The title is catchy, PHOTONS. Anyone who has ever thought about light, any physicist interested in the history of ideas, would immediately be intrigued. After all, photons—whatever they are—are wonderfully mysterious. The author, Professor Klaus Hentschel, is a well-regarded historian of science at the University of Stuttgart. Even so, to write a book about our favorite quantal phantasm is an intrepid gesture, albeit a tad premature, as we are still groping around in the metaphorical dark. Nonetheless, such a book is a welcome and worthy effort, though one destined not to come to a completely happy conclusion.

A photon, like anything else, is the sum of its manifest properties, but beyond that if a photon is some sort of quantum thing, and it certainly seems to be, that thing continues to defy our humble human imagination. “What is a photon?” is a simple enough question, even if as yet there is no satisfactory answer. Nonetheless, Prof. Hentschel has a go at it; not so much as a physicist might, but as a historian would. More than anything else, this book is a history of experiments and conclusions drawn therefrom; as its awkward subtitle proclaims, it is “The History and Mental Models of Light Quanta.”

After an engaging Preface, a brief summary of some of what's to come, we are offered Chapter 1, a bit of historiography. It is there that we get the first inkling that this is actually a book written primarily with historians in mind and methodology as the subtext rather than resembling a popularization intended for a broad audience. Readers of AJP would likely assume, as I assumed, that this book is an exploration of our contemporary 2020 understanding of the nature of light, and the history of ideas optical that got us here; PHOTONS is not quite that. It is structured less to tell a coherent story than to make several scholarly points and in so doing establish the utility of certain historical methodologies. Thus, the unfortunate term “mental model,” which seems to show up a thousand times throughout the book, is apparently professional jargon, albeit off-putting given the contemporary informal meaning of the word “mental” (crazy, insane, wacky, etc.).

Chapter 2 is mostly an erudite exposition of the roles played by two outstanding physicists, M. Planck, the reluctant progenitor of quantization, and the true parent of the photon, the bold young man at the patent office, A. Einstein. This is a remarkably detailed account of the early days of the energy quantum and its gradual metamorphosis into the photon. Anyone already familiar with blackbody radiation and the experimental techniques used to study it in the late 1800s will find this historically focused presentation fascinating. But there are a few technical flubs: on page 15, uν = 8πν2/c3 cannot possibly be energy density (units are wrong), comparing it with the expression for u on page 65. What is that mysterious v in the equation for K on page 15? It turned into a 1 in the expression for K on page 65.

Chapter 3 offers Prof. Hentschel's thesis that there are “twelve semantic layers” of meaning discernible in the historical evolution of the conceptual model of the photon. This chapter is a long, engaging exposition that runs from Newton to QED. It's full of historical tidbits, delightful quotes, and commentaries—in all an excellent read, even if it misses a few salient aspects of what the photon is. Whether you are interested in “semantic layers” or not, the author's vast knowledge of history shines through. But before we further examine those “layers,” it would be a kindness to the uninitiated reader to briefly set the stage vis-à-vis the physics.

The denizens of the atomic domain are all quantum creatures; let's called them quantum particles or better yet, quantum entities. Electrons, protons, neutrons, and the other more exotic subatomic “particles” are all quantum entities possessing the distinguishing property we call mass. The same is true for systems composed of these entities, that is, atoms and molecules. Likewise, photons are massless quantum entities, not that different in many regards from all the other “particles” that are not really particles in any traditional sense. Such “things” display a most important common defining characteristic: They all engage in the baffling process of interference/diffraction.

Since the early 1800s and the independent work of A. Fresnel and T. Young—long before J. Clerk Maxwell and electromagnetic theory—we have “understood” interference as an exclusively wave phenomenon. What kind of wave didn't much matter. Light interferes; light is a wave. And so, we've gone from elastic aether vibrations, to electromagnetic waves, to probability waves, all in the service of interference. A deeper study of the physics of interference would reveal that the classical formulation was rather naïve, even so, if there are interference bands, something must be waving—that's the gospel. Today, both two-slit and three-slit interference are actively being studied at the quantum level.

Obviously, Einstein's seemingly ridiculous (1905) notion that light itself was quantized—that it was composed of blasts of energy localized in space and time, [Lichtquant] “light quanta”—had to be foolishness, after all, corpuscles cannot interfere. Thus, out of an incomparable blend of genius and audacity was conceived what has come to be known as the photon. To put our particle-wave muddle in perspective, consider that researchers have recently (2019) produced interference fringes using beams of molecules, each composed of 2000 atoms (each of mass 25 000 amu). We do well to keep in mind that all quantum entities (from giant molecules to puffs of light) defy imagination. Given that L. de Broglie was right, you, dear reader, also have a wavelength. Back to PHOTONS.

Chapter 3, and much of the book, focuses on Hentschel's “twelve semantic layers.” Here, compressed, are those “layers,” with a bit of 21st-century illumination added for AJP readers not familiar with modern-day optical physics.

  • (1)

    “Light is corpuscular, strongly localized.” Developing that piece of the puzzle, Hentschel does a fine job of tracing corpuscular theory from Newton to Einstein. The discourse is replete with period quotations and delightful historical details. There is even mention of the effect of the force of gravity on light. Alas, “Newton's fundamental equation” of gravity on page 42 should not have been given as an equation attributed to him—Sir Isaac never wrote such an expression—the G was essentially a later gift from the Hon. H. Cavendish, one of the richest and most peculiar men in England. Hentschel's mid-page analysis of the “gravitational forces acting on particles of light” is, ironically, an anachronism.

    We know now that all quantum entities can appear in some regards to be corpuscular. Just think of all those lovely cloud chamber tracks, R. Millikan's oil drop experiments, and those giant molecules.

  • (2)

    “Light propagates at a finite but very high velocity.” Here, the author of PHOTONS briefly talks about the work done by, among others, O. Rømer, H. Fizeau, and A. Fresnel. He touches on special relativity but, surprisingly, misses an opportunity; in a treatise on light, we should learn that individual photons, being massless, only exist at c. That's an extremely important characteristic. Light, en masse, travels through material media with velocities (i.e., phase velocities) less than or even greater than c. The propagation of light at speeds other than c arises primarily from phase shifts resulting from non-resonant (ground-state) scattering by the atoms in the medium. Nevertheless, individual photons only exist at c. Photons are therefore timeless. Although we can catch light quanta that have been on the wing since the beginning of the Universe—nearly 13.8 thousand million years ago—photons themselves do not experience time. They leave now and arrive now. None of these fascinating insights are in the book.

  • (3)

    “Light is emitted and absorbed by matter.” First, it should be pointed out that along with E. Schrödinger, M. Born, L. de Broglie, W. Pauli, and countless others, we take light to be matter. This “semantic layer” could have better elucidated the nature of photon emission and absorption had it actually discussed the basic atomic processes involved.

  • (4)

    “Light transmits momentum p, therefore, radiation pressure.” Beginning with J. Kepler's astute observation concerning cometary tails and ending (1903) with a mention of the important experiments of E. Nichols and G. Hull, this “layer” is interesting but somewhat lacking in depth. There is no development of Maxwell's electromagnetic radiation pressure or of the fact that p = h/λ, which only shows up much later.

  • (5)

    “Light transfers energy E.” Here, there are a few little notational issues. The Poynting vector is a vector and so (page 54) it should be S = E × B and not S = E × B. Because the energy-momentum 4-vector contains the 3-space momentum vector p, it is appropriately written as [E/c, p] and not (page 57) as (p, mc2), wherein p is a scalar. Note that in its ability to transfer energy, a photon is different from the other quantum entities having mass; it has only KE, whereas they also possess rest energy, E0. Moreover, once a photon transfers energy, it vanishes. That's usually not the case with interacting massive quantum entities, other than in the inverse process of pair annihilation.

  • (6)

    Energy is correlated with its frequency ν: Eν.” Here, author Hentschel gives us a first-rate account of the history of the discoveries from H. Hertz to R. Millikan, but the physics is often just a little off, perhaps in translation. For example, the section begins: “The discoverer of the waves that later carried his name, Heinrich Hertz (1857–1894) observed in 1887 in his laboratory that sparks transmitted by radio waves got smaller when he held a pane of glass in the spark gap in front of the receiver …” Of course, electromagnetic (EM) waves do not “transmit” sparks. And the glass pane was placed in front of the spark gap, not in it. By the way, the first person to deliberately generate radio waves (1879–1886) was the Anglo-American, D. Hughes (1831–1900). Nonetheless, Hertz deserves credit for his far more extensive study of the behavior of EM waves.

    Because the concept of color comes up several times in this book, it should be noted that color is the human visual response to the energy of a small range, about an octave, of the electromagnetic photon spectrum. Frequency and wavelength are physical properties independent of human vision. Because they are conjugate quantities, in vacuum one can speak about the ranges of either frequency or wavelength that correspond to the colors perceived by a typical human vision system. Unfortunately, it has become commonplace to conflate color and wavelength. Beyond that, it is actually the energy of the photons (their frequencies, not their wavelengths) that determine the various colors humans see. After all, a red bathing suit is seen to be red in and out of the pool, even though the reflected wavelength is reduced (by 25%) in water—the frequency is constant and the energies of the emitted photons do not change in different linear media.

  • (7)

    Energy is quantized E = hν.” Prof. Hentschel addresses the widely held misconception that Planck was the person who quantized blackbody radiation. What Planck did in order to explain the increasingly more extensive data being produced was to quantize the energy of the oscillators emitting the radiation and not the radiation. That conservative approach—and Planck though brilliant, was conservative—was a little like dropping different sized (quantized) rocks in a pond, you still get continuous waves. The bold leap of proposing that the radiation itself comes in blasts, in light quanta, is due to young Einstein.

    Radiant energy emitted from a bound system is quantized because the energy that can be retained by such a system is itself quantized at specific values. Regarding the “oscillators,” Planck, though skeptical of his own efforts, was right. In some respects, the story actually goes from Planck to Einstein to Bohr and back to Planck. In general, however, photons (like cannonballs) can have any frequency and hence any corresponding energy. Just think of the radiation from unbound accelerating charged particles. That point could have been made somewhere in this book. Incidentally, the “vee” in the section title (page 64) should be a Greek “nu.”

  • (8)

    “The wave-particle duality applies to light.” This “semantic layer” arises because many, if not most, scholars have long believed that observable physical entities can come in only two distinct manifestations, particle or wave. And yet quantum entities are able to present both aspects independently, depending on how we choose to observe them. Whether this is simply a failure of imagination or a failure of technique remains to be seen. More likely, quantum entities are neither particles nor waves.

    Incidentally, sound waves, waves on a string, water waves, and any other matter waves you might think of are all self-sustaining disturbances of an existing medium; all are therefore basically particulate—oscillating “particles.” The notion that EM waves in vacuum, light waves, which do not propagate through an already existing electromagnetic substratum, might also be particulate is not so shocking. J. J. Thomson had implied as much, and Einstein was similarly uncomfortable with the idea of EM energy spreading out uniformly over expanding wavefronts.

    On page 67 of PHOTONS, we read “the momentum ½mv2 …” of course ½mv2 is not momentum, it's kinetic energy. Having provided very few equations thus far, on page 68, Prof. Hentschel presents us with a formidable expression for radiation pressure taken from Einstein's 1909 paper, “On the Present State of the Radiation Problem.” There Einstein was kind enough to tell his readers that the symbol f stood for a surface area, whereas Hentschel leaves it to be surmised. Unfortunately, the next equation for “the energy fluctuation” (page 68) is just a tad wrong—the c2 should obviously be c3. That aside, Prof. Hentschel provides another engaging section rich in historical details.

  • (9)

    “Light exhibits spontaneous emission and absorption.” Here, I would quibble; it isn't light exhibiting emission, it's an atom “stimulated” by light that does the emitting. This “semantic layer” treats Einstein's now famous A and B coefficients and his brilliant hypothesis of stimulated emission that made the laser possible.

  • (10)

    “Its quanta carry angular momentum (spin).” That photons possess angular momentum, spin, is a vital descriptor of what they are. This section's historical exposition of that attribute is again delightfully detailed and most enjoyable; who knew that Pauli was Mach's godson? Be that as it may, (page 80) I read the physics dealing with the electron's magnetic moment four times and still could not make much sense of it. Without first defining symbols, the equations become a meaningless puzzlement. We are told that M is the magnetic moment of the electron, and then given an equation for M in terms of S, but what S is, is left to the imagination. Perhaps it's the scalar magnitude of the spin angular momentum S. And then, just to confuse the issue, in the same paragraph we find that M is now “the anomalous momentum,” whatever that is. In the next paragraph without any explanation of what μ is supposed to be, we are presented with μ = ev/mr. Usually, μ is the symbol for magnetic moment (here at times it was M), but on the previous page, the only place a definition was offered, μ was a quantum number. So, having tried, failed, and no longer caring, I moved on.

    The very end of the section contains a charming German to English translation goof. The discussion there was about R. Beth's 1935 experiment to measure the angular momentum of circularly polarized light. He used an extremely sensitive (quartz) torsion pendulum to measure the angular momentum imparted to it by a beam of circularly polarized light and hence by the aligned spins of its constituent photons. Hentschel's all-too-brief discussion talked about the “regular” and “irregular” beams. Someone familiar with the theory of birefringent materials would recognize that those words were supposed to be the technical designations “ordinary” and “extraordinary;” old-fashioned terminology, but still used.

  • (11a)

    “Its quanta are indistinguishable, with equal energy and spin.”

  • (11b)

    “Bose-Einstein statistics applies to quanta.” As you might guess, these “semantic layers” treat the statistical behavior of the photon, and how the analysis thereof developed from attempts at understanding Planck's theory of blackbody radiation in terms of light quanta. Prof. Hentschel again brings to bear his extensive historical knowledge to produce an account brimming with little-known facts and the contributions of lesser-known researchers. He ends the segment having made the case for calling the technique Planck-Natanson-Bose-Einstein statistics. Because photons—oscillating puffs of electromagnetic energy—are spin-1 bosons, they can congregate in immense numbers to propagate altogether in the guise of continuous electromagnetic waves. That too could have been discussed and wasn't.

  • (12)

    “Photons are virtual exchange particles of quantum electrodynamics.” It is in the exposition of this last of the author's “semantic layers” that I find myself at odds with the implications of his presentation. This is a section about quantum electrodynamics (QED), which is the confluence of quantum mechanics and special relativity. QED hypothesizes a theoretical construct, a calculational device, which, for want of a better name, is called a virtual photon. The idea is that matter interacts, and it does so over an intervening distance. In the case of the electromagnetic interaction, it's easy enough to suppose that the interaction is mediated by the exchange of some sort of massless quantized electromagnetic essence; it is massless because it appears to have an infinite range. The basic idea is centuries old; Kepler proposed a similar mechanism to explain action-at-a-distance.

Since we know that light is photonic, light is electromagnetic, and photons are massless, let's call this massless theoretical construct a virtual photon. The hypothetical characteristics such virtual photons must have are not those of actual photons, the “real” quantum entities we can generate and measure. Virtual photons cannot be observed/measured and do not need to comply with the basic laws of physics.

Physics has always had its purely theoretical undetectable conceptual devices: gluons, wavefunctions, aether, Hilbert space, and so forth. As long as QED works, and it works magnificently well, we keep it; it explains and it predicts, not everything and not perfectly, but well enough; it is impressive. That does not mean for a moment, as Prof. Hentschel seems to imply, that photons are virtual exchange particles; no, no, virtual photons are virtual exchange particles. When he says, “The photon was thus re-interpreted,” he goes too far. Better put, the photon gave rise to a lookalike phantom, the virtual photon. Hentschel gets closer to it much later in the book (page 144).

Chapter 3 ends (page 92) discussing QED, with this remark, “This also explains why we are still burdened with the problem of what exactly photons are …” Well, not quite, to be honest, we're still burdened with what exactly electrons are, and protons, and quarks, and neutrons, and neutrinos, and even more so, with what positrons are. Like photons, they all have wavelengths. How is it that an electron and a positron, each with mass, can vanish into a flash of photons, matter sans mass? Prof. Hentschel's statement about being burdened with doubt continues “…or whether they [photons] really exist …” I suspect he could not find many, if any, physicists working in quantum optics who believe that the photon is anything other than an objective physical reality. By the way, I'd bet no particle physicist alive questions whether or not photons, whatever they are, are real.

Today, there is a multi-billion dollar world-wide photonics industry. You can readily buy a variety of single-photon detectors (e.g., superconducting nanowire, and/or avalanche photodiode devices). Additionally, there are several single-photon generating mechanisms in common use: atomic and ionic sources, quantum dots, spontaneous parametric down-conversion (SPDC), four-wave mixing, and diamond nitrogen-vacancy (NV) centers, each with its own virtues.

Chapter 4, Early Mental Models, stays with the evolution of the concept of photons. Here again, it is evident that writing about history is where Prof. Hentschel excels. Filled with historical gems, the chapter is a rich and enjoyable read. For example, 1921, Einstein confiding to his friend Ehrenfest, “Pondering about light quanta is driving me crazy.” We learn details about J. Stark's rabid Nazism and the response to it by his fellow physicists. J. J. Thomson (1903) described X-rays as “needle rays” because of their rectilinear propagation and the localized way they interacted with atoms in a cloud chamber. Those observations might well have influenced Einstein. In any event, Prof. Hentschel provides a rare account of J. J.'s “mental model,” (his friends called him J. J.). More “mental models”—those of Bragg, Debye, Sommerfeld, von Laue, Schrödinger, and Lewis—fill out the chapter. An aside: a wave packet propagating in material media spreads out due to dispersion of the component waves, in contemporary usage the packet doesn't “dispel,” page 117. My apologies to the otherwise most able translator.

Chapter 5, Early Reception of the Light Quantum, considers both the Compton effect (1922–1923) and Millikan's work on the photoelectric effect. It includes the wonderfully ironic quotation of Millikan's wherein he speaks of his ten-year struggle to prove Einstein's equation wrong, only to prove it right. By the way, on page 127, in the Compton equation, that β should better be a θ to agree with Fig. 5.1 and the text on that page.

Chapter 6, Light Quanta Reflected in Textbooks and Science Teaching. With only two noteworthy exceptions, Prof. Hentschel doesn't think much of our modern introductory textbooks. Had PHOTONS been a traditional physics book or even a popularization, this chapter would not have been included. Clearly, this is not a text laser-focused on elucidating the physical nature of light.

Chapter 7, TheLight Quantum’ as a Conceptual Blend, starts off with a piece of historical methodology, namely, the concept of “conceptual blending.” Prof. Hentschel then takes the notion of objects “falling into a hole” and blends it with the fact that the expression for classical gravitational force FGm/r2 blows up at r =0 and, you guessed it, comes up with the “black hole.” This he accomplishes, amazingly, without benefit of General Relativity, without the Schwarzschild metric, and without recognizing the singularity in the field at rs = 2GM/c2.

At the end of the chapter after doing some blending with QED, PHOTONS's author concludes “QED photons are partially virtualized, whereas they really exist in propagating light.” I agree with the “really exist” piece, but I have no idea what a “partially virtualized” photon might be. Not to worry, Prof. Hentschel will later on change his mind about the “really exist” part as well.

Chapter 8, Quantum Experiments with Photons Since 1945. This chapter, if done in a way that was self-explanatory would have been an entire book, an entire fat book, on quantum optics. It covers, albeit often much too superficially, a wide range of subtle topics: the Hanbury Brown and Twiss experiment; bunching and antibunching of photons; single-photon interference; Aspect's experiments; Wheeler's delayed choice suggestion; the Hong-Ou-Mandel dip; entanglement; and photon-photon scattering. A reader already well versed in quantum optics would find a lot of interesting historical tidbits strewn about. For example, the discussion of G. Taylor's 1909 low-light level experiment was the most informative I've seen in print, although the caption for Fig. 8.3 is utterly misleading.

There are some minor technical issues, for instance, “the probability of finding light at a particular location is directly proportional to the square of the light intensity at that location”—of course, it is NOT! Classically, it's proportional to the square of the amplitude of the electric field, or if you like, the square of the probability amplitude. In any event, this chapter would serve beautifully as enrichment for a reader studying the physics elsewhere.

Chapter 9, What is Today's Mental Model of the Photon? After 169 pages, Prof. Hentschel has his doubts, reminding us that the brilliant theoretician J. A. Wheeler called the photon an ephemeral “smoky dragon” (1979). We are told that another author asserted, “photons are more like coefficients in a Fourier's series” (1983). Nobel laureate W. Lamb was more unequivocal, insisting, “there is no such thing as a photon” (1984). That was not an altogether unreasonable view thirty or forty years ago. The existence of photons is “murky” claimed a teacher in the AJP (1996). And Hentschel chimed in, “photons are man-made theoretical constructs that do not describe real particles.” Are historians reporting “facts” supposed to offer opinions on such matters? In any event, the best comment of all came from 2005 Nobel laureate R. Glauber, “I don't know anything about photons, but I know one when I see one.”

Chapter 10, Summary. The book ends, not with a bang, not with any conclusions about photons, not even with the conclusion that it is still too soon to conclude. Nor does it end with a list of photon properties, which I rather hoped it would. It ends simply with a recap of Hentschel's six very scholarly historiographical mental models.

If the tenacious AJP reader, who has made it this far through the clutter of mental models, wishes to learn what we do know of the “smoky dragon” he/she will have to go elsewhere other than PHOTONS. Still, “semantic layers” aside, Prof. Hentschel has skillfully filled his pages with so much engaging history that I will surely go back to my copy again and again.

Professor Eugene Hecht (Gene) is the author of ten books, seven on physics and three on the American ceramic artist G. E. Ohr. Among his physics books is the widely used text Optics, 5th ed., published by Addison-Wesley. His professional interests are the elucidation of the foundational concepts of physics and the history of ideas therein. He has written numerous articles on energy, mass, time, relativity theory, Einstein, Newton, and Kepler, variously for AJP, TPT, and EJP. He spends much of his time teaching, studying physics, and training for his sixth-degree black belt in Tae Kwan Do.