Frank Wilczek What Is Quantum Theory?

REFERENCE FRAME

What Is Quantum Theory?

Over the period 1885-1889, Hein-

rich Hertz1 discovered that elec-

tromagnetic waves propagate through

empty space, and demonstrated

experimentally that these waves trav-

el at the speed of light and are trans-

versely polarized. This work con-

firmed predictions James Clerk

Maxwell had made 20 years earlier, in

1864. Hertz’s major papers were col-

lected in a book, Electric Waves, for

which he wrote an extensive intro-

duction. In that introduction occurs

his famous, extraordinary statement:

“To the question: ‘What is Maxwell’s

theory?’ I know of no shorter or more

definite answer than the following:

‘Maxwell’s theory is Maxwell’s system

of equations.’ ”

Superficially, this statement might

appear innocent-even ingenuous-—

but it goes deep, and in its time it

caused a sensation. There was, at the

time, a rival tradition of electromag-

netic theories, especially strong in

Germany, which advocated action-at-

a-distance formulations in preference

to fields. These theories had the

advantage of continuing the tremen-

dously successful Newtonian tradi-

tion, and of using familiar, highly

developed mathematical methods.

They also had enormous flexibility.

With velocity-dependent force laws,

most of the previously known facts

about electricity and magnetism could

readily be described using action-at-a-

distance. Arnold Sommerfeld recounts2

of his student days (1887-1889) in

Koenigsberg, “The total picture of elec-

trodynamics thus presented to us was

awkward, incoherent, and by no means

self-contained.”

Perhaps some modification would

also describe Hertz’s new results.

(Indeed, we know now that by using

retarded potentials one can reproduce

the Maxwell equations from an

action-at-a-distance theory, rather

elegantly in fact.) So Hertz sought to

forestall unproductive debates

between rival theories with identical

physical content by focusing on the

bottom-line content. Sommerfeld con-

FRANK Wuczsx is the ]. Robm Oppen-

heimer Professor in the School of Natural

Sciences at the Inszitutefor Advanced

Study in Princeton, Newjersey.

Frank Wilczck

tinues, “When I read Hertz’s great

paper, it was as though scales fell

from my eyes.”

Also, Hertz wanted to purify

Maxwell’s work itself. The point is

that Maxwell reached his equations

through a complex process of con-

structing and modifying mechanical

models of the ether and, according to

Hertz, “. . .when Maxwell composed

his great treatise, the accumulated

hypotheses of his earlier mode of con-

ception no longer suited him, or else

he discovered contradictions in them

and so abandoned them. But he did

not eliminate them completely . . .”

Yet a modern physicist, while not

contradicting it, could not rest entire-

ly satisfied with Hertz’s answer to his

question. Maxwell’s theory is much

more than Maxwell’s equations. Or, to

put it differently, merely writing down

Maxwell’s equations, and doing them

justice, are two quite different things.

Indeed, a modern physicist, asked

what is Maxwell’s theory, might be

more inclined to answer that it is spe-

cial relativity plus gauge invariance.

While not altering Maxwell’s equa-

tions, in a real sense these concepts

tell us why that superficially compli-

cated system of partial differential

equations must take precisely the

form it does, what its essential nature

is, and how it might be generalized.

This last feature bears abundant fruit

in the modern Standard Model. The

core of the Standard Model is a

mighty generalization of gauge

invariance, which provides successful

descriptions of physical phenomena

far beyond anything Maxwell or Hertz

could have imagined.

With this history as background,

let us return to the analogous ques-

tion, posed in my title, What is Quan-

tum Theory? At one level, we can

answer along the lines of Hertz.

Quantum theory is the theory you find

written down in textbooks of quantum

theory. Perhaps its definitive exposi-

tion is Dirac’s book.3 Conversely, you

can find, in the early parts of Dirac’s

book, statements very much in the

Hertzian spirit:

“The new scheme becomes a

precise physical theory when all

the axioms and rules of manip-

ulation governing the mathe-

matical quantities are specified

and when in addition certain

laws are laid down connecting

physical facts with the mathe-

matical formalism, so that from

any given physical conditions

equations between the mathe-

matical quantities may be

inferred and vice versa.”

0f course, the equations of quan-

tum theory are notoriously less

straightforward to interpret than

Maxwell’s equations. The leading

interpretations of quantum theory

introduce concepts that are extrinsic

to its equations (“observers”), or even

contradict them (“collapse of the wave

function”). The relevant literature is

famously contentious and obscure. I

believe it will remain so until someone

constructs, within the formalism of

quantum mechanics, an “observer,”

that is, a model entity whose states

correspond to a recognizable carica-

ture of conscious awareness; and

demonstrates that the perceived

interaction of this entity with the

physical world, following the equa-

tions of quantum theory, accords with

our experience. That is a formidable

project, extending well beyond what is

conventionally considered physics.

Like most working physicists, I

assume, perhaps naively, that this

project can be accomplished, and that

the equations will survive its comple-

tion unscathed. In any case, only after

its successful completion might one

legitimately claim that quantum the-

ory is defined by the equations of

quantum theory.

Stepping now toward firmer

ground, let us consider the equations

themselves. The pith of quantum the-

ory, which plays for it the central role

analogous to the role of Maxwell’s

JUNE 2000 Pil‘r'SlCS TODAY ‘.1

---

Page 2

Psist, Erwin, buddy...

Put the cat in 51 box with I/ZZZZa

a poison gas to demonstrate /

the influence of the

observer in quantum

mech8hlC3

equations in electrodynamics, is sup-

plied by the commutation relations

among dynamical variables. Specifi-

cally, it is in these commutation rela-

tions—-and, ultimately, only here-

that Planck’s constant appears. The

most familiar commutation relation,

lp, ql = —i/*1, is between linear momen-

tum and position, but there are also

different ones between spins, or

between fermion fields. In formulat-

ing these commutation relations, the

founders of quantum theory were

guided by analogy, by aesthetics,

and-ultimately—-by a complex dia-

logue with Nature, through experi-

ment. Here is how Dirac describes the

crucial step:4

“The problem of finding quan-

tum conditions is not of such a

general character . . . . It is

instead a special problem which

presents itself with each partic-

ular dynamical system one is

called upon to study. . . . a fairly

general method of obtaining

quantum conditions is the

method of classical analogy

[original italicsl.”

I think it is fair to say that one does

not find here a profound guiding prin-

ciple, comparable to the equivalence

of different observers (that inspires

both relativity theories) or of different

potentials (that implement gauge

invariance).

Those profound guiding principles

of physics are statements of symme-

try. ls it possible to phrase the equa-

tions of quantum theory as state-

ments of symmetry? A very interest-

ing but brief and inconclusive discus-

sion of this occurs in Herman Weyl’s

singular text, where he proposes [the

i2 JUNE 2000 PHYSICS TODAY

original is entire-

ly italicsllf’ “The

k in e m a t i c a 1

structure of a

physical system

is expressed by

an irreducible

unitary projec-

tive representa-

tion of abelian

rotations in Hil-

bert space.”

Naturally, I

won’t be able to

unpack this for-

mulation here,

but three com-

ments. do seem

appropriate.

First, Weyl shows

that his formula-

tion contains the

Heisenberg alge-

bra of quantum

mechanics and the quantization of

boson and fermion fields as special

cases, but also allows additional pos-

sibilities. Second, the sort of symme-

try he proposes—abelian-—is the sim-

plest possible kind. Third, his sym-

metry of quantum kinematics is

entirely separate and independent

from the other symmetries of physics.

The next level in understanding

may come when an overarching sym-

metry is found, melding the conven-

tional symmetries and Weyl’s symme-

try of quantum kinematics (made

more specific, and possibly modified)

into an organic whole. Perhaps Weyl

himself anticipated this possibility,

when he signed off his pioneering dis-

cussion with: “It seems more probable

that the scheme of quantum kinemat-

ics will share the fate of the general

scheme of quantum mechanics: to be

submerged in the concrete physical

laws of the only existing physical

structure, the actual world.”

To summarize, I feel that after sev-

enty-five years-and innumerable

successful applications-—we are still

two big steps away from understand-

ing quantum theory properly.

snort

References

1. An excellent recent brief biography of

Hertz, including an extensive selection of

his original papers and contemporary

commentary, is J. Mulligan, Heinrich

Rudolf Hertz, Garland ( New York). 1994.

2. A. Sommerfeld, Electrodynamics, Acad-

emic (London), 1964, p. 2.

3. P. A. M. Dirac, Quantum Mechanics, 4th

revised edition, Oxford (London), 1967,

p. l5.

4. Ibid., p. 84.

5. H. Weyl, The Theory of Groups and

Quantzun Mechanics, Dover (New

York)v 1950, p. 272. I