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Photon

2007 Schools Wikipedia Selection. Related subjects: General Physics

          The word "light" is defined here as "electromagnetic radiation
          of any wavelength"; thus, X-rays, gamma rays, ultraviolet light,
          microwaves, radio waves, and visible light are all forms of
          light.

                        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
    Mean lifetime:  Stable
   Electric charge: 0
        Spin:       1

   In modern physics, the photon is the elementary particle responsible
   for electromagnetic phenomena. It mediates electromagnetic interactions
   and makes up all forms of light. The photon has zero invariant mass and
   travels at the constant speed c, the speed of light in empty space.
   However, in the presence of matter, a photon can be slowed or even
   absorbed, transferring energy and momentum proportional to its
   frequency. Like all quanta, the photon has both wave and particle
   properties; it exhibits wave–particle duality.

   The modern concept of the photon was developed gradually (1905–17) 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. 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. Photons have many applications in
   technology 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.

Nomenclature

   The photon was originally called a "light quantum" (das Lichtquant) by
   Albert Einstein. The modern name "photon" derives from the Greek word
   for light, φῶς, and was coined in 1926 by the physical chemist Gilbert
   N. Lewis, who published a speculative theory 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.

   In physics, a photon is usually denoted by the symbol \gamma \! , the
   Greek letter gamma. This symbol for the photon likely derives from
   gamma rays, which were discovered and named in 1900 by Villard and
   shown to be a form of light in 1914 by Rutherford and Andrade. 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.

Physical properties

   The photon is massless, has no electric charge and does not decay
   spontaneously in empty space. A photon has two possible polarization
   states and is described by three continuous parameters: the components
   of its wave vector, which determine its wavelength \lambda \! and its
   direction of propagation. Photons are emitted in many natural
   processes, e.g., when a charge is accelerated, when an atom or a
   nucleus jumps from a higher to 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 atomic
   or nuclear transitions to a higher energy level.

   Since the photon is massless, the photon moves at c \! (the speed of
   light in empty space) 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 an
   invariant mass m \! would be 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 \!

          \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 centre 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. 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.

Historical development

   Thomas Young's double-slit experiment in 1805 showed that light can act
   as a wave, helping to defeat early particle theories of light.
   Enlarge
   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), Robert Hooke
   (1665), and Christian Huygens (1678); however, particle models remained
   dominant, chiefly due to the influence of Isaac Newton. 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. In 1865, James Clerk Maxwell's
   prediction that light was an electromagnetic wave — which was confirmed
   experimentally in 1888 by Heinrich Hertz's detection of radio waves —
   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.
   Enlarge
   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 can be provoked only by
   light of frequency higher than a certain threshold; light of lower
   frequency, no matter how intense, is incapable of exciting 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 culminated in Max
   Planck's hypothesis 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, some form of energy
   quantization must be assumed to account for the thermal equilibrium
   observed between matter and electromagnetic radiation.

   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. 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. In 1909 and 1916,
   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 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, and was solved in quantum electrodynamics and its successor, the
   Standard Model.

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.
   Enlarge
   Up to 1923, most physicists were reluctant to accept that
   electromagnetic radiation itself was quantized. Instead, they tried to
   account for photon behaviour 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. However, before Compton's experiment
   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, Planck and Millikan.) 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. 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". Nevertheless, the BKS model inspired Werner Heisenberg in
   his development of quantum mechanics.

   A few physicists persisted 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 1970's and
   1980's by elegant photon-correlation experiments. 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 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. 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. 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.
   Enlarge
   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. 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 Δx can be made arbitrarily small by reducing the
   wavelength. The momentum of the electron is uncertain, since it
   received a "kick" Δp from the light scattering from it into the
   microscope. If light were not quantized into photons, the uncertainty
   Δp 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, all the 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

          ΔnΔφ > 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. 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.
   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, 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.

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. Einstein showed that this
   modification is equivalent to assuming that photons are rigorously
   identical and that it implied a "mysterious non-local interaction", 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.

   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.

Stimulated and spontaneous emission

   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).
   Enlarge
   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 = hν could
   be derived from a kinetic rate equation. 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[ji] 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[ji] is the rate constant for absorption.

   More daringly, Einstein hypothesized that the reverse rate R[ij] 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[ij] is the rate constant for emitting a photon spontaneously,
   and B[ij] is the rate constant for emitting it in response to ambient
   photons ( induced or stimulated emission). Einstein showed that
   Planck's energy formula E = hν 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[ij] = B[ji] (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[ij] and B[ij] 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[ij] rate
   constants in 1926 using a semiclassical approach, and, in 1927,
   succeeded in deriving all the rate constants from first principles.
   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; 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.
   Similarly, Einstein hoped for a more complete theory that would leave
   nothing to chance, beginning his separation from quantum mechanics.
   Ironically, Max Born's probabilistic interpretation of the wave
   function was inspired by Einstein's later work searching for a more
   complete theory.

Second quantization

   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.
   Enlarge
   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. 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.

   In 1925, Born, Heisenberg and Jordan reinterpreted Debye's concept in a
   key way. 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.
   Enlarge
   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. 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 make infinite contributions to the sum, a problem that
   was overcome in quantum electrodynamics by 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.

The photon as a gauge boson

   The electromagnetic field can be understood as a gauge theory, i.e., as
   a field that results from requiring that a symmetry hold independently
   at every position in spacetime. 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.

   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 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. 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.

Contributions to the invariant mass of a system

   Although the photon is itself massless, it adds to the invariant mass
   of any system to which it belongs; this is true for every form of
   energy, as predicted by the special theory of relativity. For example,
   the invariant mass of a system that emits a photon is decreased by an
   amount E / c^2 upon emission, where E is the energy of the photon in
   the frame of the emitting system. Similarly, the invariant mass of a
   system that absorbs a photon is increased by a corresponding amount
   based on the energy of the photon in the frame of the absorbing system.

   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 invariant
   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.

Photons in matter

   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
   Enlarge
   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. 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 (C[20]H[28]O, 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.

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 = hν 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.

Recent research

   The fundamental nature of the photon is believed to be understood
   theoretically; the prevailing Standard Model predicts that the photon
   is a massless, chargeless boson of spin 1 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 hopes 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 nonzero 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 and mass.
   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.

   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.

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