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Quantum field theory
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QFT treats particles as excited states of the underlying physical field, so these are called field quanta. In quantum field theory, quantum mechanical interactions among particles are described by interaction terms among the corresponding underlying quantum fields and these interactions are conveniently visualized by Feynman diagrams, which are a formal tool of relativistically covariant perturbation theory, serving to evaluate particle processes. The first achievement of quantum theory, namely quantum electrodynamics, is still the paradigmatic example of a successful quantum field theory. Ordinarily, quantum mechanics cannot give an account of photons which constitute the prime case of relativistic particles, since photons have rest mass zero, and correspondingly travel in the vacuum at the speed c, a non-relativistic theory such as ordinary QM cannot give even an approximate description. Photons are implicit in the emission and absorption processes which have to be postulated, for instance, the formalism of QFT is needed for an explicit description of photons. In fact most topics in the development of quantum theory were related to the interaction of radiation and matter. However, quantum mechanics as formulated by Dirac, Heisenberg, and Schrödinger in 1926–27 started from atomic spectra, as soon as the conceptual framework of quantum mechanics was developed, a small group of theoreticians tried to extend quantum methods to electromagnetic fields. A good example is the paper by Born, Jordan & Heisenberg. The basic idea was that in QFT the electromagnetic field should be represented by matrices in the way that position. The ideas of QM were thus extended to systems having a number of degrees of freedom. The inception of QFT is usually considered to be Diracs famous 1927 paper on The quantum theory of the emission and absorption of radiation, here Dirac coined the name quantum electrodynamics for the part of QFT that was developed first. Employing the theory of the harmonic oscillator, Dirac gave a theoretical description of how photons appear in the quantization of the electromagnetic radiation field. Later, Diracs procedure became a model for the quantization of fields as well. These first approaches to QFT were further developed during the three years. P. Jordan introduced creation and annihilation operators for fields obeying Fermi–Dirac statistics and these differ from the corresponding operators for Bose–Einstein statistics in that the former satisfy anti-commutation relations while the latter satisfy commutation relations. The methods of QFT could be applied to derive equations resulting from the treatment of particles, e. g. the Dirac equation, the Klein–Gordon equation. Schweber points out that the idea and procedure of second quantization goes back to Jordan, in a number of papers from 1927, some difficult problems concerning commutation relations, statistics, and Lorentz invariance were eventually solved. The first comprehensive account of a theory of quantum fields, in particular

2.
Fermion
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In particle physics, a fermion is any subatomic particle characterized by Fermi–Dirac statistics. These particles obey the Pauli exclusion principle, fermions include all quarks and leptons, as well as any composite particle made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics, 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 theorem in any reasonable relativistic quantum field theory, particles with integer spin are bosons. Besides this spin characteristic, fermions have another specific property, they possess conserved baryon or lepton quantum numbers, therefore, what is usually 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 quantum state at any given time. If multiple fermions have the same probability distribution, then at least one property of each fermion, such as its spin. Weakly interacting fermions can also 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 Standard Model recognizes two types of elementary fermions, quarks and leptons. In all, the model distinguishes 24 different fermions, there are six quarks, and six leptons, along with the corresponding antiparticle of each of these. Mathematically, fermions come in three types - Weyl fermions, Dirac fermions, and 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 bosons or fermions depending on their constituents, more precisely, because of the relation between spin and statistics, a particle containing an odd number of fermions is itself a fermion. Examples include the following, A baryon, such as the proton or neutron, the nucleus of a carbon-13 atom contains six protons and seven neutrons and is therefore a fermion. The atom helium-3 is made of two protons, one neutron, and two electrons, and therefore it is a fermion. The number of bosons within a composite 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 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 when they become loosely bound in pairs

3.
Involution (mathematics)
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In mathematics, an involution, or an involutory function, is a function f that is its own inverse, f = x for all x in the domain of f. The identity map is an example of an involution. Common examples in mathematics of nontrivial involutions include multiplication by −1 in arithmetic, other examples include circle inversion, rotation by a half-turn, and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The number of involutions, including the identity involution, on a set with n =0,1,2. Elements is given by a recurrence relation found by Heinrich August Rothe in 1800, a0 = a1 =1, an = an −1 + an −2, for n >1. The first few terms of this sequence are 1,1,2,4,10,26,76,232, these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells. The composition g ∘ f of two involutions f and g is an if and only if they commute, g ∘ f = f ∘ g. Every involution on an odd number of elements has at least one fixed point, more generally, for an involution on a finite set of elements, the number of elements and the number of fixed points have the same parity. Basic examples of involutions are the functions, f 1 = − x, or f 2 =1 x and these are not the only pre-calculus involutions. Another in R + is, f = ln , x >0, the graph of an involution is line-symmetric over the line y = x. This is due to the fact that the inverse of any general function will be its reflection over the 45° line y = x and this can be seen by swapping x with y. If, in particular, the function is an involution, then it will serve as its own reflection, other elementary involutions are useful in solving functional equations. A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane, performing a reflection twice brings a point back to its original coordinates. Another is the reflection through the origin, this is an abuse of language as it is not a reflection. These transformations are examples of affine involutions, an involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points. Coxeter relates three theorems on involutions, Any projectivity that interchanges two points is an involution, the three pairs of opposite sides of a complete quadrangle meet any line in three pairs of an involution. If an involution has one fixed point, it has another, in this instance the involution is termed hyperbolic, while if there are no fixed points it is elliptic. Another type of involution occurring in geometry is a polarity which is a correlation of period 2

4.
Boson
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In quantum mechanics, a boson is a particle that follows Bose–Einstein statistics. Bosons make up one of the two classes of particles, the other being fermions, an important characteristic of bosons is that their statistics do not restrict the number of them that occupy the same quantum state. This property is exemplified by helium-4 when it is cooled to become a superfluid, unlike bosons, two identical fermions cannot occupy the same quantum space. Whereas the elementary particles that make up matter are fermions, the elementary bosons are force carriers that function as the glue holding matter together and this property holds for all particles with integer spin as a consequence of the spin–statistics theorem. This state is called Bose-Einstein condensation and it is believed that this property is the explanation of superfluidity. Bosons may be elementary, like photons, or composite. If it exists, a graviton must be a boson, composite bosons are important in superfluidity and other applications of Bose–Einstein condensates. This phenomenon is known as Bose-Einstein condensation and it is believed that this phenomenon is the secret behind superfluidity of liquids, Bosons differ from fermions, which obey Fermi–Dirac statistics. Two or more identical fermions cannot occupy the same quantum state, since bosons with the same energy can occupy the same place in space, bosons are often force carrier particles. Fermions are usually associated with matter Bosons are particles which obey Bose–Einstein statistics, thus fermions are sometimes said to be the constituents of matter, while bosons are said to be the particles that transmit interactions, or the constituents of radiation. The quantum fields of bosons are bosonic fields, obeying canonical commutation relations, the properties of lasers and masers, superfluid helium-4 and Bose–Einstein condensates are all consequences of statistics of bosons. Interactions between elementary particles are called fundamental interactions, the fundamental interactions of virtual bosons with real particles result in all forces we know. All known elementary and composite particles are bosons or fermions, depending on their spin, particles with spin are fermions. In the framework of quantum mechanics, this is a purely empirical observation. However, in quantum field theory, the spin–statistics theorem shows that half-integer spin particles cannot be bosons. In large systems, the difference between bosonic and fermionic statistics is only apparent at large densities—when their wave functions overlap, at low densities, both types of statistics are well approximated by Maxwell–Boltzmann statistics, which is described by classical mechanics. All observed elementary particles are fermions or bosons. The observed elementary bosons are all bosons, photons, W and Z bosons, gluons