Theoretical Practice: le - Recherche en IA spécialisée au MIT

Theoretical Practice: le
Bohm-Pines Quartet

R.I.G. Hughes
University of South Carolina

Quite rightly, philosophers of physics examine the theories of physics, theories
like Quantum Mechanics, Quantum Field Theory, the Special and General
Theories of Relativity, and Statistical Mechanics. Far fewer, cependant, exam-
ine how these theories are put to use; that is to say, little attention is paid to
the practices of theoretical physicists. In the early 1950s David Bohm and
David Pines published a sequence of four papers, collectively entitled, ‘A
Collective Description of Electron Interaction.’ This essay uses that quartet
as a case study in theoretical practice. In Part One of the essay, each of the
Bohm-Pines papers is summarized, and within each summary an overview is
given, framing a more detailed account. In Part Two theoretical practice is
broken into six elements: (un) the use of models, (b) the use of theory, (c) modes
of description and narrative, (d) the use of approximations, (e) experiment
et théorie, (F) the varied steps employed in a deduction. The last element is
the largest, drawing as it does from the earlier ones. Part Three enlarges on
the concept of ‘theoretical practice,’ and briefly outlines the subsequent theo-
retical advances which rendered the practices of Bohm and Pines obsolete, si
still respected.

Were a kind of semiotic egalitarianism to direct us to regard as so
many texts the papers that regularly appear in The Physical Review,
their literary dimension must seem deeply secondary . . .

Arthur Danto1

Preamble
My egalitarian tendencies will be all too evident throughout this essay,
dealing as it does with four papers from The Physical Review. True, I largely

1. Danto (1986, 136).

Perspectives on Science 2006, vol. 14, Non. 4
©2007 by The Massachusetts Institute of Technology

457

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Theoretical Practice: the Bohm-Pines Quartet

neglect their literary dimension, but the attention I pay to the narrative
elements in them will betray a sadly undiscriminating taste.

Pace Danto, however plebian they may be, these papers are indisputably
texts. They are written utterances that take up and respond to earlier ut-
terances in the genre, and themselves invite responses from their intended
readers. This explains why a paper can be too original for its own good,
why early papers in chaos theory, par exemple, were denied space in physics
journals.2 For although each paper is individuated by the original con-
tribution it offers, the dialogic relation in which it stands both to its pre-
decessors and its successors requires that all of them be informed by a
common set of assumptions. These shared assumptions—some method-
ological, others theoretical, some explicit, many tacit—provide a norma-
tive framework for theoretical discourse. This in turn enables us to speak
meaningfully of “theoretical practice.” The details of the framework may
vary from one sub-discipline of physics to another. They will also change
with time; under the press of theoretical advances some elements of the
framework will be jettisoned, or fall into disuse, while others become ac-
cepted in their place. Thus one should properly think of physics as involv-
ing, not theoretical practice tout court, but a set of theoretical practices in-
dexed by sub-discipline and date.

The phrase itself, “theoretical practice,” though not actually oxymor-
onic, is little used in the philosophy of physics.3 And, in describing a theo-
retical advance, physicists will rarely allude to the practices that led to it.
Par exemple, in the transcript of one of Richard Feynman’s Lectures on
Physics (1964, II, 7–7) we read,

It was ªrst observed experimentally in 1936 that electrons with en-
ergies of a few hundred to a few thousand electron volts lost energy
in jumps when scattered from or going through a thin metal foil.
This effect was not understood until 1953 when Bohm and Pines
showed that the observations could be explained in terms of the
quantum excitations of the plasma oscillations in the metal.

Feynman is here concerned only with the result that Bohm and Pines ob-
tained, not with the strategies they used to obtain it. In his lecture it ap-
pears as one illustration among others of the fact that “[le] natural reso-
nance of a plasma has some interesting effects” (ibid.). But the work that
produced it, the journey rather than the end, offers an illustration of a dif-

2. See Ruelle (1991).
3. Edmund Husserl’s The Crisis of the European Sciences is too seldom studied by philoso-

phers of science.

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Perspectives on Science

459

ferent kind. It displays an example of the theoretical practices of solid
state physics in the mid-twentieth century.

This essay has four parts. In Part One I give synopses of the four papers
in which Bohm and Pines presented a theoretical account of plasma be-
haviour; in Part Two I provide a commentary on these papers; and in Part
Three I offer some remarks concerning theoretical practice and a brief
note on the methodologies available to philosophers of science. The com-
mentary in the second (and longest) part contains discussions of the com-
ponents of theoretical practice exhibited in the quartet. They include
models, theoretical manifolds, modes of description, approximations, le
relation between theory and experiment, and deduction. Part Two can
thus be read as an introduction to the notion of “theoretical practice” that
takes the Bohm-Pines quartet for illustration. De même, Part Three illus-
trates, amongst other things, how theoretical practice evolves through
temps.

PART ONE
The Bohm-Pines Quartet

1.1 Introduction
Entre 1950 et 1953 David Bohm and David Pines published a se-
quence of four papers in The Physical Review. They were collectively enti-
tled, “A Collective Description of Electron Interactions,” and individually
subtitled, “I: Magnetic Interactions,” “II: Collective vs. Individual Parti-
cle Aspects of the Interaction,” “III: Coulomb Interactions in a Degenerate
Electron Gas," et, “IV: Electron Interaction in Metals.”4 In this quartet
of papers Bohm and Pines had two aims, one speciªc and the other gen-
eral. The speciªc aim was to provide a theoretical account of the behaviour
of electrons in metals; as their title announces, this account was to be
given in terms of a “collective description of electron interactions” rather
than “the usual individual particle description” (BP I, 625). Their more

4. I cite these papers as BP I, BP II, BP III, and P IV. Except for the abstract of BP I,
the text on each page is set out in two columns. In my citations, the letters “a” and “b” af-
ter a page number designate, respectivement, the left and right hand column. The fourth pa-
per was written by Pines alone. When the ªrst paper was published both authors were at
Princeton University, Bohm as a junior faculty member and Pines as a graduate student.
Over the period in which they were written Pines moved, ªrst to Pennsylvania State Uni-
versity and then to the University of Illinois. Between the time BP II was received by The
Physical Review (Septembre 1951) and the time it was published (Janvier 1952), Bohm
had fallen foul of Joseph McCarthy’s Un-American Activities Committee and had moved
to the University of Sao Paolo in Brazil. See the introduction to Hiley and Peat (4, 1987).

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460

Theoretical Practice: the Bohm-Pines Quartet

general aim was to explore, through this analysis, a new approach to
many-body problems.

The background to their investigation, roughly sketched, was this.
Since the work of Paul Drude at the turn of the century, the accepted ex-
planation of the high electrical conductivity of metals was that some, à
least, of the electrons in the metal were free to move through it. On the
account that emerged, the valence electrons in a metal are not attached to
speciªc atoms; instead they form an “electron gas” within a regular array
of positive ions, the “crystal lattice.” This model was modiªed by Arnold
Sommerfeld (1928). He pointed out that the electrons in a metal must
obey the Pauli exclusion principle, according to which no more than one
electron in a system can occupy a particular energy level; collectively they
need to be treated as a degenerate electron gas (hence the title of BP III).5

In the 1930s and 40s a theory of the motion of electrons in a metal was
developed using the “independent electron formulation,” otherwise re-
ferred to as the “independent electron model.” On this approach the elec-
trons are treated individually. To quote John Reitz (1955, 3),

[T]he method may be described by saying that each electron sees,
in addition to the potential of the ªxed [ionic] charges, only some
average potential due to the charge distribution of the other elec-
trons, and moves essentially independently through the system.

The independent electron theory enjoyed considerable success. The val-
ues it predicted for a number of metallic properties (electrical and thermal
conductivity among them) agreed well with experiment (Pines, 1987, 73).
Cependant, it failed badly in one important respect: The predicted cohesive
energy of the electrons in the metal was so small that, were it correct,
most metallic crystals would disintegrate into their constituent atoms.
En outre, as Pines puts it (ibid.), “[T]heoretical physicists . . . pourrait
not understand why it worked so well.”6 The challenge facing Bohm and
Pines was to formulate a theory of the behaviour of electrons in a metal
that both acknowledged the mutual interactions between electrons, et
showed why the independent electron model, which ignored them, was so
réussi.

They looked for guidance to the research on plasmas performed by
physicists like Irving Langmuir in the 1920s and 30s.7 Langmuir used the
term “plasma” to refer to an ionized gas, such as one ªnds in a ºuorescent

5. Whereas an ideal gas obeys Maxwell-Boltzmann statistics, as a result of the exclusion

principle a degenerate electron gas obeys Fermi-Dirac statistics.

6. For more details see Pines (1955, 371–74). An extended discussion of the independ-

ent electron theory is given in Reitz (1955).

7. The paper cited in BP I is Tonks and Langmuir (1929).

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Perspectives on Science

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lamp or in the ionosphere, where electrons are stripped from their parent
atoms, in one case by an electrical discharge, in the other by ultra-violet
radiation from the Sun.8 The state of matter that results resembles the one
in a metal in that a cloud of free electrons surrounds a collection of heavier,
positively charged ions. There are differences. The ions in a gas plasma
move freely, albeit much more slowly than the electrons, whereas in a
metal their motion is conªned to thermal vibrations about ªxed points in
a regular lattice. More importantly, there are about 1011 times as many
electrons per unit volume in a metal than in a gas plasma. This is the rea-
son why the electron gas in a metal should properly be treated quantum-
mechanically, as a degenerate electron gas, whereas the electron gas in a
gas plasma may be treated classically (see fn. 5).

Despite these differences, Bohm and Pines saw Langmuir’s investiga-
tions of plasmas as “offering a clue to a fundamental understanding of the
behavior of electrons in metals” (Pines, 1987, 67).9 They set out to show
that the electron gas in a metal would manifest two kinds of behavior
characteristic of gas plasmas: that high frequency “plasma oscillations”
could occur in it, and that the long range effect of an individual electron’s
charge would be “screened out” by the plasma. Paradoxically, both these
effects were attributed to the Coulomb forces between pairs of electrons,
the electrostatic forces of repulsion that exist between like charges. “Para-
doxically” because, on this account, the plasma screens out precisely those
long range effects of Coulomb forces that are responsible for plasma oscil-
lations. Note, cependant, that the screening effect, if established, would go
some way towards explaining the success of the independent electron the-
ory of metals.

Within the quartet a mathematical treatment of plasmas is interwoven
with an account of its physical signiªcance and a justiªcation of the meth-
ods used. Synopses of the four papers are given in the next four sections of
this essay. They have been written with two aims in mind: ªrst, to present
a general overview of each paper; secondly, to provide more detail about
those parts of the quartet that I comment on in Part Two of the essay. Le
two aims pull in different directions. To alleviate this tension, in each syn-
opsis I distinguish the overview from the more detailed material by en-
closing the latter in brackets (“ “ and “ ”). This allows the reader to skip
the bracketed material on a ªrst reading to attend to the overview, and to
go back later to browse on the ampliªed version.

8. Matter also enters the plasma state at the temperatures and pressures associated with
thermonuclear reactions. Interest in controlled fusion has prompted much of the research
on plasmas since the Bohm-Pines papers were published.

9. A brief summary of previous work along these lines, together with some critical ob-

servations on it, is given in BP II, 610b.

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Theoretical Practice: the Bohm-Pines Quartet

1.2 P I: Magnetic Interactions
The Introduction to BP I is an introduction to the whole quartet. Its ªrst
six paragraphs introduce the reader to the independent electron model and
its shortcomings, and to the phenomena of screening and plasma oscilla-
tion. These phenomena, the authors tell us, occur in an electron gas of
high density. In a footnote (BP I, fn. 1) they point out that “The [nega-
tively charged] electron gas must be neutralized by an approximately
equal density of positive charge.” In a metal this charge is carried by the
individual ions of the metal. But “in practice the positive charge can usu-
ally be regarded as immobile relative to the electrons, and for the most ap-
plications [sic] can also be regarded as smeared out uniformly throughout
the system.” (Ibid.) The presence of this positive charge (whether localized
or uniformly distributed) effectively screens out short range interactions
beyond a very small distance. At long range, cependant, plasma oscillations,
a collective phenomenon, can occur. These are longitudinal waves, “orga-
nized oscillations resembling sound waves” (BP I, 625un), in which local
motions are parallel to the direction of propagation of the waves. A plasma
can also transmit organized transverse oscillations, in which local motions
are at right angles to the direction of wave propagation. They may be trig-
gered by an externally applied electromagnetic ªeld, a radio wave passing
through the ionosphere, Par exemple. This applied ªeld will produce oscil-
lations of the individual electrons, each of which will in turn give rise to a
small periodic disturbance of the ªeld. Only if the cumulative effect of all
these small disturbances produces a ªeld in resonance with the original
applied ªeld will the oscillations become self-sustaining. As we learn at
the end of the Introduction (BP I, 627un), transverse oscillations like these
are the main topic of the ªrst paper.

Bohm and Pines treat these oscillations in two ways. In their ªrst treat-
ment they use the techniques of classical physics; in the second those of
quantum mechanics. So that the results of the ªrst can be carried over to
the second they use “Hamiltonian methods,” i.e. they describe the system
comprising the electrons and the ªeld by its Hamiltonian, an expression
that speciªes its total energy.10 The authors write (BP I, 626b),

This Hamiltonian may be represented schematically as

H0 (cid:2) Hpart (cid:3) Hinter (cid:3) Hªeld
where Hpart represents the kinetic energy of the electrons, Hinter rep-
resents the interaction between the electrons and the electromag-

10. I say more about the Hamiltonian and its signiªcance in Sections 2.3 et 2.4 de

this essay.

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Perspectives on Science

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netic ªeld, and Hªeld represents the energy contained in the electro-
magnetic ªeld.

This Hamiltonian is expressed in terms of the position and momentum
coordinates of the individual electrons and ªeld coordinates of the electro-
magnetic ªeld.11 Bohm and Pines then introduce a new set of coordinates,
the “collective coordinates,”12 and, as before, a distinction is made between
particle coordinates and ªeld coordinates. A mathematical transformation
replaces the original Hamilton, H0, expressed in the old variables, by an-
other, H(1) , expressed in the new ones.

Given various approximations,

[T]he Hamiltonian in the collective description can be represented
schematically as

H(1) (cid:2) H(1)

part (cid:3) Hosc (cid:3) Hpart int
where H(1)
part corresponds to the kinetic energy in these new coordi-
nates and Hosc is a sum of harmonic oscillator terms with frequen-
cies given by the dispersion relation for organized oscillations.
Hpart int then corresponds to a screened force between particles, lequel
is large only for distances shorter than the appropriate minimum
distance associated with organized oscillations. Thus we obtain ex-
plicitly in Hamiltonian form the effective separation between long
range collective oscillations, and the short-range interactions be-
tween individual particles. (Ibid.)

The “effective separation” that the authors speak of shows itself in
ce: Whereas in H0 the term Hinter contains a mixture of particle
and ªeld coordinates, in H(1) the term Hosc contains only collective
ªeld coordinates, and Hpart int contains only collective particle coordi-
nates. Bohm and Pines gather together the approximations that al-
low them to write the Hamiltonian in this way under the title “The
Collective Approximation” (BP I, 628a-b); I will say more about
them in my comments (Section 2.5). Among them the “Random
Phase Approximation” plays a particularly important role in the au-
thors’ project. The “dispersion relation” they refer to relates the fre-

11. The ªeld coordinates appear when the vector potential of the electromagnetic ªeld
is expanded as a Fourier series. For more on Fourier series in this context, see fn. 13 et le
synopses of BP II and BP III.

12. An elementary example of a move to a collective coordinate is the use of the vector
X to specify the position of the centre of gravity of a system of masses, m1, m2, . . . , mn. If
the positions of the masses are given by x1, x2, . . . , xn, then X (cid:2) (cid:4)i mixi

/(cid:4)i mi .

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Theoretical Practice: the Bohm-Pines Quartet

quency of the oscillation to its wavelength. This relation must hold
if sustained oscillations are to occur (BP I, 625b).

In both the classical and the quantum mechanical accounts, the authors
claim (BP I, 634b), the move to collective variables shows that, within an
electron gas,

[T]he effects of magnetic interactions are divided naturally into the
two components discussed earlier:

(1) The long range part [given by Hosc] responsible for the long
range organized behaviour of the electrons, leading to modiªed
transverse ªeld oscillations [ . . . ]

(2) The short-range part, [ . . . ] given by Hpart int, which does not
contribute to the organized behaviour, and represents the residual
particle-interaction after the organized behaviour of the system has
been taken into account.

BP I is essentially a preamble to the papers that follow. The magnetic
interactions that produce transverse oscillations are many orders of magni-
tude weaker than the Coulomb interactions that produce longitudinal
plasma waves, and consequently “are not usually of great physical import”
(BP I, 627un). The authors investigate them “to illustrate clearly the tech-
niques and approximations involved in our methods,” since “the canonical
treatment of the transverse ªeld is more straightforward than that of the
longitudinal ªeld” (ibid.).

1.3 P II: Collective vs. Individual Particle Aspects of the
Interaction
BP II and BP III both give theoretical treatments of longitudinal oscilla-
tion. A classical treatment in BP II is followed by a quantum mechanical
treatment in BP III. In both these papers, as in BP I, the authors’ chief
concern is the relation between the individual and the collective aspects of
electronic behaviour in plasmas.

In BP II Bohm and Pines analyze this behaviour in terms of the varia-
tion of the electron density (the number of electrons per unit volume)
within the plasma. Because of the forces of repulsion between electrons,
these variations act like variations of pressure in air, and can be transmit-
ted through the electron gas as plasma oscillations, like sound waves. À
analyze the resulting variations in electron density no transformations of
the kind used in BP I are required, since the electron density is already a
collective quantity. The authors work with Fourier components
k of this

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Perspectives on Science

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density.13 The component
gas, but the other components
tions of different wavelengths.

0 represents the mean electron density of the
k (with k (cid:5) 0) represent density ºuctua-

(( Bohm and Pines use elementary electrostatics, together with the
k /dt2
random phase approximation, to obtain an expression for d2
that can be divided into two parts (BP II, 340b). One part shows
the contribution of the interactions between electrons, et le
other the contribution of their random thermal motions.

P can occur;

k /dt2,, the expression for

The authors show that, if thermal motions are ignored, sustained
oscillations of frequency
P is the so-called plasma fre-
quency.14 If thermal motions are taken into account, the frequency w
of oscillation is no longer independent of the wave number k, as in
the case of d2
2, has two parts. For small
values of k (c'est à dire., for long ºuctuation wavelengths), the ªrst term in
2 predominates, and the electron gas displays its
the expression for
collective aspect. Inversement, for high values of k and short wave-
lengths, the second term predominates, and the system can be re-
garded as a collection of free particles. In the general case both as-
pects are involved. ))

Bohm and Pines show (BP II, 342b-343a) that in the general case each
k of the electron density can be expressed as the sum of two

component
parties:

k (cid:2) akqk

Here ak is a constant, qk is a collective coordinate that oscillates harmoni-
cally, and k describes a ºuctuation associated with the random thermal
motion of the electrons. Two general conclusions are drawn: (je) Analysis of

13. A simple example of a Fourier decomposition occurs in the analysis of musical
tones; the note from a musical instrument can be broken down into a fundamental, à-
gether with a set of overtones. More abstractly, let f(X) be any continuous function, tel
that f(x1) (cid:2) F(x2) and f(X) has only a ªnite number of maxima and minima in the interval
,x2]. Then f(X) can be represented in that interval as the sum of a set of sine waves: F(X)
[x1
kak eikx. The index k specifying each component runs over the integers, 0, 1, 2, . . . C'est
the wave number of the component of the function. c'est à dire. the number of complete wavelengths
,x2]. Thus k is inversely proportional to the wave-
of that component in the interval [x1
; the greater the wave number the shorter the wavelength (and vice versa). Mes-
length
siah (1958, 471–78) gives a useful mathematical account of Fourier transformations.

14. The plasma frequency is given by wP

2 (cid:2) 4(cid:6)ne2/m, where e and m are the electron’s
charge and mass, respectivement, and n is the electron density. An elementary classical deriva-
tion of this equation is given by Raimes (1961, 283–84).

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k shows that each individual electron is surrounded by a region from
which the other, electrons are displaced by electrostatic repulsion. That re-
gion, cependant, like the entire volume occupied by the electron gas, is as-
sumed to contain a uniformly distributed positive charge. The result is
that the Coulomb force due to the individual electron is screened off out-
side a radius of the order of lD , the so-called “Debye length.”15 In a metal
this region is about 10–8 cm. in diameter; it moves with the electron, et
can itself be regarded as a free particle.16 (ii) It turns out (BP II, 343un) que
there is a critical value kD of k such that, for k ≥ kD, rk (cid:2) hk, c'est à dire. there are
effectively no collective coordinates qk for k greater than kD. Since, par
deªnition, kD (cid:2) 1/lD , the physical import of this is that there are no
plasma oscillations of wavelengths less than lD. As in BP I, the use of col-
lective coordinates allows Bohm and Pines to predict that long range in-
teractions in the electron gas give rise to plasma oscillations (in this case
longitudinal oscillations), and that at short ranges the normal electrostatic
forces between electrons are largely screened off.

The authors go on to show how oscillations may be produced by a high-
speed electron moving through the electron gas (BP II, 344b-347a). UN
“correspondence principle argument” (on which I will comment in Sec-
tion 2.3 et 2.6) is then used (BP II, 347un) to explain the experimentally
obtained phenomenon, que, when high energy electrons pass through
thin metallic foils of the same element, they suffer losses of energy that are
all multiples of a speciªc value. These are the results cited by Feynman;
they were obtained by G. Ruthemann and by W. Lang independently in
the early 1940s (see BP II, 339un, and Pines, 1987, 77).

1.4 BP III: Coulomb Interactions in a Degenerate Electron Gas
BP III is the longest and most intricately argued paper of the four. Le
quantum mechanical analysis it presents uses the theoretical strategy de-
ployed in the latter part of BP I alongside the physical insights achieved
in BP II. As in BP I, a Hamiltonian for the system electrons-plus-ªeld in
terms of individual particle coordinates is transformed to one in terms of
collective coordinates. Dans ce cas, cependant, the collective coordinates are
those appropriate for describing longitudinal, rather than transverse, os-
cillations, and the procedure is far from straightforward. A sequence of

15. The Debye length lD was ªrst introduced in connection with screening processes in
highly ionized electrolytes (see Feynman, 1964, 7–9). It is the thickness of the ion sheath
that surrounds a large charged particle in an electrolyte (BP II, 341b, fn.).

16. A similar result appears in BP III. Here Bohm and Pines anticipate what became
standard practice in the 1950s and 1960s, whereby a particle together with its interactions
with its immediate environment was treated as an elementary system, and was referred to
as a quasi-particle. I return to this topic in Section 3.1.

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ªve modiªcations, which include coordinate transformations, algebraic
manipulations, and a variety of approximations (some relying on non-triv-
ial assumptions) takes the reader from the initial Hamiltonian (which I
will refer to as “H1”) to the ªnal Hamilton Hnew .17

H1 is itself a standard Hamiltonian for “an aggregate of electrons em-
bedded in a background of uniform positive charge” (BP III, 610un). Il
contains three terms, one for the kinetic energy of the electrons, another
for the energy due to the pair-wise electrostatic interactions between
eux, and the third for the sum of the individual self-energies of the elec-
trons. The last two are slightly adjusted to take into account the back-
ground of positive charge. The ªrst four modiªcations of the Hamiltonian
are effected in Section II of the paper. By means of them Bohm and Pines
introduce a set of “ªeld variables,” and analyze their “approximate oscilla-
tory behavior” (BP III, 611b). The ªnal modiªcation, made in Section III,
enables them “to carry out the canonical transformation to the pure collec-
tive coordinates” (ibid.).

In the ªrst modiªcation H1 is transformed into another

Hamiltonian H2, expressed in terms, not just of individual particle
coordinates, but also of the longitudinal vector potential A(X) et
the electric ªeld intensity E(X) of the electromagnetic ªeld within
the plasma. The transformation does not affect the ªrst and third
terms of H1. The difference between H1 and H2 is that the energy
that in H1 was attributed to Coulomb interactions between elec-
trons is now represented as an energy of interaction between indi-
vidual electrons and the electromagnetic ªeld engendered by the
electron gas as a whole. Like the density r in BP II, UN(X) and E(X)
are both written as Fourier series (see fn. 13), and the coefªcients of
the components pk and qk of the two series (both indexed by k) serve
as ªeld coordinates of the plasma, as in BP I. The authors claim (BP
III, 611un) that H2 will “lead to the correct equation of motion,»
when supplemented by a set of “subsidiary conditions” on the al-
lowable states (cid:7) of the system. Each member of the set has the
formulaire, Ωk(cid:7) (cid:2) 0 , where each condition Ωk isa function of a compo-
nent of the Fourier decompositions of p and r (the electron den-
ville).

17. The Hamiltonians that I refer to as H1, H2, . . . , Hnew appear in BP III as follows: H1
is the Hamiltonian on p. 610b; H2, the Hamiltonian H on p. 612un; H3, the Hamiltonian H
on p. 612un; H4 is a Hamiltonian whose last four terms appear explicitly on p. 612b, et
whose ªrst term, Hpart , is the sum of the ªrst and last terms of H3, where they represent the
kinetic and self energies of the electrons, respectivement; H5 is the Hamilton H on p. 616b;
Hnew appears on p. 618b.

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These subsidiary conditions also serve other functions. They en-
sure conformity with Maxwell’s equations, and also achieve a more
technical purpose, that of reducing the number of degrees of free-
dom allowed by the transformed description (H2 plus the condi-
tion) to that allowed by H1.18 In addition, as the expression for Ωk
makes clear,

[Ils] introduce in a simple way a relationship between the
Fourier components of the electronic density k and a set of ªeld
variables pk . [ . . . ] [T]here is in consequence a very close parallel
between the behaviour of the
haviour of our ªeld coordinates. (BP III, 611b)

k , as analysed in [BP] II, and the be-

Bohm and Pines extend the parallel further. They anticipate that,
just as in the classical theory of plasmas developed in BP II there is
a minimum wavelength lc of organized oscillations, and a corre-
sponding maximum wave number kc , so in the quantum theory a
similar (but not identical) cut-off exists (BP III, 611b). Accord-
franchement, they use a modiªed version of the transformation that yielded
H2 to obtain an operator H3. The effect of this modiªcation is to
eliminate ªeld coordinates with wave vector k greater than some
critical wave vector kc , and so conªne attention to k-values between
k

kc . Of H3, Bohm and Pines remark (BP III, 616un),

0 and k

There is a close resemblance between [ce] Hamiltonian, which de-
scribes a collection of electrons interacting via longitudinal ªelds,
and the Hamiltonian [H0] we considered in BP I, which described a
collection of electrons interacting via the transverse electromagnetic
ªelds. [ . . . ] [Ô]ur desired canonical transformation is just the lon-
gitudinal analogue of that used in BP I to treat the organized as-
pects of the transverse magnetic interactions in an electron gas.

Algebraic manipulation shows that H4 can be represented schemati-
cally as:

H4 (cid:2) Hpart (cid:3) HI (cid:3) Hosc (cid:3) Hs.r. (cid:3) U

Here Hpart represents the energy of particles (electrons) due to their
motion and self energy; HI “represents a simple interaction between
electrons and the collective ªelds”; Hosc is “the Hamiltonian appro-
priate to a set of harmonic oscillators, representing collective

18. The fact that the transformation allowed the system too many degrees of freedom

gave Bohm and Pines considerable trouble; see Pines (1987, 75).

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ªelds”; and Hs.r. “represents the short range part of the Coulomb in-
teraction between the electrons [c'est à dire., the effective Coulomb interac-
tion once the effects of screening have been taken into account]»
(BP III, 612b). The authors use (and justify) the random phase ap-
proximation to show that the remaining term U can be disregarded
(BP III, 613b-614b), and then rewrite Hosc, dubbing the result
“Hªeld”19 In this way they arrive at the Hamiltonian H5:

H5 (cid:2) Hpart (cid:3) HI (cid:3) Hªeld (cid:3) Hs.r.

The “subsidiary conditions” on the allowable states F of the system
now appear as a set of kc conditions:

Ωk(cid:7) (cid:2) 0

(k < kc) and a new expression for the operators Ωk is given (BP III, 616b). Following the strategy of BP I, Bohm and Pines apply a canoni- cal transformation to H5 and to the Ωk operators. The transforma- tion is designed to “eliminate HI to ªrst order” by distributing most of its effects among the (transformed versions of) other terms of H5, and so redescribe the system in terms of “pure collective co- ordinates.”20 That is to say, no term of the transformed Hamiltonian contains both particle and ªeld operators. The modiªcations have done their work. The ªnal Hamiltonian, which Bohm and Pines christen “Hnew,” differs markedly from the Hamiltonian H1 from which they started. It is ex- pressed schematically as: Hnew (cid:2) Helectron (cid:3) Hcoll (cid:3) Hres part. . 19. Hªeld is obtained (BP III, 616a) by replacing each occurrence of qk in Hosc, by the ex- 2 (ak (cid:8) ak*). The reason for making this move will be made clear in Sec- pression ((cid:2)/2w)1 tion 2.7. qk represents a component of the Fourier expansion for the longitudinal vector po- tential of the electromagnetic ªeld A(x), and ak and ak* are, respectively, the creation and annihilation operators for the longitudinal photon ªeld. (BP III, 616a). The equation qk (cid:2) 2 (ak (cid:8) ak*) is an identity in the operator algebra used in quantum mechanics. For ((cid:2)/2w)1 an introduction to creation and annihilation operators, see Messiah (1958), 438–39 and 963–66. For the mathematical deªnition of a transformation in this context, see the exam- ples 3, 6, and 7 in Section 2.3. 20. What is left of HI is then discarded, and the resulting Hamiltonian symbolized by “Hnew(0) “ to mark the fact that it is a lowest-order approximation. In Section III of BP III Bohm and Pines then show that this residue of the (transformed version of) HI may be ne- glected, and so the superscript “(0)” is dropped. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 470 Theoretical Practice: the Bohm-Pines Quartet Like H(0) in BP I, this Hamiltonian contains just three parts. Helectron con- tains terms referring only to individual electrons; ªeld coordinates appear only in Hcoll, which describes independent longitudinal oscillations of the ªeld; and Hres. part. represents an extremely weak “residual” electron- electron interaction; at short range the electrons are effectively screened off from each other. Thus the quantum mechanical treatment of the plasma has replicated the conclusions of the classical treatment in BP II. The oscillations described by Hcoll are independent in the sense that, under the canonical transformation, (i) the subsidiary condi- tions that guarantee conformity with Maxwell’s equations no longer relate ªeld and particle variables, and (ii) HI , which represented ªeld-particle interactions in H3, H4, and H5, has now disappeared; effectively, the ªnal transformation has distributed it between two terms of the Hamiltonian,21 Part of it reappears as Hres part ; the elec- tron-electron interaction this operator describes is negligible in comparison with the short range interaction described by Hs.r. in H4. The other part has been absorbed into Helectron, where it appears as an increase in the “effective mass” of the electron. Bohm and Pines in- terpret this (BP III, 620a-b) as “an inertial effect resulting from the fact that these electrons carry a cloud [of collective oscillations] along with them.” (See fn. 16.) In this way the program announced in BP I has been carried out. Bohm and Pines have demonstrated “explicitly in Hamiltonian form the effec- tive separation between long range collective interactions, described here in terms of organized oscillations, and the short range interactions be- tween individual particles” (BP I, 626b). 1.5 P IV: Electron Interaction in Metals In P IV, the last paper in the series, Pines applies the quantum mechanical account of plasmas developed in BP III to the behaviour of the valence electrons in metals (otherwise known as “conduction electrons”). He be- gins by drawing attention to the assumptions this involves (P IV, 626a). In BP III the system described by the initial Hamiltonian H1 consisted of cloud of electrons moving against a uniform background of positive charge. In a metal, however, the positive charge consists of positive ions localized on the nodes of the crystal lattice; in addition, this lattice under- 21. Note that in BP III the expressions Hpart and Hs.r., denoted components of H4 . Since Hnew is a transformed version of H4, it would have been better to denote the corresponding terms in Hnew by HC s.r., in order to reºect the fact that pure collective coordinates are being used. Instead, in P IV (627a) they appears, misleadingly, as Hpart part and HC and Hs.r.. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 471 goes vibrations. Thus, when the results of BP III are carried over into P IV, the ªrst assumption made is that the periodicity and the density ºuctua- tions of the positive charge in a metal can be ignored. The second is that “the only interactions for the conduction electrons in a metal are those with the other conduction electrons” (P IV, 626). An important goal throughout P IV is to show that, despite the simpliªcations and idealizations involved, the theoretical results of BP III hold for the electron interactions in (at least some) metals.22 The ªrst item on Pines’s agenda is to show how the collective account can explain why, for many purposes, the independent electron model (described here in Sec- tion 1.1) worked as well as it did. His argument (P IV, 627a-b) is very simple. He reminds the reader that two important mathematical results have been obtained in BP III: an expression for the Hamiltonian Hnew for the system, and the subsidiary conditions Ok on its states. The collective description of the electron gas that Hnew provided included a term Hcoll which summarized, so to speak, the effects of the long range Coulomb in- teractions between electrons. They were “effectively redescribed in terms of the collective oscillations of the system as a whole” (ibid.). But, given the dispersion relation for these oscillations, which relates their frequency to their wavelength, It may easily be shown that the energy of a quantum of effective os- cillations is so high [ . . . ] that these will not normally be excited in metals at ordinary temperatures, and hence may not be expected to play an important role in metals under ordinary conditions. (Ibid.) Pines goes on, The remainder of our Hamiltonian corresponds to a collection of individual electrons interacting via a comparatively weak short- range force, Hs.r.. These electrons differ from the usual “free” elec- trons in that they possess a slightly larger effective mass [see Sec- tion 1.4 of this essay], and their wave functions are subject to a set of [subsidiary] conditions. However, both of these changes are un- important qualitatively (and in some cases quantitatively). Further- more, since the effective electron-electron interaction is so greatly reduced in our collective description, we should expect that it is quite a good approximation to neglect it for many applications. 22. Pines remarks (P IV, 626), “This assumption should be quite a good one for the al- kali metals [,,,], and we may expect it to apply generally for any metallic phenomena in which the periodicity of the lattice plays no important role.” l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 472 Theoretical Practice: the Bohm-Pines Quartet Thus we are led directly to the independent electron model for a metal. Like the arguments of BP III, this argument arrives at a theoretical con- clusion about an electron gas through an examination of its theoretical de- scription, the Hamiltonian H for the system. The Hamiltonian Pines uses is a modiªed version of the Hamiltonian Hnew obtained in BP III. Whereas at the end of that paper Bohm and Pines wrote, Hnew (cid:2) Helectron (cid:3) Hcoll (cid:3) Hres part in P IV Pines rewrites Helectron as Hpart (cid:3) Hs.r., and neglects Hres part since “it will produce negligible effects compared with Hs.r., and this latter term is small.” (P IV, 627a) The resulting Hamiltonian is: H (cid:2) Hpart (cid:3) Hcoll (cid:3) Hs.r. . In the remainder of P IV Pines examines the quantitative results that this Hamiltonian yields. In Section II he uses it to calculate the energy e0 of the system in its ground state, and compares the result with those reached by other approaches. In Section III he points to a problem encoun- tered by the independent electron approach, and suggests how it could be accounted for by the BP theory. In Section IV he shows how that theory can also explain the behaviour of high energy particles as they pass through metal foils. Pines begins Section II by pointing out that, given the Hamil- tonian for the system, the direct way to obtain (cid:9)0 would be to solve the time-independent Schrödinger equation, HnewY0 (cid:2) 0Y0 in which 0 appears as the lowest eigenvalue corresponding to the eigenfunction Y0. (See fn, 33.) Instead, for ease of calculation, he discards the smallest term Hs,r. of Hnew , and works with a wave function Y0 which is both an exact eigenfunction of the resulting Hamiltonian (Hpart (cid:3) Hcoll), and an approximate eigenfunction of Hnew .23 He argues (P IV, 630a-b) that, since the contribution of the 23. Note that, even though he works with an approximate eigenfunction, Pines still denotes it by “Y0”. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 473 neglected term Hs.r. is small, it can be treated later as a small per- turbation in Hnew , giving rise to a small short-range “correlation energy” corr .24 Here the BP theory and the independent electron model part company. When (cid:9)0 is compared with the energy E as calculated on (one version of) the independent electron model, the two values dif- fer by a small amount which Pines calls a “correlation energy.” This can be broken into two parts, corresponding to a long range and a short range interaction between electrons. Symbolically, e0 (cid:8) E (cid:2) ecorr. (cid:2) ecorr l.r. (cid:3) ecorr s..r. . In Section III Pines shows that, when this “correlation energy is taken into consideration, the independent electron model faces a problem. When all electron-electron interactions are neglected, and the energy of the electron gas is taken to be the Fermi energy EF ,25 many of the results obtained using the model agree quite well with experiment. Quantum mechanics, however, decrees that one should allow for an additional “exchange energy” Eexch,26 and when the en- ergy is “corrected” from EF to EF (cid:3) Eexch , the agreement with ex- periment, far from being improved, is rendered worse than before. Pines illustrates this with two examples. The ªrst (P IV, 631a) con- cerns the speciªc heat of an electron gas. According to the “cor- rected” version of the independent electron model, this quantity will vary as T/lnT (where T is the absolute temperature).27 Experi- ment, however, shows a linear dependence on T. The second exam- ple (P IV, 632a-b) concerns the magnetic properties of an electron gas. According to the “corrected” account, the electron gas in cer- 24. I say more about perturbation theory in example 4 (and fn. 44, which accompanies it) in the discussion of the use of theory in Section 2.3, and also in Section 2.7. 25. According to the Pauli exclusion principle, no more than one electron can occupy a given state. To each state there corresponds a certain energy level. In a gas of n electrons at absolute zero the ªrst n levels will be ªlled, and the gas will have a corresponding energy. This is called the Fermi energy. Because the energy levels are close together, for modest val- ues of T the energy will not change much, since every electron that jumps to a level just above the nth level vacates a level just below it. 26. If two similar quantum systems—in this case, two electrons—are close together, quantum mechanics tells us that their combined state must be such that it would not be altered if the two systems exchanged their individual states. The effect adds a small energy Eexch to the energy of the composite system. 27. Here Pines draws on theoretical work by James Bardeen and E.P. Wohlfarth; see P IV (631a). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 474 Theoretical Practice: the Bohm-Pines Quartet tain metals—cesium, for example—would display ferromagnetic behaviour (i.e., the spins of the electrons could become aligned). But no such behaviour is observed. Neither anomaly arises with the simpler version of the independent electron model. In contrast, on the BP model, the long range Coulomb interac- tions that lead to a large exchange energy Eexch are replaced by effec- tive screened short range Coulomb interactions (see Section 1.4), with a corresponding reduction in energy contributions. The net re- sult is that the effect of exchange energy on the speciªc heat of the electron gas is comparatively slight, and the model never displays ferromagnetic behaviour (P IV, 623a-b). In the ªnal section of P IV Pines returns to a phenomenon discussed in BP II, the excitation of plasma oscillations by high energy charged parti- cles passing through a metal. To describe the motion of the particle and its interaction with the electron gas, he adds two terms to the Hamiltonian H5 of BP III; in order to rewrite these terms in collective coordinates, he then applies to them the same canonical transformation that took H5 into Hnew in BP III; lastly, he uses the random phase approximation, the disper- sion relation, and the subsidiary conditions of BP III to obtain the three- term Hamiltonian Hadd (P IV, 633a-b). The ªrst term describes “a short range screened Coulomb interaction between the charged particle and the individual electrons in the electron gas” (P IV, 633a); the second “the in- teraction between the charged particle and the collective oscillations of the system” (P IV, 633b); the third may be neglected. Analysis of the second term (P IV, 633b-634a) shows that the interaction generates forced oscil- lations in the collective ªeld, which at a certain frequency w P will pro- duce resonant oscillations in the electron gas. By the same argument as was used in BP II, since the energy associated with an oscillation of fre- P is (cid:2) P (where (cid:2) is Planck’s constant), the total energy loss suf- quency fered by a high energy particle in exciting such oscillations should be some multiple of (cid:2) P. In this way, energy is transferred from the particle to these oscillations in discrete quanta. The quantum of energy loss sus- tained by the particle in each transfer can be calculated, and the results agree well with the experimental ªndings of Ruthemann and Lang men- tioned earlier. Finally, Pines uses the ªrst term of Hadd to obtain an expression for the “stopping power” of a metal, the loss of energy by a charged particle per unit length of the distance it travels through the metal (P IV, 635a-b). He compares his results with those predicted by Aarne Bohr (1948) and H.A. Kramers (1947), each using a different theoretical approach, and with those obtained experimentally for lithium and beryllium by Bakker and l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Segré (635b).28 All four sets of results coincide, within the ranges of error of their respective theoretical and experimental practices. Perspectives on Science 475 PART TWO Observations on the Bohm-Pines Quartet 2.1 Introduction The physicist John Ziman offers this description of “the intellectual strat- egy of a typical paper in theoretical physics” (1978, 3–4): A model is set up, its theoretical properties are deduced, and exper- imental phenomena are thereby explained, without detailed refer- ence to, or criticism of, alternative hypotheses. Serious objections must be fairly stated; but the aim is to demonstrate the potentiali- ties of the theory, positively and creatively, “as if it were true.” Individually and collectively, the four Bohm-Pines papers all conform to this description; indeed it might have been written with them in mind.29 My comments on them are grouped under six headings: the use of models, the use of theory, the modes of description offered, the use of ap- proximations, the connection with experiment, and the nature of deduc- tion as it appears in the quartet. My aim is to provide descriptions, rather than evaluations, of the theoretical practices that Bohm and Pines engaged in. But before I embark on this project, a few remarks about Ziman’s cap- sule summary are called for. Like the papers in theoretical physics it de- scribes, the summary itself takes a lot for granted. A typical paper may well begin by setting up a model, but this is not done in a theoretical vac- uum. Physicists inherit the achievements of their predecessors, and yester- day’s models become today’s physical systems, waiting to be modelled in their turn. Bohm and Pines, for example, take for granted, ªrst, the ac- count of a plasma as an electron gas that contains enough positively charged ions to neutralize the electrons’ charge, and secondly, the assump- tion that a metal is a special case of such a system, whose distinguishing feature is the systematic spacing of the ions. Both assumptions are im- plicit in the references Pines makes to “the ionic ªeld,” to “laboratory plasmas,” and (in a particularly telling phrase) to “the actual plasma” in the quotations with which the next section begins. 28. Kramers used “a macroscopic description in which the electrons were treated as a continuum characterized by an effective dielectric constant.” We will meet it again in Part Three. 29. In point of fact, Ziman had read these papers carefully; see Ziman (1960, 161–68). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 476 Theoretical Practice: the Bohm-Pines Quartet 2.2 The Use of Models In the quartet Bohm and Pines reserve the word “model” for the inde- pendent electron model. Yet their own approach also relies on a highly simpliªed model of a metal, one which enabled them to treat it like a gas plasma. By 1955 Pines describes it in just those terms (1955, 371): “[W]e shall adopt a simpliªed model for a metal in which we replace the effect of the ionic ªeld by a uniform background of positive charge.” And, whereas in 1951 this theoretical move is relegated to a footnote (BP I, fn. 1), Pines subsequently accords it much more importance. He writes (1987, 68),30 In any approach to understanding the behaviour of complex sys- tems, the theorist must begin by choosing a simple, yet realistic model for the behaviour of the system in which he is interested. Two models are commonly taken to represent the behaviour of plas- mas. In the ªrst, the plasma is assumed to be a fully ionized gas; in other words as being made up of electrons and positive ions of a single atomic species. The model is realistic for experimental situa- tions in which the neutral atoms and the impurity ions, present in all laboratory plasmas, play a negligible role. The second model is still simpler; in it the discrete nature of the positive ions is ne- glected altogether. The plasma is then regarded as a collection of electrons moving in a background of uniform positive charge. Such a model can obviously only teach us about electronic behaviour in plasmas. It may be expected to account for experiments conducted in circumstances such that the electrons do not distinguish between the model, in which they interact with the uniform charge, and the actual plasma, in which they interact with positive ions. We adopt it in what follows as a model for the electronic behaviour of both classical plasmas and the quantum plasma formed by electrons in solids. This paragraph is very revealing. Pines tells us that: 1. The theorist’s task involves choosing a model. 2. In dealing with a complex system, that is the only option available. 3. The model can help us to understand the behaviour of a system; in other words, it can provide explanations of that behaviour (a claim echoed by Ziman). 30. This paragraph was comes from Pines’ contribution to a Festschrift in Bohm’s hon- our (Hiley and Peat, 1987). Fittingly, Pines goes on to describe his work on electron inter- action in metals. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 477 4. The model is to be “simple, yet realistic.” 5. The model will involve some simpliªcation; in the ªrst model Pines describes, impurities and neutral atoms are to be disre- garded.31 6. A model may misrepresent aspects of the system; a regular array of positive ions may be represented as a background of uniformly dis- tributed positive charge. 7. More than one way of modelling may be used. We may also observe that: 8. The components of the models—electrons, positive ions—would be described in standard philosophical parlance as “theoretical en- tities.” 9. The two models Pines describes are at odds with one another. 10. The model he adopts involves a greater degree of misrepresentation than the other. In the paragraph I quoted, Pines deals with a particular physical system and the ways in which it can be modelled. In contrast, among the aspects of modelling I have listed, the ªrst eight are very general, and analogues of aspects 9 and 10 appear frequently in theoretical practice. Some ampliªca- tion of these points is called for.32 Note ªrst that, in addition to the simpliªcations already mentioned, there are interactions that the jellium model cannot accommodate. Be- cause it ignores the fact that ions form a regular lattice, the model cannot take into account interactions between electrons and lattice vibrations (which are instrumental in bringing about superconductivity).33 In addi- tion, the only electron-electron interactions considered are between con- duction electrons; interactions between conduction electrons and core electrons (those immediately surrounding the positive nuclei of the ions) are assumed to have no importance.34 Despite these simpliªcations, and items 5–10 above, Pines tells us that a model should be “realistic” [4], and claims that, for certain experimental situations, the model he uses meets that criterion. A contrasting view is expressed by Conyers Herring. In commenting on a theoretical account of 31. What I have called “simpliªcation” some philosophers refer to as “abstraction.” See (e.g.) the footnote on p. 38 of Morgan and Morrison (1999). 32. In the remainder of this section a numeral placed in parentheses (e.g. “[9]”) draws attention to the speciªc point being discussed. 33. I return to this interaction in Part Three of this essay. 34. Pines himself draws attention to these simpliªcations at the start of P IV (626a). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 478 Theoretical Practice: the Bohm-Pines Quartet the surface energy of a metal (Ewald and Juretschke 1952) he observes (1952, 117), It is to be emphasized that the wave mechanical calculation of the surface energy given in the paper applies not to a real metal, but to a a ªctitious metal [ . . . ] The ªctitious metal consists, as has been explained, of a medium with a uniform distribution of positive charge—we may call it a “positive jelly”—and a compensating number of electrons. This metal [ . . . ] we may call “jellium” to distinguish it from real metals such as sodium. The same model that Pines describes as “realistic” is here described by Herring as “ªctitious.” There is no direct contradiction between these de- scriptions; works of ªction can be realistic. And, while the jellium model is not realistic in quite this sense, jellium itself is endowed by Herring with physical properties: “A rough calculation [ . . . ] has indicated that jellium of an electron density equal to that of sodium should have a bind- ing energy only about two thirds that of sodium.” (Ibid.) By using the term “realistic,” Pines marks a distinction between two types of model: the models he himself works with and analogue models like the liquid-drop model of the nucleus proposed by Niels Bohr in the late 1930s. An analogue model relies on a correspondence between the be- haviours of two otherwise radically different types of physical systems (nu- clei and liquid drops, for example). A “realistic” model, on the other hand, is deªned by taking a description of the physical system to be modelled, and modifying it by a process of simpliªcation and idealization [5, 6]. These descriptions will be in a well understood vocabulary that includes terms like “electron” and “positive ions.” The philosopher may regard such entities as merely “theoretical” [8], but when Pines used them they had been familiar and accepted elements of the physicist’s world for forty years. But why are such models necessary, as Pines insists [1, 2]? They are needed because without them it would be impossible to apply our theories to the physical world. In the ªrst place, given a “complete description” of a natural system, we still need a principle by which irrelevancies can be winnowed out from salient information. Secondly, from Galileo onwards, our theories of physics have treated only ideal entities (point masses, rigid bodies, frictionless planes), items that are absent from the physical world. As we have seen, a realistic model of the kind Pines envisages is an entity deªned in terms of simpliªcations and idealizations [5, 6]. Effectively, the model’s deªnition allows it to act as a principle of selection, and the ideal- izations built into it make it amenable to treatment by our theories. A l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 479 model of this kind functions as an essential intermediary between the the- ories of physics and the physical world.35 How does a model help us to understand the behaviour of a physical system [3]? The short answer is that models are things we can play with. A model’s resources are gradually made available to us as we come to see how much can be deduced from its theoretical deªnition, and how aspects of its behaviour are interlinked. As a variety of phenomena are successfully represented by the model, it becomes progressively more familiar to us; increasingly, we come to see the physical system in terms of the model, and vice versa.36 Because a model of this kind is a deªned entity, pragmatic consider- ations can inºuence what simpliªcations and idealizations are made [6]. The theorist has to make choices [7, 9]. An additional idealization may make his problems more tractable, but at the cost of making his predic- tions less accurate. Luckily these choices are not irrevocable. In making that idealization, the theorist will be opting for the model that involves the greater degree of misrepresentation [10], but he can always rescind it in the search for greater empirical adequacy. Pines did just that. Through- out the quartet he and Bohm opted for the jellium model, and treated the positive charge of metallic ions as uniformly distributed. Three years later Pines (1956) treated the charge more realistically, as one that varied regu- larly within the metal, in order to make more accurate estimates of the en- ergy loss of high energy particles in metal foils. 2.3 The Use of Theory After a model has been set up, Ziman tells us, “its theoretical properties can be deduced.” But, as we learn from Pines, the same model can be adopted “as a model for the electronic behaviour of both classical plasmas and the quantum plasma formed by electrons in solids.” Which is to say (i) that the behaviour of the model is assumed to be governed by a founda- tional theory, and (ii) that in some cases it is appropriate to use classical mechanics, and in others, quantum mechanics. We have seen this happen, ªrst in BP I, which provided both classical and quantum mechanical treat- ments of magnetic interactions between electrons, and then in BP II and BP III, which provided, respectively, classical and quantum mechanical 35. This point has been made by many writers, including Nancy Cartwright (1983 and 1999), Ernan McMullin (1985), Ronald Giere (1985) and (1988), Mauricio Suarez (1999), and Margaret Morrison and Mary Morgan (1999). Morrison and Morgan go on to empha- size that models act as mediating instruments in a great variety of ways in addition to the one I discuss here. 36. For an extended account along these lines, see Hughes (1993). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 480 Theoretical Practice: the Bohm-Pines Quartet accounts of electrostatic (Coulomb) interactions. In addition, the use of a “correspondence principle argument” in BP II can be seen as an appeal to the old (pre-1925) quantum theory. (See Section 1.3.) A wholly classical analysis is given of the energy loss suffered by a high velocity charged par- ticle when it excites plasma oscillations; this analysis is then supple- mented by the assumption that the energy loss is quantized, and that the energy E per quantum is given by the Planck formula, E (cid:2) (cid:2)w. Here (cid:2) is Planck’s constant, and w is the frequency of the plasma oscillation which it excites. Early quantum theory was notorious for its reliance on ad hoc proce- dures. To quote Max Jammer (1966, 196), In spite of its high sounding name [ . . . ] quantum theory, and es- pecially the quantum theory of polyelectronic systems, prior to 1925 was, from the methodological point of view, a lamentable hodgepodge of hypotheses, principles, theorems, and computational recipes rather than a logical consistent theory. The introduction in BP II of the Planck formula within an otherwise clas- sical treatment of plasmas is a case in point. The formula functions simply as a useful tool in the theoretician’s workshop. In contrast, the theories the authors use in BP III are post-1925 orthodox quantum mechanics and (oc- casionally) standard electromagnetic theory and quantum ªeld theory. Each of these theories is a foundational theory, in the sense that it is under- girded by a powerful mathematical theory.37 While not conceptually wrin- kle-free,38 none of them would normally be described as “a lamentable hodgepodge.” Nonetheless, in one respect, each of them resembles early quantum theory in the ªrst quarter of the 20th century. They too provide sets of ready-to-hand tools for the theoretician’s use. To extend the meta- phor, the difference is that all the tools from a given theory now come from the same tray of the toolkit.39 I will illustrate this use of theory with nine examples from the Bohm- Pines papers, the ªrst very general, the rest speciªc. In all four papers foundational theory provides a template for the mathematical description 37. A paradigm example is the mathematical theory set out in John von Neumann’s Mathematical Foundations of Quantum Mechanics ([1932] 1955). 38. Orthodox quantum mechanics was not a wholly uniªed theory, and no-one has yet solved the “measurement problem.” 39. I was ªrst introduced to the image of a theory as a set of tools by Paul Teller, in con- versation. It is drawn by Nancy Cartwright, Toªc Shomar, and Mauricio Suarez in the pa- per, “The Tool-Box of Science” (1994). The earliest use of it that I can trace is by Pierre Duhem in the series of articles in the Revue de Philosophie in 1904 and 1905 that later be- came his Aim and Structure of Physical Theory (Duhem [1914] 1991, 24). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 481 of a system, and speciªes how that form can be given content. Whether provided by classical physics or quantum mechanics, these descriptions are given by the Hamiltonian H for the system, which represents its total en- ergy.40 H may be written as the sum H1 (cid:3) H2 (cid:3) . . . (cid:3) Hn of terms, each of them representing a different source of energy (kinetic energy, electro- static potential energy, and so on). The mathematical nature of H is not the same in the two theories; in classical physics H is a function, in quan- tum mechanics it is an operator.41 There are, however, standard procedures for obtaining a quantum mechanical Hamiltonian from a classical one. To take a particular case, wherever a momentum p appears in a classical Hamiltonian function, one substitutes the operator (cid:8)i(cid:2)( / x) to obtain the corresponding quantum mechanical operator; thus the kinetic energy term represented classically by the function p2/2m appears as the quantum mechanical operator (cid:8)((cid:2)2/2m) 2/ x2 . Speciªc instances of the use of theoretical tools—results, strategies, and technical manoeuvres—which theory provides and whose use needs no justiªcation, are supplied by eight examples from BP III. Here as else- where, Bohm and Pines are considering “an aggregate of electrons embed- ded in a background of uniform positive charge” (610b).42 1. (610b) The authors write down the Hamiltonian H1 for the system. It contains three terms: The ªrst is the operator provided by the the- ory to express the kinetic energy of the electrons; the second and third are standard expressions for the energy due to Coulomb attrac- tions between electrons, and for their self energy; each of them is slightly modiªed to take into account the uniform background of positive charge, and the second term is expressed as a Fourier series (see fn. 13). 2. (610b-611a) When this Hamiltonian is rewritten in terms of the longitudinal vector potential A(x) and the electric ªeld intensity 40. In classical physics an alternative mode of description, in terms of forces, may be used, but this is not available in quantum mechanics. 41. In classical physics the Hamiltonian H of a system is a function H: R, where , where is the set of states of the system. For any state w (cid:2) , the number H(w) is the total energy of the system in that state. In quantum mechanics the Hamiltonian H of a system is an op- erator H: is the set of states of the system. A state in quantum mechanics , there is a state yH (not necessar- is represented by a wave function y. For any state y ily distinct from y) such that H(y) yH . If y is an eigenfunction of H, then there is a real Ey (which is to say, E is an eigenvalue of H), and E is the en- number E such that H(y) ergy of the system in state y (as in the synopsis of P IV). A fuller comparison of classical physics and quantum mechanics is given in Hughes (1989, ch. 2). Jordan (1969) provides a comprehensive yet concise treatment of the operators used in quantum mechanics. 42. All page references in this subsection are to BP III. With one exception, the synop- sis of BP III in Section 1.4 shows where each of the tools was used. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 482 Theoretical Practice: the Bohm-Pines Quartet E(x), each of these quantities is expressed as a Fourier series involv- ing one of the ªeld coordinates qk or pk (for the position and the mo- mentum associated with the ªeld). Both of these series are supplied by electromagnetic theory. 3. (611a-b) To show that the resulting Hamiltonian H2 is equivalent to H1, Bohm and Pines use the method quantum mechanics prescribes: they display a unitary operator S, such that H2 (cid:2) SH1S 1.43 As they note (611a, fn.), this operator is supplied by Gregor Wentzel’s text- book, Quantum Theory of Wave Fields. 4. (614b) The “perturbation theory” of quantum mechanics is called on to estimate what corrections would have to be applied to com- pensate for the neglect of the terms U and HI in H 4.44 5. (616a) As I noted in example 2, at an early stage of BP III Bohm and Pines introduce ªeld coordinates qk and pk . Now, “in order to point up the similarity [between the transformations used in BP III and those used in BP I] and to simplify the commutator calculus,” they help themselves to the fact that in quantum mechanics these opera- tors can be expressed in terms of creation and annihilation operators ak and ak* (see fn. 19). 6. (616b-617a) The last of the transformations performed on the Hamiltonian in BP III takes H5 into Hnew. Like the transformation of H1 into H2 mentioned in example 3, this transformation is per- formed by a unitary operator. The authors’ goal is to ªnd a unitary operator U such that U(cid:8)1H5U (cid:2) Hnew . A basic theorem of the alge- bra of operators is that any unitary operator U is expressible as an exponential function of another (non-unitary) operator S, the so- called generator of U; we may write U (cid:2) exp(iS/(cid:2)).45 Hence Bohm and Pines set out to obtain U by ªnding a suitable generator S. 43. For more on unitary operators see Jordan (1969, 18–22). S is a unitary operator if S(cid:8)1S, where I is the identity operator: for all y. S(cid:8)1 is the inverse of S; for any wave function y1, if S(y1) (cid:2) y2, then S(cid:8)1(y2 ) there is an operator S(cid:8)1 such that S S(cid:8)1 (cid:2) I y, I(y) (cid:2) y1. 44. The use of perturbation techniques in physics goes back to Isaac Newton. In quan- tum mechanics, perturbation theory is invoked when a term that makes a very small con- tribution to the Hamiltonian is neglected to simplify calculations. Its function is to esti- mate the correction that would have to be applied to the result obtained in order to allow for the effect of that term. A concise account of the principles of ªrst and second order per- turbation theory in quantum mechanics is given by Cassels (1970, 780–80); for more de- tails see Messiah (1958, chs. XVI–XVII). To estimate corrections for U and HI in H3, Bohm and Pines use second order perturbation theory. 45. See Jordan (1968, 52). Also, two remarks on notation: (a) In example 3, I used the letter “S” for the unitary operator that transforms H1 into H2, but in this example I use it for the generator of the unitary operator that transforms H4 into Hnew, rather than the unitary l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 483 This was a perfectly orthodox strategy to employ; in a footnote (616b) the authors cite the second edition of P.A.M. Dirac’s classic text, The Principles of Quantum Mechanics. As it turned out, implementing the strategy was a different matter. The authors tell us (617a), The problem of ªnding the proper form of S to realize our program was solved by a systematic study of the equations of motion. We do not have the space to go into the details of this study here but conªne ourselves to giving the correct transformation below.46 Evidently, in this instance no ready-to-hand tool was available that would do the job. 7. (617b) To continue the narrative of example 6: Given the generator S, H5 can be transformed into Hnew by the rule used in example 3: Hnew (cid:2) UH5U 1, where U (cid:2) exp(iS/(cid:2)), and U 1 (cid:2) exp((cid:8)iS/(cid:2)). A cer- tain amount of algebraic drudgery would then yield an expression for Hnew. Bohm and Pines obtain considerable simpliªcations, how- ever, by using a standard mathematical tool, another theorem of the algebra of operators:47 Given (self-adjoint) operators A, A’ , and S, if A’ (cid:2) UAU 1, where U (cid:2) exp(iS) , then A’ (cid:2) A (cid:8) i[A,S] (cid:8) 1 2[A,S],S] (cid:3) (i/3!)[A,S],S]S ] (cid:3) . . . Here [A,S] is the commutator of A and S, i.e. [A,S] (cid:2) AS (cid:8) SA .48 8. (623b) The ªnal example comes from Appendix I to BP III, in which Bohm and Pines develop a quantum-mechanical version of the approach used in BP II, based on ºuctuations of the charge den- sity in the plasma. Here the recourse to the toolbox of theory is ex- plicit: We use the electron ªeld second-quantization formalism [ . . . ]. Following the usual treatments,* we describe electrons operator itself. (b) I write “U” for the second unitary operator, even though the same letter is used elsewhere to denote a term in the Hamiltonian H3. Within this example, “U” al- ways denotes a unitary operator and “S” its generator. (See item 7 in the list of theoretical tools.) In both cases, (a) and (b), I am following the usage of Bohm and Pines. 46. The reader is given no further information about this “systematic study,” and is left to conjecture that it involved a sophisticated form of trial and error. 47. While the theorem itself is standard, and its proof is straightforward, its use in this context required considerable ingenuity. 48. Since both the product and the difference of two operators are also operators (hence l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 484 Theoretical Practice: the Bohm-Pines Quartet by the ªeld quantities Y(x) which satisfy the anti-commutation relations. At the point marked with an asterisk the authors again cite G. Wentzel, Quantum Theory of Wave Fields (1949). These tools are not all of one kind. They include mathematical identi- ties (examples 6 and 7), existing results within electromagnetic theory, quantum ªeld theory and orthodox quantum mechanics (example 2, ex- amples 3 and 8, and example 5, respectively), well-established perturba- tion techniques of approximation (example 4), and standard mathematical expressions for physical quantities (example 1). As my footnotes and the authors’ own citations show, all these tools are to be found in well-known textbooks. If, as I suggested in the Preamble to this essay, the example of the Bohm-Pines quartet can be generalized, a preliminary account of theoreti- cal practice emerges.49 To apply, say, quantum mechanics to a particular physical situation—in Kuhnian terms, to work within the paradigm that quantum mechanics provides—a physicist must have a working knowl- edge of how and when to use the following elements: the simple models that the theory deals with; the mathematical representations of their be- haviour that the theory prescribes; the mathematical theory within which these representations are embedded (in this case the theory of Hilbert spaces, which includes as a sub-theory the algebra of operators); and the perturbation techniques and methods of approximation associated with the theory. This congeries of elements I call a theoretical manifold. I do not claim that my list of elements is exhaustive. An additional requirement, ªfty years after the Bohm-Pines papers were published, is a knowledge of how to use computer methods to assist the work of—or to replace—one or more of the above. Phrases like “that the theory deals with” and “that the theory pre- scribes,” which qualify some of the items in this catalogue, might suggest that underpinning every theoretical manifold there is a theory separable from the manifold it supports. From the perspective of the practitioner of physics, however, that assumption would be a mistake. As Gilbert Ryle said in another context (1963, 18), The same mistake would be made by a child witnessing the march- past of a division, who, having had pointed out to him such and such battalions, batteries, squadrons, etc., asked when the division the expression “the algebra of operators”), all the expressions on the series on the right hand side of the equation are well-formed. 49. I augment this account in Part Three of this essay. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 485 was going to appear. He would be supposing that a division was a counterpart to the units already seen, partly similar to them and partly unlike them. He would be shown his mistake by being told that in watching the battalions, batteries and squadrons marching past he had been watching the division marching past. The march- past was not a parade of battalions, batteries, squadrons and a divi- sion; it was a parade of the battalions, batteries and squadrons of a division. Analogously, all the items in the theoretical manifold may be thought of as elements of the theory; to use a theory is to use one or more of the ele- ments of the theoretical manifold. On this reading the terms “theory” and “theoretical manifold” are co-extensive. Call this the broad reading of “theory”. Customary usage, however, suggests that some elements of the theoret- ical manifold are more central than others. In the list given, the phrase “associated with the theory” carries the connotation that the items it qualiªes, the theory’s “perturbation techniques and methods of approxi- mation” occupy positions peripheral to the manifold’s central core. Central to the manifold, on this account, would be the elements to which the per- turbations and approximations are applied, namely the simple models of the theory and the mathematical representations of their behaviour. We may reserve the term “theory” for this central cluster of elements, while pointing out that without the other elements of the manifold very little could be achieved in the way of theoretical practice. Call this the narrow reading of “theory”. I have so far restricted my use of the term “theory” to foundational the- ories, theories like classical mechanics, classical electromagnetic theory, or quantum mechanics that have a wide range of applicability. But there is also a more local, but perfectly respectable, use of the term, as when we talk of “the Bohm-Pines theory of electronic behaviour in metals.” Indeed, Ziman gives this local use primacy. He writes (1964, v), “A theory is an analysis of the properties of a hypothetical model.”50 A local theory will em- ploy the theoretical manifold of an existing foundational theory, but will conªne its application to a single model, here the jellium model. But in addition to the standard methods of approximation and perturbation the- ory that come with the foundational manifold, the local theory may also introduce approximative techniques of its own. Again, the Bohm-Pines theory is a case in point, as we shall see. 50. The quotation comes from Ziman’s preface to his Principles of the Theory of Solids (1964). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 486 Theoretical Practice: the Bohm-Pines Quartet 2.4 Modes of Description The title of the B-P quartet promises a “Collective Description of Electron Interactions.” In fact, two modes of description, physical and mathemati- cal, are presented. The physical descriptions are in English, augmented by the vocabulary of physics. Couched in terms of “electrons”, “ªelds”, and so on, they are, strictly speaking, descriptions of the jellium model, but in the authors’ discourse the distinction between model and physical system is rarely observed. In Pines’s words, the model chosen is to be “simple, yet realistic,” and the physical descriptions are clearly intended to be con- strued realistically. The mathematical descriptions of the model are provided by its Hamiltonian, as we have seen. A Hamiltonian is standardly expressed as a sum of individual terms, each of them either a standard issue Hamiltonian from the tool-kit of theory, or obtainable from one by minor modiªcat- ions. Recall the ªrst example of the use of theoretical tools in BP III, in which Bohm and Pines wrote down a Hamiltonian for their model, “an aggregate of electrons embedded in a background of uniform positive charge.” The Hamiltonian contained three terms; the ªrst denoted the ki- netic energy of the electrons, the second the energy due to the Coulomb interactions between them, and the third their self-energy. The last two were both slightly modiªed to take into account the uniform background of positive charge. The requirement that the model be “simple, yet realis- tic” can be read with such procedures in mind. The simplicity of the model may be judged by the ease with which the Hamiltonian can be con- structed from the standard expressions for energy that textbooks provide, its realism by the degree to which the energies associated with the model match those of the system that it represents. Once the Hamiltonian has been rewritten in collective coordinates, a comparable procedure takes place in the reverse direction. The trans- formed Hamiltonian is manipulated and approximations are applied until, like the original Hamiltonian, it appears as the sum of recognizable ele- ments, each of them capable of physical interpretation. Thus, as we saw in the synopsis of BP I, the collective approximation allows the transformed Hamiltonian H(1) to be expressed as the sum of three parts. Bohm and Pines write (BP I, 626b), H(1) (cid:2) H(1) part (cid:3) Hosc (cid:3) Hpart int , and interpret H(1) part as “the kinetic energy in these new coordinates,” Hosc as “a sum of harmonic oscillator terms with frequencies given by the dis- persion relation for organized oscillations,” and Hpart int as a term that “cor- responds to a screened force between particles” (ibid.). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 487 Likewise, in BP III the authors write (BP III, 612b), Let us [ . . . ] neglect U, a procedure which we have called the ran- dom phase approximation [ . . . ]. With this approximation we see that the third and fourth terms in our [transformed] Hamiltonian reduce to Hosc (cid:2) (cid:8)1 2 (cid:4)k < k(c)(pk p k (cid:3) wp 2 qk q k) the Hamiltonian appropriate to a set of harmonic oscillators, repre- senting collective ªelds, with a frequency wp . In similar vein the other three terms in the Hamiltonian H4 are inter- preted as representing “the kinetic energy of the electrons,” “a simple in- teraction between the electrons and the collective ªeld,” and “the short range part of the Coulomb interactions between the electrons” (ibid.; see the synopsis of BP III). The two modes of description, the physical and the mathematical, while in many ways autonomous, are each responsive to the demands made by the other. They are not totally intertranslatable. Not every de- scribable model can be represented mathematically; not every mathemati- cal description can be given a physical interpretation. Models may need to be simpliªed; Hamiltonians may need to be mathematically massaged. In both cases the aim is the same: to bring the description closer to the ca- nonical examples supplied by physics textbooks and vice versa. These ex- amples license movement from one mode of description to the other: the rendering of physical descriptions in mathematical terms, and the inter- pretation of mathematical expressions in physical terms.51 This reliance on a comparatively small repertoire of examples imposes a severe constraint on theoretical practice. Or so one might think. But working within constraints may bring its own beneªts. Was sonata form an impediment to Mozart? Or the form of the sonnet a hindrance to Petrarch? In the case at hand, the constraint is positively beneªcial—in two ways. In the ªrst place, it solves the Meno problem: “How will you en- quire, Socrates, into that which you do not know? [ . . . ] And if you ªnd what you want, how will you ever know that this is the thing which you did not know?”52 Bohm and Pines know very well what they are looking 51. The role of these canonical models as “bridge principles” has been stressed by Nancy Cartwright (1983, Essay 7). She emphasizes the need for them in what she calls “theory entry,” or, in my vocabulary, the move to the mathematical mode of description. As the BP quartet shows, they are also needed for moves in the opposite direction, from the mathematical to the physical mode. For Cartwright (1983, 139), the prime virtue of these examples is explanatory. 52. Plato, Meno 80d. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 488 Theoretical Practice: the Bohm-Pines Quartet for. They seek a Hamiltonian of a recognizable kind, and the standard ex- amples provide aids to recognition. Secondly, the canonical status of these examples allows them to be bearers of meaning. For the habitual reader of The Physical Review the interpretations of these mathematical expressions do not have to be re-established ab initio on each occasion of their use. Fur- thermore the expressions remain meaningful even when they are modiªed to ªt speciªc circumstances, as in BP III, for example, when the terms in H1 for the electrons’ self-energy and the energy due to Coulomb interac- tions are modiªed to take into account the uniform background of posi- tive charge. Each of the physical descriptions I have so far considered is isomorphic to a corresponding mathematical description. It contains as many clauses as there are terms in the Hamiltonian, each attributing a particular type of energy to the model. But Bohm and Pines also furnish physical descrip- tions of another kind. Within the quartet they use the adjective “physi- cal,” together with its cognate adverb “physically” a dozen times, four times in BP II, seven times in BP III, and once in P IV. On only three oc- casions is it used in a way that tallies with the account I have given of the interplay between mathematical and physical descriptions of the model. In BP III (619a), for instance, we read, The physical consequences of our canonical transformation follow from the lowest-order Hamiltonian Hnew set of subsidiary conditions on our system wave function. (0) [ . . . ] and the associated More often—in fact on seven of the eleven occasions when the adjective is used in BP II and BP III—it appears in the phrase “physical picture.” The picture in question is described in the concluding section of BP II (350b- 51a). In conclusion we give a brief summary of our results in terms of a physical picture of the behavior of the electron gas. As we have seen, the density ºuctuations can be split into two approximately independent components, associated, respectively, with the collec- tive and individual particle aspects of the assembly. The collective component, which is present only for wavelengths (cid:5) D , repre- sents organized oscillations brought about by the long range part of the Coulomb interaction. When such an oscillation is excited, each individual particle suffers a small perturbation of its velocity and position arising from the combined potential of all the other parti- cles. The contribution to the density ºuctuations resulting from these perturbations is in phase with the potential producing it, so that in an oscillation we ªnd a small organized wave-length pertur- l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 489 bation superposed on the much larger random thermal motion of the particle. The cumulative potential of all the particles may, how- ever, be considerable because the long range of the force permits a very large number of particles to contribute to the potential at a given point. The individual particles component of the density ºuctuation is associated with the random thermal motion of the particles and shows no collective behavior. It represents the individual particles surrounded by comoving clouds which screen their ªeld within a distance (cid:10) D . Thus it describes an assembly of effectively free par- ticles interacting only through the short-range part of the Coulomb force. The screening of the ªeld is actually brought about by the Coulomb repulsion which leads to a deªciency of electrons in the immediate neighborhood of the particle. This same process also leads to a large reduction in the random ºuctuations of the density in the electron gas for wavelength larger than D . A third paragraph examines further the interactions of an individual elec- tron with the electron gas as a whole. What is offered here, the authors tell us, is “a brief summary of [their] results in terms of a physical picture of the behavior of the electron gas.” The “physical picture” is thus a supplement to the theoretical investiga- tions that have been the main task of the paper. It provides, in a vocabu- lary markedly different from the one used in descriptions obtained from the system’s Hamiltonian, a summary of results already achieved. Orga- nized oscillations are brought about by the long range part of the Coulomb interaction. Each individual particle suffers small perturbations arising from the combined potential of the other particles. Comoving clouds screen the ªeld of each particle. This screening is brought about by the Coulomb re- pulsion, which leads to a deªciency of electrons in the neighbourhood of the particle. The same process also leads to a large reduction in the random ºuctuations of the density in the gas at large wavelengths. Though it appears late in the second paragraph, the key word here is “process.” The behaviour of the electron gas is described in terms of causal processes, whereby one thing brings about, or leads to another. As I have said, the description is presented as a supplement to the theoretical inves- tigations pursued in the quartet. Although its themes are anticipated in the central, theoretical sections of BP II, only once (348a) does the phrase “physical picture” occur in those sections. Elsewhere in BP II it appears only in the Abstract, in Section I: Introduction, and Section VII: Conclusion; similarly, in BP III it appears twice in the ªrst section, once in the last, but nowhere else. This points to the fact that the picture itself is relatively l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 490 Theoretical Practice: the Bohm-Pines Quartet independent of the theoretical approaches taken. Though drawn from the classical account of the electron gas given in BP II, it nevertheless holds good alongside the quantum mechanical account in BP III. Descriptions of this kind—”narrative descriptions,” as I will call them—are used throughout physics. Although their closest afªliation is with the theoreti- cal manifolds of classical physics, within theoretical practice they are effec- tively independent of “high theory,” and, as in the present case, can co-ex- ist alongside theoretical manifolds of many different persuasions. And for obvious reasons, they are a large part of the lingua franca of experimental practice.53 2.5 The Use of Approximations In just ªve pages of BP I (Sections II C and II D and Section III B) the verb “neglect”, variously conjugated, appears no fewer than twelve times. In these sections Bohm and Pines are examining the results of moving from individual to collective coordinates, ªrst in the classical case and then in the quantum case. In Sections II A and III A, respectively, the transformations that will effect these moves have been speciªed with mathematical exactness. After they are applied to the original Hamilton- ian H0, however, approximative strategies are brought into play, so that terms that make only small contributions to the Hamiltonian disappear. In this way the otherwise unwieldy expression for the transformed Hamil- tonian is presented in a form that allows each of its components to be given a physical interpretation. Similar approximations are used throughout the quartet. Each involves the assumption that some aspect of the physical situation guarantees that corresponding terms in its mathematical description can be neglected without grossly affecting the results of the analysis. These assumptions are brought together and discussed in Section II B of BP I (628a-b) under the title “The Collective Approximation.” There are four of them; I will com- ment brieºy on each in turn. 1. Electron-ion and electron-electron collisions are ignored. I drew at- tention to this simpliªcation in Section 2.2. It is of the same kind as the simpliªcation (some philosophers of physics call it “abstrac- tion”) whereby the effect of air resistance on the motion of a pendu- lum is neglected (see fn. 31). In each case the effect of the neglected element is to produce a small damping of the oscillations of the sys- 53. The importance of narrative descriptions has been emphasized by several authors. Stephan Hartmann (1999) has drawn attention to the contribution they make to hadron physics. He calls them simply “stories.” l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 491 tem. The neglect of electron-ion collisions is not one of the approxi- mations used in BP I to simplify the transformed Hamiltonian; rather it is implicit in the authors’ choice of the jellium model, in which the effect of the plasma ions is represented by a background of uniformly distributed positive charge. 2. The organized oscillations in the electron gas are assumed to be small. The “customary linear approximation, appropriate for small oscillations” (BP I, 628b) is then used, allowing quadratic ªeld terms (i.e., products of ªeld terms) in the equations of motion of the system to be neglected. 3. In BP I, the velocity v of the electrons is assumed to be small com- pared with c, the velocity of light, so that terms involving v2/c2 are negligible. (This approximation appears repeatedly; it accounts for eight of the twelve occurrences of “neglect” that I mentioned ear- lier.) In BP III, as we have seen, the assumption is made that there is a maximum wave number kc (equivalently, a lowest wavelength lc) for organized oscillations. These assumptions are presented as two versions of a single assumption, that (k.v)/w is small, appropriate for transverse and longitudinal oscillations, respectively. These ªrst three approximations are straightforward, the last, the “random phase approximation,” less so. 4. Bohm and Pines write (BP I, 628b), We distinguish between two kinds of response of the electrons to a wave. One of these is in phase with the wave, so that the phase difference between the particle response and the wave producing it is independent of the position of the particle. This is the response which contributes to the organized behaviour of the system. The other response has a phase difference with the wave producing it which depends on the position of the parti- cle. Because of the general random location of the particles, this second response tends to average out to zero when we consider a large number of electrons, and we shall neglect the contribu- tions arising from this. This procedure we call the “random phase approximation.” The claim here is not that out-of-phase responses are individually negligible—they may well be of the same order of magnitude as the in- phase responses. The assumption made is that, taken collectively, they cancel each other out. Bohm and Pines justify the r.p.a. mathematically in the classical treatment of longitudinal oscillations given in BP II (349– l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 492 Theoretical Practice: the Bohm-Pines Quartet 50), and in “a more qualitative and physical fashion” in the quantum me- chanical treatment given in BP III (621). The second of these approximative strategies and the ªrst part of the third are standard fare, and I will say more about them in Section 2.7. I call them “strategies” because their application takes different forms in different theoretical contexts. None of the four are speciªc to the particu- lar theoretical manifolds that the authors use. That is why the authors need to provide an extended discussion of them. As they point out (BP I, 628b), this four-part collective approximation differs from an orthodox perturbation theory in that the latter would not allow for the fact that small changes in the ªeld arising from each particle may add up to a large change in the net ªeld—which, on the authors’ account, is precisely what generates plasma waves in metals. 2.6 Experiment and the Bohm-Pines Theory Within the Bohm-Pines quartet little attention is paid to experimental ªndings, none at all to experimental practice. The ªrst paper deals with the possibility of transverse oscillations due to electromagnetic interac- tions between moving electrons, the other three with the possibility of longitudinal oscillations due to electrostatic (Coulomb) interactions. The role of BP I is “to illustrate clearly the techniques and approximations in- volved in [the authors’] methods” (BP I, 627a). Virtually no mention is made of observed phenomena or experimental results. Bohm and Pines tell us, The electromagnetic interactions [which give rise to transverse os- cillations] are weaker than the corresponding Coulomb interactions by a factor of v2/c2 [where c is the velocity of light] and, conse- quently, are not usually of great physical interest. (Ibid.) Within the papers that follow, discussion of observable phenomena and experimental results is conªned to BP II and P IV. BP III is entirely theo- retical; the only mention of a possibly observable phenomenon comes in an aside (BP III, 610a-b), where the authors cite the work of two other theorists (Kronig and Kramer) who “treated the effects of electron-elec- tron interaction on the stopping power of a metal for fast charged particles.” In BP II and P IV, a related phenomenon, the loss of energy suffered by a high energy electron when it passes through a metal ªlm, is adduced as evidence to support the authors’ own analysis. In a paragraph of the Intro- duction to BP II the authors predict the excitation of collective oscilla- tions in a metal by high energy particles passing through it; the paragraph ends with a laconic reference to experimental conªrmation: l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 493 Experiments by Ruthemann and Lang, on the bombardment of thin metallic ªlms by fast electrons tend to verify our theoretical predictions concerning this type of oscillation. (BP II, 339a.) This remark is ampliªed in Section IV of the paper, where the link be- tween theory and experiment is provided by a “correspondence principle argument” (see Sections 1.3 and 2.3 of this essay). In Section IV of P IV Pines uses quantum mechanics to provide an explanation of this phenome- non, which is summarized in the penultimate paragraph of Section 1.5 of this essay. Analysis of the interaction of the high-energy electrons with the collective ªeld shows that only at one particular frequency w, very close to the plasma frequency wP , will the oscillations of the ªeld become self-sus- taining (P IV 634a). Pines points out (P IV 634b) that for aluminium and beryllium ªlms the predicted values of energy lost (in multiples of (cid:2)w) by the electrons agree very well with those obtained experimentally by Ruthemann and by Lang. Furthermore, the calculated mean free path (the average distance travelled by the high energy particle between excitations) matches an empirical value based on Lang’s data for aluminium ªlms. When gold, copper, or nickel ªlms are used, however, the spectrum of energy loss is not discrete. Pines attributes this failure to the fact that “the valence electrons in these metals are not sufªciently free to take part in undamped collective motion” (ibid.), i.e., that the jellium model, in which there is no interaction between valence electrons and the individual ions of the metal, is inadequate for these elements. He notes, “Experiments have not yet been performed on the alkali metals, where we should expect to ªnd collective oscillation and the appearance of discrete energy losses.”54 (Ibid.) There are four other places in P IV where empirical evidence is referred to. Three instances occur in Section III, where Pines is discussing the sur- prising fact that, if the independent electron model of electron behaviour is “corrected” to allow for the so-called “exchange energy” between elec- trons, the result is to worsen the agreement with experiment rather than to improve it. (See Section 1.5 of this essay.) One instance involves the magnetic properties of the electron gas, another the way its speciªc heat varies with temperature. In neither case does Pines cite the experiments that yielded the relevant empirical results. Nor does he in the third in- stance (P IV, 631a). In this instance Pines describes an ad hoc theoretical move made by P.T. Landsberg. In order to account for an observed feature of the x-ray emission spectrum for sodium, Landsberg had found it neces- sary to introduce a screened Coulomb interaction between electrons. Since 54. Alkali metals (lithium, sodium, potassium, etc.) appear in Group Ia of the Periodic Table. They have just one valence electron per atom. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 494 Theoretical Practice: the Bohm-Pines Quartet a screening effect of this kind is predicted by the B-P theory, one might expect Pines to present Landsberg’s manoeuvre as an indirect conªrma- tion of that theory. Instead, Pines points to a result by E.P. Wohlfahrt. Landsberg had proposed that the screened Coulomb interaction could be mathematically modelled by writing (e2/rij)exp[(cid:8)(rij /l)] for the electron- electron interaction potential in place of the standard expression e2/rij , and was of the order of 10–8 cm. Wohlfahrt that the “screening radius” showed that, if both of these proposals were accepted, then the unfortu- nate effect of “correcting” the independent electron model to allow for the exchange energy would be greatly reduced, along with the error in its pre- dictions concerning speciªc heat. The comparison implicit in Pines’s dis- cussion of Landsberg’s and Wohlfahrt’s work is this: On the independent electron account of a plasma there are results that can be purchased only by making ad hoc assumptions concerning the electron-electron interac- tion potential within the plasma. On the BP approach, in contrast, they come for free. Here, as throughout the ªrst three sections of P IV, Pines is more con- cerned to compare the B-P theory with other theoretical approaches than with experimental results. In addition to the work of the theorists I have already mentioned, Pines points to two separate treatments of an electron gas by Eugene Wigner (1934 and 1938), and shows how their results match those obtained using the collective approach. This form of valida- tion, whereby a new theoretical approach gains credence by reproducing the results of an earlier one, has received little attention from philosophers of science, though instances of it are not far to seek. For example, when Einstein ªrst proposed his general theory of relativity, one of the earliest tasks he set himself was to show how the theory could recapture, to a ªrst approximation, the classical principle of conservation of energy as it ap- plied to an orbiting planet.55 In the present case, a general sense that, for Bohm and Pines, empirical justiªcation was something of an afterthought is reinforced by an acknowledgement they offer at the end of BP II (351a): “The authors wish to thank Dr. Conyers Herring for informing us of the experiments of Ruthemann and Lang.” Be that as it may, it is certainly true that, in the B-P quartet, theory and experiment exist as almost independent disciplines, one of them barely glanced at. We read that both Ruthemann and Lang conducted ex- periments in which high energy electrons were directed at thin metallic ªlms and their energy losses in passing through the ªlms were measured. 55. It appears in Einstein (1915), the paper in which he explained the anomalous ad- vance of the perihelion of Mercury. This paper is the third in a quartet of papers, all pub- lished in November, 1915, in which Einstein introduced the general theory. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 495 We are told (P IV, 634b) that in Lang’s experiments the initial energy of the electrons was 7.6 kev, and that the energy losses were multiples of 14.7 ev in aluminium ªlms and 19.0 ev in beryllium ªlms. (The calcu- lated values are 15.9 ev and 18.8 ev respectively.) But we are told neither how these beams of electrons were prepared, not how the electrons’ initial energies and losses in energy were established. It seems that the practices involved in preparation and measurement belong exclusively to the guild of experimenters, and are not the province of the worshipful company of theoreticians. But this cannot be the whole story. While the two sets of practices, experimental and theoretical, may be different, they cannot be wholly disjoint, for two reasons. The ªrst is that experimental practices include theoretical practices—a truism which has the virtue of being true. The energy of an electron may be measurable, but it is not observable, and every measuring instrument more sophisticated than a meter rule is a ma- terial realization of a theory. When a theoretical result is compared with an experimental result, the practices involved in the theoretical side of things may or may not belong within the same theoretical manifold as those on the experimental side. Although two different theoretical mani- folds are used in BP II and P IV, in both cases the conclusions reached are compared with the results of the same experiments; thus in at least one of these cases the theoretical manifold used by Bohm and Pines differs from that used by Lang. Nonetheless—and here I come to the second point—if empirical conªrmation is to be possible, in both cases the set of practices on the theoretical side must mesh with the set of practices on the empiri- cal side. Theoretician and experimenter must agree that a theoretical en- ergy loss of 18.8 ev can be meaningfully compared with an experimental energy loss of 19.0 ev. In both the literal and the Kuhnian sense, the two sets of practices cannot be incommensurable. In Galison’s phrase (1997, passim), the two parties need to establish a trading zone. How this might be done is beyond the scope of this paper. The fourth and last instance of an occasion when empirical evidence is at issue is an instance of a slightly peculiar kind. It occurs very early in P IV (626–27) where, as in the third instance, Pines is comparing the collec- tive account of electron behaviour with the account given by the inde- pendent electron model. On these occasions Pines is in the position of one who needs to explain the curious incident of the dog in the night time. As readers of Conan Doyle will recall, what made the incident curious was that the dog did nothing in the night time. In like manner Pines must ex- plain the fact that the collective behaviour that he and Bohm ascribe to metallic plasmas has very little effect on the plasma’s properties and be- haviour, so little that the independent electron model, in which “the mo- tion of a given electron is assumed to be independent of all the other elec- l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 496 Theoretical Practice: the Bohm-Pines Quartet trons,” enjoys widespread success. He makes this explanation his ªrst order of business: a qualitative explanation is given in the paper’s intro- duction, and a quantitative account follows in its central sections. (See Section 1.5.) In certain circumstances the absence of a phenomenon may stand as much in need of explanation as would the phenomenon itself. 2.7 Deduction in the Bohm-Pines Quartet I introduced Part Two of this essay by quoting John Ziman’s delineation of the intellectual strategy of a typical paper in theoretical physics: “A model is set up, its theoretical properties are deduced and experimental phenomena are thereby explained.” Taking my cue from the ªrst clause of that account, I began my observations on the B-P quartet by describing the role played by models and modelling. Turning my attention to the second clause, I will end them by examining the notion of deduction, as it appears in the third of these papers. A curious feature of Chapter 3 of Ernest Nagel’s The Structure of Science (1961) is that, although it bears the title “The Deductive Pattern of Ex- planation,” there is no mention in it of the steps by which conclusions fol- low from premises. In that chapter the actual process of deduction is taken for granted, witness this paragraph (op. cit., 32). [A] deductive scientiªc explanation, whose explanans is the occur- rence of some event or the possession of some property by a given object, must satisfy two logical conditions. The premises must con- tain at least one universal, whose inclusion in the premises is essen- tial for the deduction of the explanandum. And the premises must also contain a suitable number of initial conditions. This is a beautiful example of what, borrowing the phrase from Nancy Cartwright (1999, 247), I will call the “vending machine” view of theoriz- ing Originally, Cartwright used the phrase in criticizing a particular ac- count of theory entry, the process by which, in Ziman’s words, “a model [of a physical system] is set up”: The theory is a vending machine: you feed it input in certain pre- scribed forms for the desired output; it gurgitates for a while; then it drops out the sought-for representation; plonk, on the tray, fully formed, as Athena from the brain of Zeus. (Cartwright, 1999, 247). The feature of the machine that Cartwright challenges is its mode of in- put, which must accord with “certain prescribed forms.”56 (Ibid.) I will use 56. She continues, “Producing a model of a new phenomenon such as superconductivity is an incredibly difªcult and creative activity. It is how Nobel prizes are born.” (Ibid.) The l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 497 the same metaphor to characterize the next stage of Ziman’s narrative, where “theoretical properties of the model are deduced and experimental phenomena are thereby explained.” In this usage, the salient feature of the machine will be the process of gurgitation. Nagel has talked in the previous chapter (albeit unknowingly) about that phase of the machine’s working (1961, 21): A type of explanation commonly encountered in the natural sci- ences [ . . . ] has the formal structure of a deductive argument, in which the explanandum is a logically necessary consequence of the ex- planatory premises. [My emphasis.] Here “logical necessity” includes “mathematical necessity,” as Nagel points out a few lines later. This kind of deduction we may call strict deduc- tion; it is the kind of deduction that Nagel subsequently takes for granted.57 Pace Nagel, adherence to strict deduction is not “commonly encoun- tered in the natural sciences.” An insistence on strictness would mean, for example, that the explanation of the behaviour of a system consisting of more than two mutually attracting bodies would be beyond the scope of classical mechanics. Starting with Newton, physicists have become adept in ªnding ways around this problem, and Bohm and Pines are no excep- tion. To illustrate how they go about it, I will examine the extended argu- ment in BP III that transforms the Hamilton for the jellium model from H1 to Hnew , and in so doing makes manifest the collective properties of the electron gas. I will ªrst give a précis of the argument, and then list the in- dividual moves within it. The order in which they are listed, however, is determined, not by the order of their occurrence in BP III, but by the kinds of justiªcation offered for them. First in the list are moves justiªable by strict deduction; next come moves that have obvious, but not strictly deductive, justiªcations; at the end of the list are moves that are highly pragmatic, and are peculiar to the investigation under way. By now virtu- ally all these moves will be familiar to the reader. My aim in setting them as I do is to draw attention to the gulf between the concept of “deduc- tion,” as the word is used is used by Ziman, a working physicist, and as it is used by Nagel, a mid-twentieth century philosopher of science. First the précis, in six steps: essay from which this quotation is taken deals at length with the Bardeen, Cooper, and Shrieffer theory of superconductivity (for which they were indeed awarded the Nobel prize). I give a brief account of their achievement in Section 3.1. 57. See, for instance, the extended footnote on p. 353 of The Structure of Science. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 498 Theoretical Practice: the Bohm-Pines Quartet Step (1) Bohm and Pines present a standard Hamiltonian H1 for a cloud of electrons against a background of uniform positive charge. Step (1 2) HI is then claimed to be equivalent to H2, given a fam- ily of subsidiary conditions Ok on the state of the system. The claim is justiªed by displaying a unitary operator S such that H2 (cid:2) SH1S 1. Two central terms in H2 are expressed as Fourier series, in- dexed by k (as was the term for the energy due to Coulomb interac- tions in H1), and each of the subsidiary conditions corresponds to a component k of the Fourier decompositions. Step (1 3) The procedure of Step (1 2) is now repeated, using a modiªed operator S , so that the two Fourier series in the terms of the resulting Hamilton, H3, are truncated; neither of them has an index higher than kC . Correspondingly, there are just kC subsidiary conditions Ok . Step (3 4) Algebraic manipulation of H3 yields the Hamilton H4, which contains ªve terms. Step (4 5) This step contains just two moves. One of the terms of H4, U†, is shown to be negligible, and is therefore deleted;58 an- other, Hosc , is rewritten to produce Hªeld . The combined result is H5. Step (5 new) Bohm and Pines then apply a “canonical transforma- tion” to H5 (and to the operators Ok), and obtain the ªnal Hamil- ton, Hnew . Now for the list of moves, categorized by justiªcation; the page numbers cited are all from BP III: Trivial Moves Step (3 4) involves just one move, an elementary example of strict deduction, in which H4 is obtained by rearranging the components of the terms of H3. 58. Recall that within BP III Bohm and Pines are not consistent in their use of the let- ters “U”´and “S”. In Step (5→new) the letters refer to a unitary operator and its generator, respectively. The authors have, however, previously used “U” to refer to a component of the Hamiltonian H4, and “S” to refer to a unitary operator in Steps (1→2) and (1→3). In this section I will resolve one of these ambiguities by writing “U†” for the component of H4. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Strictly Mathematical Moves Perspectives on Science 499 1. In Step (1) the authors “have used the fact that the Coulomb interac- tion between the ith and jth electrons may be expanded as a Fourier series in a box of unit volume” (610a). Though their subject matter is the physical world, the “fact” they use is a mathematical fact, be- longing to the branch of mathematics known as analysis. 2. Likewise, in Step (4 5), when the ªeld operators qk and pk are ex- pressed in terms of creation and annihilation operators ak and ak* (616a), although this move has a physical interpretation, it is a per- fectly permissible formal move, made “to simplify the commutator calculus.” 3. As we have seen , Steps (1 2), (1 3), and (5 new) each involves a unitary transformation of a Hamilton Ha into another, Hb, In es- sence, a transformation of this kind is equivalent to a move from one system of coordinates to another in the (abstract) Hilbert space on which these operators are deªned. Formally, these transformations employ a mathematical identity in the operator algebra of that Hilbert space of the kind Hb (cid:2) UHaU 1. (See example 6 in Section 2.3.) 4. In Step (5 new), after a move of type 3, Bohm and Pines use an- other theorem of the operator algebra to rewrite the right hand side of that identity: Hb (cid:2) UHaU 1 (cid:2) Ha (cid:8) i[Ha,S] (cid:8) 1 2[[Ha,S],S] (cid:3) (i/3!)[[[Ha,S],S],S ] (cid:3) . . . Here S is the generator of U: U (cid:2) exp(iS/h ); and [A,B] is an abbrevi- ation for the commutator of operators A and B: [A,B] (cid:2) AB - BA. (See example 7 in Section 2.3; the identity appears on p. 617a.) The term “[Ha,S]” is a commutator of the ªrst order, “[[Ha,S],S]” a com- mutator of the second order, and so on. Standard Approximations Physics, in particular the physics of many-body systems, is not an exact science. Approximations (in the literal sense: numerical results “very close” to the “real” values) are not only tolerated but inevitable. If a for- mula for a physical quantity Q contains a term t whose value is small in comparison with the value of Q, then a term t2 of the second order will of- ten be neglected. In quantum mechanics, a similar procedure applies to the operator representing Q. Two of the four types of approximation dis- cussed in BP I (and in Section 2.5 above) conform to this pattern: ne- l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 500 Theoretical Practice: the Bohm-Pines Quartet glected are quadratic ªeld terms and the term v2/c2, when v is small com- pares with c. Step (5 new) in BP III provides an example of the former kind (618a): “In obtaining [equation (54)] we have neglected a number of terms which are quadratic in the ªeld variables and are multiplied by a phase factor with a non-vanishing argument.” Another example from the same step involves the mathematical move we have just encountered, whereby a unitary transformation is unpacked as a series of commutators involving the generator S of the relevant unitary operator: Hb (cid:2) UHaU 1 (cid:2) Ha (cid:8) i[Ha,S] (cid:8) 1 2[[Ha,S],S] (cid:3) (i/3!)[[[Ha,S],S],S ] (cid:3) . . . Paradoxical though it may seem, the use of this identity leads to consider- able simpliªcations. In preparation for that move Bohm and Pines have deªned an expansion parameter, (cid:11), which is a measure of the strength of “the coupling between the ªeld and electrons” (615b). They expected (cid:11) to be small; in fact they described it as “the measure of the smallness” of a term in the system’s Hamiltonian (ibid.). If we replace Ha in the identity above by HI (one of the terms of H4) it transpires that “the effects of the ªeld- particle interaction (up to order (cid:11)) are contained in the ªrst correction term, (i/(cid:2))[HI, S] . The higher order commutators will be shown to be ef- fects of order (cid:11)2, [ . . . ] and may hence be neglected.” (618a; my emphasis; (cid:2) is Planck’s constant.)59 To anticipate: In the vanishing case, when [Ha,S] (cid:2) 0, then Hb (cid:2) Ha, and Ha is unaffected by the transformation. Perturbation Theory In quantum mechanics the treatment of small terms is codiªed by pertur- bation theory. A basic element of quantum theory is the (time-independent) Schrödinger equation, which relates the Hamiltonian to a spectrum of val- ues of energy. The values are determined by the solutions of the equation. A problem arises when a Hamiltonian has the form Ha (cid:3) Hb , and Hb is small compared with Ha , because in that case the Schrödinger equation may not have exact solutions. Perturbation theory then provides a proce- dure whereby an approximate solution may be found. Step (4 5) provides an example. Bohm and Pines were faced with a Hamiltonian (H4) which contained ªve terms: Hpart (cid:3) HI (cid:3) H osc (cid:3) H s.r. (cid:3) U†. Since both U† and HI were known to be small compared with the other three terms, Bohm and Pines used second order perturbation theory to de- 59. A similar strategy was adopted in BP I. At the corresponding point in their quan- tum mechanical treatment of transverse oscillations Bohm and Pines tell us that “higher or- der commutators can be neglected if we restrict our attention to the lowest order terms in v/c.” (632b) I will come back to the commutator (i/(cid:2))[HI , S,] later. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 501 termine just how small those effects were. Bohm and Pines concluded that they were “justiªed in neglecting completely the term U†” (BP III, 614b), but that, although they were “justiªed in neglecting HI in order to obtain a qualitative and rough quantative understanding of the behaviour of [their] system, [ . . . ] the effects arising from HI should be taken into ac- count in a careful quantitative treatment” (op. cit., 615a).60 Note that per- turbation theory does not go beyond orthodox quantum theory; it merely provides a recipe whereby selected terms in the Schrödinger equation are discarded; in ªrst order perturbation theory terms of second order are dis- carded, in second order perturbation theory, terms of third and higher or- ders. Formal Analogies I turn now to analogical reasoning, speciªcally to the use within BP III of arguments and results from the two papers that came before it. BP I con- tained both a classical and a quantum mechanical treatment of transverse plasma waves, and BP II a classical treatment of longitudinal plasma waves. BP III gives a quantum mechanical treatment of longitudinal plasma waves, and so marries the second of the theories used in BP I to the phenomena discussed in BP II. It is therefore not surprising that the corre- spondences drawn between BP III and BP I are different in kind from those drawn between BP III and BP II. In BP III Bohm and Pines reserve the words “analog” and “analogous” for the correspondences between that paper and the quantum mechanical treatment provided in BP I, and, on each of the eight occasions when BP I is mentioned, one at least of those words occurs. The analogies the authors refer to are formal analogies, correspondences between mathematical for- mulae. For instance, “HI and Hªeld are [ . . . ] analogous to the transverse terms encountered in BP I, and we may expect that many of the results obtained there may be directly transposed to the longitudinal case.” (617a) Likewise, the operator S which is instrumental in the canonical transformation of H5 to Hnew in Step (5 new) “may be seen to be just the longitudinal analog of the ‘transverse’ generating function given in BP I” (ibid.). Non-Standard Approximations As we have seen, in Step (4 5), Bohm and Pines use second order pertur- bation theory to justify the neglect of the term U† in H4. They had previ- ously used another argument to the same end (612b): 60. Recall that the transformation that took H5 into Hnew was designed to distribute the effects of HI among those terms that remained. See Section 2.4 and fn. 20. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 502 Theoretical Practice: the Bohm-Pines Quartet U† [ . . . ] always depends on the electron coordinates, and since these are distributed over a wide range of positions, there is a strong tendency for the various terms entering into U† to cancel out. Let us for the time being neglect U†, a procedure we have called the random phase approximation, and which we shall pres- ently justify. Physical Analogies Though Bohm and Pines had used two distinct theories in BP I, in each case their approach had been the same. The starting point of each was a Hamiltonian for a collection of electrons in a transverse electromagnetic ªeld, in which the electrons were represented by the momentum pI and position xI of the individual charges. A similar approach was taken in BP III; the only difference was that the longitudinal, rather than the trans- verse components of the electromagnetic ªeld are included in the Hamil- tonian. In both these papers, only after a canonical transformation was ap- plied to the Hamiltonian was the system described in terms of collective variables. In contrast, in BP II Bohm and Pines wrote (BP II, 340a), Our approach to the equations of motion is aimed at making use of the simplicity of the collective behavior as a starting point for a trac- table solution. As a ªrst step, we study the ºuctuations in the parti- cle density, because [ . . . ] their behavior provides a good measure of the applicability of a collective description. [My emphases.] In other words, since the particle density is a collective property, nothing corresponding to the canonical transformations used in BP I is needed in BP II. And while it is true that in BP III Bohm and Pines develop “a di- rect quantum mechanical extension of the methods used in Paper II,” this extension is relegated to a brief appendix (623–24), and is used in Section V of the paper only to resolve some complications involving the subsidiary conditions on the quantum states of the system (see fn. 18). In the theoret- ical core of BP III the mathematical aspects of BP II are set to one side. In- stead, Bohm and Pines emphasize the “physical picture” the paper presented: In the preceding paper [BP II] we developed a detailed physical picture of the electronic behavior [of a dense electron gas]. Al- though the electron gas was treated classically, we shall see that most of the conclusion reached there are also appropriate (with cer- tain modiªcations) in the quantum domain. Let us review brieºy the physical picture we developed in [BP II], since we shall have to make frequent use of it in this paper. (609a) l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 503 As we have seen, the most striking feature of the picture was that, within it, [T]he electron gas displayed both collective and individual particle aspects. [ . . . ] The collective behaviour [i.e., the plasma oscilla- tions] of the electron gas is decisive for phenomena involving dis- tances greater than the Debye length, while for smaller distances the electron gas is best considered as a collection of individual par- ticles which interact weakly by means of a screened Coulomb.force. (609a-b) I will focus on just two of the correspondences between BP III and BP II that Bohm and Pines rely on. The ªrst is very general: it involves the de- marcation marked in the classical case by the Debye length. The second is more speciªc: it concerns the dispersion relation, the relation between the frequency and the wavelength of plasma oscillations. 1. Some prefatory remarks are needed. In BP II (341a-b) Bohm and Pines present a criterion for the applicability of a collective descrip- tion. The criterion involves an equation (labelled “(9)”), which has been deduced from the principles of electrostatics with the help of the random phase approximation. In this equation d2rk/dt2 is equated with the sum of two terms,61 one representing the random thermal motion of the individual particles, the other the collective oscillations of the electron gas. Thus the “rough criterion for the ap- plicability of a collective description” is that “for most particles the collective [second] term in (9) be much greater than the term aris- ing from the random thermal individual particle motions” (BP II, 341a-b). It turns out that, if k is sufªciently small, the effect of the ªrst term can be neglected, and a straightforward derivation shows that the quantitative form of this condition can be written as (cid:8)2 , where D is “the well known Debye length” (ibid.).62 k2 (cid:12)(cid:12) 4(cid:6)n D ) col- Conversely, for high wave number k (and small wavelength lective behaviour (i.e. plasma oscillations) will not be generated. This result is carried over into Step (1 3) of BP III (611b-612a). We found in Paper II that in the classical theory there is a min- imum wave wavelength C (which classically is the Debye length), and hence a maximum wave vector kC , beyond which l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 61. Recall that is the electron density and k is the wave number of the oscillations. It appears when Bohm and Pines represent the electron density r by the Fourier decomposi- tion r k ak rk. See Section 1.3. 62. D is deªned by the equation: D (cid:2) T/4 ne2 , where is Boltzmann’s constant and T is the absolute temperature of the electron gas. 504 Theoretical Practice: the Bohm-Pines Quartet organized oscillation is not possible. We may anticipate that in the quantum theory a similar (but not identical) limit arises, so that there is a corresponding limit on the extent to which we can introduce collective coordinates to describe the electron gas. [ . . . ] The number of collective coordinates, n(cid:13), will then correspond to the number of k values lying between k (cid:2) 0 and k (cid:2) kC . [ . . . ] The modiªcation of [H1] to include only terms involving (pk , qk ) with k (cid:12) kC , may be conveniently carried out by applying a unitary transformation similar to [the transfor- mation which took H1 to H2 ], but involving only [position co- ordinates] qk for which k (cid:12) kC . By this means H1 is transformed into H3, and the information about plasma behaviour obtained by classical means becomes encoded in a quantum theoretic Hamiltonian. This correspondence, however, is not quantitative. As the quotation above tells us, Bohm and Pines anticipate that, in the quantum case, the limit kC will be “similar, but not identical” to the classical limit, where it is the reciprocal of the Debye length D . 2. The dispersion relation of a plasma relates the frequency w of a plasma oscillation to its wave number k (the reciprocal of its wavelength). The relation is presented on p. 618b, in the penultimate move of Step (5 new). Bohm and Pines show that, because of the form it takes, signiªcant simpliªcations can be effected.63 They write (619b), This dispersion relation plays a key role in our collective de- scription, since it is only for w(k) which satisfy it that we can eliminate the unwanted terms in the Hamiltonian [ . . . ] and the unwanted ªeld terms in the subsidiary condition. The fact that a particular choice for the dispersion relation leads to desirable results does not, however, justify that choice, as Bohm and Pines acknowledge. They continue, The frequency of these collective [longitudinal] oscillations is given by the dispersion relation [ . . . ], which is the appropri- ate quantum mechanical generalization of the classical disper- sion relation derived in BP II, as well being the longitudinal 63. To be precise, several terms in the ªrst version of Hnew disappear if the dispersion re- lation takes the form suggested (618b). The terms in question are the ªrst two terms of the commutator (i/(cid:2))[HI, S,] and the transformed version of one of the terms of Hªeld in H5 . l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 505 analog of the quantum-dispersion relation for organized trans- verse oscillation, which we obtained in BP I. This is a double-barrelled justiªcation, appealing as it does both to a physical and to a formal analogy. Causal Inferences In this catalogue of argumentative moves, all but one of those under the headings Non-Standard Approximations, Physical Analogies, and Causal Infer- ences (that is to say, those moves furthest away from the canons of strict de- duction) are buttressed by more than one argument.64 The two arguments under the heading Causal Inference both involve the term Hs.r. , which ap- pears in H4 and H5 and represents “the short-range part of the Coulomb in- teraction between the electrons” (612b). In each case Bohm and Pines want to establish that the canonical transformation which takes H5 to Hnew has a negligible effect on Hs.r.; or, in mathematical terms, if U is the uni- tary operator associated with the transformation, then to a good approxi- mation, UHs.r.U(cid:8)1 (cid:2) Hs.r . . . 1. Example 4 from the strictly mathematical moves shows that the de- sired conclusion holds, provided that the commutator [S,Hs.r..] is negligible (where S is the generator of U). Bohm and Pines present “a typical ªrst order term arising from [S, Hs.r.],” and label it “(74).” They point out that “the structure of (74) is quite similar to that of U† [in H4]” (621a). As we have seen, U† was shown to be negligible by the use of second order perturbation theory; in treating (74), however, the authors opt for a different strategy (621a-b). Because of the analytic difªculties involved [in the use of per- turbation theory] we prefer to justify our neglect of (74) in a more qualitative and physical fashion. We see that (74) describes the effect of the collective oscilla- tions on the short-range collisions between the electrons, and conversely, the effect of the short-range collisions on the collec- tive oscillations. [ . . . ] The short-range electron-electron colli- sions arising from Hs.r. will act to damp the collective oscilla- 64. In the non-standard approximation the neglect of U† was justiªed by the r.p.a., but further justiªcation was provided, as we have seen, by second order perturbation theory ap- plied to the single operator U† (614); furthermore, justiªcation of the r.p.a. in the classical case appeared in Section VI of BP II, and although it was tailor-made for a speciªc example readers were invited to regard it as a template to be used elsewhere. And as I pointed out at the end of the discussion of the second physical analogy, the choice of dispersion relation had a twofold justiªcation. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 506 Theoretical Practice: the Bohm-Pines Quartet tions [ . . . ]. A test for the validity of our approximation in neglecting terms like (74) is that the damping time from the collisions be small compared with the period of a collective os- cillation. In this connection we may make the following re- marks: 1. Electron-electron collisions are comparatively ineffective in damping the oscillations, since momentum is conserved in such collisions, so that to a ªrst approximation such collisions pro- duce no damping. [ . . . ] 2. The exclusion principle will further reduce the cross section for electron-electron collision. 3. If HI [which represents “a simple interaction between the elec- trons and the collective ªeld”] is neglected, collisions have no effect on the collective oscillations. This means that the major part of the collective energy is unaffected by these short-range collisions, since only that part coming from HI (which is of or- der [the expansion parameter] relative to wp ) can possibly be inºuenced. Thus at most 20 percent of the collective energy can be damped in a collision process. All of these factors combine to reduce the rate of damping, so that we believe this rate is not more that 1 percent per period of an oscil- lation and probably quite a bit less. [ . . . ] It is for these reasons that we feel justiªed in neglecting the effects of our canonical transfor- mation on H s.r.. We have been given two reasons for taking (74) (and hence [S,Hs.r..] ) to be negligible compared with other terms in H5: ªrst, Bohm and Pines drew attention to the existence of formal similarities between (74) and U† (a term already known to be negligible); secondly, they presented a causal argument to show that the effects, represented by (74), of interactions be- tween long-range collective oscillations and short-range collisions of elec- trons would be small.65 Even before the canonical transformation took place (616b-618b), a condensed version of this argument had been presented (616a): From [the expressions for the components of H5] we see that if we neglect HI , the collective oscillations are not affected at all by Hs.r.. Thus Hs.r. can inºuence the qk only indirectly through HI . But, as we shall see, the direct effects of HI on the collective oscillations are 65. In addition, the authors draw attention to a classical treatment of damping of col- lective oscillation by electron-electron collisions by Bohm and Gross (1949). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 507 small. Thus, it may be expected that the indirect effects of Hs.r. on the qk through HI are an order of magnitude smaller and may be ne- glected in our treatment which is aimed at approximating the ef- fect of HI . (The variable qk that appears here has been introduced in H2 . It represents a generic component of the Fourier expansion of A, the electromagnetic vector potential in the plasma, and will be supplanted by a collective vari- able when the canonical transformation takes place.) I call this argument a “causal inference” because of the vocabulary em- ployed: “Hs.r. can inºuence the qk only indirectly”; “the direct effects of HI”; “the indirect effects of Hs.r.”; “collective oscillations are not affected at all by Hs.r.” (emphases mostly mine). The agents portrayed as bringing about ef- fects, directly or indirectly, are components of the Hamilton H5. But here we should listen to the stern voice of Pierre Duhem ([1914] 1991, 20), “These mathematical symbols have no connection of an intrinsic nature with the properties they represent; they bear to the latter only the relation of sign to thing signiªed.” In what sense, then, can a theoretical term be a causal agent? The short answer is that in this argument Bohm and Pines have moved into a ªgurative discourse, within which the terms, HI and Hs.r. play metonymic roles. The long answer, which may shed light on the short one, requires us to follow Duhem’s lead, and undertake an analysis of theoretical representation as it appears in BP III, a project that will take us to the end of Part Two of this essay. In the BP quartet we may distinguish two layers of theoretical repre- sentation. The ªrst is the representation of a metal by a simpliªed and ide- alized model, in this case the jellium model, consisting of a gas of elec- trons moving against a uniform background of positive charge. The second involves a foundational theory. The model is represented in two ways, as a classical system and as a quantum mechanical system;66 the quantum mechanical treatment is applied in BP III. As we have seen, be- cause the behaviour of a quantum system is governed by its Hamiltonian, to represent it the theoretician needs to know the system’s energy. In the jellium model it comes from three sources: the kinetic energy of the elec- trons, the Coulomb energy due to the electrostatic repulsion between them, and their self-energy. These physical quantities are denoted (at the second level of representation) in the language of quantum mechanics by standard formulas (slightly modiªed to take into account the effect of the positively charged background). The sum of these formulas, signiªed by “H1” represents the total energy of the model. By a series of mathematical 66. The choice of a foundational theory need not involve an extra layer of representa- tion; instead, we can simply distinguish two jellium models at the ªrst level. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 508 Theoretical Practice: the Bohm-Pines Quartet transformations and judicious approximations Bohm and Pines demonstrate that H1 is effectively equivalent to another cluster of formulas, Hnew , which can be interpreted in physical terms: Helectron contains terms referring only to individual electrons; “ªeld coordinates appear only in Hcoll, and thus describe a set of uncoupled ªelds which carry out real independent longitudinal oscillations” (619b); and Hres part “describes an extremely weak velocity-dependent electron-electron interaction” (620b). The two-stage denotation of the physical systems by H1 and the inter- pretation in physical terms of Hnew mark off the two boundaries of the demonstration phase. Within this phase, however, there is a continuous interplay between the formulas of the second level and the physical quan- tities they represent. Indeed, it would not be an overstatement to say that this interplay drives the deduction from H1 to Hnew . Accompanying the steps that lead from H1 to Hnew is a commentary on their physical implica- tions. Only one of these steps is purely formal: Step (3 4) consists solely of the Trivial Move. All the others have physical import. Even in Step (1), when Bohm and Pines write the energy for the Coulomb interactions be- tween electrons as a Fourier series, the choice has two rationales. The an- nounced rationale is that the choice allows them to “take into account the uniform background of positive charge” by the simple device of excluding from the summation over k in the Fourier series the index k(cid:2)0 (610b). The long term rationale is that the new idiom is ideal for expressing oscil- latory phenomena. The subsidiary conditions on the wave function intro- duced in Step (1 2) guarantee that Maxwell’s equations are satisªed. Step (1 3) is prefaced by an argument which justiªes by Physical Analogy the truncation of the Fourier series of H2 . The remaining steps yield the Hamiltonians H4, H5, and Hnew . Each of these – not only Hnew—is broken down into clauses that are given physical interpretations. H4and H5 share three clauses, Hpart , HI , and Hs.r. , and a fourth clause of H4 (U †) is dis- carded on both theoretical and physical grounds (Perturbation Theory and the Random Phase Approximation). (The replacement of the last clause in H4 (Hosc ) by Hªeld in H5 has no physical signiªcance; it relies on the second Strictly Mathematical Move.) Step (5 new) involves not only the canonical transformation of H5 into Hnew (an example of the third Strictly Mathemati- cal Move) but the assumption that a particular equation gives us the dispeersion relation (the relation between the frequencies of the collective os- cillations and their wavelengths, justiªed by Physical Analogy, The clauses HI and Hs.r. in H4 are, of course, the formulas which were en- dowed with causal powers in the second Causal Inference. We are now in a position to make sense of that endowment. Bohm and Pines interpret HI as “a simple interaction between the electrons and the collective ªeld” (612b), and Hs.r. as “the short-range part of the Coulomb interaction be- l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 509 tween the electrons” (ibid.). We may reverse the metonymy that the sec- ond Causal Inference relied on, and replace HI and Hs.r. by their interpreta- tions to obtain a straightforwardly causal argument. From [the expressions for the components of H5] we see that if we neglect a simple interaction between the electrons and the collective ªeld, the collective oscillations are not affected at all by the short-range part of the Coulomb interaction between the electrons. Thus this short-range part of the Coulomb interaction can inºuence the qk only indirectly through the interaction between the electrons and the collective ªeld. But, as we shall see, the direct effects of the interaction between the electrons and the collective ªeld on the collective oscillations are small. Thus, it may be expected that the indirect effects of the short-range part of the Coulomb interactions” on the qk through “the interaction between the elec- trons and the collective ªeld are an order of magnitude smaller and may be neglected in our treatment which is aimed at approximat- ing the effect of as a simple interaction between the electrons and the col- lective ªeld. Notice that the argument is now couched in the narrative mode illustrated in Section 2.4. The ªrst topic of this section was the vending machine account of de- duction; the last one the role of representation in theoretical physics. In Nagel’s version of the vending machine, the input and the outut are state- ments, and the mechanisms which transform one into the other are lin- guistic. Its components are statements, and its workings are governed by the laws of logic. If the machine is to work, then although the referring terms are divided into two disjoint classes, observational terms and theo- retical terms, and correspondence rules are needed to link the two, all these elements must belong to a single language.67 Bohm. Pines, and their contemporaries took it for granted that, impurities aside, a metal in the solid state consisted of crystals, each comprising a regular lattice of ions surrounded by a gas of electrons, And given this description, the physicist faced a true “Theoretician’s Dilemma.” In Pines’s words (1987, 68), which I quoted very early in this part of the essay, In any approach to understanding the behaviour of complex sys- tems, the theorist must begin by choosing a simple, yet realistic model for the behaviour of the system in which he is interested. 67. The difªculties of carrying out the project are acknowledged by Nagel in ch.5 of The Structures of Science, and (very clearly) by Carl Hempel in his essay, “The Theoretician’s Dilemma” ([1958], 1965). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 510 Theoretical Practice: the Bohm-Pines Quartet The dilemma resides in the phrase “a simple, yet realistic model.” For verisimilitude begets complexity, and complexity begets intractability. The dilemma is not new; the “many-body problem” was faced by Newton, for whom “many” was “three.” He supplemented the deductive resources of Apollonian geometry with the ªrst perturbation techniques of the mod- ern age. (Ptolemy’s epicycles, equants, and eccentrics were his counter- parts in the ancient world.) Today it is neither surprising nor reprehensi- ble that physicists should supplement the resources of strict deduction, and that these supplements should be adopted by others. The random phase approximation, for example, is arguably the most valuable contribu- tion to the deductive practices of solid-state physics directly traceable to Bohm and Pines. Though they arrived too late for Bohm and Pines to make use of them, two other supplements that appeared around 1950 should be noted. One emerged within physics, the other from elsewhere. In 1949 Feynman in- troduced what were soon referred to as “Feynman diagrams,” and by the mid-1950s they were part of the toolkit of many theoretical physicists— including those working in solid state physics.68 And, as early as 1956, a paper on solid state physics by Neville Mott (1956, 1205) began as fol- lows, The use of electronic computers has made it possible to calculate the electronic wave functions of simple molecules with any degree of accuracy desired. From our present perspective we can hardly regard the use of computer techniques as a “supplement” to deductive practice. Rather, virtually all the work performed in the demonstration phases of theoretical representa- tion is out-sourced to computer programs. Paradoxically, the greatest change of theoretical practice in the last half century did not involve any change of theory. PART THREE On Theoretical Practice 3.1 Theoretical Practice and the Bohm-Pines Quartet My chief aim in this essay has been to provide a case study of the theoreti- cal practices of physics. I chose the quartet of papers that David Bohm and David Pines published in the early 1950s, and in this section of the essay I 68. I say more about Feynman diagrams in Part Three. We may note that in 1967 Richard Mattuck published A Guide to Feynman Diagrams in the Many Body Problem, a book that was entirely devoted to applications of Feynman diagrams. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 511 will use this example to summarize what theoretical practice involves, and to sketch the ways in which this practice changes though time. Initially this will involve some recapitulation of familiar material, but some new themes will be sounded as I go on. For most readers of the BP quartet, its importance lay in its third and fourth papers, in which the authors used the resources of orthodox quan- tum mechanics to investigate the behaviour of the conduction electrons in metals.69 The mathematical foundations of that theory had been laid twenty years earlier by Paul Dirac and John von Neumann; Dirac’s The Principles of Quantum Mechanics was published in 1930, and von Neu- mann’s Mathematische Grundlagen der Quantenmechanik in 1932. The greater part of Dirac’s work presented an abstract formalism of which both Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics (pub- lished in 1925 and 1926 respectively) were realizations;70 von Neumann’s work absorbs Dirac’s “transformation theory” within the mathematics of “Hilbert spaces.” (Both terms were coined by von Neumann.) Each of these publications marked a major advance in mathematical physics, but I mention them only to emphasize that, on two counts, none of them pro- vided an example of theoretical practice, as I use the phrase. The ªrst is that, even though a progression can be traced from Heisenberg to von Neumann, all four were too original to be regarded as part of an estab- lished practice; rather, they were the innovations around which practices would coalesce. The second is that theoretical practices emerge when a theory is put to work; that is to say, when a mathematical theory is applied to a particular phenomenon or system, and must be supplemented by ad- ditional procedures and techniques. Twenty years separated the publication of the Bohm-Pines quartet from the work of Dirac and von Neumann, In that period not only did physi- cists become thoroughly familiar with the intricacies of quantum mechan- ics, but they also devised a variety of stratagems and artiªces to comple- ment the bare mathematics of the theory. Bohm and Pines in their turn contributed a number of original techniques, as we shall see, but it is the unoriginal part of their work, the miscellany of theoretical elements which they inherited from previous investigators, and with which they and their peers were fully conversant, that allows us to speak of a “theoretical prac- tice” shared by a community of practitioners.71 A partial inventory will show just how diverse the elements of this 69. The exception was Stanley Raimes, whose book (1961), aimed at students of experi- mental physics, presented the classical account of plasma oscillations given in BP II. 70. Schrödinger had previously (1926) shown the equivalence of the two approaches. 71. The emphasis on a community of practitioners immediately raises the question: Is it a solecism to speak of “Newton’s theoretical practice”? It is not. But Newton is a unique l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 512 Theoretical Practice: the Bohm-Pines Quartet practice were. From the mathematical part of orthodox quantum theory Bohm and Pines took standard results; some came from the algebra of Hilbert space operators, others from the standard perturbation techniques of the theory. On the physical side, the basic structure of the system they investigated was well known. Metals in the solid state were crystalline; within each crystal, the positive ions of the metal formed a regular array, the “crystal lattice,” and were surrounded by a cloud of valence electrons. To apply quantum mechanics to this system Bohm and Pines needed to model it in a way that adequately satisªed the competing desiderata of verisimilitude and simplicity. The criteria for the latter rested on the exi- gencies of quantum theory, in particular, on how amenable the model was to representation by the stock Hamiltonians of quantum mechanics.72 In BP III, by choosing a model in which the discrete nature of the individual ions was ignored, the authors were enabled to write down a comparatively simple Hamiltonian for the system, H1. It contained just three terms. Each was a standard expression for a particular component of the energy of an aggregate of electrons, slightly modiªed to allow for a background of positive charge. The term for the Coulomb potential was expressed as a Fourier series, and when H1 was rewritten in terms of the longitudinal vector potential A(x) and the electric ªeld intensity E(x) of the electro- magnetic ªeld within the plasma, each of them was also decomposed by Fourier analysis into a series of sinusoidal waves. Both moves were well es- tablished. Indeed, the latter was the most venerable procedure used in the quartet: Jean Baptiste Fourier was born in 1768, a year before Napoleon Bonaparte. Again, of the four kinds of approximations which Bohm and Pines drew attention to in BP I (see Section 2.5), only the fourth, the random phase approximation, was original. The ªrst approximation allowed elec- tron-electron collisions and electron-ion collisions to be ignored. The ab- straction which ignored electron-electron collisions was analogous to that which ignored atom-atom collisions in a simpliªed 19th century derivation of the ideal gas law, and (as I pointed out in Section 2.5) individual elec- tron-ion collisions were effectively ignored by all theoreticians who used the jellium model. The second approximation relied on the assumption that the organized oscillations in the plasma were small, in order that the authors could use “the customary linear approximation appropriate for small oscillations” (my emphasis); the third, the neglect of terms involving v2/c2 case. No other theoretical physicist has simultaneously articulated a truly original theory and applied it to as many diverse phenomena as did Newton. 72. This point has been emphasized by Cartwright (1999, 268–78) in analyzing the Hamiltonian used by Bardeen, Cooper, and Schrieffer in their theory of superconductivity. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 513 on the assumption that the velocity v of the electron is small compared with the speed of light, c, was hardly a novel manoeuvre (see Section 2.7). The elements of theoretical practice are often easily identiªed; their use is so widespread that they have acquired names. The jellium model is an ex- ample. In that instance the name is descriptive of the element, but more often the name attached to an element is the name of its originator, as in “a Fourier series.” In BP III alone we ªnd eleven such names attached to thirteen elements of ªve different kinds: to physical phenomena [Coulomb interactions (609a, passim)]; to theoretical models [a Fermi gas (610b), a Bose ªeld (625a)]; to magnitudes associated with such models [the Debye length (611b), the Bohr radius (615a)]; to mathematical entities [a Fourier series (610b), the nth Hermite polynomial (613a), the Slater determinant (613a), the Fermi distribution (615a)]; and to mathematical representations of physical systems [Maxwell’s equations (611a), the Heisenberg representation (623b), Fermi statistics (623b), and, of course, the Hamiltonian, which appears on virtually every page]. Two of these phrases are used in describing the kind of theoretical approach taken by others; the rest are woven into the au- thors’ own arguments. There are no footnotes to provide glosses on them, nor are they needed. The phrases are part of the vernacular of the readers of The Physical Review, and the elements they denote are part of a physicist’s stock-in-trade, not least because of their versatility. The “Debye length,” for example, entered the physicist’s lexicon in 1926, as the thickness of the ion sheath that surrounds a large charged particle in a highly ionized elec- trolyte. As we have seen, twenty ªve years later Bohm and Pines used it in their treatment of conduction electrons in a metal, at a scale several orders of magnitude smaller than the phenomenon it was originally designed to model. Because of this versatility, when Bohm and Pines wrote in the In- troduction to BP II, “For wavelengths greater than a certain length D the ºuctuations are primarily collective,” this parenthetical aside conveyed two messages. It gave the reader information, and it told him that he was on familiar ground. So much for the techniques that Bohm and Pines inherited. In turn, the quartet bequeathed a number of useful theoretical tools to future investi- gators. The most obvious is the “random phase approximation.” Another is the practice of treating a system consisting of a particle and its immedi- ate environment as a quasi-particle.73 In BP III the authors observed that, when an electron is surrounded by a cloud of collective oscillations, it be- 73. Nearly a decade later, Pines included a section (1962, 31–34) on the deªnition of quasi-particle in the lecture notes that form the ªrst part of The Many Body Problem. The re- mainder of the volume is an anthology of papers from previous dozen years on the physics of many-body problems. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 514 Theoretical Practice: the Bohm-Pines Quartet haves as though its mass has increased, and they treated it accordingly. (See the penultimate paragraph of Section 1.4.) Subsequent examples of quasi-particles include (a) a conduction electron, which consists of an electron moving within the periodic potential provided by ªxed lattice ions; (b) a polaron, which is an electron moving in an insulating polar crystal, and (c) a quasi-nucleon, which is a proton or a neutron surrounded by a cloud of other nucleons.74 Another bequest was made by Pines in 1956, when he proposed that the quantum of energy lost by a high energy electron when it is scattered within a metal foil should be regarded as a particle: a plasmon. He summarized “the evidence, both experimental and theoretical, which points to the plasmon as a well-deªned entity in nearly all solids.” (1956, 184b). Plasmons are analogous to phonons. Both are quanta of en- ergy in condensed matter, but while phonons are stored as thermal energy by atoms as they vibrate about their mean positions, plasmons are stored by the valence electrons of a metal either individually, or collectively in plasma oscillations. But arguably the most signiªcant legacy of Bohm and Pines’ work was the contribution it made to the theory of superconductivity published by Bardeen, Cooper, and Schrieffer in 1957. The overall strategy of their pa- per resembled that used in BP III. To echo Ziman, in both papers a model was set up, its Hamiltonian was prescribed and its theoretical properties deduced, and experimental phenomena were thereby explained. The BCS model was the more complicated of the two. Whereas the BP model repre- sented the charge of the positive ions as a uniform background of charge, the BCS model not only allowed for the fact that the ions formed a regular lattice, but took into account the lattice vibrations as well. These vibra- tions (and here I switch into narrative mode) are quantized into phonons, which mediate an attractive interaction between two valence electrons (the so-called “Cooper pairs”). Since the repulsive force of the Coulomb in- teraction between these electrons is attenuated by the screening effect pre- dicted by Bohm and Pines, the net force between the members of a Cooper pair may be attractive, causing the metal to become a superconductor.75 74. See Mattuck (1967, 15). 75. This “criterion for the occurrence of a superconducting phase” (BCS, 1176b) had been put forward by Bardeen and Pines (1955). (Note the connecting link between the BP quartet and the BCS paper.) The achievement of the BCS paper was to show how the crite- rion could account for the “main facts” about superconductivity, and the authors list (BCS, 1175a): “(1) a second order phase transition at the critical temperature, Tc [ . . . ], (2) an electronic speciªc heat varying as exp((cid:8)T0/T) near T 0oK [ . . . ], (3) the fact that a super- conductor exhibits perfect diamagnetic behaviour, (4) effects associated with inªnite con- ductivity (E(cid:2)0), and (5) the dependence of Tc on isotopic mass, Tc(cid:14)M const.” In 1972 all three authors were awarded the Nobel Prize for their work on superconductivity. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 515 The Hamiltonian for the BCS model reºects these two opposing inter- actions. It contains four terms (BCS, 1179a), of which the third, Hcoul , represents the energy due to the screened Coulomb interactions between electrons, and the fourth (oddly referred to as H2) the energy due to the phonon interaction with the Cooper pairs. For an expression for Hcoul the authors go to the BP theory. Recall from Sections 1.4 and 1.5 that what had started out in BP III as the Hamiltonian H1 for a dense electron gas was successively modiªed until it appeared in P IV in its ªnal version: H (cid:2) Hpart (cid:3) Hcoll (cid:3) Hs.r.. In that ªnal Hamiltonian the effects of Coulomb interactions were divided into two parts. The long-range effects were sum- marized in the term Hcoll , where they were “effectively redescribed in terms of collective oscillations of the system as a whole” (P IV, 627a). In contrast, the term Hs.r. corresponded to “a collection of individual elec- trons interacting with a comparatively short-range force” (ibid.). Since the high energies associated with collective oscillations do not occur in a su- perconductor, the result is that the term Hcoul in the BCS Hamiltonian need only represent short range effects, i.e. the screened interaction repre- sented by Hs.r.. The kind of theoretical practice exempliªed by the BP theory of plas- mas and the BCS theory of superconductivity did not last. From the per- spective of the early twenty-ªrst century, these theories appear as two of the last constructionist contributions to solid state physics. I borrow the term “constructionist” from Philip Anderson (1972), who contrasts two hypotheses: constructionist and reductionist.76 The latter is a hypothesis about the physical world. On the reductionist hypothesis the behaviour of a macroscopic system is ultimately determined by the behaviour of its sub-microscopic constituents, which in turn is governed by simple funda- mental laws. The constructionist hypothesis is bolder; it is a hypothesis about the reach eventually attainable by our fundamental theories of phys- ics. It suggests that, as and when our “ªnal theory” has established the on- tology of fundamental particles and the laws which they obey, we will have the theoretical resources to explain all the phenomena of nature. This hypothesis Anderson rejects (op. cit., 393): “(T)he reductionist hypothesis does not by any means imply a ‘constructionist’ one: the ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe.” To amplify this he contin- ues,”[I]t seems to me that we may array the sciences linearly according to the idea: The elementary entities of Science X obey the laws of Y,” and he then sketches a hierarchy whose ªrst three entries are these: 76. Incidentally, in 1958 Anderson published a paper which drew on the work, both of Bohm and Pines and also of Bardeen, Cooper and Schrieffer. The title was “Random-Phase Approximation in the Theory of Superconductivity.” l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 516 Theoretical Practice: the Bohm-Pines Quartet X Y Solid state or many body physics elementary particle physics chemistry many body physics molecular biology chemistry . . And he quickly adds, “But this hierarchy does not imply that science X is ‘just applied Y.’ At each stage entirely new laws, concepts and generaliza- tions are necessary [ . . . ].” I extend the usage of the adjective “constructionist” to cover theoretical endeavours that would, if successful, conªrm the constructionist hypothe- sis. In this sense, the approach taken by Bohm and Pines was construc- tionist in many ways. This is beautifully illustrated by the transformations that I alluded to two paragraphs ago, whereby the Hamiltonian H1 in BP III became H in P IV. Bohm and Pines started with elementary particles; the components of H1 dealt with the properties and interactions of elec- trons: their individual kinetic energies, their individual self-energies, and the energy due to the pair-wise Coulomb forces between them. As we saw, the ªnal Hamiltonian is the sum of three terms, Hpart (cid:3) Hcoll (cid:3) Hs.r.. The ªrst term and third terms still deal only with electrons. Hpart represents their kinetic energy and self energy (albeit as those were modiªed by the environment) and Hs.r. “corresponds to a collection of individual electrons interacting via a comparatively weak short range force (P IV, 627b).” Hcoll, however, is expressed entirely in collective variables: “The long range part of the Coulomb interaction has effectively been redescribed in terms of the collective oscillations of the system as a whole.” (P IV, 627a) In this way the emergent behaviour of the system is made explicit, and the micro- scopic and the macroscopic are accommodated under one roof.77 But, as we saw in Section 3.2, the collective variable approach to many- body problems had severe limitations. It was never taken up outside Da- vid Bohm’s immediate circle at Princeton. Even the authors of the BCS paper had reservations about its use; in the conclusion of their paper they wrote (BCS, 1198a), “An improvement of the general formulation of the theory is desirable,” and listed half a dozen items which could be im- 77. Subsequently Pines predicted the value of an emergent property of the plasma, its speciªc heat (P IV, 632). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 517 proved on, the BP “collective model” being one. From the middle of the 1950s on attention moved away from constructionist treatments of high density plasmas in favour of macroscopic treatments. For example, in 1954 Lindhard described the behaviour of plasma entirely in terms of a macroscopic property, its dielectric constant, and his example was quickly followed by others. Intertwined with the rejection of a constructionist methodology was a major change in theoretical practice: the orthodox quantum mechanics used by Bohm and Pines was supplanted by quantum ªeld theory.78 Throughout the 1930s and 1940s quantum ªeld theory had been highly problematic. It had a number of successes, like the prediction of the positron, but it was prone to divergencies; that is to say, seemingly in- nocuous calculations went to inªnity.79 In 1949 the problem was resolved from two directions. Sin-Ituro Tomonaga and Julian Schwinger used a generalization of operator methods, which was theoretically impeccable but very difªcult to work with; Richard Feynman, on the other hand, used a “propagator approach,” and showed how it could be pictorially repre- sented by simple diagrams.80 While the Tomonaga-Schwinger approach made quantum ªeld theory respectable, the Feynman approach made it easy to apply. Later in the year Freeman Dyson showed the two approaches to be equivalent, and by 1955 quantum ªeld theory had established itself as the core of a new theoretical practice, largely because the Feynman dia- gram had shown itself to be one of the most remarkable theoretical tools of the twentieth century. In particular, there were two major reasons why physicists found the theory well suited for treating dense electron gases. The ªrst was “[t]he realization, that there exists a great formal similarity between the quantum theory of a large number of Fermi particles and quantum ªeld theory” (Hugenholtz and Pines, 1959, 489/332); the sec- ond was the theoretical economy afforded by Feynman diagrams, and was as important as the ªrst.81 Eugene Gross wrote (1987, 47), 78. Nevertheless, despite the change of methodology and of foundational theory, the various approaches to high density plasmas in the 1950s were by no means incommensura- ble. For example, Philippe Nozières and Pines bridged the gap between the collective vari- able approach and the alternatives in papers like “Electron Interactions” (1958b), which was subtitled, “Collective Approach to the Dielectric Constant.” 79. For a succinct account of the theory’s successes and failures, see Howard Georgi (1989, 449). The seriousness of the problem can be judged by the language used to de- scribe it. Michio Kaku (1993, 4) describes quantum ªeld theory as “plagued with inªni- ties,” and for Georgi that was its “tragic ºaw.” 80. Not all physicists welcomed the Feynman diagram. In fact Schwinger is reputed to have forbidden his graduate students from using it, on the grounds that its use represented the triumph of theft over honest toil. 81. In an earlier version of this paper, when I discussed alternative approaches to elec- l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 518 Theoretical Practice: the Bohm-Pines Quartet Feynman’s introduction of diagrams freed the imagination of theo- retical physicists to deal with what had been depressingly compli- cated formalisms in quantum ªeld theory and many-body physics. Concerning the latter, Gross wrote from experience. Like Pines, he was one of Bohm’s students, and he too had collaborated with Bohm in papers that took the collective approach. 3.2 A Very Brief Note on Methodology There are two respectable ways in which a philosopher of physics can ap- proach theoretical physics. The ªrst is to examine speciªc theories with an eye to philosophical issues. Examples are the usual suspects—statistical mechanics, general relativity, quantum theory—but the theories investi- gated need neither have an extensive domain of application nor be widely accepted; Bohmian quantum mechanics and Heinrich Hertz’s version of classical mechanics are cases in point. Some of the issues will be metaphys- ical: The problem of The Direction of Time; What kind of Being does space-time have? Others will be internal to the theory: What is measurement in quantum theory?82 Though a theory may be associated with a particular physicist, as Einstein is with the special and general theories of relativity, that is irrele- vant to the philosophical issues the theory raises. A theory achieves a life of its own, so to speak. The second approach consists in examining how physicists use these theories, and is more empirical in nature. Its object is, ªrst of all, to give accurate descriptions of theoretical practices, something signally absent until recently in philosophical circles. On this approach, the philosopher regards each speciªc application of the theory as uniquely tied to the theo- rist (or theorists) who made it. The application may involve standard tech- niques, like the use of the Bloch Hamiltonian or the random phase ap- proximation, but they are the choices the authors made. By its nature the material examined by the philosopher are the words, equations, and dia- grams appearing in a paper in a physics journal. To amplify a theme from the Preamble to this essay, such a philosopher regards those works as texts, and her task to be analogous to that of a literary critic. Like a good literary critic, the philosopher who uses this methodology draws the reader’s at- tention to the elements of a text and how they ªt together. In particu- tron plasmas between 1954 and 1958, I cited fourteen papers from that period. Feynman diagrams appeared in nine of them. 82. These three issues are section headings in Lawrence Sklar’s admirable Philosophy of Physics (1992). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / e d u p o s c / a r t i c e - p d l f / / / / / 1 4 4 4 5 7 1 7 8 9 3 9 7 p o s c . 2 0 0 6 1 4 4 4 5 7 p d . . . . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Perspectives on Science 519 lar, she shows the reader what kind of text it is and the nature of its suc- cess. This essay is an endeavour of the second kind. References Bohm, David, and David Pines (1951). “A Collective Description of Elec- tron Interactions. I. Magnetic Interactions.” The Physical Review 82, 625–34. (Referred to as BP I in text). Pines, David, and David Bohm (1952). “A Collective Description of Elec- tron Interactions. 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