physical Review Letters,

outlining his approach and suggesting ‘. . . new ways of attacking a number

of problems in both classical and quantum gravity.’ In January of the fol-

lowing year he relocated to Santa Barbara to act as the coordinator for a

six-month workshop on quantum gravity held at the National Science

Foundation Institute for Theoretical Physics at the University of California.

Trading an east coast for a west coast winter might be regarded by

many as a smart move. Despite having only just joined the faculty at Yale,

Smolin had managed to persuade his department to allow him to attend

the Santa Barbara workshop. On arrival, he recruited two colleagues,

Paul Renteln from Harvard University and Theodore Jacobson from the

University of Maryland, to help him push forward the programme that

Sen and Ashtekar had started.

Just a month later, in February 1986, Smolin and Jacobson made a

breakthrough. They had sat in a small classroom scrawling up on a black-

board possible solutions to the simpler version of the Wheeler–DeWitt

equation that had resulted from Sen and Ashtekar’s reformulation.

Something quite remarkable had happened:

All of a sudden we realized that our second or third guess, which we had

written on the blackboard in front of us, solved the equations exactly. We

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394

tried to compute a term that would measure how much our results were in

error, but there was no error term. At ﬁ rst we looked for our mistake, then

all of a sudden we saw that the expression we had written on the blackboard

was spot on: an exact solution of the full equations of quantum gravity.

The moment was etched on Smolin’s memory. He remembered that it

was a sunny day, and that Jacobson was wearing a T-shirt. ‘. . . then again,

it is always sunny in Santa Barbara and Ted always wears a T-shirt.’

They had found solutions to the Wheeler–DeWitt equation based

on ‘Wilson loops’, named for American theoretician Kenneth Wilson.

Wilson loops had been introduced into quantum theory in the 1970s in

attempts to ﬁ nd an exactly solvable (i.e. non-perturbative) formulation of

QCD. They can be thought of as closed loops of force. We tend to think

of the magnetic ﬁ eld around a bar magnet in terms of the ‘lines of force’

drawn between its poles. These can be revealed in the laboratory by

sprinkling iron ﬁ lings on a piece of paper held above the magnet. In the

absence of charged particles the ﬁ eld excitations become closed loops.

Wilson had constructed quantized loops of force on a space–time that

was not continuous, but composed of a discrete lattice of extremely small

dimensions. Particles could exist only at the nodes (intersections) of the

lattice and ﬁ eld lines only along the lattice edges. In essence, according

to Wilson’s analysis, space–time is the lattice: there is no space–time

between the lattice nodes and edges.

Smolin and Jacobson had dispensed with the lattice and had worked

only with the loops as the fundamental ‘observables’ of the theory. What

they discovered was that, provided they did not intersect or possess sharp

kinks, any one of an inﬁ nite number of these quantized loops could serve

as solutions to the Wheeler–DeWitt equation. There was no need for a

background space–time framework. This was not a theory of loops exist-

ing in space and time. Rather, the relationships between the loops deﬁ ne

space. Smolin wrote:

. . . even at the start we knew that we had in our grasp a quantum theory

of gravity that could do what no theory before it had done—it gave us

an exact description of the physics of the Planck scale in which space is

constructed from nothing but the relationships among a set of discrete

elementary objects.

quanta of space and time

395

After making this second breakthrough, Smolin discovered to his dismay that the loop rep-

resentation they had discovered had already been applied to QCD (but not quantum gravity) in

1980 by Uruguayan physicist Rodolfo Gambini and his collaborator Antoni Trias.

There was just one more hurdle to clear.

In order to qualify as a fully background-independent theory, the solu-

tions that Smolin and Jacobson had obtained had to be shown to be solu-

tions of a second set of equations, called diffeomorphism constraints.

These are the equations that could demonstrate that the solutions con-

formed to the general covariance requirements of general relativity. They

would demonstrate that the solutions were completely independent of

any choice of coordinate system. In general relativity, all coordinate sys-

tems must be equivalent.

This was meant to be the easy part, but it proved stubbornly resistant.

Working back at Yale, Smolin and his colleagues grappled with the dif-

feomorphism constraints and drew a blank. The theory depended not

only on the loops themselves but also on the ﬁ eld ﬂ owing around the

loops, and this made for an intractable problem.

They were getting nowhere fast. Then, in October 1987, a young Italian

theorist called Carlo Rovelli arrived at Yale. Rovelli had gained his PhD

at the University of Padua a year earlier. He and Smolin had met towards

the end of the Santa Barbara workshop, and Rovelli had subsequently

written asking if he could come to Yale and spend some time collabo-

rating with Smolin on quantum gravity. On his arrival, Smolin had to

explain to his visitor that there was nothing to be done, as they had

become completely stuck.

There was an awkward silence. Then Smolin suggested that they go

sailing instead. Happily, Rovelli was a keen sailor.

Smolin saw nothing of Rovelli the next day. On the third day, Rovelli

appeared at the door to Smolin’s ofﬁ ce and declared: ‘I’ve found the

answer to all the problems.’

He had reformulated the theory once more, using the loops them-

selves as ‘basis’ states. This removed the dependence on the ﬁ eld, leaving

only the loops. The result was a theory that satisﬁ ed the diffeomorphism

constraints and was genuinely background-independent.

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396

In this formulation of quantum gravity, the loops are the quanta of

space. As Rovelli later wrote:

The loops are space because they are the quantum excitations of the gravi-

tational ﬁ eld, which is the physical space. It therefore makes no sense to

think of a loop being displaced by a small amount in space. There is only

sense in the relative location of a loop with respect to other loops, and the

location of a loop with respect to the surrounding space is only determined

by the other loops it intersects. A state of space is therefore described by a

net of intersecting loops. There is no location of the net, but only location

on the net itself; there are no loops on space, only loops on loops.

Space is a result of the way loops knot and link together, forming a

‘weave’, stitching together the very fabric of the universe. No longer was

the play of nature just about the actors strutting and fretting their hour

upon the stage. It was also as much about the stage, and the set; the back-

ground against which the play is acted out.

Rovelli gave the ﬁ rst seminar on loop quantum gravity at Syracuse

University. Ashtekar was very appreciative. The ﬁ rst public announce-

ment was made at an international conference on gravitation and cos-

mology held in Goa, India, in December 1987.

As they continued to work on the theory, Smolin and Rovelli real-

ized that they had stumbled across a structure already familiar. They

discovered parallels between the networks of loops and a mathemati-

cal structure developed by Roger Penrose in 1971 called spin networks. At

its simplest, a spin network is a diagram representing the states of, and

interactions between, particles and ﬁ elds.

In Penrose’s original formulation, each vertex or ‘node’ of the network

represents an event, such as two particles colliding or a single particle

breaking up into two or more fragments. Each line joining one event

to another is labelled with the spin number, a measure of the angular

momentum of the particle (in units of ħ/2), with even spin numbers

for bosons and odd numbers for fermions. The particles then interact

in ways in which the total angular momentum is conserved. Penrose’s

motivation in developing spin networks was to ﬁ nd a way to eliminate

the space–time background and deal only with relationships between the

particle events.

quanta of space and time

397

Initially rather nervous about using spin networks, Smolin visited

Penrose in 1994 and learned directly how to use them from their inventor.

He took what he had learned back to Rovelli in Verona, Italy, and to-

gether they spent the summer of 1994 working out how to apply spin

networks in loop quantum gravity. What they found was that each spin

network, formed from a speciﬁ c combination of loops, represents a

quantum state of geometry. For each of these states, the nodes are volume

elements, characterized by the number of quanta of volume, and the

links between them are elements of area, characterized by the number of

quanta of area.

By solving the equation for the mathematical ‘operator’ for area, they

were able to determine the ‘eigenstates’ of area and so deduce the dimen-

sions of a single quantum. What they found is that every area is com-

posed of an integral number of fundamental quanta with dimensions

about the square of the Planck length. Similarly, by solving the equation

for the ‘operator’ for volume, they were also able deduce the dimensions

of a single quantum of volume. They found that every volume is com-

posed of an integral number of fundamental quanta given by the cube

of the Planck length.

These are unimaginably small dimensions. About 10

quanta of vol-

ume will ﬁ t inside a single proton.

At these dimensions space–time is no

longer continuous. There is no area smaller than a quantum of area, no

volume smaller than a quantum of volume.

And, of course, the interconnectedness of space and time means that

time, viewed as the rearrangement of links between the loops or evolution

of the spin network, must be similarly quantized, with the dimensions of

the quanta determined by the Planck time. It had been believed for some

time that the Planck scale should somehow represent an ultimate limit,

beyond which it is no longer possible to ask sensible questions about

the nature of space and time. Now loop quantum gravity had furnished

the reason.

There is no restriction on the size of a spin network. As Smolin

explained: ‘If we could draw a detailed picture of the quantum state of

That’s 10 raised to the power 65, or 1 followed by 65 zeros.

the quantum story

398

our universe—the geometry of its space, as curved and warped by the

gravitation of galaxies and black holes and everything else—it would be

a gargantuan spin network of unimaginable complexity, with approxi-

mately 10

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nodes.’

There was still plenty to do. Loop quantum gravity was recognized

as a strong candidate for a quantum theory of gravity. At this time, it

was not considered as a candidate theory of everything, like superstring

theory. Although it had provoked little interest at the time, the fact that

superstring theory accommodated a particle with the expected proper-

ties of the graviton held much appeal for particle physicists. They liked

particles. In loop quantum gravity, however, the graviton appeared to

have gone missing.

And loop quantum gravity was a theory of space and time. The familiar

particles of the Standard Model had yet to be introduced to it.

399

The Standard Model could not be advanced beyond the position it had reached in the

late 1970s. The Higgs boson remained out of reach of CERN and Fermilab physicists.

Grand uniﬁ ed theories had come, and gone. Supersymmetry looked promising, but pro-

liferated supersymmetric partners for every particle in the Standard Model; particles for

which there was no experimental evidence. For a time, Hawking had declared N = 8

supergravity to be the real deal, the physicists’ holy grail. But it was not.

Superstring theory, meanwhile, had suffered an embarrassment of riches. In addition

to Type I string theory, based on the symmetry group SO(32), Type IIA, Type IIB, and

heterotic string theories, a ﬁ fth version was formulated. This was a variant of heterotic

string theory, also based on SO(32). As theories proliferated, interest waned.

Witten then made a bold conjecture. A number of interesting dualities between string

theory and quantum ﬁ eld theories had emerged in the early-mid 1990s.

At a superstring

theory conference at the University of Southern California in March 1995, Witten specu-

lated that the duality relations between the ﬁ ve string theories meant that they might all

be subsumed into one, overarching structure. He called it M-theory. He was not speciﬁ c

on the meaning or signiﬁ cance of ‘M’.

Crisis? What Crisis?

Durham, Summer 1994

For example, S-duality, or strong–weak duality, allows states with a coupling constant g in

one type of theory to be mapped to states with coupling 1/g in its dual theory. This means that

the techniques of perturbation theory, normally applicable only in quantum ﬁ eld theories for

systems of low energy and weak coupling, can be applied to the high-energy, strong-coupling

regimes of string theory.