Bell, John Stewart
BELL, JOHN STEWART
(b. Belfast, Northern Ireland, United Kingdom, 28 July 1928; d. Geneva, Switzerland, 1 October 1990)
physics, elementary particle theory, foundations of quantum mechanics, accelerator design.
Bell was a physicist of extraordinary depth and scope. As a member of the Theoretical Division of the Conseil Européen pour la Recherche Nucléaire (CERN) he made many theoretical analyses of the experiments performed there and formulated important general ideas regarding elementary particles and quantum field theory. Most important of these is his discovery (with Roman Jackiw) of quantum mechanical symmetry breaking: the BellJackiwAdler anomaly. In addition he applied elegant mathematics to the design of accelerators.
He worked privately, outside of CERN, on hidden variables (HV) interpretations of quantum mechanics (QM), on quantum nonlocality (which was authoritatively established by Bell’s theorem and experiments inspired by it), and on the measurement problem. He is widely considered the most important investigator of these subjects in the latter half of the twentieth century and the person most responsible for reopening questions about these topics, which once were considered to have been settled by the founders of quantum mechanics.
Background and Career. The biographical information for the present article was largely drawn from an essay by Andrew Whitaker (2002) of Queen’s University in Belfast, a Royal Society biographical memoir by Philip Burke and Ian Percival (1999), and an interview by Jeremy Bernstein (1991).
John Bell was the son of John and Annie Bell, both from workingclass families, strongly Protestant but on friendly terms with Catholic neighbors. Although none of Bell’s ancestors were academically distinguished, his parents valued education for practical reasons. Because secondary schooling in Belfast was not free and the family was poor, only John was educated beyond elementary school, but his brothers were selftaught and one became a professor of electrical engineering in a Canadian college and wrote several textbooks. John read avidly as a boy, expressed his intention at age eleven to become a scientist, and was called “the Prof” in his family. He won a scholarship to the Belfast Technical High School, where he did excellent work in academic and practical subjects.
Upon graduation in 1944 he was hired as a technician in the physics department of Queen’s University of Belfast, being too young and impecunious to enter as a student. Karl Emeléus and Reader Richard Sloane recognized his ability, loaned him books, and allowed him to attend lectures. The following year he received a small grant from the Cooperative Society and entered the university as a student. He was able to take a BSc degree with first class honors in experimental physics in only three years, and in a fourth year took a BSc degree with first class honors in mathematical physics.
Bell’s training in classical and atomic physics at Queen’s University was excellent, and the emphasis on experimental physics undoubtedly influenced his later work as a theoretician. He was discontented, however, with Sloane’s exposition of the principles of QM and challenged, sometimes angrily, Sloane’s advocacy of the Copenhagen interpretation. In Bell’s extra year as a student at Queen’s University he was the beneficiary of the arrival of the great crystallographer Paul Peter Ewald, a refugee from Nazi Germany. Ewald was very open to discussions of matters of principle in physics and was undoubtedly more sophisticated about subtleties in quantum theory than Bell’s other teachers.
For financial reasons Bell went from Queen’s University into the Scientific Civil Service without a PhD. Following Ewald’s advice he applied successfully to the Atomic Energy Research Establishment at Harwell, from which he was recruited by an accelerator design group at Malvern that was later incorporated into the Theoretical Physics Division back at Harwell. He was recognized as a brilliant solver of theoretical and practical problems of accelerator design.
At Harwell he met Mary Ross, a Scottish mathematical physicist also working on accelerator design, whom he married in 1951. Their collaboration on accelerators continued until the end of his life, and she also was an adviser in his research on the foundations of QM. The partnership of John and Mary Bell was exemplary in the history of science.
In 1953 and 1954 John Bell was granted a year’s leave of absence to work in the Department of Mathematical Physics at the University of Birmingham. There he demonstrated the CTP theorem (invariance of physical processes under the combination of charge conjugation, time reversal, and space inversion) under the direction of Rudolf Peierls. This result was part of his PhD thesis, but was independently discovered by Gerhart Lüders (1954). The experience at Birmingham enabled Bell to do important research in quantum field theory and nuclear physics upon his return to Harwell. However, organizational changes at Harwell made it an unsuitable locus for the interests of both John and Mary Bell, and therefore in 1960 they resigned tenured positions there for untenured positions at CERN in Geneva, where they remained except for occasional leaves until the end of their careers.
At CERN John Bell worked on a wide variety of problems connected with nuclear and highenergy particle physics. He also continued work on accelerator design, often in collaboration with Mary Bell (1991). With Jon Magne Leinaas (1987) he demonstrated that a circular variant of the Unruh effect, which in its original version relates the linear acceleration of a charged particle to the effective temperature of the blackbody radiation that it encounters, is responsible for the depolarization of electrons circularly accelerated in storage rings—remarkably showing that a nuisance to experimentalists is actually a beautiful phenomenon in relativistic quantum field theory.
During Bell’s sojourn at CERN his obsession with the principles of QM did not diminish, but it remained his “hobby” which he pursued in evenings and on holidays, since it was not part of his professional responsibilities. Josef Jauch at the University of Geneva was a knowledgeable and enthusiastic partner in discussions of the foundations of QM. John Bell was personally modest and seemed to be surprised by the recognition given to him by laymen as well as physicists. Bell’s theorem and the experiments inspired by it, which seem unavoidably to imply the existence of nonlocal causal relations in nature, has intrigued philosophers, theologians, devotees of science fiction, and lovers of mysteries. Bell accepted this renown with good nature, amusement, and perhaps with some satisfaction.
Field Theory and Elementary Particle Physics. John Bell’s public profession (also his employment) from the mid1950s was that of a theoretical nuclear and later particle physicist, a progression that reflects the historical development of the subject. The framework within which he approached subatomic phenomena was always field theory, even when competing ideas such as Smatrix/Regge theory in the early years and string theory in later years became popular among particle theorists. (This section was largely drawn from the corresponding more detailed discussion in Jackiw and Shimony, 2002.)
His first paper in this area, “Time Reversal in Field Theory,” published in 1955, followed five years of research in accelerator physics. Drawn from his doctoral thesis at Birmingham University, it established the CTP theorem, one of the most basic precepts of particle physics. For example, it can be utilized to show that particles and their antiparticles must have equal masses.
Unknown to Bell this result had already been shown the year before by Lüders using an argument markedly different from Bell's. Yet Bell was not often credited for his independent derivation, perhaps because he was not in the circle of formal field theorists (Wolfgang Pauli, Eugene Wigner, Julian Schwinger, Res Jost, etc.) who dominated this topic. However, today Bell’s “elementary derivation” is more accessible than the formal field theoretic arguments.
The subject of time reversal remained an important theme in his subsequent research (1957), especially when it became clear that time inversion (T) (unaccompanied by space inversion and charge conjugation) is not a symmetry of nature, and neither is space inversion conjoined with charge conjugation (PC) (unaccompanied by time inversion). Together with his friend, the experimentalist Jack Steinberger, he wrote an influential review on the phenomenology of PCviolating experiments (1966), and with Perring (1964) proposed a “simple model” theory to explain that effect. Though speculative, the suggestion was truly physical, hence falsifiable, and it was soon ruled out by further experiments. Nevertheless, it remains a bold and beautiful idea that continues to intrigue theorists.
In 1960, Bell joined CERN and remained there to the end of his life. Although primarily a center for particle physics accelerator experiments, CERN houses Europe’s preeminent particle theory group, and Bell became active in that field. Typically particle theorists are divided into phenomenologists—people who pay close attention to experimental results and interact professionally with experimentalists—and formalists, who explore the mathematical and other properties of theoretical models, propose new ideas for model building, and usually are somewhat removed from the reality of the experiments. Although Bell’s timereversal paper belongs forcefully in the formalist category, at CERN he was very much also a particle physics phenomenologist, drawing on his previous experience with nuclear physics. Indeed, with characteristic conscientiousness, Bell felt an obligation to work on subjects related to the activities of the laboratory, such as the analysis of the first neutrino experiments performed there in 1963. But his readiness to discuss and study any topic in physics ensured that he would pursue highly theoretical and speculative issues as well.
In a crucial paper in 1967 Bell argued that weak interactions should be described using a gauge theory, a type of mathematical formalism that provides a unified framework in which to describe quantum field theories of electromagnetism. Theoretical understanding of particle physics at that time had been hampered by the absence of a single reliable model for the fundamental interactions of elementary particles, and competing models could not be assessed because of their extremely complicated dynamical equations. To overcome these difficulties, Murray GellMann (1964) had proposed current algebra, which is a particle physics/quantum field theory reprise of an old technique of atomic physics: the ThomasReicheKuhn sum rule, and the Bethe energy loss sum rule (see Bethe and Jackiw, 1967).
Bell (1967a) contributed to the program of current algebra by demonstrating for a solvable model (Lee model) that using current algebra relations sometimes led to sum rules that disagreed with explicitly calculated amplitudes. To improve the situation, Bell (1967b, 1968) invoked the nonAbelian gauge principle, at the heart of YangMills theory, to derive the desired current commutation relations. (For a review, see the section on field theory in Jackiw and Shimony, 2002.)
Bell found, however, some failures in his generally successful version of current algebra, notably that current algebra and PCAC (partially conserved axialvector current) seemed to forbid the decay of the etameson into three pimesons; yet this process was seen experimentally (Bell and Sutherland, 1968). Investigating this failure led to Bell’s most farreaching contribution to particle physics: a novel method of symmetry breaking, in which symmetries of an unquantized theory do not survive quantization. This led to Bell’s most famous field theory paper, coauthored with Roman Jackiw in 1969, in which is described what is now called the BellJackiwAdler anomaly (also known as the chiral anomaly): a mechanism that would explain physical phenomena such as neutral pion decay in terms of a socalled anomalous term. This work resolved the failures of current algebra and also provided important support for the color theory of quarks. (For details of the analysis and for applications see Bell and Jackiw, 1969; Treiman et al., 1985; Holstein, 1993; and Jackiw and Shimony, 2002.
Another investigation by Bell provides yet another example of his far ranging interests and research style. On a visit to India, R. Rajaraman told Bell about a peculiar effect seen in quantum field theory, according to which the number operator of an electron moving in the background of a topological soliton, such as that created by the domain wall of a solidstate substance, would be a fraction of an integer. It was alleged that this effect could be physically realized in a polymer. But is the observed fraction an expectation value, or a sharp observable without fluctuations? Only in the latter case would this represent a truly novel and unexpected phenomenon.
Bell doubted that the fraction could be a sharp observable. Nevertheless, he wanted to find out, and in two papers with Rajaraman (1982, 1983) he established that, perhaps somewhat counter to his intuition, the fractions were indeed eigenvalues of a number operator though one which implied a “somewhat sophisticated definition of charge.” This lastmentioned work illustrates well John Bell’s attitude to his research on fundamental physical questions. Rather than merely advancing new theoretical models, his publications are infused with the desire to know and explain existing structures, preferably in “simple terms”, in a “simple model”—phrases that occur frequently in his papers.
Research on Foundations of Quantum Mechanics. As a student at Queen’s University Bell sharply questioned Sloane, the reader who taught the course on QM, about Heisenberg’s uncertainty principle and Bohr’s quantum mechanical epistemology (Whitaker, 2002, pp. 14–15). Bell’s skepticism about the justification of the “Copenhagen” interpretation of QM, and even the meaningfulness of Bohr’s terms, such as “complementarity” (Bohr, 1958), continued in his later career (e.g., Bell, 1989, reprinted 2004).
Bell’s skepticism regarding Bohr’s rationalizations attracted him to the general program of hidden variables(HV), according to which the QM state of any system S describes it incompletely and only statistically, while a complete description of S would also require a specification of the HV. Bell’s interest in HV was further stimulated by reading David Bohm’s pair of papers of 1952, proposing a specific HV model that agreed with the statistical predictions of QM.
Bell first examined the demonstration in von Neumann (1955) of the impossibility of an HV interpretation agreeing with all the predictions of QM. He realized that although von Neumann’s theorem was mathematically rigorous, it was physically unconvincing because it depended upon the premise
where A and B are physical quantities of a system S— represented in QM by arbitrary selfadjoint operators on the Hilbert space associated with S—and “Exp” designates the expectation value calculated by means of a probability distribution over the space of HV. Equation (1) is evidently true in a theory obeying the standard logic of classical physics, and it is also true in QM. There is, however, no general reason for asserting its truth in HV if A and B are quantum mechanically represented by noncommuting operators, because then the procedures for measuring A and B cannot be combined into a single procedure (1966, reprinted 2004, pp. 4–5). Bell then constructed a counterexample to von Neumann’s theorem in the case of Hilbert space of dimension two if equation (1) is assumed only for commuting A and B (1966, reprinted 2004, pp. 2–4).
Having undercut von Neumann’s impossibility proof he turned around and gave a new physically definitive proof of the impossibility of a HV interpretation of QM for Hilbert spaces of dimension three or greater, assuming equation (1) only for commuting A and B (1966, reprinted 2004, pp. 7–8). Actually, this theorem is a corollary of a previous deep and difficult theorem published by Andrew Gleason (1957). (Bell remarked once that he realized he either had to understand Gleason’s theorem or produce a simpler one of his own, and it was easier to do the latter.) It should be noted that essentially the same result as Bell’s was proved independently and published somewhat later by Simon Kochen and Ernst P. Specker (1967).
Bell surprisingly introduced a new consideration which transformed the conceptual significance of all the results previously achieved by himself and others concerning HV: he broadened the conception of HV by arguing for the physical acceptability of contextual theories.
That so much follows from such apparently innocent assumptions leads us to question their innocence. … It was tacitly assumed that measurement of an observable must yield the same value independently of what other measurements may be made simultaneously. … The result of an observation may reasonably depend not only on the state of the system (including hidden variables) but also on the complete disposition of the apparatus.” (1966, p. 451, reprinted 2004, pp. 8–9).
There is a wonderful irony in Bell’s innovation: he rescued the entire program of HV from Bohr’s philosophical objections by insisting along with Bohr on the inseparability of a physical quantity of a system from the apparatus used to measure that quantity. In HV that are contextual (incidentally, not Bell’s own terminology), the value of a quantity A of a physical system S has the functional dependence A(λ, B, C, .…, Z), where λ is a full specification of the HV, and B, C, …, Z is a set of physical quantities of S which are measured together with A, thus constituting the context of the measurement.
Bell reflected on two peculiarities of Bohm’s HV model: that it is contextual and also nonlocal, in that the trajectory assigned by the theory to a particle is instantaneously changed by varying the magnetic field in an arbitrarily distant region. Bell intensively studied variants of Bohm’s model in order to remove this feature, which was undesirable because of its conflict with the locality of special relativity theory, but without success. This failure provided the heuristics for proving that nonlocality is necessary for agreement between HV interpretations and the predictions of QM concerning systems with spatially separated constituents, a proposition now referred to as Bell’s theorem. The theorem marks a rare case of experimental metaphysics in that it allows the transformation of the famous thought experiment presented by Albert Einstein, Boris Podolsky, and Nathan Rosen (1935), known as EPR, and later elaborated on by Bohm (1951), into a definite setup—a setup that led to a series of experiments (the first carried out by Freedman and Clauser in 1972 and improved by Alain Aspect et. al. in 1981). The nonlocality it has exposed was at first considered a paradox, but in recent years, with the development of quantum information science, physicists regard it more as an important resource for outperforming classical information processing.
The original version of Bell’s theorem (1964, reprinted 2004, pp. 15–19) focused on a pair of spin onehalf particles, in the singlet quantum state. A(a ) is the outcome of measurement of σ _{1}a , the component along a (a unit vector in Euclidean space) of the Pauli spin of particle 1 and B(b ) is the analogous quantity for particle 2. The possible values of each of these two quantities are +1 and 1. The expectation value of the product of these two quantities in the quantum state is
where α is the angle between the a and b vectors. The HV expectation value of the same product when ρ is the distribution over the space of HV is
In order to avoid the nonlocality exhibited by the Bohm model Bell assumed that the integrand satisfies the following factorization:
which asserts that A(a ) does not depend on b , a parameter of a quantity measured on the distant particle 2, and likewise B(b ) does not depend on a . Furthermore, the context for particle 1 contains no quantities of particle 1 other than A(a ) itself (with the trivial exceptions of A(a ) and scalar multiples of the identity operator, scalar multiples of A(a ), and linear combinations thereof), since for a spin onehalf particle no others commute with it; and likewise regarding the context of particle 2. Because the spins of particles 1 and 2 are strictly anticorrelated in the singlet state, the HV model can recover the statistical predictions of QM only if
for all values of λ (except a set of measure zero). Straightforward reasoning from equations (3), (4), and (5) yields
Inequality (6) is the pioneering example of formulas called Bell’s inequalities. It is easily checked that if the HV expectation values E(a,b ), etc. agree with the expectation values of QM expressed in equation (2), then inequality (5) is violated for some choices of the unit vectors a ,b , c , for examplea along the x axis, c along the y axis, and b in the xy plane at a 45^{o} angle to both x and y .
Hence no HV theory in which all quantities recognized by QM are assigned definite values, and which furthermore satisfies Bell’s locality condition, can agree with all of the statistical predictions of QM. (Further comments on Bell’s argument are given in Jackiw and Shimony, 2002.)
Bell (1971, reprinted 2004, pp. 37–38) strengthened his original result by considering a much wider class of HV (sometimes called “stochastic theories”), in which λ does not assign a definite value to each selfadjoint operator but rather assigns probabilities to all of the eigenvalues of the operator. He considered theories which are local in the sense that the probability of a specified outcome of a measurement on particle 1 is independent of the choice of a measurement on particle 2 and also of the outcome of this measurement (and likewise with 1 and 2 interchanged). He then demonstrated that no local stochastic HV theory can agree with all of the statistical predictions of QM.
A striking feature of Bell’s theorem is the light it throws on of the thesis of the classical paper of Einstein, Podolsky, and Rosen (1935), hereafter referred to as EPR. These authors had maintained that hidden variables are needed in order to eliminate the apparent nonlocality in certain twoparticle quantum correlations, but Bell showed that nonlocality recurs in any HV theory agreeing with the quantum theory of correlated systems. Numerous experiments have been performed to test Bell’s inequalities and are summarized in essays in the book edited by Bertlmann and Zeilinger (2002) and also in Shimony’s “Bell’s Theorem” (2004). Since these experiments overwhelmingly confirm the predictions of QM and violate Bell’s Inequalities, the program of EPR of postulating HV to rescue relativistic locality from its conflict with QM does not seem to be salvageable.
Several authors (Eberhard, 1978; Ghirardi, Rimini, and Weber, 1980; Page, 1982) have tried to reconcile the apparent nonlocality of QM with the locality of special relativity by demonstrating that the nonlocal causal connections exhibited in certain quantum correlations cannot be used to send superluminal messages. It is not surprising that Bell, who throughout his career had insisted on a nonanthropocentric understanding of QM, rejected this attempt at reconciliation of the two great physical theories.
Do we then have to fall back on “no signaling faster than light” as the expression of the fundamental causal structure of contemporary physics? That is hard for me to accept. For one thing we have lost the idea that correlations can be explained, or at least this idea awaits reformulation. More importantly, the “no signaling…” notion rests on concepts which are desperately vague, or vaguely applicable. (1990b, reprinted 2004, p. 245)
Because of the evidence against local HV one cannot accept EPR’s initially attractive explanation of the measurement process, which would show how the physical part of a measurement can conclude with a definite value of the indexical quantity of the measuring apparatus. What are the alternative explanations of definite measurement results? Bell regrettably died before the flowering of the currently influential “consistent histories” interpretation of QM by Griffiths (2002), Omnès (1999), and GellMann and Hartle (1990). However, it is reasonable to conjecture that he would assess their proposals as he did that of Kurt Gottfried (Bell, 1987, reprinted 2004, pp. 221–222), who claimed that the exact statistical operator ρ of a system plus apparatus can be replaced by a convenient ρ' by dropping interference terms involving pairs of macroscopically different states. Bell granted that for all practical purposes, for which he used the skeptical acronym FAPP, the interference between macroscopically different states is elusive. To go beyond practicality to true theoretical justification Bell clearly felt that an appropriate set of real physical attributes is essential.
… to avoid the vague “microscopic” “macroscopic” distinction—again a shifty split—I think one would be led to introduce variables which have values even on the smallest scale. If the exactness of the Schrödinger equation is maintained, I see this leading towards the picture of de Broglie and Bohm.” (1987, reprinted 2004, pp. 224–225)
But note that this criticism of Gottfried, which plausibly would be directed also to the consistent history theorists, ends with a conditional: “if the exactness of the Schrödinger equation is maintained.” In conjecturing about promising solutions to the measurement problem Bell was strongly attracted to a program in which that exactness is not maintained: that of GianCarlo Ghirardi, Alberto Rimini, and Tullio Weber (1986).
In the GRW project, the standard time evolution of the quantum state is modified on rare occasions and in small amounts by stochastic reductions: the wave function, which is the quantum state in the position representation, is truncated to a region of atomic size. The retention of the spread of the wave function in this small region has the effect of preserving quantum dynamics for atoms and their constituents, with all the richness of phenomenology that has been discovered in the twentieth century. But the truncation has the consequence of making pointer needles and other indexical entities have quite definite values on a human scale. Bell was enthusiastic about this program, even though it has so far not been confirmed experimentally (1987, reprinted 2004, pp. 202–204).
John S. Bell died of a stroke at the age of sixtytwo. He was still at the height of his powers. If physicists come in two types, those who try to read the book of nature and those who try to write it, Bell belonged to the first category. He was conservative when it came to speculative and unconventional suggestions: he would prefer that unexpected contradictions not arise, that ideas flow along clearly delineated channels. But this would not prevent him from establishing what exactly is the case and accepting, albeit reluctantly, even puzzling results. Even in his quantum mechanical investigations, Bell would have preferred to side with the rational and clearly spoken Einstein rather than with the murky pronouncements of Bohr. But once he convinced himself where the truth lies, he would not allow his investigations to be affected by his inclinations, even if he remained disturbed by their outcome. Such a commitment to “truth”—as he saw it—marked John Bell’s activity in science and in life.
Bell was honored fairly early by election to the Royal Society in 1972. In the 1980s many more honors came his way including the Reality Foundation Prize (with John Clauser), the Dirac Medal, the Dannie Heineman Prize for Mathematical Physics and the Hughes Medal of the Royal Society.
BIBLIOGRAPHY
Reprints of Bell’s major papers are given in Mary Bell et al., 1995, and in John Bell, 2004. A full bibliography is contained in Burke and Percival, 1999.
WORKS BY BELL
“Time Reversal in Field Theory.” Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 231 (1955): 479–495.
“Time Reversal in BetaDecay.” Proceedings of the Physical Society of London Section A 70 (1957): 552–553.
“On the EinsteinPodolskyRosen Paradox.” Physics 1 (1964): 195–200. Reprinted in Bell, 2004.
With J. K. Perring. “2π Decay of the K_{2}^{0} Meson.” Physical Review Letters 13 (1964): 348–349.
With J. Steinberger. “Weak Interactions of Kaons.” In Oxford International Conference on Elementary Particles, 19/25 September 1965, Proceedings, pp. 195–222. Chilton, U.K.: Rutherford High Energy Laboratory, 1966.
“On the Problem of Hidden Variables in Quantum Mechanics.” Reviews of Modern Physics 38 (1966): 447–452. Reprinted in Bell, 2004.
“EqualTime Commutator in a Solvable Model.” Nuovo cimento A 47 (1967a): 616–625.
“Current Algebra and Gauge Variance.” Nuovo cimento A 50 (1967b): 129–134.
“Current and Density Algebra and Gauge Invariance.” In Selected Topics in Particle Physics, edited by J. Steinberger. Società italiana di fisica, course 41. New York: Academic Press, 1968.
With D. G. Sutherland. “Current Algebra and ηη→ 3π.” Nuclear Physics B4 (1968): 315–325.
With Roman Jackiw. “A PCAC Puzzle: π^{0} →γγ in the σ Model.” Nuovo cimento della Società italiana di fisica A— Nuclei Particles and Fields 60 (1969): 47–61.
“Introduction to the HiddenVariable Question.” In Foundations of Quantum Mechanics, edited by Bernard d’Espagnat, pp. 171–181. Società italiana di fisica, course 49. New York: Academic Press, 1971. Reprinted in Bell, 2004, pp. 28–39.
With R. Rajaraman. “On Solitons with Half Integral Charge.” Physics Letters B 116 (1982): 151–154.
With R. Rajaraman. “On States, on a Lattice, with HalfIntegral Charge.” Nuclear Physics B 220 (1983): 1–12.
With Jon Magne Leinaas. “The Unruh Effect and Quantum Fluctuations of Electrons in Storage Rings.” Nuclear Physics B284 (1987): 488–508.
“Six Possible Worlds of Quantum Mechanics.” In Possible Worlds in Humanities, Arts, and Sciences: Proceedings of Nobel Symposium 65, edited by Sture Allén. Berlin: W. de Gruyter, 1989. Reprinted in Bell, 2004.
“Against ‘Measurement.’” In SixtyTwo Years of Uncertainty, edited by Arthur I. Miller. New York: Plenum, 1990a. Reprinted in Bell, 2004.
“La Nouvelle Cuisine.” In Between Science and Technology, edited by Andries Sarlemijn and Peter Kroes, pp. 232–248. Amsterdam: Elsevier, 1990b. Reprinted in Bell, 2004.
With Mary Bell, Kurt Gottfried, Martinus Veltman, eds.
Quantum Mechanics, High Energy Physics and Accelerators: Selected Papers of John S. Bell, with Commentary. Singapore: World Scientific, 1995. Represents John Bell’s work in all of his fields of research, including some, like accelerators, which are scantly treated in the present article.
Speakable and Unspeakable in Quantum Mechanics: Collected
Papers on Quantum Philosophy, first ed. Cambridge, U.K.: Cambridge University Press, 1987; revised ed., 2004. The revised edition is a collection of all of Bell’s papers on the foundations of quantum mechanics, except for some near duplicates; it is referred to in the text as “Bell (2004).” The first edition does not contain his late papers “Against ‘Measurement’” and “La Nouvelle Cuisine.”
OTHER SOURCES
Aspect, Alain, Philippe Grangier, and Gérard Roger. “Experimental Tests of Realistic Local Theories via Bell’s Theorem.” Physical Review Letters 47 (1981): 460–463.
Bell, Mary. “John Bell and Accelerator Physics.” Europhysics News 22 (1991): 72.
Bernstein, Jeremy. Quantum Profiles. Princeton, NJ: Princeton University Press, 1991.
Bertlmann, Reinhold A., and Anton Zeilinger, eds. Quantum [Un]speakables: From Bell to Quantum Information. Berlin: Springer, 2002.
Bethe, H. A., and R. Jackiw. Intermediate Quantum mechanics. 3^{rd} ed. Reading, MA: Addison Wesley, 1997.
Bohm, David. Quantum Theory. Englewood Cliffs, NJ: PrenticeHall, 1951.
———. “A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables,” parts I and II. Physical Review85 (1952): 166–179, 180–193.
Bohr, Niels. Atomic Physics and Human Knowledge. New York: Wiley, 1958.
Burke, Philip G., and Ian C. Percival. “John Stewart Bell: 28 July 1928–1 October 1990.” Biographical Memoirs of Fellows of the Royal Society 45 (November 1999): 2–17. Includes a full bibliography.
Eberhard, Philippe. “Bell’s Theorem and Different Concepts of Locality.” Nuovo cimento della Società italiana di fisica BGeneral Physics Relativity Astronomy and Mathematical Physics and Methods 46 (1978): 392–419.
Einstein, Albert, Boris Podolsky, and Nathan Rosen. “Can QuantumMechanical Description of Physical Reality Be Considered Complete?” Physical Review 47 (1935): 777–780.
Freedman, Stuart J., and John F. Clauser. “Experimental Test of Local Hidden Variable Theories.” Physical Review Letters28 (1972): 938–941.
GellMann, Murray. “The Symmetry Group of Vector and AxialVector Currents.” Physics 1 (1964): 63–75.
———, and J. Hartle. “Quantum Mechanics in the Light of Quantum Cosmology.” In Complexity, Entropy, and the Physics of Information, edited by Wojciech Zurek. Redwood City, CA: Addison Wesley, 1990.
Ghirardi, Gian Carlo, A. Rimini, and T. Weber. “General Argument against Superluminal Transmission through the Quantum Mechanical Measurement Process.” Lettere al Nuovo cimento 27 (1980): 293–298.
———, A. Rimini, and T. Weber. “Unified Dynamics for Microscopic and Macroscopic Systems.” Physical Review D34 (1986): 470–491.
Gleason, Andrew. “Measures on the Closed Subspaces of a Hilbert Space.” Journal of Mathematics and Mechanics 6 (1957): 885–893.
Griffiths, Robert B. Consistent Quantum Theory. Cambridge, U.K.: Cambridge University Press, 2002.
Holstein, B. “Anomalies for Pedestrians.” American Journal of Physics 61 (1993): 142–147.
Jackiw, Roman, and A. Shimony. “The Depth and Breadth of John Bell’s Physics.” Physics in Perspective 4 (2002): 78–116.
Kochen, Simon, and Ernst Specker. “The Problem of Hidden Variables in Quantum Mechanics.” Journal of Mathematics and Mechanics 17 (1967): 59–88.
Lüders, Gerhart. “On the Equivalence of Invariance under Time Reversal and under ParticleAntiparticle Conjugation for Relativistic Field Theories.” Matematiskfysiske meddelelser det Kongelige Danske videnskabernes selskab 5 (1954): 1–17.
Omnès, Roland. Quantum Philosophy. Princeton, NJ: Princeton University Press, 1999). Translation by Arturo Sangalli of Philosophie de la science contemporaine. Paris: Gallimard, 1994.
Page, Don. “The EinsteinPodolskyRosen Physical Reality Is Completely Described by Quantum Mechanics.” Physics Letters A 91 (1982): 57–60.
Shimony, Abner. “Bell’s Theorem.” Updated 2004. In Stanford Encyclopedia of Philosophy (Fall 2006 Edition), edited by Edward N. Zalta. Available at http://plato.stanford.edu/archives/fall2006/entries/belltheorem/.
Treiman, Sam, Roman Jackiw, Bruno Zumino, and Edward Witten. Current Algebra and Anomalies. Princeton, NJ: Princeton University Press, 1985.
von Neumann, John. Mathematische Grundlagen der Quantenmechanik. Berlin: Springer, 1932. English translation, Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton University Press, 1955.
Whitaker, Andrew. “John Bell and the Most Profound Discovery of Science.” Physics World(December 1998). Available from http://physicsweb.org/articles/world/11/12/8.
———.“John Bell in Belfast: Education and Early Years.” In Bertlmann and Zeilinger, 2002.
R. Jackiw
A. Shimony
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Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.