For other uses see Photon (disambiguation).
| Photon | |
| Composition: | Elementary particle |
| Family: | Boson |
| Group: | Gauge boson |
| Interaction: | Electromagnetic |
| Theorized: | Albert Einstein |
| Symbol: | \gamma h\nu |
| Mass: | 0 |
| Mean Lifetime: | Stable |
| Electric Charge: | 0 |
| Spin: | 1[1] |
In physics, the photon is an elementary particle, the quantum of the electromagnetic field and the basic unit of light and all other forms of electromagnetic radiation.It is also the force carrier for the electromagnetic force.This force's easily visible human-scale effects and applications, from sunlight to radiotelephones, are because the photon has no mass and thus can produce interactions at long distances. Like all elementary particles, the photon is governed by quantum mechanics and so exhibits wave-particle duality: that is, it exhibits both wave and particle properties.For example, a single photon may undergo refraction by a lens or exhibit wave interference, but also act as a particle giving a definite result when its location is measured.
The modern concept of the photon was developed gradually by Albert Einstein to explain experimental observations that did not fit the classical wave model of light. In particular, the photon model accounted for the frequency dependence of light's energy, and explained the ability of matter and radiation to be in thermal equilibrium. It also accounted for anomalous observations, including the properties of black body radiation, that other physicists, most notably Max Planck, had sought to explain using semiclassical models, in which light is still described by Maxwell's equations, but the material objects that emit and absorb light are quantized. Although these semiclassical models contributed to the development of quantum mechanics, further experiments proved Einstein's hypothesis that light itself is quantized; the quanta of light are photons.
In the modern Standard Model of particle physics, photons are described as a necessary consequence of physical laws having a certain symmetry at every point in spacetime. The intrinsic properties of photons, such as charge, mass and spin, are determined by the properties of this gauge symmetry.
The photon concept has led to momentous advances in experimental and theoretical physics, such as lasers, Bose–Einstein condensation, quantum field theory, and the probabilistic interpretation of quantum mechanics. It has been applied to photochemistry, high-resolution microscopy, and measurements of molecular distances. Recently, photons have been studied as elements of quantum computers and for sophisticated applications in optical communication such as quantum cryptography.
In 1900 Max Planck was working on black-body radiation and suggested that the energy in electromagnetic waves could only be released in "packets" of energy, he called these quanta (singular quantum). Later, in 1905 Albert Einstein[2] went further by suggesting that EM waves could only exist in these discrete wave-packets. He called such a wave-packet the light quantum (German: das Lichtquant). The name photon derives from the Greek word for light, (transliterated phôs), and was coined in 1926 by the physical chemist Gilbert N. Lewis, who published a speculative theory[3] in which photons were "uncreatable and indestructible". Although Lewis' theory was never accepted—being contradicted by many experiments—his new name, photon, was adopted immediately by most physicists. Isaac Asimov credits Arthur Compton with defining quanta of energy as photons in 1927.[4] [5]
In physics, a photon is usually denoted by the symbol γ (the Greek letter gamma). This symbol for the photon probably derives from gamma rays, which were discovered and named in 1900 by Villard[6] [7] and shown to be a form of electromagnetic radiation in 1914 by Ernest Rutherford and Edward Andrade.[8] In chemistry and optical engineering, photons are usually symbolized by hν, the energy of a photon, where h is Planck's constant and the Greek letter ν (nu) is the photon's frequency. Much less commonly, the photon can be symbolized by hf, where its frequency is denoted by f.
See also: Special relativity.
The photon is massless,[9] has no electric charge,[10] and does not decay spontaneously in empty space. A photon has two possible polarization states and is described by exactly three continuous parameters: the components of its wave vector, which determine its wavelength λ and its direction of propagation. The photon is the gauge boson for electromagnetism,[11] and therefore all other quantum numbers—such as lepton number, baryon number, and flavour quantum numbers are zero.[12]
Photons are emitted in many natural processes. For example, when a charge is accelerated it emits synchrotron radiation. During a molecular, atomic or nuclear transition to a lower energy level, photons of various energy will be emitted, from infrared light to gamma rays. A photon can also be emitted when a particle and its corresponding antiparticle are annihilated (see Electron-positron annihilation for an example).
In empty space, the photon moves at c (the speed of light) and its energy and momentum are related by , where p is the magnitude of the momentum vector p. For comparison, the corresponding equation for particles with a mass m is:[13]
E2=c2p2+m2c4.
The energy and momentum of a photon depend only on its frequency ν or equivalently, its wavelength λ:
E=\hbar\omega=h\nu=
| hc | |
| λ |
p=\hbark,
where k is the wave vector (with the wave number k = 2π/λ as its magnitude),
\omega=2\pi\nu
\hbar=h/2\pi
Notice that p points in the direction of the photon's propagation. Consequently, the magnitude of the momentum is
p=\hbark=
| h\nu | |
| c |
=
| h | |
| λ |
.
The photon also carries spin angular momentum that does not depend on its frequency.[15] The magnitude of its spin is
\sqrt{2}\hbar
\pm\hbar
To illustrate the significance of these formulae, the annihilation of a particle with its antiparticle must result in the creation of at least two photons for the following reason. In the center of mass frame, the colliding antiparticles have no net momentum, where as a single photon always has momentum (since it is determined, as we have seen, only by the photon's frequency or wavelength - which cannot be zero). Hence, conservation of momentum requires that at least two photons are created, with zero net momentum. The energy of the two photons—or, equivalently, their frequency—may be determined from conservation of four-momentum. Seen another way, the photon can be considered as its own antiparticle. The reverse process, pair production, is the dominant mechanism by which high-energy photons such as gamma rays lose energy while passing through matter.[17]
The classical formulae for the energy and momentum of electromagnetic radiation can be re-expressed in terms of photon events. For example, the pressure of electromagnetic radiation on an object derives from the transfer of photon momentum per unit time and unit area to that object, since pressure is force per unit area and force is the change in momentum per unit time.[18]
See main article: Light. In most theories up to the eighteenth century, light was pictured as being made up of particles. One of the earliest particle theories was described in the Book of Optics (1021) by Alhazen, who held light rays to be streams of minute particles that "lack all sensible qualities except energy." Since particle models cannot easily account for the refraction, diffraction and birefringence of light, wave theories of light were proposed by René Descartes (1637),[19] Robert Hooke (1665),[20] and Christian Huygens (1678);[21] however, particle models remained dominant, chiefly due to the influence of Isaac Newton.[22] In the early nineteenth century, Thomas Young and August Fresnel clearly demonstrated the interference and diffraction of light and by 1850 wave models were generally accepted.[23] In 1865, James Clerk Maxwell's prediction[24] that light was an electromagnetic wave—which was confirmed experimentally in 1888 by Heinrich Hertz's detection of radio waves[25] —seemed to be the final blow to particle models of light.
The Maxwell wave theory, however, does not account for all properties of light. The Maxwell theory predicts that the energy of a light wave depends only on its intensity, not on its frequency; nevertheless, several independent types of experiments show that the energy imparted by light to atoms depends only on the light's frequency, not on its intensity. For example, some chemical reactions are provoked only by light of frequency higher than a certain threshold; light of frequency lower than the threshold, no matter how intense, does not initiate the reaction. Similarly, electrons can be ejected from a metal plate by shining light of sufficiently high frequency on it (the photoelectric effect); the energy of the ejected electron is related only to the light's frequency, not to its intensity.[26]
At the same time, investigations of blackbody radiation carried out over four decades (1860–1900) by various researchers[27] culminated in Max Planck's hypothesis[28] [29] that the energy of any system that absorbs or emits electromagnetic radiation of frequency
\nu
E=h\nu
Since the Maxwell theory of light allows for all possible energies of electromagnetic radiation, most physicists assumed initially that the energy quantization resulted from some unknown constraint on the matter that absorbs or emits the radiation. In 1905, Einstein was the first to propose that energy quantization was a property of electromagnetic radiation itself.[2] Although he accepted the validity of Maxwell's theory, Einstein pointed out that many anomalous experiments could be explained if the energy of a Maxwellian light wave were localized into point-like quanta that move independently of one another, even if the wave itself is spread continuously over space.[2] In 1909[31] and 1916,[32] Einstein showed that, if Planck's law of black-body radiation is accepted, the energy quanta must also carry momentum
p=h/λ
Einstein's 1905 predictions were verified experimentally in several ways in the first two decades of the 20th century, as recounted in Robert Millikan's Nobel lecture.[35] However, before Compton's experiment[33] showing that photons carried momentum proportional to their frequency (1922), most physicists were reluctant to believe that electromagnetic radiation itself might be particulate. (See, for example, the Nobel lectures of Wien,[27] Planck[29] and Millikan.[35] ). Instead, there was a widespread belief that energy quantization resulted from some unknown constraint on the matter that absorbs or emits radiation. Attitudes changed over gradually. In part, the change can be traced to experiments such as Compton scattering, where it was much more difficult not to ascribe quantization to light itself to explain the observed results.
Even after Compton's experiment, Bohr, Hendrik Kramers and John Slater made one last attempt to preserve the Maxwellian continuous electromagnetic field model of light, the so-called BKS model.[36] To account for the then-available data, two drastic hypotheses had to be made:
However, refined Compton experiments showed that energy-momentum is conserved extraordinarily well in elementary processes; and also that the jolting of the electron and the generation of a new photon in Compton scattering obey causality to within 10 ps. Accordingly, Bohr and his co-workers gave their model "as honorable a funeral as possible".[34] Nevertheless, the failures of the BKS model inspired Werner Heisenberg in his development of matrix mechanics.[37]
A few physicists persisted[38] in developing semiclassical models in which electromagnetic radiation is not quantized, but matter appears to obey the laws of quantum mechanics. Although the evidence for photons from chemical and physical experiments was overwhelming by the 1970s, this evidence could not be considered as absolutely definitive; since it relied on the interaction of light with matter, a sufficiently complicated theory of matter could in principle account for the evidence. Nevertheless, all semiclassical theories were refuted definitively in the 1970s and 1980s by photon-correlation experiments.[39] Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven.
See also: Wave–particle duality, Squeezed coherent state and Uncertainty principle.
Photons, like all quantum objects, exhibit both wave-like and particle-like properties. Their dual wave–particle nature can be difficult to visualize. The photon displays clearly wave-like phenomena such as diffraction and interference on the length scale of its wavelength. For example, a single photon passing through a double-slit experiment lands on the screen with a probability distribution given by its interference pattern determined by Maxwell's equations.[40] However, experiments confirm that the photon is not a short pulse of electromagnetic radiation; it does not spread out as it propagates, nor does it divide when it encounters a beam splitter. Rather, the photon seems to be a point-like particle since it is absorbed or emitted as a whole by arbitrarily small systems, systems much smaller than its wavelength, such as an atomic nucleus (≈10–15 m across) or even the point-like electron. Nevertheless, the photon is not a point-like particle whose trajectory is shaped probabilistically by the electromagnetic field, as conceived by Einstein and others; that hypothesis was also refuted by the photon-correlation experiments cited above. According to our present understanding, the electromagnetic field itself is produced by photons, which in turn result from a local gauge symmetry and the laws of quantum field theory (see the Second quantization and Gauge boson sections below).
A key element of quantum mechanics is Heisenberg's uncertainty principle, which forbids the simultaneous measurement of the position and momentum of a particle along the same direction. Remarkably, the uncertainty principle for charged, material particles requires the quantization of light into photons, and even the frequency dependence of the photon's energy and momentum. An elegant illustration is Heisenberg's thought experiment for locating an electron with an ideal microscope.[41] The position of the electron can be determined to within the resolving power of the microscope, which is given by a formula from classical optics
\Deltax\sim
| λ | |
| \sin\theta |
where
\theta
\Deltax
λ
\Deltap
\Deltap
\Deltap\simpphoton\sin\theta=
| h | |
| λ |
\sin\theta
giving the product
\Deltax\Deltap\simh
The analogous uncertainty principle for photons forbids the simultaneous measurement of the number
n
\phi
\Deltan\Delta\phi>1
See coherent state and squeezed coherent state for more details.
Both photons and material particles such as electrons create analogous interference patterns when passing through a double-slit experiment. For photons, this corresponds to the interference of a Maxwell light wave whereas, for material particles, this corresponds to the interference of the Schrödinger wave equation. Although this similarity might suggest that Maxwell's equations are simply Schrödinger's equation for photons, most physicists do not agree.[43] [44] For one thing, they are mathematically different; most obviously, Schrödinger's one equation solves for a complex field, whereas Maxwell's four equations solve for real fields. More generally, the normal concept of a Schrödinger probability wave function cannot be applied to photons.[45] Being massless, they cannot be localized without being destroyed; technically, photons cannot have a position eigenstate
|r\rangle
\Deltax\Deltap>h/2
See main article: Bose gas, Bose–Einstein statistics and Spin-statistics theorem.
In 1924, Satyendra Nath Bose derived Planck's law of black-body radiation without using any electromagnetism, but rather a modification of coarse-grained counting of phase space.[50] Einstein showed that this modification is equivalent to assuming that photons are rigorously identical and that it implied a "mysterious non-local interaction",[51] [52] now understood as the requirement for a symmetric quantum mechanical state. This work led to the concept of coherent states and the development of the laser. In the same papers, Einstein extended Bose's formalism to material particles (bosons) and predicted that they would condense into their lowest quantum state at low enough temperatures; this Bose–Einstein condensation was observed experimentally in 1995.[53]
The modern view on this is that photons are, by virtue of their integer spin, bosons (as opposed to fermions with half-integer spin). By the spin-statistics theorem, all bosons obey Bose–Einstein statistics (whereas all fermions obey Fermi-Dirac statistics).
See main article: Stimulated emission and Laser.
In 1916, Einstein showed that Planck's radiation law could be derived from a semi-classical, statistical treatment of photons and atoms, which implies a relation between the rates at which atoms emit and absorb photons. The condition follows from the assumption that light is emitted and absorbed by atoms independently, and that the thermal equilibrium is preserved by interaction with atoms. Consider a cavity in thermal equilibrium and filled with electromagnetic radiation and atoms that can emit and absorb that radiation. Thermal equilibrium requires that the energy density
\rho(\nu)
\nu
Einstein began by postulating simple proportionality relations for the different reaction rates involved. In his model, the rate
Rji
\nu
Ej
Ei
Nj
Ej
\rho(\nu)
Rji=NjBji\rho(\nu)
where
Bji
Rij
\nu
Ei
Ej
Rij=NiAij+NiBij\rho(\nu)
where
Aij
Bij
Rji
Rij
Ni
Nj
gi/gj\exp{(Ej-Ei)/kT)},
gi,j
Ei,j
giBij=gjBji
Aij=
| 8\pih\nu3 | |
| c3 |
Bij.
Einstein could not fully justify his rate equations, but claimed that it should be possible to calculate the coefficients
Aij
Bji
Bij
Bij
Einstein was troubled by the fact that his theory seemed incomplete, since it did not determine the direction of a spontaneously emitted photon. A probabilistic nature of light-particle motion was first considered by Newton in his treatment of birefringence and, more generally, of the splitting of light beams at interfaces into a transmitted beam and a reflected beam. Newton hypothesized that hidden variables in the light particle determined which path it would follow.[22] Similarly, Einstein hoped for a more complete theory that would leave nothing to chance, beginning his separation[34] from quantum mechanics. Ironically, Max Born's probabilistic interpretation of the wave function[63] [64] was inspired by Einstein's later work searching for a more complete theory.[65]
See main article: Quantum field theory.
In 1910, Peter Debye derived Planck's law of black-body radiation from a relatively simple assumption.[66] He correctly decomposed the electromagnetic field in a cavity into its Fourier modes, and assumed that the energy in any mode was an integer multiple of
h\nu
\nu
In 1925, Born, Heisenberg and Jordan reinterpreted Debye's concept in a key way.[67] As may be shown classically, the Fourier modes of the electromagnetic field—a complete set of electromagnetic plane waves indexed by their wave vector k and polarization state—are equivalent to a set of uncoupled simple harmonic oscillators. Treated quantum mechanically, the energy levels of such oscillators are known to be
E=nh\nu
\nu
E=nh\nu
n
h\nu
Dirac took this one step further.[58] [59] He treated the interaction between a charge and an electromagnetic field as a small perturbation that induces transitions in the photon states, changing the numbers of photons in the modes, while conserving energy and momentum overall. Dirac was able to derive Einstein's
Aij
Bij
Dirac's second-order perturbation theory can involve virtual photons, transient intermediate states of the electromagnetic field; the static electric and magnetic interactions are mediated by such virtual photons. In such quantum field theories, the probability amplitude of observable events is calculated by summing over all possible intermediate steps, even ones that are unphysical; hence, virtual photons are not constrained to satisfy
E=pc
In modern physics notation, the quantum state of the electromagnetic field is written as a Fock state, a tensor product of the states for each electromagnetic mode
| |n | |
| k0 |
| \rangle ⊗ |n | |
| k1 |
| \rangle ⊗ ... ⊗ |n | |
| kn |
\rangle...
where
| |n | |
| ki |
\rangle
| n | |
| ki |
ki
ki
| |n | |
| ki |
\rangle →
| |n | |
| ki |
+1\rangle
See main article: Gauge theory.
The electromagnetic field can be understood as a gauge theory, i.e., as a field that results from requiring that symmetry hold independently at every position in spacetime.[70] For the electromagnetic field, this gauge symmetry is the Abelian U(1) symmetry of a complex number, which reflects the ability to vary the phase of a complex number without affecting real numbers made from it, such as the energy or the Lagrangian.
The quanta of an Abelian gauge field must be massless, uncharged bosons, as long as the symmetry is not broken; hence, the photon is predicted to be massless, and to have zero electric charge and integer spin. The particular form of the electromagnetic interaction specifies that the photon must have spin ±1; thus, its helicity must be
\pm\hbar
In the prevailing Standard Model of physics, the photon is one of four gauge bosons in the electroweak interaction; the other three are denoted W+, W− and Z0 and are responsible for the weak interaction. Unlike the photon, these gauge bosons have invariant mass, owing to a mechanism that breaks their SU(2) gauge symmetry. The unification of the photon with W and Z gauge bosons in the electroweak interaction was accomplished by Sheldon Glashow, Abdus Salam and Steven Weinberg, for which they were awarded the 1979 Nobel Prize in physics.[71] [72] [73] Physicists continue to hypothesize grand unified theories that connect these four gauge bosons with the eight gluon gauge bosons of quantum chromodynamics; however, key predictions of these theories, such as proton decay, have not been observed experimentally.[74]
See main article: Quantum chromodynamics.
According to quantum chromodynamics, a real photon can interact both as a point-like particle, or as a collection of quarks and gluons, i.e., like a hadron. The structure of the photon is determined not by the traditional valence quark distributions as in a proton, but by fluctuations of the point-like photon into a collection of partons.[75]
See also: Mass in special relativity and General relativity. The energy of a system that emits a photon is decreased by the energy
E
{E}/{c2}
{E}/{c2}
This concept is applied in key predictions of quantum electrodynamics (QED, see above). In that theory, the mass of electrons (or, more generally, leptons) is modified by including the mass contributions of virtual photons, in a technique known as renormalization. Such "radiative corrections" contribute to a number of predictions of QED, such as the magnetic dipole moment of leptons, the Lamb shift, and the hyperfine structure of bound lepton pairs, such as muonium and positronium.[77]
Since photons contribute to the stress-energy tensor, they exert a gravitational attraction on other objects, according to the theory of general relativity. Conversely, photons are themselves affected by gravity; their normally straight trajectories may be bent by warped spacetime, as in gravitational lensing, and their frequencies may be lowered by moving to a higher gravitational potential, as in the Pound-Rebka experiment. However, these effects are not specific to photons; exactly the same effects would be predicted for classical electromagnetic waves.[78]
See also: Group velocity and Photochemistry.
Light that travels through transparent matter does so at a lower speed than c, the speed of light in a vacuum. For example, photons suffer so many collisions on the way from the core of the sun that radiant energy can take about a million years to reach the surface;[79] however, once in open space, a photon takes only 8.3 minutes to reach Earth. The factor by which the speed of light is decreased is called the refractive index of the material. In a classical wave picture, the slowing can be explained by the light inducing electric polarization in the matter, the polarized matter radiating new light, and the new light interfering with the original light wave to form a delayed wave. In a particle picture, the slowing can instead be described as a blending of the photon with quantum excitations of the matter (quasi-particles such as phonons and excitons) to form a polariton; this polariton has a nonzero effective mass, which means that it cannot travel at c.
Alternatively, photons may be viewed as always traveling at c, even in matter, but they have their phase shifted (delayed or advanced) upon interaction with atomic scatters: this modifies their wavelength and momentum, but not speed. [80] A light wave made up these photons does travel slower than the speed of light. In this view the photons are "bare", and are scattered and phase shifted, while in the view of the preceding paragraph the photons are "dressed" by their interaction with matter, and move without scattering or phase shifting, but at a lower speed.
Light of different frequencies may travel through matter at different speeds; this is called dispersion. In some cases, it can result in extremely slow speeds of light in matter. The effects of photon interactions with other quasi-particles may be observed directly in Raman scattering and Brillouin scattering.[81]
Photons can also be absorbed by nuclei, atoms or molecules, provoking transitions between their energy levels. A classic example is the molecular transition of retinal (C20H28O, Figure at right), which is responsible for vision, as discovered in 1958 by Nobel laureate biochemist George Wald and co-workers. As shown here, the absorption provokes a cis-trans isomerization that, in combination with other such transitions, is transduced into nerve impulses. The absorption of photons can even break chemical bonds, as in the photodissociation of chlorine; this is the subject of photochemistry.[82] [83]
Photons have many applications in technology. These examples are chosen to illustrate applications of photons per se, rather than general optical devices such as lenses, etc. that could operate under a classical theory of light. The laser is an extremely important application and is discussed above under stimulated emission.
Individual photons can be detected by several methods. The classic photomultiplier tube exploits the photoelectric effect: a photon landing on a metal plate ejects an electron, initiating an ever-amplifying avalanche of electrons. Charge-coupled device chips use a similar effect in semiconductors: an incident photon generates a charge on a microscopic capacitor that can be detected. Other detectors such as Geiger counters use the ability of photons to ionize gas molecules, causing a detectable change in conductivity.[84]
Planck's energy formula
E=h\nu
Under some conditions, an energy transition can be excited by two photons that individually would be insufficient. This allows for higher resolution microscopy, because the sample absorbs energy only in the region where two beams of different colors overlap significantly, which can be made much smaller than the excitation volume of a single beam (see two-photon excitation microscopy). Moreover, these photons cause less damage to the sample, since they are of lower energy.[86]
In some cases, two energy transitions can be coupled so that, as one system absorbs a photon, another nearby system "steals" its energy and re-emits a photon of a different frequency. This is the basis of fluorescence resonance energy transfer, a technique that is used in molecular biology to study the interaction of suitable proteins.
Several different kinds of hardware random number generator involve the detection of single photons. In one example, for each bit in the random sequence that is to be produced, a photon is sent to a beam-splitter. In such a situation, there are two possible outcomes of equal probability. The actual outcome is used to determine whether the next bit in the sequence is "0" or "1".
See also: Quantum optics.
The fundamental nature of the photon is believed to be understood theoretically; the prevailing Standard Model predicts that the photon is a gauge boson of spin 1, without mass (at rest[87] ) and without charge, that results from a local U(1) gauge symmetry and mediates the electromagnetic interaction. However, physicists continue to check for discrepancies between experiment and the Standard Model predictions, in the hope of finding clues to physics beyond the Standard Model. In particular, experimental physicists continue to set ever better upper limits on the charge and mass of the photon; a non-zero value for either parameter would be a serious violation of the Standard Model. However, all experimental data hitherto are consistent with the photon having zero charge[10] and mass[88] .[89] The best universally accepted upper limits on the photon charge and mass are (or ) and (, or ), respectively.[90]
Much research has been devoted to applications of photons in the field of quantum optics. Photons seem well-suited to be elements of an extremely fast quantum computer, and the quantum entanglement of photons is a focus of research. Nonlinear optical processes are another active research area, with topics such as two-photon absorption, self-phase modulation and optical parametric oscillators. However, such processes generally do not require the assumption of photons per se; they may often be modeled by treating atoms as nonlinear oscillators. The nonlinear process of spontaneous parametric down conversion is often used to produce single-photon states. Finally, photons are essential in some aspects of optical communication, especially for quantum cryptography.[91]
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. Robert Hooke. 1667. London (UK). Royal Society of London. Micrographia: or some physiological descriptions of minute bodies made by magnifying glasses with observations and inquiries thereupon....
. Christian Huygens. 1678. Traité de la lumière. . An English translation is available from Project Gutenberg
. Isaac Newton. 1730. 1952. Opticks. 4th. Book II, Part III, Propositions XII–XX; Queries 25–29. true. Dover (NY). Dover Publications. 0-486-60205-2.
. Abraham Pais. 1982. Subtle is the Lord: The Science and the Life of Albert Einstein. Oxford University Press. 0-198-53907-X.
. Hendrik Anthony Kramers. 1958. Quantum Mechanics. North-Holland. Amsterdam. .
. David Bohm. 1954. 1989. Quantum Theory. Dover Publications. 0-486-65969-0.
. Abraham Pais. 1986. Inward Bound: Of Matter and Forces in the Physical World. Oxford University Press. 0-198-51997-4. Specifically, Born claimed to have been inspired by Einstein's never-published attempts to develop a "ghost-field" theory, in which point-like photons are guided probabilistically by ghost fields that follow Maxwell's equations.
By date of publication:
. Abraham Pais. 1982. Subtle is the Lord: The Science and the Life of Albert Einstein. Oxford University Press.