Fields are believed to be the fundamental constituents of the universe, particles being excitations, or quanta, of fields. There are quantum fluctuations of fields even in a space with no particles. The photon number, for example, can be zero while the electric and magnetic fields fluctuate about their zero averages. The most important effect, historically, of these vacuum field fluctuations was discovered in the spectrum of the hydrogen atom. The energy levels of an electron in a Coulomb potential can be calculated exactly, but they do not exactly reproduce the observed hydrogen spectrum because the Coulomb interaction is not the whole story: the electron also interacts with the vacuum field. Experiments in the late 1940s revealed that the 2s1/2 and 2p1/2 levels of the hydrogen atom, which should be equal according to the solution of the Dirac equation for the Coulomb potential alone, differ by about 1058 MHz, the famous Lamb shift.

Another effect attributable to vacuum fields was predicted at around the same time by Hendrik Casimir. In a vacuum, there is energy associated with the field fluctuations, much like the zero-point energy of the ground state of the simple harmonic oscillator in quantum mechanics. Casimir considered the change in this energy caused by two perfectly conducting, uncharged parallel plates. When the plates are separated by a finite distance only certain field modes are possible, unlike when the plates are infinitely far apart. The vacuum energy in both cases is infinite, but Casimir derived a finite value for the energy difference that implied an attractive force between the plates. Experiments many years later verified Casimir's prediction, and recently a “dynamical Casimir effect,” in which photons are created when one of the plates is very rapidly displaced, has been observed.

The possibility of creating particles out of a vacuum has been known for a long time. It was shown by Sauter in 1931 that electron-positron pair production can occur in a uniform electric field. This “sparking of the vacuum,” or Schwinger pair production, requires field strengths on the order of 1018 V/m and for this reason has not yet been directly observed. But other phenomena in which particles are created by interactions with vacuum fields have been observed in high-intensity laser experiments, for instance. Feynman diagrams suggest that the fluctuating vacuum can be imagined to consist of virtual particle-antiparticle pairs that are continually created and annihilated. Schwinger pair production is often interpreted heuristically as a pulling apart by the field of virtual electron-positron pairs. Hawking radiation has been described as a result of virtual particles near the event horizon of a black hole falling in before annihilation can occur, their antiparticles then being free to escape as radiation.

Weatherall writes very readably, and without any equations or mystifying jargon, about the different conceptions of empty space in Newtonian physics, relativity, and quantum field theory, especially as they relate to what it means for there to be something rather than nothing. He wastes few words on the recent, heated debates about whether “the physics of nothing” can answer the age-old question of why there is something rather than nothing. Although his book is primarily a very gentle introduction to the physics of the vacuum and how our understanding of it has evolved, he includes many interesting notes and references to the research literature, and in an epilogue he explains the difficulties involved in constructing a quantum theory of gravity and touches briefly on the string landscape and other speculations.

Newton broke with Aristotelian ideas that motion could only result from pushing and pulling by something, and therefore was not possible in a vacuum, and with ideas about the structure of space that led Descartes to imagine a “plenum” filling all of space. Newton's laws of motion described absolute, not relative motion, and he believed that absolute motion implied absolute space (and time), independent of any observer or point of reference, and with no plenum. In his famous rotating bucket experiment, he interpreted the concave shape of the water surface when there was no relative motion between the water and the bucket as a consequence of motion with respect to absolute space. Leibniz disagreed. His metaphysics led him to regard all motion as relational and to believe that there could be no absolute space.

After revisiting these old ideas about the structure of space, Weatherall describes work by Howard Stein to the effect that Newton erred in thinking that his laws required absolute space, and that in his bucket experiment Newton only showed “that there are physical consequences to a body's accelerating.” Lacking both first-hand knowledge of the Principia and familiarity with Stein's work, I can say nothing further about that. But I was somewhat surprised that nothing was said about Mach's rejection of Newtonian absolute space and his influential conjecture that the curvature of the water surface in the bucket experiment is due to rotation with respect to “the mass of the earth and the other celestial bodies.” (Mach's principle, that local inertial frames are determined by such large-scale distributions of mass, was taken seriously by Einstein.)

The discussions of the aether, Maxwell's electromagnetism, special relativity, and Minkowski space-time, though covering territory very familiar to readers of this journal, are brisk and entertaining. Here and throughout the book, there are anecdotes about some of the characters in the story, the main character here being, of course, Einstein. Most relevant to “the physics of nothing” are the discussions about vacuum solutions of Einstein's field equation relating the curvature of space-time to the energy-momentum tensor. De Sitter found that the equation allows space-time curvature even in the absence of any “stuff.” Weatherall explains why Einstein “hated” de Sitter's solution, and how he “ultimately accepted the possibility of empty space-times with complex structure.” As for the gravitational waves he discovered with his field equation, Einstein had periodic misgivings. At one point, he and Nathan Rosen submitted a paper to the Physical Review in which they concluded that gravitational waves do not exist. When the paper was rejected after a negative report by a referee, Einstein wrote an angry letter to the editor, saying he and Rosen had submitted their paper “for publication, not for review.” Howard Robertson, evidently without divulging that he had been the referee, later convinced Einstein that the arguments in the paper were incorrect, and in a revised paper Einstein and Rosen argued that gravitational waves were possible. Weatherall writes that “It is not clear [Einstein] ever reached a stable view on the matter,” and concludes the chapter with a paragraph on the recent confirmation of the existence of gravitational waves.

Perhaps, as John Wheeler said, “No point is more central than this, that empty space is not empty.” Empty space is certainly not “empty” in quantum field theory, which is generally thought, with good reason, to be one of the most successful theories in the history of science. The third and last chapter of this book focuses mainly on the development of the most accurately tested quantum field theory, quantum electrodynamics, beginning with some of the most “non-classical” features of quantum theory itself. The contributions of Dirac, Jordan, Feynman, Schwinger, and others are described, along with remarks about their personalities and opinions about the mathematical foundations of quantum field theory and its handling of infinities, which Dirac felt was “just not sensible mathematics.” Feynman, however, is said to have “dismissed worries about the mathematical rigor” of the theory. (I have a copy of a letter from Feynman to Wheeler, dated 19 May 1966, that reads, in its entirety, “Dear John, I am not interested in what todays' mathematicians find interesting. Kind regards. Sincerely yours, Richard P. Feynman,” signed “Dick.”) The importance of the first accurate estimate of the Lamb shift by Hans Bethe is appropriately emphasized, but its characterization as the first calculation of an observable consequence of vacuum polarization is inaccurate. Bethe's result could be interpreted as an effect of the fluctuating vacuum field on the atom, or the emission and absorption of virtual photons, but he did not account for vacuum polarization, which was later found by others to contribute only about 27 MHz to the Lamb shift in hydrogen. I was somewhat surprised by the absence in the text of any discussion of the Casimir effect, which is arguably the most palpable effect attributable to the vacuum electromagnetic field and one that has been of particular interest in recent years. (Actually, the Casimir force can be explained, like Bethe's result for the Lamb shift, without explicit reference to the vacuum field. But I digress.)

The vacuum has recently been the subject of quite a few books. This one might be the most congenial to readers who are interested in “the physics of nothing” as it has come to be understood, but not in any of its more mathematical aspects. Its main message is that the physics of the vacuum is strange but fundamental, and it makes a good case for that.

Peter W. Milonni is a Fellow of the Los Alamos National Laboratory and a Research Professor of Physics at the University of Rochester. He has worked in areas relating to quantum fluctuations of electromagnetic fields.