## Introduction

A photon is an elementary particle describing the quantum nature of light and all other forms of electromagnetic radiation. It is the force carrier for the electromagnetic force. The effects of this force are easily observed at both the microscopic, and macroscopic level resulting from the fact that the photon has zero rest mass; allowing for long distance interactions. Photons are best explained through quantum mechanics like all other elementary particles that exhibit wave–particle duality.

Albert Einstein was the first to develop the modern concept of a photon through experimental observations. This new concept did not agree with the classical wave model of light. In contrast with the wave theory of light, 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 with each other. The new photon model also accounted for anomalous observations, such as black-body radiation, a phenomenon that other physicists, such as Max Planck, had sought to explain using semiclassical models. The “Compton scattering experiment” of single photons by electrons, first carried out in 1923, validated Einstein’s hypothesis that light itself is quantised.

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. This gauge symmetry allows us to determine the intrinsic properties of photons, such as charge, mass and spin. Moreover, the photon concept has led to various advances in experimental and theoretical physics, such as lasers, Bose-Einstein condensation, quantum field theory, and the probabilistic interpretation of quantum mechanics. As we shall see later on, photons are also being applied in 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.

## What are the basic physical properties of a photon?

A photon has zero mass, no electric charge [1] and has two possible polarisation states. In quantum field theory the momentum representation is preferred. This representation shall be used to derive said properties about photons. In this representation, a photon is described by its wave vector, which determines its wavelength and its direction of propagation. A photon’s wave vector may not be zero and can be represented either as a spatial three-vector or as a (relativistic) four-vector usually represented by the light cone shown in Figure 1.

Figure 1. 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 polarisation. [29]

Photons are emitted in many naturally occurring radiative processes. During a molecular, atomic or nuclear transition to a lower energy level, photons of various energy are emitted. These range from radio waves to gamma rays. A photon can also be emitted when a particle and its corresponding antiparticle are annihilated. In deriving equations for the energy of a photon and other particles such as electrons, concepts from quantum mechanics and special relativity are used. Special relativity is useful since it predicts the momentum and wavelength of the “particle”. Together with quantum mechanics these statistics yield the general equations given below where *m* is the rest mass and *E* the total energy. [2] It is important to note for now that photons have zero rest mass. The reason for this shall be explained later on. 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 [3]:

Moreover the energy and momentum of a photon depend only on its frequency or inversely, its wavelength :

where *k* is the wavenumber and is the reduced Planck constant. [4] 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. [5] The magnitude of its spin is and the component measured along its direction of motion, its helicity (a combination of the spin and the linear motion of a subatomic particle), must be . [6]

## The photon as a gauge boson

In particle physics, a gauge boson is a force carrier, a bosonic particle that carries any of the fundamental interactions of nature.

The Standard Model of particle physics recognises four kinds of gauge bosons: photons, which carry the electromagnetic interaction; W and Z bosons, which carry the weak interaction; and gluons, which carry the strong interaction (refer to Figure 2). These elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of gauge bosons. In a quantised gauge theory, gauge bosons are quanta of the gauge fields. Consequently, there are as many gauge bosons as there are generators of the gauge field.

Figure 2. The Standard Model of elementary particles, with the gauge bosons in the fourth column in red. [30]

The electromagnetic field can be understood as a gauge field, a field that results from requiring that gauge symmetry holds independently at every position in spacetime. In quantum electrodynamics, the gauge group is the Abelian U(1) symmetry group 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 only gauge boson in this field is the photon. For technical reasons involving gauge invariance, gauge bosons are required to be massless particles. The other gauge bosons on the other hand have mass, owing to a mechanism that breaks their SU(2) gauge symmetry. [7, 8, 9] From this we deduce that all other quantum numbers of the photon such as lepton number, baryon number, flavour quantum numbers as well as charge take on zero values. The only quantum number to have integer values is spin, having values of . From spin we can further derive the helicity of the photon which is found to take on values of . These two spin components correspond to the classical concepts of right-handed and left-handed circularly polarised light. However, the transient virtual photons of quantum electrodynamics may also adopt unphysical polarisation states. [10]

## Wave–particle duality and uncertainty principles

Photons, like all quantum objects, exhibit both wave-like and particle-like properties. The photon displays clearly wave-like phenomena such as diffraction and interference on the length scale of its wavelength. Young’s double slit experiment is clear evidence of this. To account for the particle interpretation we make use of probability distribution functions which behave in accordance to Maxwell’s equations. [11]

Experiments have shown that the photon is not a short pulse of electromagnetic radiation since it does not spread out as it propagates, nor does it divide once it encounters a beam splitter. [12] On the other hand, the photon seems to be a point-like particle since it is absorbed or emitted as a whole by arbitrarily small systems, having relatively very large wavelengths. Such systems include the atomic nucleus which is approximately m across and even the point-like electron.

Nevertheless, the photon is not a point-like particle whose trajectory is shaped probabilistically by the electromagnetic field. The important distinction between particles such as electrons and photons is that electrons obey Fermi-Dirac statistics, whereas photons obey Bose-Einstein statistics. A consequence of Fermi-Dirac statistics is that no two electrons in the same interacting system can be in the same state, that is, have precisely the same physical properties. Bose-Einstein statistics impose no such prohibition, hence identical photons with the same energy, momentum, and polarisation can occur together in large numbers as they do for example in a laser cavity.

## Does the photon have mass?

Photons are said to be massless but the mass of the photon is set to zero in order to satisfy conditions of special relativity as we shall see in this section. Consider an isolated system of one particle which is being accelerated to some velocity *v*. Using Newtonian mechanics we get that the particle’s momentum *p* must be proportional to *v* where the proportionality constant is simply the particle’s mass , such that . [13, 14]

In special relativity, it turns out that although *p* and *v* still point in the same direction, these two vectors are no longer proportional. To solve this, the best alternate way to define *p* is through the particle’s “relativistic mass” . This yields:

which tells us that when the particle is at rest, its relativistic mass has a minimum value called the “rest mass”, . As the particle is then accelerated to higher speeds, its relativistic mass increases indefinitely. It also turns out that in special relativity, we are able to define the concept of energy *E* such that it has simple and well-defined properties similar to those used in Newtonian mechanics. [13] [14] In fact, when a particle is accelerated such that it has some momentum *p* (the length of the vector *p* and relativistic mass , then its energy *E* is,

There are two interesting cases resulting from the last equation (which is just a slightly different equation to the one given in the previous section).

- If the particle is at rest, then , and .
- If we set the rest mass equal to zero (regardless of whether or not that’s a reasonable thing to do), then so that there is no distinction between a photon’s total energy and its kinetic energy. [12]

In reality photons cannot be brought to rest, so the idea of rest mass doesn’t really apply to them. We can thus bring these “particles” of light into the fold of equation (1) by considering them to have no rest mass. By so doing, equation (1) gives the correct expression for light, , and it can now be used as a fully general equation. [13] [14]

The big question is whether we can verify the rest mass of a photon to be zero through experiment. The truth is that such an experiment is unattainable. The best we can do is to place limits on the maximum value of the rest mass of a photon. A non-zero rest mass would introduce a small damping factor in the inverse square Coulomb law of electrostatic forces. That means the electrostatic force would be weaker over very large distances. [13][14]

## Heisenberg’s thought Experiment

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 quantisation of light into photons and even the frequency dependence of the photon’s energy and momentum. [15] Heisenberg’s thought experiment goes as follows: 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 . 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 quantised 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 - a violation of 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. Within the same analogy, the uncertainty principle for locating a photon forbids the simultaneous measurement of the number of photons in an electromagnetic wave and the phase (\phi) of that wave such that

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. [16, 17] For one thing, they are mathematically different. 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. [18] Being massless, they cannot be localised without being destroyed. Technically, photons cannot have a position eigenstate r and thus, the normal Heisenberg uncertainty principle does not pertain to photons.

## Einstein model of a photon gas

Max Planck (1858-1947), in a paper presented on December 14, 1900, discovered that the observed spectrum of the radiation emitted from a black-body could not be explained in terms of classical electromagnetic theory. It was not a minor problem: Classical theory predicted an infinite energy of radiation, a disagreement so gross it was called the “ultraviolet catastrophe”. [19] This led to the new definition of a “black-body”. A black-body is a body which absorbs all types of radiation incident on it. This discovery marked the very beginning of quantum mechanics.

The most common example of black-body radiation is a photon gas, whereby the energy distribution is established by the interaction of photons with matter, usually with the walls of the container. In such a system the photons and the container are in thermal equilibrium with each other. [20] A Bose-Einstein condensate (BEC) is a state of matter in which separate atoms or subatomic particles, cooled to near absolute zero, coalesce into a single quantum mechanical entity which can be described by a wave function on a near-macroscopic scale. [21] This form of matter was predicted in 1924 by Albert Einstein and seen for the first time at room temperature for photons trapped in a dye between mirrors in 2010. [20]

## Photons in matter

Any “explanation” of how photons travel through matter has to take into account why different arrangements of matter are transparent or opaque at different wavelengths and why individual photons behave in the same way as large groups. Explanations that invoke “absorption” and “re-emission” have to provide an answer for the directionality of the photons (diffraction, reflection) and further illustrate how entangled photon pairs can travel through matter without their quantum state collapsing.

The simplest explanation is that light traveling through transparent matter does so at a speed lower than *c*, the speed of light in a vacuum. In addition, light can also undergo scattering and absorption. The factor by which the speed of light is decreased in a material is called the refractive index of the material. 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 their frequency. Photons can also be absorbed by nuclei, atoms or molecules, provoking transitions between their energy levels. It is important to note though that since photons are electrically neutral, they do not steadily lose energy via Coulombic interactions with atomic electrons as charged particles do. [26]

## Technological applications and recent research

Photons have many applications in technology. One of the most well-known if these is the laser. There are various methods one can use to detect individual photons. For example, the classic photomultiplier tube exploits the photoelectric effect in order to detect photons. The 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 ionise gas molecules, causing a detectable change in conductivity. [27]

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 current focus of research. The study of nonlinear optical processes is also another active research area. Photons are also essential in some aspects of optical communication, especially for quantum cryptography. [28]

## Conclusion

Although the nature of a photon has been the subject of much debate throughout the past, the photon is now thought of as an elementary particle describing the quantum nature of light and all other forms of electromagnetic radiation moving at a constant speed of *c* = 2.998 x 10^{8} m/s. It has been deduced that the photon has zero mass and zero rest energy in order to confirm with laws of special relativity. It also has no electric charge and carries spin angular momentum that does not depend on its frequency. Nowadays photons have many applications in technology and are used extensively in research.

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[4] Davison E. Soper, Electromagnetic radiation is made of photons, Institute of Theoretical Science, University of Oregon

[5] This property was experimentally verified by Raman and Bhagavantam in 1931: Raman, C.V.; Bhagavantam, S. (1931). “Experimental proof of the spin of the photon” (PDF). Indian Journal of Physics 6: 353. E.g., section 1.3.3.2 in Burgess, C.; Moore, G. (2007). The Standard

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[28] Introductory-level material on the various sub-fields of quantum optics can be found in Fox, M. (2006). Quantum Optics: An Introduction. Oxford University Press.

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