Van Huyssteen J.W. (editor) Encyclopedia of Science and Religion

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PHYSICS,CLASSICAL

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According to Aristotle (384–322 B.C.E.), who

was for nearly two millennia taken to be the au-

thority on these matters, motion in the terrestrial

realm required the continuous application of a

cause. Remove the cause, and motion would

cease. When a horse ceases to pull a cart, for in-

stance, the cart comes to a halt. In Newton’s for-

mulation, however, what requires an active cause

is not motion itself, but acceleration—any change

in the speed or direction of motion. In effect, New-

ton’s First Law of Motion asserts that the natural

motion of things is uniform motion, straight-line

motion at constant speed. Any deviation from

this—any acceleration, that is—would require a

cause. The name for this cause is force—specifi-

cally, the force exerted on one object by interac-

tion with another. Expressed more traditionally,

Newton’s First Law states that unless acted upon

by an applied force, an object will continue in a

state of rest or uniform motion.

What happens when a force is applied to an

object? The answer to that question is the subject

of Newton’s Second Law of Motion: When acted

upon by an applied force, an object will accelerate;

the resultant acceleration will be in the same di-

rection as the applied force, and its magnitude will

be directly proportional to the magnitude of the

applied force and inversely proportional to the ob-

ject’s mass. Stated more succinctly, acceleration is

proportional to force divided by mass. This state-

ment, more than any other, functions as the core of

Newtonian dynamics, Newton’s formulation of the

fundamental cause-effect relationship for motion.

Force is the cause; acceleration is the effect. For a

substantial class of motions, with exceptions to be

noted later, this formulation continues to provide a

fruitful way to predict or account for acceleration

in response to applied forces.

Newton’s Third Law of Motion is a statement

about the character of the applied forces men-

tioned in the first two laws. All such forces occur in

pairs and are the result of two bodies interacting

with one another. When two bodies interact, says

Newton, each exerts a force on the other. When

bodies A and B interact, the force exerted on A by

B is equal in magnitude and opposite in direction

to the force exerted on B by A. This is sometimes

abbreviated to read, “action equals reaction,” but

the meanings of action and reaction must be very

carefully specified.

Among the various types of forces that con-

tribute to the acceleration of terrestrial objects is

the force of gravity—the force that causes apples,

for example, to fall to the ground, or to “accelerate

earthward.” It was the genius of Newton that al-

lowed him to consider the possibility that the or-

bital motion of the moon, which entails an accel-

eration toward the Earth, might also be a

consequence of the Earth’s gravitational attraction.

This suggestion required a remarkable break

with Aristotelian tradition. According to Aristotle,

the natural motion of the moon, of the planets, or

of any other member of the celestial realm was en-

tirely different from the terrestrial motions consid-

ered so far. The natural motion of celestial bodies

was neither rest nor uniform straight-line motion.

Rather, the motion of celestial bodies would nec-

essarily be based on uniform circular motion, mo-

tion at constant speed on a circular path. In the

spirit of this assumption, Claudius Ptolemy in the

second century crafted a remarkably clever combi-

nation of uniform circular motions with which to

describe the motions of the sun, moon, and plan-

ets relative to the central Earth.

However, building on the fruitful contributions

of astronomers Nicolaus Copernicus (1473–1543),

Galileo Galilei (1564–1642), and Johannes Kepler

(1571–1630), Newton was able to demonstrate that

Kepler’s sun-centered model for planetary motions

could be seen as but one more illustration of New-

ton’s theory regarding the cause-effect relationship

for motion. The moon was steered in its orbit

around the Earth in response to a force exerted by

the Earth on the moon. The Earth and the other

planets orbited the sun in response to a force ex-

erted on them by the sun. What was the force op-

erating in these celestial motions? The same kind of

force that caused apples to accelerate earthward—

the universal gravitational force.

It was helpful to recognize gravity as a force

exerted by one object on another. It was excep-

tionally insightful for Newton to propose that every

pair of objects everywhere in the universe exerted

gravitational forces on one another. Gone was the

confusion of two kinds of natural motions. Gone

was the even greater distinction between terres-

trial and celestial realms—one characterized by im-

perfection and change, the other characterized by

perfection and constancy. The cosmos is one sys-

tem, not two. The world is a universe made of one

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set of substances and behaving according to one

set of patterns. Classical mechanics provided the

means to study all motions, both terrestrial and ce-

lestial, with one and the same methodology.

Classical electromagnetism

Classical electromagnetism provided a systematic

account of numerous phenomena involving the in-

teraction of electric charges and currents. Electric

charges at rest were considered to be the source of

electric fields—modifications in the nature of space

that cause other charges to experience a force.

Electric charges in motion, giving rise to an electric

current, were considered to be the source of mag-

netic fields, modifications in the nature of space

that could be detected by a magnetic compass and

caused other electric currents to experience a

force. Given any static distribution of electric

charge, the configuration of the resultant electric

field could be computed. Given any distribution of

electric currents, the configuration of the resultant

magnetic field could be computed. Given these

electric and magnetic field configurations, the

forces on all electric charges and currents could be

predicted.

In addition to phenomena involving static

charge distributions and steady electric currents,

another important category of phenomena arises

from dynamically changing configurations of

charge or current. When charge or current config-

urations change, the resultant electric and mag-

netic fields will also change. However, changes in

these field configurations must propagate at a finite

speed—now called the speed of light, approxi-

mately 300,000 kilometers per second. Electromag-

netic radiation is the phenomenon of traveling

variations, or waves, in electric and magnetic field

strength caused by accelerated electric charges.

The electromagnetic spectrum spans the full range

of wavelength values from very short to very

long—from gamma rays, X-rays, and ultraviolet to

visible light, infrared, microwaves and radio waves.

Maxwell’s equations—four mathematical state-

ments that systematically integrated the work of

predecessors like Charles-Augustin de Coulomb

(1736–1806), Hans Christian Oersted (1777–1851),

Michael Faraday (1791–1867), and André-Marie

Ampère (1775–1836)—were taken to be the com-

plete specification of all electromagnetic phenom-

ena, including electromagnetic radiation.

Limitations of classical physics

Until the early twentieth century, classical physics

appeared to be adequate to account for all ob-

served phenomena. But new discoveries soon

demonstrated that, although classical physics

would continue to provide a convenient and pow-

erful means of dealing with many phenomena, it

needed to be supplemented with other theoretical

strategies based on differing sets of assumptions

regarding the fundamental character of the physi-

cal universe. In the arena of electromagnetism, for

instance, classical physics assumed that electro-

magnetic energy could be continuously varied in

value and that its transmission could be fully de-

scribed in terms of traveling electromagnetic

waves. However, in order to account for such phe-

nomena as blackbody radiation (electromagnetic

energy radiated by any warm object) and the pho-

toelectric effect (electrons ejected from the surface

of a metal illuminated by light), physicists had to

propose and accept the idea that electromagnetic

energy was transmitted in particle-like quanta of

energy, now called photons. Phenomena in which

the photon character of electromagnetic radiation

plays a central role requires the employment of

quantum physics in place of classical physics.

Quantum physics is also needed to account for

the behavior of extremely small systems like atoms

and molecules. The motion of electrons relative to

atomic nuclei cannot be adequately described in

the language of classical mechanics. Contrary to

Newtonian expectations, the energy of atoms and

molecules is not continuously variable, but is

quantized—restricted to certain specific values.

And, contrary to the expectations of classical elec-

tromagnetism, electrons in motion relative to

atomic nuclei do not radiate energy continuously,

but only when making a transition from one stable

energy state to another of lower energy value.

Consistent with the Principle of Conservation of

Energy, the amount of energy lost by the atom is

exactly equal to the energy carried away by the

emitted photon.

A second shortcoming of classical physics be-

comes evident when Newtonian mechanics at-

tempts to deal with things that are moving at very

high speed relative to an observer. When this

speed becomes a substantial fraction of the speed

of light, several Newtonian expectations require

modification. Many of these modifications are ac-

counted for by the Special Theory of Relativity

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proposed by Albert Einstein (1879–1955) in 1905.

The relationship between kinetic energy (energy

associated with motion) and speed must be modi-

fied. Distance and time intervals once thought to

be invariant become dependent on relative mo-

tion. Even the mass of an object is measured dif-

ferently by different observers. Other modifications

are accounted for by Einstein’s General Theory of

Relativity, published in 1916, which deals with the

interaction of mass and the geometry of space. The

General Theory describes the force of gravity in a

manner very different from Newton’s and is able to

account for several discrepancies between obser-

vation and Newtonian predictions.

Religious concerns and classical physics

Classical physics gave support to the idea that the

world was fundamentally deterministic. Given full

information about the configuration and motion of

some system today, its entire future could, in prin-

ciple, be computed. Its future was considered to

be fully determined by its present. But is there

room in such a universe for contingency or choice?

The apparent absence of choice presents difficul-

ties for religious concepts like human responsibil-

ity and human accountability to God for obedience

to revealed standards for moral action.

Another religious concern arises when one in-

quires about the character and role of divine action

in the universe. When Newton considered the fu-

ture motions of the planets in the solar system, for

instance, he judged that this set of orbital motions

was inherently unstable and would, from time to

time, need to be adjusted by God to restore the de-

sired array of orbits. This introduction of occa-

sional supernatural interventions may be consid-

ered a form of the God of the gaps approach to

divine action: the universe is presumed to lack

some quality or capability that must be compen-

sated for by direct divine action. In the case of

planetary motions, for example, Newton consid-

ered the universe to lack the capability of main-

taining a stable set of orbits. This “capability gap”

could, however, be bridged with occasional acts of

supernatural intervention. Eventually, however, it

was demonstrated that the system of planetary or-

bits was, in fact, stable, thereby removing the need

for occasional gap-bridging interventions. When a

“gap” of this sort becomes filled, the God of the

gaps becomes superfluous. For this reason, many

contemporary theologians are inclined to see di-

vine action, not as a supernatural compensation

for capability gaps in the universe, but as an es-

sential aspect of an enriched concept of what takes

place naturally.

See also ARISTOTLE; DETERMINISM; DIVINE ACTION;

OD OF THE GAPS; GRAVITATION; NEWTON, ISAAC;

HYSICS, QUANTUM; RELATIVITY, GENERAL THEORY

; RELATIVITY, SPECIAL THEORY OF; WAV E-

PARTICLE DUALITY

Bibliography

Bernal, J. D. History of Classical Physics. New York:

Barnes and Noble, 1997.

Pullman, Bernard. The Atom in the History of Human

Thought, trans. Axel R. Reisinger. Oxford: Oxford

University Press, 1998.

Feynman, Richard P.; Leighton, Robert B.; and Sands,

Matthew L. The Feynman Lectures on Physics. Boston:

Addison-Wesley, 1994.

Goldstein, Herbert; Poole, Charles; and Safko, John L.

Classical Mechanics, 3rd edition. Upper Saddle River,

N.J.: Prentice Hall, 2002.

Griffiths, David J. Introduction to Electrodynamics, 3rd

edition. Upper Saddle River, N.J.: Prentice Hall, 1998.

Halliday, David; Resnick, Robert; and Walker, Jearl. Funda-

mentals of Physics, 6th edition. New York: Wiley, 2000.

Jackson, John David. Classical Electrodynamics, 3rd edi-

tion. New York: Wiley, 1998.

Symon, Keith R. Mechanics, 3rd edition. Boston: Addison-

Wesley, 1971.

HOWARD J. VAN TILL

PHYSICS,PARTICLE

The thought that the bewildering variety of the

world might be the result of many different

arrangements of certain simple kinds of basic stuff

is a very old one. In the sixth century

B.C.E., vari-

ous pre-Socratic philosophers explored such ideas.

Thales thought that the fundamental entity might

be water, while Anaximenes favored air. Such at-

tempts were both insightful and hopelessly prema-

ture. A more sophisticated notion was the atomism

introduced by Democritus a century later and pro-

moted with considerable literary skill by the Latin

poet Lucretius in the first century

B.C.E.

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Progress of a recognizably modern kind began

with chemist John Dalton’s (1766–1844) atomic

theory, which introduced in 1803 the notion of

atomic weights, derived principally from the prop-

erties of gases. Chemist and physician William

Prout’s (1785–1850) observation in 1815 that most

of these weights were near integer multiples of the

atomic weight of hydrogen led to what one might

call the first true theory of elementary particles,

with hydrogen as the conjectured fundamental

building block.

Twentieth-century developments

In 1897, physicist Joseph Thomson (1856–1940)

convincingly demonstrated that there are light,

electrically negative particles (subsequently called

electrons) that are constituents of what, until then,

had been considered to be the indivisible atom. In

1911, physicist Ernest Rutherford (1871–1937) suc-

cessfully interpreted experiments in which projec-

tiles called alpha particles were significantly de-

flected by a thin gold foil as showing that the

positive charge in the atom was concentrated at its

center. Rutherford had discovered the nucleus.

In the rest of the twentieth century there fol-

lowed a series of discoveries, each of which led in

turn to a yet deeper conception of the structure of

matter, expressed in terms of still smaller con-

stituents playing the role of “elementary” particles.

Each phase of these investigations, often pictured

metaphorically as peeling another layer off the nu-

clear “onion,” had a sequential form. The process

of discovery took place in two parts. The first half

consisted in the revelation of an increasing prolif-

eration of putative elementary entities. An example

would be the varieties of different nuclei generat-

ing the chemical properties of the ninety-two ele-

ments of the periodic table. There is a strong con-

viction in the human mind (exemplified as much

by the pre-Socratic philosophers as by twentieth-

century physicists) that the fundamental structure

of matter should take a simple form, elegant and

economical in its character. Proliferation threatens

this conviction, but rescue comes in the second

half of the process of discovery. Patterns are dis-

cerned linking together the proliferating elements,

and these patterns are interpreted as reflecting the

ways in which a small number of yet more funda-

mental constituents can be combined. In this way

the next level of structure is revealed. It seems fun-

damental enough until, in turn, it too begins to

proliferate, and the cycle begins again.

Thus, nuclei were first recognized as being

made up of two kinds of nuclear particles, protons

and neutrons. Then experimentalists began to dis-

cover many short-lived cousins of these nuclear

particles and a proliferation began to threaten.

However, the association of these different forms of

nuclear matter into certain patterns (called the eight-

fold way by its most insightful investigator, Murray

Gell-Mann [b. 1929]) eventually led to the identifi-

cation of the quark level in the structure of matter.

Consideration of symmetry provides an impor-

tant mathematical tool for the understanding of

pattern formation. For example, the beautiful pat-

tern of a snowflake is due to the sixfold symmetry

that leaves it unchanged under a rotation of sixty

degrees. It turned out that the patterns of nuclear

matter were also generated by symmetry princi-

ples, though principles of a more abstract kind

than those given by simple rotations in space. Gell-

Mann identified the relevant symmetry as being as-

sociated with what mathematicians call the group

SU(3). The SU(3) structure involves certain kinds of

transformation applied to a set of three basic ob-

jects. Such a mathematical fact did not necessarily

imply a physical counterpart but, if it did, the cor-

responding physical entities would generate the

next layer in the nuclear onion. Gell-Mann named

these entities quarks.

There was initially doubt about the physical

reality of quarks. The theory requires them to have

fractional electric charge (



; −



), and no such par-

ticles have ever been observed in nature. How-

ever, when indirect evidence of their existence

came to light, it proved to be very convincing. The

experiments involved what is called deep inelastic

scattering. This is the analogue of the experiments

that enabled Rutherford to discover the nucleus,

but conducted at much higher energy. Projectiles,

such as electrons, when scattered off protons and

neutrons, were discovered sometimes to “bounce

back” in just the way that they would if they were

hitting pointlike quarks lying within these nuclear

particles. Physicists could eventually understand

why projectiles behaved this way, but in the case

of quarks there was a new feature without any

precedent in physical experience. However strong

the impact of the projectiles, it never proved pow-

erful enough to actually eject a single quark. Even-

tually, physicists were forced to conclude that

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quarks were “confined,” that is to say, the forces

that bound them inside protons and neutrons were

always strong enough to overcome the effect of

the impact, however great that might be. No one

has ever seen an individual quark.

The forces that produce quark confinement are

generated by the exchange of further particles that,

in the relentlessly jokey terminology endemic in

particle physics, are called gluons. Further discov-

eries of exotic kinds of nuclear matter increased

the number of types of quark from three to six.

These ideas, together with others of a more tech-

nical character, constitute what has come to be

called the Standard Model.

Only one piece in the jigsaw that defines the

standard model is still missing. This is the particle

proposed by Peter Higgs as the source of mass

within the theory. Particle accelerators can yield

energies that are just on the border of where this

Higgs particle (as it is called) may be expected to

show up. Establishing its existence would be ex-

tremely satisfying.

The Standard Model describes very well the

properties of subnuclear matter but, with its six va-

rieties of quark and with other somewhat inelegant

elaborations, there is an air of proliferation about

it. Most physicists, therefore, do not feel a final sat-

isfaction with the Standard Model. There are two

ways in which one might hope eventually to go

beyond it. One is the discovery of a Grand Unified

Theory (GUT).

In terms of directly observed phenomena there

seem to be four basic forces of nature: strong nu-

clear forces (holding nuclei together); electromag-

netic forces (holding atoms and bulk matter to-

gether); weak nuclear forces (causing matter to

decay); and gravity. One of the triumphs of the

Standard Model was to show that two of these

forces, electromagnetic and weak, are in reality as-

pects of a single phenomenon, a fact that becomes

clear experimentally at very high energies. Physi-

cists believe that at even higher energies (such as

would be present in the very early universe) these

two forces would unite with the strong nuclear

force to give a GUT. The detailed form this theory

might be expected to take has not been established.

At higher energies still, there is the possibility

that gravity and the GUT unite. For technical rea-

sons, however, a theory of this super-unified kind

is even harder to formulate than a GUT. The best

speculative prospect appears to be superstring the-

ory (or its generalizations), in which quarks and

electrons are pictured as modes of vibration of ex-

tremely tiny strings oscillating in many dimensions,

all but four of which (space and time) are “rolled

up” out of empirical sight.

Lessons for theology

Particle physics is methodologically the most re-

ductionist form of physics. It encourages the

thought of constituent reductionism, implying that

were a human being to be decomposed into bits

and pieces, the ultimate result would be an im-

mense collection of quarks, gluons, and electrons.

This observation, however, by no means proves

that human beings are nothing but collections of

elementary particles, since such a decomposition

would kill the person. In fact, quantum physics en-

courages an antireductionist stance, because it has

been shown that there is a counterintuitive mutual

entanglement of quantum entities, even when they

are spatially separated (the EPR effect). It does not

seem that even the subatomic world can be treated

purely atomistically.

An important technique of discovery in funda-

mental physics has proved to be the search for

equations endowed with the unmistakable quality

of mathematical beauty. Paul Dirac (1902–1984),

one of the founding figures of quantum mechan-

ics, once expressed the opinion that it was more

important to have mathematical beauty in one’s

equations than to have them fit experiment. Of

course, he did not mean that empirical adequacy is

irrelevant to physics, but apparent failure to fit ex-

periment might be due to a number of reasons,

such as making an incorrect approximation in solv-

ing the equations, or even to the experimental re-

sults themselves being wrong. But if the equations

were ugly, there was really no hope, for ugliness

ran counter to everything that experience of fun-

damental physical theory had led one to expect.

Dirac made his own significant discoveries through

just such a quest for mathematical beauty, and the

same principle is the guiding strategy followed by

the bold proponents of superstring theory. It seems

that the physical world is not only rationally trans-

parent to our enquiry, it is also rationally beautiful.

Beneath the vast variety of everyday objects, at the

subatomic level there is a fundamental structure

that is intellectually exciting in its simplicity and

profoundly satisfying in the elegance and economy

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of its order. The reward for doing particle physics

is the sense of wonder at its discoveries. The the-

istic religious believer will readily see the mind of

the Creator behind the rationally beautiful order of

the physical world.

Another lesson one may learn from particle

physics is that human powers of rational prevision

are severely limited. Time and again, nature has

proved surprising as it resists our prior expecta-

tions. In the 1950s, particle physicists who were at-

tempting to make sense of certain weak decays

faced profound difficulties. After much fruitless

struggle, the situation was transformed and made

intelligible when in 1956 two physicists, Tsung

Dao Lee and Chen Ning Yang, proposed the aban-

donment of what had been a cherished belief of

particle physicists. Until then, it had been an article

of faith that there could be no intrinsic handedness

in nature, meaning that fundamental processes

should show no preference for right-handed ver-

sions over left-handed versions or—putting it an-

other way—that the laws of physics seen in a mir-

ror should look exactly the same as the laws of

physics observed directly. This supposed property

was called the conservation of parity, and it was

believed to be a self-evident truth about nature.

Lee and Yang showed that this was not so, a dis-

covery for which they rightly and promptly re-

ceived the Nobel Prize.

Particle physics teaches us that the physical

world is extremely surprising. It would be strange

if that were not also true of human encounter with

the much deeper mystery of divine reality. Physi-

cists do not favor the question “Is it reasonable?”

with its tacit presumption that one knows before-

hand what form rationality should take. Rather,

they ask the more open question “What makes you

think this might be the case?” Theology too can

benefit from seeking belief motivated by experi-

ence rather than by a priori expectation.

Finally, particle physicists believe in unseen

realities (quarks) because such a belief makes

sense of great swathes of physical experience. For

them, it is intelligibility that affords the clue to ex-

istence. This does not seem altogether different

from the reasons for theology’s belief in the un-

seen reality of God.

See also FORCES OF NATURE; GRAND UNIFIED THEORY;

TRING THEORY; SUPERSTRINGS; SYMMETRY

Bibliography

Greene, Bryan. The Elegant Universe: Superstrings, Hidden

Dimensions, and the Quest for an Ultimate Theory.

London: Jonathan Cape, 1999.

Pagels, Heinz R. The Cosmic Code: Quantum Physics and

the Language of Nature. London: Michael Joseph,

1982.

Pais, Abraham. Inward Bound: Of Matter and Forces in the

Physical World. Oxford: Oxford University Press, 1986.

Polkinghorne, John. The Particle Play: An Account of the

Ultimate Constituents of Matter. Oxford: W. H. Free-

man, 1979.

Polkinghorne, John. Rochester Roundabout: The Story of

High Energy Physics. Harlow, UK: Longman; New

York: W. H. Freeman, 1989.

Weinberg, Steven. Dreams of a Final Theory: The Search

for the Fundamental Laws of Nature. London:

Hutchinson, 1993.

JOHN POLKINGHORNE

PHYSICS,QUANTUM

Quantum theory is one of the most successful the-

ories in the history of physics. The accuracy of its

predictions is astounding. The breath of its appli-

cation is impressive. Quantum theory is used to ex-

plain how atoms behave, how elements can com-

bine to form molecules, how light behaves, and

even how black holes behave. There can be no

doubt that there is something very right about

quantum theory.

But at the same time, it is difficult to under-

stand what quantum theory is really saying about

the world. In fact, it is not clear that quantum the-

ory gives any consistent picture of what the physi-

cal world is like. Quantum theory seems to say

that light is both wavelike and particlelike. It seems

to say that objects can be in two places at once, or

even that cats can be both alive and dead, or nei-

ther alive nor dead, or—what? There can be no

doubt that there is something troubling about

quantum theory.

Early research

Quantum theory, more or less as it is known at the

beginning of the twenty-first century, was devel-

oped during the first quarter of the twentieth cen-

tury in response to several problems that had

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arisen with classical mechanics. The first is the

problem of blackbody radiation. A blackbody is

any physical body that absorbs all incident radia-

tion. As the blackbody continues to absorb radia-

tion, its internal energy increases until, like a

bucket full of water, it can hold no more and must

re-emit radiation equal in energy to any additional

incident radiation. The problem is, most simply,

that the classical prediction for the energy of the

emitted radiation as a function of its frequency is

wrong. The problem was well known but unsolved

until the German physicist Max Planck (1858–1947)

proposed in 1900 the hypothesis that the energy

absorbed and emitted by the blackbody could

come only in discrete amounts, multiples of some

constant, finite, amount of energy. While Planck

himself never felt satisfied with this hypothesis as

more than a localized, phenomenological descrip-

tion of the behavior of blackbodies, others eventu-

ally accepted Planck’s hypothesis as a revolution, a

claim that energy itself can come in only discrete

amounts, the quanta of quantum theory.

A second problem with classical mechanics

was the challenge of describing the spectrum of

hydrogen, and eventually, other elements. Atomic

spectra are most easily understood in light of a

fundamental formula linking the energy of light

with its frequency: E = hν, where E is the energy of

light, h is a constant (Planck’s constant, as it turns

out), and ν is the frequency of the light (which de-

termines the color of the visible light).

Suppose, now, that the energy of some atom

(for example, an atom of hydrogen) is increased. If

the atom is subsequently allowed to relax, it re-

leases the added energy in the form of (electro-

magnetic) radiation. The relationship E = hν re-

veals that the frequency of the light depends on

the amount of energy that the atom emits as it re-

laxes. Prior to the development of quantum theory,

the best classical theory of the atom was Ernest

Rutherford’s (1871–1937), according to which neg-

atively charged electrons orbit a central positively

charged nucleus. The energy of a hydrogen atom

(which has only one electron) corresponds to the

distance of the electron from the nucleus. (The fur-

ther the electron is, the higher its energy is.)

Rutherford’s model predicts that the radiation emit-

ted by a hydrogen atom could have any of a con-

tinuous set of possible energies, depending on the

distance of its electron from the nucleus. Hence a

large number of hydrogen atoms with energies

randomly distributed among them will emit light of

many frequencies. However, in the nineteenth cen-

tury it was well known that hydrogen emits only a

few frequencies of visible light.

In 1913, Niels Bohr (1885–1962) introduced the

hypothesis that the electrons in an atom can be

only certain distances from the nucleus; that is,

they can exist in only certain “orbits” around the

nucleus. The differences in the energies of these

orbits correspond to the possible energies of the

radiation emitted by the atom. When an electron

with high energy “falls” to a lower orbit, it releases

just the amount of energy that is the difference be-

tween the energies of the higher and lower orbits.

Because only certain orbits are possible, the atom

can emit only certain frequencies of light.

The crucial part of Bohr’s proposal is that elec-

trons cannot occupy the space between the orbits,

so that when the electron passes from one orbit to

another, it “jumps” between them without passing

through the space in between. Thus, Bohr’s model

violates the principle of classical mechanics that

particles always follow continuous trajectories. In

other words, Bohr’s model left little doubt that

classical mechanics had to be abandoned.

Over the next twelve years, the search was on

for a replacement. By 1926, as the result of con-

siderable experimental and theoretical work on

the part of numerous physicists, two theories—

experimentally equivalent—were introduced,

namely, Werner Heisenberg’s (1901–1976) matrix

mechanics and Erwin Schrödinger’s (1887–1961)

wave mechanics.

Matrix mechanics. Heisenberg’s matrix me-

chanics arose out of a general approach to

quantum theory advocated already by Bohr and

Wolfgang Pauli (1900–1958), among others. In

Heisenberg’s hands, this approach became a com-

mitment to remove from the theory any quantities

that cannot be observed. Heisenberg took as his

“observable” such things as the transition probabil-

ities of the hydrogen atom (the probability that an

electron would make a transition from a higher to

a lower orbit). Heisenberg introduced operators

that, in essence, represented such observable

quantities mathematically. Soon thereafter, Max

Born (1882–1970) recognized Heisenberg’s opera-

tors as matrices, which were already well under-

stood mathematically.

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Heisenberg’s operators can be used in place of

the continuous variables of Newtonian physics. In-

deed, one can replace Newtonian position and

momentum with their matrix “equivalents” and ob-

tain the equations of motion of quantum theory,

commonly called (in this form) Heisenberg’s equa-

tions. The procedure of replacing classical (New-

tonian) quantities with the analogous operators is

known as quantization. A complete understanding

of quantization remains elusive, due primarily to

the fact that quantum-mechanical operators can be

incompatible, which means in particular that they

cannot be comeasured.

Wave mechanics. Schrödinger’s wave mechan-

ics arose from a different line of reasoning, prima-

rily due to Louis de Broglie (1892–1987) and Albert

Einstein (1879–1955). Einstein had for some time

expressed a commitment to a physical world that

can be adequately described causally, which meant

that it could be described in terms of quantities that

evolve continuously in time. Einstein, who was pri-

marily responsible for showing that light has both

particlelike and wavelike properties, hoped early

on for a theory that somehow “fused” these two

aspects of light into a single consistent theory.

In 1923, de Broglie instituted the program of

wave mechanics. He was impressed by the Hamil-

ton-Jacobi approach to classical physics, in which

the fundamental equations are wave equations, but

the fundamental objects of the theory are still par-

ticles, whose trajectories are determined by the

waves. Recalling this formalism, de Broglie sug-

gested that the particlelike and wavelike properties

of light might be reconcilable in similar fashion.

Einstein’s enthusiasm for de Broglie’s ideas—both

because de Broglie’s waves evolved continuously

and because the theory fused the wavelike and

particlelike properties of light and matter—stimu-

lated Schrödinger to work on the problem from

that point of view, and in 1926 Schrödinger pub-

lished his wave mechanics.

It was quickly realized that matrix mechanics

and wave mechanics are experimentally equiva-

lent. Shortly thereafter, in 1932, John von Neumann

(1903–1957) showed their equivalence rigorously

by introducing the Hilbert space formalism of

quantum theory. The Uncertainty Principle serves

to illustrate the equivalence. The Uncertainty Prin-

ciple follows immediately from Heisenberg’s ma-

trix mechanics. Indeed, in only a few lines of ar-

gument, one can arrive at the mathematical

statement of the Uncertainty Principle for any op-

erators (physical quantities) A and B: ∆A∆B ≥ Kh,

where K is a constant that depends on A and B,

and h is Planck’s constant. The symbol ∆A means

“root mean square deviation of A” and is a meas-

ure of the statistical dispersion (uncertainty) in a

set of values of A. So the Uncertainty Principle says

that the statistical dispersion in values of A times

the statistical dispersion in values of B are always

greater than or equal to some constant. If (and

only if) A and B are incompatible (see above) then

this constant is greater than zero, so that it is im-

possible to measure a both A and B on an ensem-

ble of physical systems in such a way as to have no

dispersion in the results.

Schrödinger’s wave mechanics gives rise to the

same result. It is easiest to see how it does so in

the context of the classic example involving posi-

tion and momentum, which are incompatible

quantities. In the context of Schrödinger’s wave

mechanics, the probability of finding a particle at a

given location is determined by the amplitude

(height) of the wave at that location. Hence, a par-

ticle with a definite position is represented by a

“wave” that is zero everywhere except at the loca-

tion of the particle. On the other hand, a particle

with definite momentum is represented by a wave

that is flat (i.e., has the same amplitude at all

points), and, conversely to position, momentum

becomes more and more “spread” as the wave be-

comes more sharply peaked. Hence the more pre-

cisely one can predict the location of a particle, the

less precisely one can predict its momentum. A

more quantitative version of these considerations

leads, again, to the Uncertainty Principle.

Quantum field theory. Perhaps the major de-

velopment after the original formulation of quan-

tum theory by Heisenberg and Schrödinger (with

further articulation by many others) was the exten-

sion of quantum mechanics to fields, resulting in

quantum field theory. Paul Dirac (1902–1984) and

others extended the work to relativistic field theo-

ries. The central idea is the same: The quantities of

classical field theory are quanticized in an appro-

priate way. Work on quantum field theory is on-

going, a central unresolved issue being how one

can incorporate the force of gravity, and specifi-

cally Einstein’s relativistic field theory of gravity,

into the framework of relativistic quantum field

theory. A related, though even more speculative,

area of research is quantum cosmology, which is,

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more or less, the attempt to discern how Big Bang

theory (itself derived from Einstein’s Theory of

Gravity) will have to be modified in the light of

quantum gravity.

Contemporary research

Contemporary research in the interpretation of

quantum theory focuses on two key issues: the

“measurement problem” and locality (Bell’s

Theorem).

Schrödinger’s cat. Although the essence of the

measurement problem was clear to several re-

searchers even before 1925, it was perhaps first

clearly stated in 1935 by Schrödinger. In his fa-

mous example, Schrödinger imagines a cat in the

following unfortunate situation. A box, containing

the cat, also contains a sample of some radioactive

substance that has a probability of 1/2 to decay

within one hour. Any decay is detected by a Geiger

counter, which releases poison into the box if it

detects a decay. At the end of an hour, the state of

the cat is indeterminate between “alive” and

“dead,” in much the same way that a state of defi-

nite position is indeterminate with regard to

momentum.

The cat is said to be in a superposition of the

alive state and the dead state. In standard quantum

theory, such a superposition is interpreted to mean

that the cat is neither determinately alive, nor de-

terminately dead. But, says Schrödinger, while one

might be able to accept that particles such as elec-

trons are somehow indeterminate with respect to

position or momentum, one can hardly accept in-

determinacy in the state of a cat.

More generally, Schrödinger’s point is that in-

determinacy at the level of the usual objects of

quantum theory (electrons, protons, and so on)

can easily be transformed into indeterminacy at the

level of everyday objects (such as cats, pointers on

measuring apparatuses, and so on) simply by cou-

pling the state of the everyday object to the state of

the quantum object. Such couplings are exactly the

source of our ability to measure the quantum ob-

jects in the first place. Hence, the problem that

Schrödinger originally raised with respect to the

cat is now called the measurement problem: Every-

day objects such as cats and pointers can, accord-

ing to standard quantum theory, be indeterminate

in state. For example, a cat might be indeterminate

with respect to whether it is alive. A pointer might

be indeterminate with respect to its location (i.e., it

is pointing in no particular direction).

Approaches to the measurement problem.

Thus, the interpretation of quantum theory faces a

serious problem, the measurement problem, to

which there have been many approaches. One ap-

proach, apparently advocated by Einstein, is to

search for a hidden-variables theory to underwrite

the probabilities of standard quantum theory. The

central idea here is that the indeterminate descrip-

tion of physical systems provided by quantum the-

ory is incomplete. Hidden variables (so-called be-

cause they are “hidden” from standard quantum

theory) complete the quantum-mechanical de-

scription in a way that renders the state of the sys-

tem determinate in the relevant sense. The most fa-

mous example of a successful hidden-variables

theory is the 1952 theory of David Bohm

(1917–1992), itself an extension of a theory pro-

posed by Louis de Broglie in the 1920s. In the

Broglie-Bohm theory, particles always have deter-

minate positions, and those positions evolve deter-

ministically as a function of their own initial posi-

tion and the initial positions of all the other

particles in the universe. The probabilities of stan-

dard quantum theory are obtained by averaging

over the possible initial positions of the particles,

so that the probabilities of standard quantum the-

ory are due to ignorance of the initial conditions,

just as in classical mechanics. According to some,

the problematic feature of this theory is its nonlo-

cality—the velocity of a given particle can depend

instantaneously on the positions of particles arbi-

trarily far away.

Other hidden-variables theories exist, both de-

terministic and indeterministic. They have some

basic features in common with the de Broglie-

Bohm theory, although they do not all take posi-

tion to be “preferred”—some choose other pre-

ferred quantities. In the de Broglie-Bohm theory,

position is said to be “preferred” because all parti-

cles always have a definite position, by stipulation.

There are other approaches to solving the

measurement problem. One set of approaches in-

volves so-called Many-worlds interpretations, ac-

cording to which each of the possibilities inherent

in a superposition is in fact actual, though each in

its own distinct and independent “world.” There is

a variant, the Many-minds theory, according to

which each observer observes each possibility,

though with distinct and independent “minds.”

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These interpretations have a notoriously difficult

time reproducing the probabilities of quantum the-

ory in a convincing way. A slightly more technical,

but perhaps even more troubling, issue arises from

the fact that any superposition can be “decom-

posed” into possibilities in an infinity of ways. So,

for example, a superposition of “alive” and “dead”

can also be decomposed into other pairs of possi-

bilities. It is unclear how Many-worlds interpreta-

tions determine which decomposition is used to

define the “worlds,” though there are various pro-

posals.

Yet another set of approaches to the measure-

ment problem is loosely connected to the Copen-

hagen Interpretation of quantum theory. According

to these approaches, physical quantities have

meaning only in the context of an experimental

arrangement designed to measure them. These ap-

proaches insist that the standard quantum-me-

chanical state is considered to describe our igno-

rance about which properties a system has in cases

where the possible properties are determined by

the experimental context. Only those properties

that could be revealed in this experimental context

are considered “possible.” In this way, these inter-

pretations sidestep the issue of which decomposi-

tion of a superposition one should take to describe

the possibilities over which the probabilities are

defined. Once a measurement is made, the super-

position is “collapsed” to the possibility that was in

fact realized by the measurement. In this context,

the collapse is a natural thing to do, because the

quantum mechanical state represents our igno-

rance about which experimental possibility would

turn up. The major problem facing these ap-

proaches is to define measurement and experi-

mental context in a sufficiently rigorous way.

Another set of approaches are the realistic col-

lapse proposals. Like the Copenhagen approaches,

they take the quantum-mechanical state of a sys-

tem to be its complete description, but unlike

them, these approaches allow the meaningfulness

of physical properties even outside of the appro-

priate experimental contexts. The issue of how to

specify when collapse will occur is thus somewhat

more pressing for these approaches because the

collapse represents not a change in our knowl-

edge, but a physical change in the world. There

are several attempts to provide an account of when

collapse will occur, perhaps the two most famous

being observer-induced collapse and spontaneous

localization theories. According to the former, no-

tably advocated by Eugene Wigner (1902–1995),

the act of observation by a conscious being has a

real effect on the physical state of the world, caus-

ing it to change from a superposition to a state

representing the world as perceived by the con-

scious observer. This approach faces the very sig-

nificant problem of explaining why there should

be any connection between the act of conscious

observation and the state of, for example, some

electron in a hydrogen atom.

The spontaneous-localization theories define

an observer-independent mechanism for collapse

that depends, for example, on the number of par-

ticles in a physical system. For low numbers of

particles the rate of collapse is very slow, whereas

for higher values, the rate of collapse is very high.

The collapse itself occurs continuously, by means

of a randomly distributed infinitesimal deformation

of the quantum state. The dynamics of the collapse

are designed to reproduce the probabilities of

quantum theory to a very high degree of accuracy.

The problem of nonlocality. The other major

issue facing the interpretation of quantum theory is

nonlocality. In 1964, John Bell (1928–1990) proved

that, under natural conditions, any interpretation of

quantum theory must be nonlocal. More precisely,

in certain experimental situations, the states of

well-separated pairs of particles are correlated in a

way that cannot be explained in terms of a com-

mon cause. One can think, here, of everyday cases

to illustrate the point. Suppose you write the same

word on two pieces of paper and send them to

two people, who open the envelopes simultane-

ously and discover the word. There is a correlation

between these two events (they both see the same

word), but the correlation is easily explained in

terms of a common cause, you.

Under certain experimental circumstances, par-

ticles exhibit similar correlations in their states, and

yet those correlations cannot be explained in terms

of a common cause. It seems, instead, that one

must invoke nonlocal explanations, explanations

that resort to the idea that something in the vicin-

ity of one of the particles instantaneously influ-

ences the state of the other particle, even though

the particles are far apart.

On the face of it, nonlocality contradicts spe-

cial relativity. According to standard interpretations

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