Nuclear force
The nuclear force is a force that acts between the protons and neutrons of atoms. Neutrons and protons, both nucleons, are affected by the nuclear force identically. Since protons have charge +1 e, they experience an electric force that tends to push them apart, but at short range the attractive nuclear force is strong enough to overcome the electromagnetic force; the nuclear force binds nucleons into atomic nuclei. The nuclear force is powerfully attractive between nucleons at distances of about 1 femtometre, but it decreases to insignificance at distances beyond about 2.5 fm. At distances less than 0.7 fm, the nuclear force becomes repulsive. This repulsive component is responsible for the physical size of nuclei, since the nucleons can come no closer than the force allows. By comparison, the size of an atom, measured in angstroms, is five orders of magnitude larger; the nuclear force is not simple, since it depends on the nucleon spins, has a tensor component, may depend on the relative momentum of the nucleons.
The strong nuclear force is one of the fundamental forces of nature. The nuclear force plays an essential role in storing energy, used in nuclear power and nuclear weapons. Work is required to bring charged protons together against their electric repulsion; this energy is stored when the protons and neutrons are bound together by the nuclear force to form a nucleus. The mass of a nucleus is less than the sum total of the individual masses of the protons and neutrons; the difference in masses is known as the mass defect, which can be expressed as an energy equivalent. Energy is released; this energy is the electromagnetic potential energy, released when the nuclear force no longer holds the charged nuclear fragments together. A quantitative description of the nuclear force relies on equations that are empirical; these equations model the internucleon potential energies, or potentials. The constants for the equations are phenomenological, that is, determined by fitting the equations to experimental data.
The internucleon potentials attempt to describe the properties of nucleon–nucleon interaction. Once determined, any given potential can be used in, e.g. the Schrödinger equation to determine the quantum mechanical properties of the nucleon system. The discovery of the neutron in 1932 revealed that atomic nuclei were made of protons and neutrons, held together by an attractive force. By 1935 the nuclear force was conceived to be transmitted by particles called mesons; this theoretical development included a description of the Yukawa potential, an early example of a nuclear potential. Mesons, predicted by theory, were discovered experimentally in 1947. By the 1970s, the quark model had been developed, by which the mesons and nucleons were viewed as composed of quarks and gluons. By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. While the nuclear force is associated with nucleons, more this force is felt between hadrons, or particles composed of quarks.
At small separations between nucleons the force becomes repulsive, which keeps the nucleons at a certain average separation if they are of different types. This repulsion arises from the Pauli exclusion force for identical nucleons. A Pauli exclusion force occurs between quarks of the same type within nucleons, when the nucleons are different. At distances larger than 0.7 fm the force becomes attractive between spin-aligned nucleons, becoming maximal at a center–center distance of about 0.9 fm. Beyond this distance the force drops exponentially, until beyond about 2.0 fm separation, the force is negligible. Nucleons have a radius of about 0.8 fm. At short distances, the attractive nuclear force is stronger than the repulsive Coulomb force between protons. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm.
The nuclear force has a spin-dependent component. The force is stronger for particles with their spins aligned than for those with their spins anti-aligned. If two particles are the same, such as two neutrons or two protons, the force is not enough to bind the particles, since the spin vectors of two particles of the same type must point in opposite directions when the particles are near each other and are in the same quantum state; this requirement for fermions stems from the Pauli exclusion principle. For fermion particles of different types, such as a proton and neutron, particles may be close to each other and have aligned spins without violating the Pauli exclusion principle, the nuclear force may bind them, since the nuclear force is much stronger for spin-aligned particles, but if the particles' spins are anti-aligned the nuclear force is too weak to bind them if they are of different types. The nuclear force has a tensor component which depends on the interaction between the nucleon spins and the angular momentum of the nucleons, leading to deformation from a simple spherical shape
Neutrino
A neutrino is a fermion that interacts only via the weak subatomic force and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is so small that it was long thought to be zero; the mass of the neutrino is much smaller than that of the other known elementary particles. The weak force has a short range, the gravitational interaction is weak, neutrinos, as leptons, do not participate in the strong interaction. Thus, neutrinos pass through normal matter unimpeded and undetected. Weak interactions create neutrinos in one of three leptonic flavors: electron neutrinos, muon neutrinos, or tau neutrinos, in association with the corresponding charged lepton. Although neutrinos were long believed to be massless, it is now known that there are three discrete neutrino masses with different tiny values, but they do not correspond uniquely to the three flavors. A neutrino created with a specific flavor is in an associated specific quantum superposition of all three mass states.
As a result, neutrinos oscillate between different flavors in flight. For example, an electron neutrino produced in a beta decay reaction may interact in a distant detector as a muon or tau neutrino. Although only differences of squares of the three mass values are known as of 2016, cosmological observations imply that the sum of the three masses must be less than one millionth that of the electron. For each neutrino, there exists a corresponding antiparticle, called an antineutrino, which has half-integer spin and no electric charge, they are distinguished from the neutrinos by having opposite signs of lepton chirality. To conserve total lepton number, in nuclear beta decay, electron neutrinos appear together with only positrons or electron-antineutrinos, electron antineutrinos with electrons or electron neutrinos. Neutrinos are created by various radioactive decays, including in beta decay of atomic nuclei or hadrons, nuclear reactions such as those that take place in the core of a star or artificially in nuclear reactors, nuclear bombs or particle accelerators, during a supernova, in the spin-down of a neutron star, or when accelerated particle beams or cosmic rays strike atoms.
The majority of neutrinos in the vicinity of the Earth are from nuclear reactions in the Sun. In the vicinity of the Earth, about 65 billion solar neutrinos per second pass through every square centimeter perpendicular to the direction of the Sun. For study, neutrinos can be created artificially with nuclear reactors and particle accelerators. There is intense research activity involving neutrinos, with goals that include the determination of the three neutrino mass values, the measurement of the degree of CP violation in the leptonic sector. Neutrinos can be used for tomography of the interior of the earth; the neutrino was postulated first by Wolfgang Pauli in 1930 to explain how beta decay could conserve energy and angular momentum. In contrast to Niels Bohr, who proposed a statistical version of the conservation laws to explain the observed continuous energy spectra in beta decay, Pauli hypothesized an undetected particle that he called a "neutron", using the same -on ending employed for naming both the proton and the electron.
He considered that the new particle was emitted from the nucleus together with the electron or beta particle in the process of beta decay. James Chadwick discovered a much more massive neutral nuclear particle in 1932 and named it a neutron leaving two kinds of particles with the same name. Earlier Pauli had used the term "neutron" for both the neutral particle that conserved energy in beta decay, a presumed neutral particle in the nucleus; the word "neutrino" entered the scientific vocabulary through Enrico Fermi, who used it during a conference in Paris in July 1932 and at the Solvay Conference in October 1933, where Pauli employed it. The name was jokingly coined by Edoardo Amaldi during a conversation with Fermi at the Institute of Physics of via Panisperna in Rome, in order to distinguish this light neutral particle from Chadwick's heavy neutron. In Fermi's theory of beta decay, Chadwick's large neutral particle could decay to a proton and the smaller neutral particle: n0 → p+ + e− + νeFermi's paper, written in 1934, unified Pauli's neutrino with Paul Dirac's positron and Werner Heisenberg's neutron–proton model and gave a solid theoretical basis for future experimental work.
The journal Nature rejected Fermi's paper, saying that the theory was "too remote from reality". He submitted the paper to an Italian journal, which accepted it, but the general lack of interest in his theory at that early date caused him to switch to experimental physics. By 1934 there was experimental evidence against Bohr's idea that energy conservation is invalid for beta decay: At the Solvay conference of that year, measurements of the energy spectra of beta particles were reported, showing that there is a strict limit on the energy of electrons from each type of beta decay; such a limit is not expected if the conservation of energy is invalid, in which case any amount of energy would be statistically available in at least a few decays. The natural explanation of the beta decay spectrum as first measured in 1934 was that only a limited amount of en
Speed of light
The speed of light in vacuum denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299,792,458 metres per second, it is exact because by international agreement a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 second. According to special relativity, c is the maximum speed at which all conventional matter and hence all known forms of information in the universe can travel. Though this speed is most associated with light, it is in fact the speed at which all massless particles and changes of the associated fields travel in vacuum; such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. In the special and general theories of relativity, c interrelates space and time, appears in the famous equation of mass–energy equivalence E = mc2; the speed at which light propagates through transparent materials, such as glass or air, is less than c.
The ratio between c and the speed v at which light travels in a material is called the refractive index n of the material. For example, for visible light the refractive index of glass is around 1.5, meaning that light in glass travels at c / 1.5 ≈ 200,000 km/s. For many practical purposes and other electromagnetic waves will appear to propagate instantaneously, but for long distances and sensitive measurements, their finite speed has noticeable effects. In communicating with distant space probes, it can take minutes to hours for a message to get from Earth to the spacecraft, or vice versa; the light seen from stars left them many years ago, allowing the study of the history of the universe by looking at distant objects. The finite speed of light limits the theoretical maximum speed of computers, since information must be sent within the computer from chip to chip; the speed of light can be used with time of flight measurements to measure large distances to high precision. Ole Rømer first demonstrated in 1676 that light travels at a finite speed by studying the apparent motion of Jupiter's moon Io.
In 1865, James Clerk Maxwell proposed that light was an electromagnetic wave, therefore travelled at the speed c appearing in his theory of electromagnetism. In 1905, Albert Einstein postulated that the speed of light c with respect to any inertial frame is a constant and is independent of the motion of the light source, he explored the consequences of that postulate by deriving the theory of relativity and in doing so showed that the parameter c had relevance outside of the context of light and electromagnetism. After centuries of precise measurements, in 1975 the speed of light was known to be 299792458 m/s with a measurement uncertainty of 4 parts per billion. In 1983, the metre was redefined in the International System of Units as the distance travelled by light in vacuum in 1/299792458 of a second; the speed of light in vacuum is denoted by a lowercase c, for "constant" or the Latin celeritas. In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used c for a different constant shown to equal √2 times the speed of light in vacuum.
The symbol V was used as an alternative symbol for the speed of light, introduced by James Clerk Maxwell in 1865. In 1894, Paul Drude redefined c with its modern meaning. Einstein used V in his original German-language papers on special relativity in 1905, but in 1907 he switched to c, which by had become the standard symbol for the speed of light. Sometimes c is used for the speed of waves in any material medium, c0 for the speed of light in vacuum; this subscripted notation, endorsed in official SI literature, has the same form as other related constants: namely, μ0 for the vacuum permeability or magnetic constant, ε0 for the vacuum permittivity or electric constant, Z0 for the impedance of free space. This article uses c for the speed of light in vacuum. Since 1983, the metre has been defined in the International System of Units as the distance light travels in vacuum in 1⁄299792458 of a second; this definition fixes the speed of light in vacuum at 299,792,458 m/s. As a dimensional physical constant, the numerical value of c is different for different unit systems.
In branches of physics in which c appears such as in relativity, it is common to use systems of natural units of measurement or the geometrized unit system where c = 1. Using these units, c does not appear explicitly because multiplication or division by 1 does not affect the result; the speed at which light waves propagate in vacuum is independent both of the motion of the wave source and of the inertial frame of reference of the observer. This invariance of the speed of light was postulated by Einstein in 1905, after being motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous aether, it is only possible to verify experimentally that the two-way speed of light is frame-independent, because it is impossible to measure the one-way speed of light without some convention as to how clocks at the source and at the detector should be synchronized. However
Electronvolt
In physics, the electronvolt is a unit of energy equal to 1.6×10−19 joules in SI units. The electronvolt was devised as a standard unit of measure through its usefulness in electrostatic particle accelerator sciences, because a particle with electric charge q has an energy E = qV after passing through the potential V. Like the elementary charge on which it is based, it is not an independent quantity but is equal to 1 J/C √2hα / μ0c0, it is a common unit of energy within physics used in solid state, atomic and particle physics. It is used with the metric prefixes milli-, kilo-, mega-, giga-, tera-, peta- or exa-. In some older documents, in the name Bevatron, the symbol BeV is used, which stands for billion electronvolts. An electronvolt is the amount of kinetic energy gained or lost by a single electron accelerating from rest through an electric potential difference of one volt in vacuum. Hence, it has a value of one volt, 1 J/C, multiplied by the electron's elementary charge e, 1.6021766208×10−19 C.
Therefore, one electronvolt is equal to 1.6021766208×10−19 J. The electronvolt, as opposed to volt, is not an SI unit, its derivation is empirical, which means its value in SI units must be obtained by experiment and is therefore not known unlike the litre, the light-year and such other non-SI units. Electronvolt is a unit of energy; the SI unit for energy is joule. 1 eV is equal to 1.6021766208×10−19 J. By mass–energy equivalence, the electronvolt is a unit of mass, it is common in particle physics, where units of mass and energy are interchanged, to express mass in units of eV/c2, where c is the speed of light in vacuum. It is common to express mass in terms of "eV" as a unit of mass using a system of natural units with c set to 1; the mass equivalent of 1 eV/c2 is 1 eV / c 2 = ⋅ 1 V 2 = 1.783 × 10 − 36 kg. For example, an electron and a positron, each with a mass of 0.511 MeV/c2, can annihilate to yield 1.022 MeV of energy. The proton has a mass of 0.938 GeV/c2. In general, the masses of all hadrons are of the order of 1 GeV/c2, which makes the GeV a convenient unit of mass for particle physics: 1 GeV/c2 = 1.783×10−27 kg.
The unified atomic mass unit, 1 gram divided by Avogadro's number, is the mass of a hydrogen atom, the mass of the proton. To convert to megaelectronvolts, use the formula: 1 u = 931.4941 MeV/c2 = 0.9314941 GeV/c2. In high-energy physics, the electronvolt is used as a unit of momentum. A potential difference of 1 volt causes an electron to gain an amount of energy; this gives rise to usage of eV as units of momentum, for the energy supplied results in acceleration of the particle. The dimensions of momentum units are LMT−1; the dimensions of energy units are L2MT−2. Dividing the units of energy by a fundamental constant that has units of velocity, facilitates the required conversion of using energy units to describe momentum. In the field of high-energy particle physics, the fundamental velocity unit is the speed of light in vacuum c. By dividing energy in eV by the speed of light, one can describe the momentum of an electron in units of eV/c; the fundamental velocity constant c is dropped from the units of momentum by way of defining units of length such that the value of c is unity.
For example, if the momentum p of an electron is said to be 1 GeV the conversion to MKS can be achieved by: p = 1 GeV / c = ⋅ ⋅ = 5.344286 × 10 − 19 kg ⋅ m / s. In particle physics, a system of "natural units" in which the speed of light in vacuum c and the reduced Planck constant ħ are dimensionless and equal to unity is used: c = ħ = 1. In these units, both distances and times are expressed in inverse energy units (while energy and mass are expressed in the same units, see mas
Electron
The electron is a subatomic particle, symbol e− or β−, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, are thought to be elementary particles because they have no known components or substructure; the electron has a mass, 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum of a half-integer value, expressed in units of the reduced Planck constant, ħ; as it is a fermion, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light; the wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy. Electrons play an essential role in numerous physical phenomena, such as electricity, magnetism and thermal conductivity, they participate in gravitational and weak interactions.
Since an electron has charge, it has a surrounding electric field, if that electron is moving relative to an observer, it will generate a magnetic field. Electromagnetic fields produced from other sources will affect the motion of an electron according to the Lorentz force law. Electrons absorb energy in the form of photons when they are accelerated. Laboratory instruments are capable of trapping individual electrons as well as electron plasma by the use of electromagnetic fields. Special telescopes can detect electron plasma in outer space. Electrons are involved in many applications such as electronics, cathode ray tubes, electron microscopes, radiation therapy, gaseous ionization detectors and particle accelerators. Interactions involving electrons with other subatomic particles are of interest in fields such as chemistry and nuclear physics; the Coulomb force interaction between the positive protons within atomic nuclei and the negative electrons without, allows the composition of the two known as atoms.
Ionization or differences in the proportions of negative electrons versus positive nuclei changes the binding energy of an atomic system. The exchange or sharing of the electrons between two or more atoms is the main cause of chemical bonding. In 1838, British natural philosopher Richard Laming first hypothesized the concept of an indivisible quantity of electric charge to explain the chemical properties of atoms. Irish physicist George Johnstone Stoney named this charge'electron' in 1891, J. J. Thomson and his team of British physicists identified it as a particle in 1897. Electrons can participate in nuclear reactions, such as nucleosynthesis in stars, where they are known as beta particles. Electrons can be created through beta decay of radioactive isotopes and in high-energy collisions, for instance when cosmic rays enter the atmosphere; the antiparticle of the electron is called the positron. When an electron collides with a positron, both particles can be annihilated, producing gamma ray photons.
The ancient Greeks noticed. Along with lightning, this phenomenon is one of humanity's earliest recorded experiences with electricity. In his 1600 treatise De Magnete, the English scientist William Gilbert coined the New Latin term electrica, to refer to those substances with property similar to that of amber which attract small objects after being rubbed. Both electric and electricity are derived from the Latin ēlectrum, which came from the Greek word for amber, ἤλεκτρον. In the early 1700s, Francis Hauksbee and French chemist Charles François du Fay independently discovered what they believed were two kinds of frictional electricity—one generated from rubbing glass, the other from rubbing resin. From this, du Fay theorized that electricity consists of two electrical fluids and resinous, that are separated by friction, that neutralize each other when combined. American scientist Ebenezer Kinnersley also independently reached the same conclusion. A decade Benjamin Franklin proposed that electricity was not from different types of electrical fluid, but a single electrical fluid showing an excess or deficit.
He gave them the modern charge nomenclature of negative respectively. Franklin thought of the charge carrier as being positive, but he did not identify which situation was a surplus of the charge carrier, which situation was a deficit. Between 1838 and 1851, British natural philosopher Richard Laming developed the idea that an atom is composed of a core of matter surrounded by subatomic particles that had unit electric charges. Beginning in 1846, German physicist William Weber theorized that electricity was composed of positively and negatively charged fluids, their interaction was governed by the inverse square law. After studying the phenomenon of electrolysis in 1874, Irish physicist George Johnstone Stoney suggested that there existed a "single definite quantity of electricity", the charge of a monovalent ion, he was able to estimate the value of this elementary charge e by means of Faraday's laws of electrolysis. However, Stoney could not be removed. In 1881, German physicist Hermann von Helmholtz argued that both positive and negative charges were divided into elementary parts, each of which "behaves like atoms of electricity".
Stoney coined the term
Fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. These particles obey the Pauli exclusion principle. Fermions include all quarks and leptons, as well as all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons. A fermion can be an elementary particle, such as the electron, or it can be a composite particle, such as the proton. According to the spin-statistics theorem in any reasonable relativistic quantum field theory, particles with integer spin are bosons, while particles with half-integer spin are fermions. In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is referred to as the spin statistics relation is in fact a spin statistics-quantum number relation; as a consequence of the Pauli exclusion principle, only one fermion can occupy a particular quantum state at any given time. If multiple fermions have the same spatial probability distribution at least one property of each fermion, such as its spin, must be different.
Fermions are associated with matter, whereas bosons are force carrier particles, although in the current state of particle physics the distinction between the two concepts is unclear. Weakly interacting fermions can display bosonic behavior under extreme conditions. At low temperature fermions show superfluidity for uncharged particles and superconductivity for charged particles. Composite fermions, such as protons and neutrons, are the key building blocks of everyday matter; the name fermion was coined by English theoretical physicist Paul Dirac from the surname of Italian physicist Enrico Fermi. The Standard Model recognizes two types of elementary fermions: leptons. In all, the model distinguishes 24 different fermions. There are six quarks, six leptons, along with the corresponding antiparticle of each of these. Mathematically, fermions come in three types: Weyl fermions, Dirac fermions, Majorana fermions. Most Standard Model fermions are believed to be Dirac fermions, although it is unknown at this time whether the neutrinos are Dirac or Majorana fermions.
Dirac fermions can be treated as a combination of two Weyl fermions. In July 2015, Weyl fermions have been experimentally realized in Weyl semimetals. Composite particles can be fermions depending on their constituents. More because of the relation between spin and statistics, a particle containing an odd number of fermions is itself a fermion, it will have half-integer spin. Examples include the following: A baryon, such as the proton or neutron, contains three fermionic quarks and thus it is a fermion; the nucleus of a carbon-13 atom is therefore a fermion. The atom helium-3 is made of two protons, one neutron, two electrons, therefore it is a fermion; the number of bosons within a composite particle made up of simple particles bound with a potential has no effect on whether it is a boson or a fermion. Fermionic or bosonic behavior of a composite particle is only seen at large distances. At proximity, where spatial structure begins to be important, a composite particle behaves according to its constituent makeup.
Fermions can exhibit bosonic behavior. This is the origin of superconductivity and the superfluidity of helium-3: in superconducting materials, electrons interact through the exchange of phonons, forming Cooper pairs, while in helium-3, Cooper pairs are formed via spin fluctuations; the quasiparticles of the fractional quantum Hall effect are known as composite fermions, which are electrons with an number of quantized vortices attached to them. In a quantum field theory, there can be field configurations of bosons which are topologically twisted; these are coherent states which behave like a particle, they can be fermionic if all the constituent particles are bosons. This was discovered by Tony Skyrme in the early 1960s, so fermions made of bosons are named skyrmions after him. Skyrme's original example involved fields which take values on a three-dimensional sphere, the original nonlinear sigma model which describes the large distance behavior of pions. In Skyrme's model, reproduced in the large N or string approximation to quantum chromodynamics, the proton and neutron are fermionic topological solitons of the pion field.
Whereas Skyrme's example involved pion physics, there is a much more familiar example in quantum electrodynamics with a magnetic monopole. A bosonic monopole with the smallest possible magnetic charge and a bosonic version of the electron will form a fermionic dyon; the analogy between the Skyrme field and the Higgs field of the electroweak sector has been used to postulate that all fermions are skyrmions. This could explain why all known fermions have baryon or lepton quantum numbers and provide a physical mechanism for the Pauli exclusion principle
Parity (physics)
In quantum mechanics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it can refer to the simultaneous flip in the sign of all three spatial coordinates: P: ↦, it can be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity; the weak interaction thus provides a means for probing chirality in physics. In interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions. A matrix representation of P has determinant equal to −1, hence is distinct from a rotation, which has a determinant equal to 1. In a two-dimensional plane, a simultaneous flip of all coordinates in sign is not a parity transformation. In quantum mechanics, wave functions which are unchanged by a parity transformation are described as functions, while those which change sign under a parity transformation are odd functions.
Under rotations, classical geometrical objects can be classified into scalars and tensors of higher rank. In classical physics, physical configurations need to transform under representations of every symmetry group. Quantum theory predicts that states in a Hilbert space do not need to transform under representations of the group of rotations, but only under projective representations; the word projective refers to the fact that if one projects out the phase of each state, where we recall that the overall phase of a quantum state is not an observable a projective representation reduces to an ordinary representation. All representations are projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states; the projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. For example, projective representations of the 3-dimensional rotation group, the special orthogonal group SO, are ordinary representations of the special unitary group SU.
Projective representations of the rotation group that are not representations are called spinors, so quantum states may transform not only as tensors but as spinors. If one adds to this a classification by parity, these can be extended, for example, into notions of scalars and pseudoscalars which are rotationally invariant. Vectors and axial vectors. One can define reflections such as V x: ↦, which have negative determinant and form a valid parity transformation. Combining them with rotations one can recover the particular parity transformation defined earlier; the first parity transformation given does not work in an number of dimensions, because it results in a positive determinant. In odd number of dimensions only the latter example of a parity transformation can be used. Parity forms the abelian group Z 2 due to the relation P ^ 2 = 1 ^. All Abelian groups have only one-dimensional irreducible representations. For Z 2, there are two irreducible representations: one is under parity, P ^ ϕ = + ϕ, the other is odd, P ^ ϕ = − ϕ.
These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any phase. Newton's equation of motion F → = m a → equates two vectors, hence is invariant under parity; the law of gravity involves only vectors and is therefore, invariant under parity. However, angular momentum L → is an axial vector, L → = r → × p → {\displaystyle =\times {\vec