A
photon is an
elementary particle, the
quantum of
light and all other forms of
electromagnetic radiation. It is the
force carrier for the
electromagnetic force, even when
static via
virtual photons. The effects of this
force are easily observable at the
microscopic and at the
macroscopic level, because the photon has zero
rest mass; this allows long distance
interactions. Like all elementary particles, photons are currently best explained by
quantum mechanics and exhibit
wave–particle duality, exhibiting properties of
waves and of
particles. For example, a single photon may be
refracted by a
lens or exhibit
wave interference with itself, but also act as a particle giving a definite result when its
position is measured.
The modern photon concept 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 do so in amounts of energy that are
quantized (i.e., they change energy only by certain particular discrete amounts and cannot change energy in any arbitrary way). Although these semiclassical models contributed to the development of quantum mechanics, many further experiments
[2][3] starting with
Compton scattering of single photons by electrons, first observed in 1923, validated Einstein's hypothesis that
light itself is
quantized. In 1926 the optical physicist Frithiof Wolfers and the chemist
Gilbert N. Lewis coined the name
photon for these particles, and after 1927, when
Arthur H. Compton won the Nobel Prize for his scattering studies, most scientists accepted the validity that
quanta of light have an independent existence, and the term
photon for light quanta was accepted.
In the
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 applications in
optical imaging and
optical communication such as
quantum cryptography.
Nomenclature
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. In his 1901 article
[4] in
Annalen der Physik he called these packets "energy elements". The word
quanta (singular
quantum) was used even before 1900 to mean particles or amounts of different
quantities, including
electricity. Later, in 1905,
Albert Einstein went further by suggesting that electromagnetic waves could only exist in these discrete wave-packets.
[5] He called such a
wave-packet
the light quantum (German:
das Lichtquant).
[Note 1] The name
photon derives from the
Greek word for light,
φῶς (transliterated
phôs).
Arthur Compton used
photon in 1928, referring to
Gilbert N. Lewis.
[6] The same name was used earlier, by the American physicist and psychologist
Leonard T. Troland, who coined the word in 1916, in 1921 by the Irish physicist
John Joly and in 1926 by the French physiologist
René Wurmser (1890-1993) and by the French physicist
Frithiof Wolfers (ca. 1890-1971).
[7] The name was suggested initially as a unit related to the illumination of the eye and the resulting sensation of light and was used later on in a physiological context.
Although Wolfers's and Lewis's theories were never accepted, as they were contradicted by many experiments, the new name was adopted very soon by most physicists after Compton used it.
[7][Note 2]
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 in 1900 by
Paul Villard,
[8][9] named by
Ernest Rutherford in 1903, and shown to be a form of
electromagnetic radiation in 1914 by Rutherford and
Edward Andrade.
[10] 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.
Physical properties
The cone shows possible values of wave 4-vector of a photon. The "time" axis gives the angular frequency (
rad⋅s−1) and the "space" axes represent the angular wavenumber (rad⋅m
−1). Green and indigo represent left and right polarization
A photon is
massless,
[Note 3] has no
electric charge,
[11] and is
stable. A photon has two possible
polarization states. In the
momentum representation, which is preferred in quantum field theory, a photon is described by its
wave vector, which determines its wavelength
λ and its direction of propagation. A photon's wave vector may not be zero and can be represented either as a
spatial 3-vector or as a (relativistic)
four-vector; in the latter case it belongs to the
light cone (pictured). Different signs of the four-vector denote different
circular polarizations, but in the 3-vector representation one should account for the polarization state separately; it actually is a
spin quantum number. In both cases the space of possible wave vectors is three-dimensional.
The photon is the
gauge boson for
electromagnetism,
[12] and therefore all other quantum numbers of the photon (such as
lepton number,
baryon number, and
flavour quantum numbers) are zero.
[13]
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
radio waves to
gamma rays. A photon can also be emitted when a particle and its corresponding
antiparticle are
annihilated (for example,
electron–positron annihilation).
In empty space, the photon moves at
c (the
speed of light) and its
energy and
momentum are related by
E = pc, where
p is the
magnitude of the momentum vector
p. This derives from the following relativistic relation, with
m = 0:
[14]
The energy and momentum of a photon depend only on its
frequency (
ν) or inversely, its
wavelength (
λ):
where
k is the
wave vector (where the wave number
k = |k| = 2π/λ),
ω = 2πν is the
angular frequency, and
ħ = h/2π is the
reduced Planck constant.
[15]
Since
p points in the direction of the photon's propagation, the magnitude of the momentum is
The photon also carries
spin angular momentum that does not depend on its frequency.
[16] The magnitude of its spin is
and the component measured along its direction of motion, its
helicity, must be ±ħ. These two possible helicities, called right-handed and left-handed, correspond to the two possible
circular polarization states of the photon.
[17]
To illustrate the significance of these formulae, the annihilation of a particle with its antiparticle in free space 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, whereas 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 (or equivalently,
translational invariance) requires that at least two photons are created, with zero net momentum. (However, it is possible if the system interacts with another particle or field for annihilation to produce one photon, as when a positron annihilates with a bound atomic electron, it is possible for only one photon to be emitted, as the nuclear Coulomb field breaks translational symmetry.) 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.
[18] That process is the reverse of "annihilation to one photon" allowed in the electric field of an atomic nucleus.
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.
[19]
Experimental checks on photon mass
Current commonly accepted physical theories imply or assume the photon to be strictly massless, but this should be also checked experimentally. If the photon is not a strictly massless particle, it would not move at the exact speed of light in vacuum,
c. Its speed would be lower and depend on its frequency. Relativity would be unaffected by this; the so-called speed of light,
c, would then not be the actual speed at which light moves, but a constant of nature which is the maximum speed that any object could theoretically attain in space-time.
[20] Thus, it would still be the speed of space-time ripples (
gravitational waves and
gravitons), but it would not be the speed of photons.
If a photon did have non-zero mass, there would be other effects as well.
Coulomb's law would be modified and the electromagnetic field would have an extra physical degree of freedom. These effects yield more sensitive experimental probes of the photon mass than the frequency dependence of the speed of light. If Coulomb's law is not exactly valid, then that would cause the presence of an
electric field inside a hollow conductor when it is subjected to an external electric field. This thus allows one to
test Coulomb's law to very high precision.
[21] A null result of such an experiment has set a limit of
m ≲ 10
−14 eV/c
2.
[22]
Sharper upper limits have been obtained in experiments designed to detect effects caused by the galactic
vector potential. Although the galactic vector potential is very large because the galactic
magnetic field exists on very long length scales, only the magnetic field is observable if the photon is massless. In case of a massive photon, the mass term
would affect the galactic plasma. The fact that no such effects are seen implies an upper bound on the photon mass of
m <
6973300000000000000♠3×10−27 eV/c2.
[23] The galactic vector potential can also be probed directly by measuring the torque exerted on a magnetized ring.
[24] Such methods were used to obtain the sharper upper limit of 10
−18eV/c
2 (the equivalent of
6973107000000000000♠1.07×10−27 atomic mass units) given by the
Particle Data Group.
[25]
These sharp limits from the non-observation of the effects caused by the galactic vector potential have been shown to be model dependent.
[26] If the photon mass is generated via the
Higgs mechanism then the upper limit of
m≲10
−14 eV/c
2 from the test of Coulomb's law is valid.
Photons inside
superconductors do develop a nonzero
effective rest mass; as a result, electromagnetic forces become short-range inside superconductors.
[27]
Historical development
In most theories up to the eighteenth century, light was pictured as being made up of particles. Since
particle models cannot easily account for the
refraction,
diffraction and
birefringence of light, wave theories of light were proposed by
René Descartes (1637),
[28] Robert Hooke (1665),
[29] and
Christiaan Huygens (1678);
[30] however, particle models remained dominant, chiefly due to the influence of
Isaac Newton.
[31] 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.
[32] In 1865,
James Clerk Maxwell's
prediction[33] that light was an electromagnetic wave—which was confirmed experimentally in 1888 by
Heinrich Hertz's detection of
radio waves[34]—seemed to be the final blow to particle models of light.
In 1900,
Maxwell's theoretical model of light as oscillating
electric and
magnetic fields seemed complete. However, several observations could not be explained by any wave model of
electromagnetic radiation, leading to the idea that light-energy was packaged into
quanta described by E=hν. Later experiments showed that these light-quanta also carry momentum and, thus, can be considered
particles: the
photon concept was born, leading to a deeper understanding of the electric and magnetic fields themselves.
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.
[35][Note 4]
At the same time, investigations of
blackbody radiation carried out over four decades (1860–1900) by various researchers
[36] culminated in
Max Planck's
hypothesis[4][37] that the energy of
any system that absorbs or emits electromagnetic radiation of frequency
ν is an integer multiple of an energy quantum
E = hν. As shown by
Albert Einstein,
[5][38] some form of energy quantization
must be assumed to account for the thermal equilibrium observed between matter and
electromagnetic radiation; for this explanation of the
photoelectric effect, Einstein received the 1921
Nobel Prize in physics.
[39]
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.
[5] 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.
[5] In 1909
[38] and 1916,
[40] Einstein showed that, if
Planck's law of black-body radiation is accepted, the energy quanta must also carry
momentum p = h/λ, making them full-fledged
particles. This photon momentum was observed experimentally
[41] by
Arthur Compton, for which he received the
Nobel Prize in 1927. The pivotal question was then: how to unify Maxwell's wave theory of light with its experimentally observed particle nature? The answer to this question occupied
Albert Einstein for the rest of his life,
[42] and was solved in
quantum electrodynamics and its successor, the
Standard Model (see
Second quantization and
The photon as a gauge boson, below).
Einstein's light quantum
Unlike Planck, Einstein entertained the possibility that there might be actual physical quanta of light—what we now call photons. He noticed that a light quantum with energy proportional to its frequency would explain a number of troubling puzzles and paradoxes, including an unpublished law by Stokes, the
ultraviolet catastrophe, and of course the
photoelectric effect. Stokes's law said simply that the frequency of fluorescent light cannot be greater than the frequency of the light (usually ultraviolet) inducing it. He eliminated the ultraviolet catastrophe by imagining a gas of photons behaving like a gas of electrons that he had previously considered. He was advised by a colleague to be careful how he wrote up this paper, in order to not challenge Planck too directly, as he was a powerful figure, and indeed the warning was justified, as Planck never forgave him for writing it.
[43]
Early objections
Up to 1923, most physicists were reluctant to accept that light itself was quantized. Instead, they tried to explain photon behavior by quantizing only
matter, as in the
Bohr model of the
hydrogen atom (shown here). Even though these semiclassical models were only a first approximation, they were accurate for simple systems and they led to
quantum mechanics.
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.
[44] However, before
Compton's experiment[41] showing that photons carried
momentum proportional to their
wave number (or frequency) (1922), most physicists were reluctant to believe that
electromagnetic radiation itself might be particulate. (See, for example, the Nobel lectures of
Wien,
[36] Planck[37] and Millikan.
[44]) 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 time. 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.
[45]
Even after Compton's experiment,
Niels 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.
[46] To account for the data then available, two drastic hypotheses had to be made:
- Energy and momentum are conserved only on the average in interactions between matter and radiation, not in elementary processes such as absorption and emission. This allows one to reconcile the discontinuously changing energy of the atom (jump between energy states) with the continuous release of energy into radiation.
- Causality is abandoned. For example, spontaneous emissions are merely emissions induced by a "virtual" electromagnetic field.
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".
[42] Nevertheless, the failures of the BKS model inspired
Werner Heisenberg in his development of
matrix mechanics.
[47]
A few physicists persisted
[48] 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.
[Note 5] Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven.
Wave–particle duality and uncertainty principles
Photons, like all quantum objects, exhibit 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 exhibiting interference phenomena but only if no measurement was made on the actual slit being run across. To account for the particle interpretation that phenomenon is called
probability distribution but behaves according to
Maxwell's equations.
[49] 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.
[50] 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.
[51] 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
where
is the
aperture angle of the microscope. Thus, the position uncertainty
can be made arbitrarily small by reducing the wavelength λ. The momentum of the electron is uncertain, since it received a "kick"
from the light scattering from it into the microscope. If light were
not quantized into photons, the uncertainty
could be made arbitrarily small by reducing the light's intensity. In that case, since the wavelength and intensity of light can be varied independently, one could simultaneously determine the position and momentum to arbitrarily high accuracy, violating the
uncertainty principle. By contrast, Einstein's formula for photon momentum preserves the uncertainty principle; since the photon is scattered anywhere within the aperture, the uncertainty of momentum transferred equals
giving the product
, which is Heisenberg's uncertainty principle. Thus, the entire world is quantized; both matter and fields must obey a consistent set of quantum laws, if either one is to be quantized.
[52]
The analogous uncertainty principle for photons forbids the simultaneous measurement of the number
of photons (see
Fock state and the
Second quantization section below) in an electromagnetic wave and the phase
of that wave
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.
[53][54] 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.
[55] Being massless, they cannot be localized without being destroyed; technically, photons cannot have a position eigenstate
, and, thus, the normal Heisenberg uncertainty principle
does not pertain to photons. A few substitute wave functions have been suggested for the photon,
[56][57][58][59] but they have not come into general use. Instead, physicists generally accept the second-quantized theory of photons described below,
quantum electrodynamics, in which photons are quantized excitations of electromagnetic modes.
Another interpretation, that avoids duality, is the
De Broglie–Bohm theory: knowned also as the
pilot-wave model, the photon in this theory is both, wave and particle.
[60] "This idea seems to me so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored",
[61] J.S.Bell.
Bose–Einstein model of a photon gas
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.
[62] Einstein showed that this modification is equivalent to assuming that photons are rigorously identical and that it implied a "mysterious non-local interaction",
[63][64] 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.
[65] It was later used by
Lene Hau to slow, and then completely stop, light in 1999
[66] and 2001.
[67]
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).
[68]
Stimulated and spontaneous emission
Stimulated emission (in which photons "clone" themselves) was predicted by Einstein in his kinetic analysis, and led to the development of the
laser. Einstein's derivation inspired further developments in the quantum treatment of light, which led to the statistical interpretation of quantum mechanics.
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
of photons with frequency
(which is proportional to their
number density) is, on average, constant in time; hence, the rate at which photons of any particular frequency are
emitted must equal the rate of
absorbing them.
[69]
Einstein began by postulating simple proportionality relations for the different reaction rates involved. In his model, the rate
for a system to
absorb a photon of frequency
and transition from a lower energy
to a higher energy
is proportional to the number
of atoms with energy
and to the energy density
of ambient photons with that frequency,
where
is the
rate constant for absorption. For the reverse process, there are two possibilities: spontaneous emission of a photon, and a return to the lower-energy state that is initiated by the interaction with a passing photon.
Following Einstein's approach, the corresponding rate
for the emission of photons of frequency
and transition from a higher energy
to a lower energy
is
where
is the rate constant for
emitting a photon spontaneously, and
is the rate constant for emitting it in response to ambient photons (
induced or stimulated emission). In thermodynamic equilibrium, the number of atoms in state i and that of atoms in state j must, on average, be constant; hence, the rates
and
must be equal.
Also, by arguments analogous to the derivation of
Boltzmann statistics, the ratio of
and
is
where
are the
degeneracy of the state i and that of j, respectively,
their energies, k the
Boltzmann constant and T the system's
temperature. From this, it is readily derived that
and
The A and Bs are collectively known as the
Einstein coefficients.
[70]
Einstein could not fully justify his rate equations, but claimed that it should be possible to calculate the coefficients
,
and
once physicists had obtained "mechanics and electrodynamics modified to accommodate the quantum hypothesis".
[71] In fact, in 1926,
Paul Dirac derived the
rate constants in using a semiclassical approach,
[72] and, in 1927, succeeded in deriving
all the rate constants from first principles within the framework of quantum theory.
[73][74] Dirac's work was the foundation of quantum electrodynamics, i.e., the quantization of the electromagnetic field itself. Dirac's approach is also called
second quantization or
quantum field theory;
[75][76][77] earlier quantum mechanical treatments only treat material particles as quantum mechanical, not the electromagnetic field.
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.
[31] Similarly, Einstein hoped for a more complete theory that would leave nothing to chance, beginning his separation
[42] from quantum mechanics. Ironically,
Max Born's
probabilistic interpretation of the
wave function[78][79] was inspired by Einstein's later work searching for a more complete theory.
[80]
Second quantization and high energy photon interactions
Different
electromagnetic modes (such as those depicted here) can be treated as independent
simple harmonic oscillators. A photon corresponds to a unit of energy E=hν in its electromagnetic mode.
In 1910,
Peter Debye derived
Planck's law of black-body radiation from a relatively simple assumption.
[81] 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
, where
is the frequency of the electromagnetic mode. Planck's law of black-body radiation follows immediately as a geometric sum. However, Debye's approach failed to give the correct formula for the energy fluctuations of blackbody radiation, which were derived by Einstein in 1909.
[38]
In 1925,
Born,
Heisenberg and
Jordan reinterpreted Debye's concept in a key way.
[82] 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
, where
is the oscillator frequency. The key new step was to identify an electromagnetic mode with energy
as a state with
photons, each of energy
. This approach gives the correct energy fluctuation formula.
Dirac took this one step further.
[73][74] 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
and
coefficients from first principles, and showed that the Bose–Einstein statistics of photons is a natural consequence of quantizing the electromagnetic field correctly (Bose's reasoning went in the opposite direction; he derived
Planck's law of black-body radiation by
assuming B–E statistics). In Dirac's time, it was not yet known that all bosons, including photons, must obey Bose–Einstein statistics.
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
, and may have extra
polarization states; depending on the
gauge used, virtual photons may have three or four polarization states, instead of the two states of real photons. Although these transient virtual photons can never be observed, they contribute measurably to the probabilities of observable events. Indeed, such second-order and higher-order perturbation calculations can give apparently
infinite contributions to the sum. Such unphysical results are corrected for using the technique of
renormalization.
Other virtual particles may contribute to the summation as well; for example, two photons may interact indirectly through virtual
electron–
positron pairs.
[83] In fact, such photon-photon scattering (see
two-photon physics), as well as electron-photon scattering, is meant to be one of the modes of operations of the planned particle accelerator, the
International Linear Collider.
[84]
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
where
represents the state in which
photons are in the mode
. In this notation, the creation of a new photon in mode
(e.g., emitted from an atomic transition) is written as
. This notation merely expresses the concept of Born, Heisenberg and Jordan described above, and does not add any physics.
The hadronic properties of the photon
Measurements of the interaction between energetic photons and hadrons show that the interaction is much more intense than expected by the interaction of merely photons with the hadron's electric charge. Furthermore, the interaction of energetic photons with protons is similar to the interaction of photons with neutrons
[85] in spite of the fact that the electric charge structures of protons and neutrons are substantially different.
A theory called
Vector Meson Dominance (VMD) was developed to explain this effect. According to VMD, the photon is a superposition of the pure electromagnetic photon (which interacts only with electric charges) and vector meson.
[86]
However, if experimentally probed at very short distances, the intrinsic structure of the photon is recognized as a flux of quark and gluon components, quasi-free according to asymptotic freedom in
QCD and described by the
photon structure function.
[87][88] A comprehensive comparison of data with theoretical predictions is presented in a recent review.
[89]
The photon as a gauge boson
The electromagnetic field can be understood as a
gauge field, i.e., as a field that results from requiring that a gauge symmetry holds independently at every position in
spacetime.
[90] 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
observables or
real valued functions 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
. These two spin components correspond to the classical concepts of
right-handed and left-handed circularly polarized light. However, the transient
virtual photons of
quantum electrodynamics may also adopt unphysical polarization states.
[90]
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 Z
0 and are responsible for the
weak interaction. Unlike the photon, these gauge bosons have
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.
[91][92][93] 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.
[94]
Contributions to the mass of a system
The energy of a system that emits a photon is
decreased by the energy
of the photon as measured in the rest frame of the emitting system, which may result in a reduction in mass in the amount
. Similarly, the mass of a system that absorbs a photon is
increased by a corresponding amount. As an application, the energy balance of nuclear reactions involving photons is commonly written in terms of the masses of the nuclei involved, and terms of the form
for the gamma photons (and for other relevant energies, such as the recoil energy of nuclei).
[95]
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.
[96]
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.
[97]
Photons in matter
Any 'explanation' of how photons travel through matter has to explain why different arrangements of matter are transparent or opaque at different wavelengths (light through carbon as diamond or not, as graphite) and why individual photons behave in the same way as large groups. Explanations that invoke 'absorption' and 're-emission' have to provide an explanation for the directionality of the photons (diffraction, reflection) and further explain how entangled photon pairs can travel through matter without their quantum state collapsing.
The simplest explanation is that light that travels through transparent matter does so at a lower speed than
c, the speed of light in a vacuum. In addition, light can also undergo
scattering and
absorption. There are circumstances in which heat transfer through a material is mostly radiative, involving emission and absorption of photons within it.
An example would be in the
core of the Sun. Energy can take about a million years to reach the surface.
[98]
However, this phenomenon is distinct from scattered radiation passing diffusely through matter, as it involves local equilibrium between the radiation and the temperature. Thus, the time is how long it takes the
energy to be transferred, not the
photons themselves. Once in open space, a photon from the Sun takes only 8.3 minutes to reach Earth. The factor by which the speed of light is decreased in a material 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 excitation 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.
[99] A light wave made up of 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.
[100]
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 C
20H
28O, which is responsible for
vision, as discovered in 1958 by Nobel laureate
biochemist George Wald and co-workers. 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.
[101][102] Analogously,
gamma rays can in some circumstances dissociate atomic nuclei in a process called
photodisintegration.
Technological applications
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.
[103]
Planck's energy formula
is often used by engineers and chemists in design, both to compute the change in energy resulting from a photon absorption and to predict the frequency of the light emitted for a given energy transition. For example, the
emission spectrum of a
fluorescent light bulb can be designed using gas molecules with different electronic energy levels and adjusting the typical energy with which an electron hits the gas molecules within the bulb.
[Note 6]
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.
[104]
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.
[105]
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".
[106][107]
Recent research
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,
modulational instability 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.
[Note 7]