Photon

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Photon
Photons emitted in a coherent beam from a laser
Composition:	Elementary particle
Family:	Boson
Group:	Gauge boson
Interaction:	Electromagnetic
Theorized:	Albert Einstein (1905–17)
Symbol:	$\gamma\$ or $\ h\nu$
Mass:	0^[1]
Mean lifetime:	Stable^[2]
Electric charge:	0
Spin:	1^[1]

In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. It is the carrier of electromagnetic radiation of all wavelengths, including gamma rays, X-rays, ultraviolet light, visible light, infrared light, microwaves, and radio waves. The photon differs from many other elementary particles, such as the electron and the quark, in that it has zero rest mass;^[3] therefore, it travels (in vacuum) at the speed of light, c. Like all quanta, the photon has both wave and particle properties (“wave–particle duality”). Photons show wave-like phenomena, such as refraction by a lens and destructive interference when reflected waves cancel each other out; however, as a particle, it can only interact with matter by transferring the amount of energy

$E = \frac{hc}{\lambda},$

where h is Planck's constant, c is the speed of light, and $λ$ is its wavelength. This is different from a classical wave, which may gain or lose arbitrary amounts of energy. For visible light the energy carried by a single photon is around a tiny $4\times10^{-19}$ joules; this energy is just sufficient to excite a single molecule in a photoreceptor cell of an eye,^{[citation needed]} thus contributing to vision.^[4]

Apart from energy a photon also carries momentum and has a polarization. It follows the laws of quantum mechanics, which means that often these properties do not have a well-defined value for a given photon. Rather, they are defined as a probability to measure a certain polarization, position, or momentum. For example, although a photon can excite a single molecule, it is often impossible to predict beforehand which molecule will be excited.

The above description of a photon as a carrier of electromagnetic radiation is commonly used by physicists. However, in theoretical physics, a photon can be considered as a mediator for any type of electromagnetic interactions, including magnetic fields and electrostatic repulsion between like charges.

The modern concept of the photon was developed gradually (1905–17) by Albert Einstein^[5]^[6]^[7]^[8] 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. Other physicists sought to explain these anomalous observations by 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.

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. According to the Standard Model of particle physics, photons are responsible for producing all electric and magnetic fields, and are themselves the product of requiring that physical laws have 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 concept of photons is applied to many areas such as 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.

1 Nomenclature
2 Physical properties
3 Historical development
4 Early objections
5 Wave–particle duality and uncertainty principles
6 Bose–Einstein model of a photon gas
7 Stimulated and spontaneous emission
8 Second quantization
9 The photon as a gauge boson
10 Photon structure
11 Contributions to the mass of a system
12 Photons in matter
13 Technological applications
14 Recent research
15 See also
16 References and footnotes
17 Additional references

[edit] Nomenclature

The photon was originally called a “light quantum” (das Lichtquant) by Albert Einstein.^[5] The modern name “photon” derives from the Greek word for light, φῶς, (transliterated phôs), and was coined in 1926 by the physical chemist Gilbert N. Lewis, who published a speculative theory^[9] 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 light as photons in 1927.^[10]^[11]

In physics, a photon is usually denoted by the symbol $\gamma\!$ , the Greek letter gamma. This symbol for the photon probably derives from gamma rays, which were discovered and named in 1900 by Villard^[12]^[13] and shown to be a form of electromagnetic radiation in 1914 by Rutherford and Andrade.^[14] In chemistry and optical engineering, photons are usually symbolized by $h \nu \!$ , the energy of a photon, where $h \!$ is Planck's constant and the Greek letter $\nu \!$ (nu) is the photon's frequency. Much less commonly, the photon can be symbolized by hf, where its frequency is denoted by f.

[edit] Physical properties

A Feynman diagram of the exchange of a virtual photon (symbolized by a wavy-line and a gamma, $\gamma \,$ ) between a positron and an electron.

See also: Special relativity

The photon is massless,^[3] has no electric charge^[15] 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 $\lambda \!$ and its direction of propagation. The photon is the gauge boson for electromagnetism, and therefore all other quantum numbers — such as lepton number, baryon number, or strangeness — are exactly zero.

Photons are emitted in many natural processes, e.g., when a charge is accelerated, during a molecular, atomic or nuclear transition to a lower energy level, or when a particle and its antiparticle are annihilated. Photons are absorbed in the time-reversed processes which correspond to those mentioned above: for example, in the production of particle–antiparticle pairs or in molecular, atomic or nuclear transitions to a higher energy level.

In empty space, the photon moves at $c \!$ (the speed of light) and its energy $E \!$ and momentum $\mathbf{p}$ are related by $E = c \, p \!$ , where $p \!$ is the magnitude of the momentum. For comparison, the corresponding equation for particles with a mass $m \!$ is $E^{2} = c^{2} p^{2} + m^{2} c^{4} \!$ , as shown in special relativity.

The energy and momentum of a photon depend only on its frequency $\nu \!$ or, equivalently, its wavelength $\lambda \!$

$E = \hbar\omega = h\nu = \frac{h c}{\lambda}$

$\mathbf{p} = \hbar\mathbf{k}$

and consequently the magnitude of the momentum is

$p = \hbar k = \frac{h}{\lambda} = \frac{h\nu}{c}$

where $\hbar = h/2\pi \!$ (known as Dirac's constant or Planck's reduced constant); $\mathbf{k}$ is the wave vector (with the wave number $k = 2\pi/\lambda \!$ as its magnitude) and $\omega = 2\pi\nu \!$ is the angular frequency. Notice that $\mathbf{k}$ points in the direction of the photon's propagation. The photon also carries spin angular momentum that does not depend on its frequency. The magnitude of its spin is $\sqrt{2} \hbar$ and the component measured along its direction of motion, its helicity, must be $\pm\hbar$ . These two possible helicities correspond to the two possible circular polarization states of the photon (right-handed and left-handed).

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, whereas a single photon always has momentum. 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.

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.

[edit] Historical development

Main article: Light

Thomas Young's double-slit experiment in 1805 showed that light can act as a wave, helping to defeat early particle theories of light.

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),^[16] Robert Hooke (1665),^[17] and Christian Huygens (1678);^[18] however, particle models remained dominant, chiefly due to the influence of Isaac Newton.^[19] 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.^[20] In 1865, James Clerk Maxwell's prediction^[21] that light was an electromagnetic wave — which was confirmed experimentally in 1888 by Heinrich Hertz's detection of radio waves^[22] — 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.

At the same time, investigations of blackbody radiation carried out over four decades (1860–1900) by various researchers^[23] culminated in Max Planck's hypothesis^[24]^[25] that the energy of any system that absorbs or emits electromagnetic radiation of frequency $ν$ is an integer multiple of an energy quantum $E = h ν$ . As shown by Albert Einstein,^[5]^[6] 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.

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^[6] and 1916,^[8] 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^[26] 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,^[27] and was solved in quantum electrodynamics and its successor, the Standard Model.

[edit] Early objections

Up to 1923, most physicists were reluctant to accept that electromagnetic radiation itself was quantized. Instead, they tried to account for photon behavior by quantizing matter, as in the Bohr model of the hydrogen atom (shown here). Although all semiclassical models have been disproved by experiment, these early atomic models led to quantum mechanics.

Einstein's 1905 predictions were verified experimentally in several ways within the first two decades of the 20th century, as recounted in Robert Millikan's Nobel lecture.^[28] However, before Compton's experiment^[26] 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,^[23] Planck^[25] and Millikan.^[28]) This reluctance is understandable, given the success and plausibility of Maxwell's electromagnetic wave model of light. Therefore, most physicists assumed rather that energy quantization resulted from some unknown constraint on the matter that absorbs or emits radiation. Niels Bohr, Arnold Sommerfeld and others developed atomic models with discrete energy levels that could account qualitatively for the sharp spectral lines and energy quantization observed in the emission and absorption of light by atoms; their models agreed excellently with the spectrum of hydrogen, but not with those of other atoms. It was only the Compton scattering of a photon by a free electron (which can have no energy levels, since it has no internal structure) that convinced most physicists that light itself was quantized.

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.^[29] To account for the then-available data, 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“.^[27] Nevertheless, the BKS model inspired Werner Heisenberg in his development^[30] of quantum mechanics.

A few physicists persisted^[31] in developing semiclassical models in which electromagnetic radiation is not quantized, but matter obeys 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 elegant photon-correlation experiments.^[32] Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven.

[edit] Wave–particle duality and uncertainty principles

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.^[33] 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 like 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.^[32] 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).

Heisenberg's thought experiment for locating an electron (shown in blue) with a high-resolution gamma-ray microscope. The incoming gamma ray (shown in green) is scattered by the electron up into the microscope's aperture angle θ. The scattered gamma ray is shown in red. Classical optics shows that the electron position can be resolved only up to an uncertainty Δx that depends on θ and the wavelength λ of the incoming light.

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.^[34] 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

$\Delta x \sim \frac{\lambda}{\sin \theta}$

where $θ$ is the aperture angle of the microscope. Thus, the position uncertainty $Δ x$ can be made arbitrarily small by reducing the wavelength. The momentum of the electron is uncertain, since it received a “kick” $Δ p$ from the light scattering from it into the microscope. If light were not quantized into photons, the uncertainty $Δ p$ 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

$\Delta p \sim p_{\mathrm{photon}} \sin\theta = \frac{h}{\lambda} \sin\theta$

giving the product $\Delta x \Delta p \, \sim \, h$ , 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.

The analogous uncertainty principle for photons forbids the simultaneous measurement of the number $n$ of photons (see Fock state and the Second quantization section below) in an electromagnetic wave and the phase $φ$ of that wave

Δ n Δφ > 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.^[35]^[36] 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.^[37] Being massless, they cannot be localized without being destroyed; technically, photons cannot have a position eigenstate $|\mathbf{r} \rangle$ , and, thus, the normal Heisenberg uncertainty principle $\Delta x \Delta p \, > \, h/2$ does not pertain to photons. A few substitute wave functions have been suggested for the photon,^[38]^[39]^[40]^[41] 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.

[edit] Bose–Einstein model of a photon gas

Main articles: 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.^[42] Einstein showed that this modification is equivalent to assuming that photons are rigorously identical and that it implied a “mysterious non-local interaction”,^[43]^[44] 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.^[45]

Photons must obey Bose–Einstein statistics if they are to allow the superposition principle of electromagnetic fields, the condition that Maxwell's equations are linear. All particles are divided into bosons and fermions, depending on whether they have integer or half-integer spin, respectively. The spin-statistics theorem shows that all bosons obey Bose–Einstein statistics, whereas all fermions obey Fermi-Dirac statistics or, equivalently, the Pauli exclusion principle, which states that at most one particle can occupy any given state. Thus, if the photon were a fermion, only one photon could move in a particular direction at a time. This is inconsistent with the experimental observation that lasers can produce coherent light of arbitrary intensity, that is, with many photons moving in the same direction. Hence, the photon must be a boson and obey Bose–Einstein statistics.

[edit] Stimulated and spontaneous emission

Main articles: Stimulated emission and Laser

Stimulated emission (in which photons “clone” themselves) was predicted by Einstein in his kinetic derivation of E=hν, and led to the development of the laser. Einstein's derivation also provoked further developments in the quantum treatment of light, the semiclassical model and quantum electrodynamics (see below).

In 1916, Einstein showed that Planck's quantum hypothesis $E = h ν$ could be derived from a kinetic rate equation.^[7] Consider a cavity in thermal equilibrium and filled with electromagnetic radiation and systems that can emit and absorb that radiation. Thermal equilibrium requires that the number density $ρ(ν)$ of photons with frequency $ν$ is constant in time; hence, the rate of emitting photons of that frequency must equal the rate of absorbing them.

Einstein hypothesized that the rate $R j i$ for a system to absorb a photon of frequency $ν$ and transition from a lower energy $E j$ to a higher energy $E i$ was proportional to the number $N j$ of molecules with energy $E j$ and to the number density $ρ(ν)$ of ambient photons with that frequency

$R_{ji} = N_{j} B_{ji} \rho(\nu) \!$

where $B j i$ is the rate constant for absorption.

More daringly, Einstein hypothesized that the reverse rate $R i j$ for a system to emit a photon of frequency $ν$ and transition from a higher energy $E i$ to a lower energy $E j$ was composed of two terms:

$R_{ij} = N_{i} A_{ij} + N_{i} B_{ij} \rho(\nu) \!$

where $A i j$ is the rate constant for emitting a photon spontaneously, and $B i j$ is the rate constant for emitting it in response to ambient photons (induced or stimulated emission). Einstein showed that Planck's energy formula $E = h ν$ is a necessary consequence of these hypothesized rate equations and the basic requirements that the ambient radiation be in thermal equilibrium with the systems that absorb and emit the radiation and independent of the systems' material composition.

This simple kinetic model was a powerful stimulus for research. Einstein was able to show that $B i j = B j i$ (i.e., the rate constants for induced emission and absorption are equal) and, perhaps more remarkably,

$A_{ij} = \frac{8 \pi h \nu^{3}}{c^{3}} B_{ij}.$

Einstein did not attempt to justify his rate equations but noted that $A i j$ and $B i j$ should be derivable from a “mechanics and electrodynamics modified to accommodate the quantum hypothesis”. This prediction was borne out in quantum mechanics and quantum electrodynamics, respectively; both are required to derive Einstein's rate constants from first principles. Paul Dirac derived the $B i j$ rate constants in 1926 using a semiclassical approach,^[46] and, in 1927, succeeded in deriving all the rate constants from first principles.^[47]^[48] 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;^[49]^[50]^[51] the earlier quantum mechanics (the quantization of material particles moving in a potential) represents the “first quantization”.

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.^[19] Similarly, Einstein hoped for a more complete theory that would leave nothing to chance, beginning his separation^[27] from quantum mechanics. Ironically, Max Born's probabilistic interpretation of the wave function^[52]^[53] was inspired by Einstein's later work searching for a more complete theory.^[54]

[edit] Second quantization

Main article: Quantum field theory

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.^[55] 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 $h\nu \!$ , where $\nu \!$ 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.^[6]

In 1925, Born, Heisenberg and Jordan reinterpreted Debye's concept in a key way.^[56] As may be shown classically, the Fourier modes of the electromagnetic field — a complete set of electromagnetic plane waves indexed by their wave vector $\mathbf{k}$ 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 $E = nh\nu \!$ , where $\nu \!$ is the oscillator frequency. The key new step was to identify an electromagnetic mode with energy $E = nh\nu \!$ as a state with $n \!$ photons, each of energy $h\nu \!$ . This approach gives the correct energy fluctuation formula.

In quantum field theory, probabilities of events are computed by summing over all possible ways in which they can happen, as in the Feynman diagram shown here.

Dirac took this one step further.^[47]^[48] 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 $A_{ij} \!$ and $B_{ij} \!$ 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 BE statistics). In Dirac's time, it was not yet known that all bosons, including photons, must obey BE 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 $E = p \, c \!$ , 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.

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

$|n_{k_0}\rangle\otimes|n_{k_1}\rangle\otimes\dots\otimes|n_{k_n}\rangle\dots$

where $|n_{k_i}\rangle$ represents the state in which $\, n_{k_i}$ photons are in the mode $\, k_i$ . In this notation, the creation of a new photon in mode $\, k_i$ (e.g., emitted from an atomic transition) is written as $|n_{k_i}\rangle \rightarrow |n_{k_i}+1\rangle$ . This notation merely expresses the concept of Born, Heisenberg and Jordan described above, and does not add any physics.

[edit] The photon as a gauge boson

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.^[57] 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 $\pm \hbar$ . 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.^[57]

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.^[58]^[59]^[60] 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.

[edit] Photon structure

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.^[61]

[edit] Contributions to the mass of a system

See also: Mass in special relativity and Gravitation

The energy of a system that emits a photon is decreased by the energy $E$ of the photon as measured in the rest frame of the emitting system, which may result in a reduction in mass in the amount $E / c 2$ . Similarly, the mass of a system that absorbs a photon is increased by a corresponding amount.

This concept is applied in a key prediction of QED, the theory of quantum electrodynamics begun by Dirac (described above). QED is able to predict the magnetic dipole moment of leptons to extremely high accuracy; experimental measurements of these magnetic dipole moments have agreed with these predictions perfectly. The predictions, however, require counting the contributions of virtual photons to the mass of the lepton. Another example of such contributions verified experimentally is the QED prediction of the Lamb shift observed in the hyperfine structure of bound lepton pairs, such as muonium and positronium.

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.

[edit] Photons in matter

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 years to reach the surface; however, once in open space, a photon only takes 8.3 minutes to reach Earth. The factor by which the speed 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. Light of different frequencies may travel through matter at different speeds; this is called dispersion. The polariton propagation speed $v$ equals its group velocity, which is the derivative of the energy with respect to momentum.

$v = \frac{d\omega}{dk} = \frac{dE}{dp}$

Retinal straightens after absorbing a photon γ of the correct wavelength

where, as above, $E$ and $p$ are the polariton's energy and momentum magnitude, and $ω$ and $k$ are its angular frequency and wave number, respectively. In some cases, the dispersion 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.

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.

[edit] 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.

Planck's energy formula $E = h ν$ 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.

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.

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, which is used to measure molecular distances.

[edit] Recent research

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 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^[15] and mass.^[62] The best universally accepted upper limits on the photon charge and mass are 5×10⁻⁵² C (or 3×10⁻³³ times the elementary charge) and 1.8×10⁻⁵⁰ kg, respectively. ^{[citation needed]}

Much research has been devoted to applications of photons in the field of quantum optics. Photons seem well-suited to be elements of an ultra-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.