1.
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.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
3.
Special orthogonal group
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Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, an orthogonal matrix is a real matrix whose inverse equals its transpose. An important subgroup of O is the orthogonal group, denoted SO. This group is called the rotation group, because, in dimensions 2 and 3. In low dimension, these groups have been studied, see SO, SO and SO. This is a subgroup of the linear group GL given by O = where QT is the transpose of Q and I is the identity matrix. This article mainly discusses the groups of quadratic forms that may be expressed over some bases as the dot product, over the reals. Over the reals, for any quadratic form, there is a basis. Thus the orthogonal group depends only on the numbers of 1 and of −1, and is denoted O, for details, see indefinite orthogonal group. The derived subgroup Ω of O is an often studied object because, the Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form. The determinant of any orthogonal matrix is either 1 or −1, the orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O known as the special orthogonal group SO, consisting of all proper rotations. By analogy with GL–SL, the group is sometimes called the general orthogonal group and denoted GO. The term rotation group can be used to either the special or general orthogonal group. When this distinction is to be emphasized, the groups may be denoted O and O, reserving n for the dimension of the space. The letters p or r are also used, indicating the rank of the corresponding Lie algebra, in odd dimension the corresponding Lie algebra is s o, while in even dimension the Lie algebra is s o. In two dimensions, O is the group of all rotations about the origin and all reflections along a line through the origin, SO is the group of all rotations about the origin. These groups are related, SO is a subgroup of O of index 2. More generally, in any number of dimensions an even number of reflections gives a rotation, therefore, the rotations define a subgroup of O, but the reflections do not define a subgroup. A reflection through the origin may be generated as a combination of one reflection along each of the axes, the reflection through the origin is not a reflection in the usual sense in even dimensions, but rather a rotation
4.
Special unitary group
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In mathematics, the special unitary group of degree n, denoted SU, is the Lie group of n×n unitary matrices with determinant 1. The group operation is matrix multiplication, the special unitary group is a subgroup of the unitary group U, consisting of all n×n unitary matrices. As a compact group, U is the group that preserves the standard inner product on Cn. It is itself a subgroup of the linear group, SU ⊂ U ⊂ GL. The SU groups find application in the Standard Model of particle physics, especially SU in the electroweak interaction. The simplest case, SU, is the group, having only a single element. The group SU is isomorphic to the group of quaternions of norm 1, since unit quaternions can be used to represent rotations in 3-dimensional space, there is a surjective homomorphism from SU to the rotation group SO whose kernel is. SU is also identical to one of the groups of spinors, Spin. The special unitary group SU is a real Lie group and its dimension as a real manifold is n2 −1. Topologically, it is compact and simply connected, algebraically, it is a simple Lie group. The center of SU is isomorphic to the cyclic group Zn and its outer automorphism group, for n ≥3, is Z2, while the outer automorphism group of SU is the trivial group. A maximal torus, of rank n −1, is given by the set of matrices with determinant 1. The Weyl group is the symmetric group Sn, which is represented by signed permutation matrices, the Lie algebra of SU, denoted by su, can be identified with the set of traceless antihermitian n×n complex matrices, with the regular commutator as Lie bracket. Particle physicists often use a different, equivalent representation, the set of traceless hermitian n×n complex matrices with Lie bracket given by −i times the commutator, the Lie algebra su can be generated by n2 operators O ^ i j, i, j=1,2. N, which satisfy the commutator relationships = δ j k O ^ i ℓ − δ i ℓ O ^ k j for i, j, k, ℓ =1,2, N, where δjk denotes the Kronecker delta. Additionally, the operator N ^ = ∑ i =1 n O ^ i i satisfies =0, which implies that the number of independent generators of the Lie algebra is n2 −1. We also take ∑ c, e =1 n 2 −1 d a c e d b c e = n 2 −4 n δ a b as a normalization convention. In the -dimensional adjoint representation, the generators are represented by × matrices, whose elements are defined by the structure constants themselves, SU is the following group, S U =, where the overline denotes complex conjugation
5.
Quantum chromodynamics
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QCD is a type of quantum field theory called a non-abelian gauge theory with symmetry group SU. The QCD analog of electric charge is a property called color, gluons are the force carrier of the theory, like photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics, a large body of experimental evidence for QCD has been gathered over the years. QCD enjoys two peculiar properties, Confinement, which means that the force between quarks does not diminish as they are separated. Although analytically unproven, confinement is widely believed to be true because it explains the consistent failure of free quark searches, asymptotic freedom, which means that in very high-energy reactions, quarks and gluons interact very weakly creating a quark–gluon plasma. This prediction of QCD was first discovered in the early 1970s by David Politzer, Frank Wilczek, for this work they were awarded the 2004 Nobel Prize in Physics. The phase transition temperature between two properties has been measured by the ALICE experiment to be well above 160 MeV. Below this temperature, confinement is dominant, while above it, american physicist Murray Gell-Mann coined the word quark in its present sense. It originally comes from the phrase Three quarks for Muster Mark in Finnegans Wake by James Joyce, Gell-Mann, however, wanted to pronounce the word to rhyme with fork rather than with park, as Joyce seemed to indicate by rhyming words in the vicinity such as Mark. Gell-Mann got around that by supposing that one ingredient of the line Three quarks for Muster Mark was a cry of Three quarts for Mister, earwickers pub, a plausible suggestion given the complex punning in Joyces novel. The three kinds of charge in QCD are usually referred to as color charge by loose analogy to the three kinds of color perceived by humans, other than this nomenclature, the quantum parameter color is completely unrelated to the everyday, familiar phenomenon of color. Since the theory of charge is dubbed electrodynamics, the Greek word χρῶμα chroma color is applied to the theory of color charge. With the invention of bubble chambers and spark chambers in the 1950s, experimental particle physics discovered a large and it seemed that such a large number of particles could not all be fundamental. First, the particles were classified by charge and isospin by Eugene Wigner and Werner Heisenberg, then, in 1953, according to strangeness by Murray Gell-Mann and Kazuhiko Nishijima. To gain greater insight, the hadrons were sorted into groups having similar properties and masses using the way, invented in 1961 by Gell-Mann. The problem considered in this preprint was suggested by Nikolay Bogolyubov, in the beginning of 1965, Nikolay Bogolyubov, Boris Struminsky and Albert Tavkhelidze wrote a preprint with a more detailed discussion of the additional quark quantum degree of freedom. This work was presented by Albert Tavchelidze without obtaining consent of his collaborators for doing so at an international conference in Trieste. A similar mysterious situation was with the Δ++ baryon, in the quark model, han and Nambu noted that quarks might interact via an octet of vector gauge bosons, the gluons
6.
Gauge theory
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In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations. An invariant is a model that holds no matter the mathematical procedure applied to it. This is the concept behind gauge invariance, the idea of fields as described by Michael Faraday in his study of electromagnetism led to the postulate that fields could be described mathematically as scalars and vectors. When a field is transformed, but the result is not, applying gauge theory creates a unification which describes mathematical formulas or models that hold good for all fields of the same class. The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian, the transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators, for each group generator there necessarily arises a corresponding field called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations, when such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, the theory is referred to as non-abelian. Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups, when they are invariant under a transformation identically performed at every point in the spacetime in which the physical processes occur, they are said to have a global symmetry. Local symmetry, the cornerstone of gauge theories, is a stricter constraint, in fact, a global symmetry is just a local symmetry whose groups parameters are fixed in spacetime. Gauge theories are important as the field theories explaining the dynamics of elementary particles. Quantum electrodynamics is a gauge theory with the symmetry group U and has one gauge field. The Standard Model is a gauge theory with the symmetry group U×SU×SU and has a total of twelve gauge bosons. Gauge theories are important in explaining gravitation in the theory of general relativity. Its case is unusual in that the gauge field is a tensor. Theories of quantum gravity, beginning with gauge gravitation theory, also postulate the existence of a gauge boson known as the graviton, both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, gauge theory gravity, replaces the principle of covariance with a true gauge principle with new gauge fields. Historically, these ideas were first stated in the context of classical electromagnetism, however, the modern importance of gauge symmetries appeared first in the relativistic quantum mechanics of electrons – quantum electrodynamics, elaborated on below
7.
Particle physics
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Particle physics is the branch of physics that studies the nature of the particles that constitute matter and radiation. By our current understanding, these particles are excitations of the quantum fields that also govern their interactions. The currently dominant theory explaining these fundamental particles and fields, along with their dynamics, is called the Standard Model, in more technical terms, they are described by quantum state vectors in a Hilbert space, which is also treated in quantum field theory. All particles and their interactions observed to date can be described almost entirely by a field theory called the Standard Model. The Standard Model, as formulated, has 61 elementary particles. Those elementary particles can combine to form composite particles, accounting for the hundreds of species of particles that have been discovered since the 1960s. The Standard Model has been found to agree with almost all the tests conducted to date. However, most particle physicists believe that it is a description of nature. In recent years, measurements of mass have provided the first experimental deviations from the Standard Model. The idea that all matter is composed of elementary particles dates from at least the 6th century BC, in the 19th century, John Dalton, through his work on stoichiometry, concluded that each element of nature was composed of a single, unique type of particle. Throughout the 1950s and 1960s, a variety of particles were found in collisions of particles from increasingly high-energy beams. It was referred to informally as the particle zoo, the current state of the classification of all elementary particles is explained by the Standard Model. It describes the strong, weak, and electromagnetic fundamental interactions, the species of gauge bosons are the gluons, W−, W+ and Z bosons, and the photons. The Standard Model also contains 24 fundamental particles, which are the constituents of all matter, finally, the Standard Model also predicted the existence of a type of boson known as the Higgs boson. Early in the morning on 4 July 2012, physicists with the Large Hadron Collider at CERN announced they had found a new particle that behaves similarly to what is expected from the Higgs boson, the worlds major particle physics laboratories are, Brookhaven National Laboratory. Its main facility is the Relativistic Heavy Ion Collider, which collides heavy ions such as gold ions and it is the worlds first heavy ion collider, and the worlds only polarized proton collider. Its main projects are now the electron-positron colliders VEPP-2000, operated since 2006 and its main project is now the Large Hadron Collider, which had its first beam circulation on 10 September 2008, and is now the worlds most energetic collider of protons. It also became the most energetic collider of heavy ions after it began colliding lead ions and its main facility is the Hadron Elektron Ring Anlage, which collides electrons and positrons with protons
8.
AdS/CFT correspondence
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On one side are anti-de Sitter spaces which are used in theories of quantum gravity, formulated in terms of string theory or M-theory. On the other side of the correspondence are conformal field theories which are quantum field theories, the duality represents a major advance in our understanding of string theory and quantum gravity. It also provides a toolkit for studying strongly coupled quantum field theories. This fact has been used to study aspects of nuclear. The AdS/CFT correspondence was first proposed by Juan Maldacena in late 1997, important aspects of the correspondence were elaborated in articles by Steven Gubser, Igor Klebanov, and Alexander Markovich Polyakov, and by Edward Witten. By 2015, Maldacenas article had over 10,000 citations and our current understanding of gravity is based on Albert Einsteins general theory of relativity. Formulated in 1915, general relativity explains gravity in terms of the geometry of space and time and it is formulated in the language of classical physics developed by physicists such as Isaac Newton and James Clerk Maxwell. The other nongravitational forces are explained in the framework of quantum mechanics, developed in the first half of the twentieth century by a number of different physicists, quantum mechanics provides a radically different way of describing physical phenomena based on probability. Quantum gravity is the branch of physics that seeks to describe gravity using the principles of quantum mechanics, currently, the most popular approach to quantum gravity is string theory, which models elementary particles not as zero-dimensional points but as one-dimensional objects called strings. In the AdS/CFT correspondence, one typically considers theories of quantum gravity derived from string theory or its modern extension, in everyday life, there are three familiar dimensions of space, and there is one dimension of time. Thus, in the language of physics, one says that spacetime is four-dimensional. The quantum gravity theories appearing in the AdS/CFT correspondence are typically obtained from string and this produces a theory in which spacetime has effectively a lower number of dimensions and the extra dimensions are curled up into circles. A standard analogy for compactification is to consider an object such as a garden hose. Thus, an ant crawling inside it would move in two dimensions, the application of quantum mechanics to physical objects such as the electromagnetic field, which are extended in space and time, is known as quantum field theory. In particle physics, quantum field theories form the basis for our understanding of elementary particles, quantum field theories are also used throughout condensed matter physics to model particle-like objects called quasiparticles. In the AdS/CFT correspondence, one considers, in addition to a theory of quantum gravity and this is a particularly symmetric and mathematically well behaved type of quantum field theory. In the AdS/CFT correspondence, one considers string theory or M-theory on an anti-de Sitter background and this means that the geometry of spacetime is described in terms of a certain vacuum solution of Einsteins equation called anti-de Sitter space. It is closely related to space, which can be viewed as a disk as illustrated on the right
9.
Condensed matter physics
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Condensed matter physics is a branch of physics that deals with the physical properties of condensed phases of matter, where particles adhere to each other. Condensed matter physicists seek to understand the behavior of these phases by using physical laws, in particular, they include the laws of quantum mechanics, electromagnetism and statistical mechanics. The field overlaps with chemistry, materials science, and nanotechnology, the theoretical physics of condensed matter shares important concepts and methods with that of particle physics and nuclear physics. A variety of topics in physics such as crystallography, metallurgy, elasticity, magnetism, etc. were treated as distinct areas until the 1940s, when they were grouped together as solid state physics. Around the 1960s, the study of properties of liquids was added to this list, forming the basis for the new. The Bell Telephone Laboratories was one of the first institutes to conduct a program in condensed matter physics. References to condensed state can be traced to earlier sources, as a matter of fact, it would be more correct to unify them under the title of condensed bodies. One of the first studies of condensed states of matter was by English chemist Humphry Davy, Davy observed that of the forty chemical elements known at the time, twenty-six had metallic properties such as lustre, ductility and high electrical and thermal conductivity. This indicated that the atoms in John Daltons atomic theory were not indivisible as Dalton claimed, Davy further claimed that elements that were then believed to be gases, such as nitrogen and hydrogen could be liquefied under the right conditions and would then behave as metals. In 1823, Michael Faraday, then an assistant in Davys lab, successfully liquefied chlorine and went on to all known gaseous elements, except for nitrogen, hydrogen. By 1908, James Dewar and Heike Kamerlingh Onnes were successfully able to hydrogen and then newly discovered helium. Paul Drude in 1900 proposed the first theoretical model for an electron moving through a metallic solid. Drudes model described properties of metals in terms of a gas of free electrons, the phenomenon completely surprised the best theoretical physicists of the time, and it remained unexplained for several decades. Drudes classical model was augmented by Wolfgang Pauli, Arnold Sommerfeld, Felix Bloch, Pauli realized that the free electrons in metal must obey the Fermi–Dirac statistics. Using this idea, he developed the theory of paramagnetism in 1926, shortly after, Sommerfeld incorporated the Fermi–Dirac statistics into the free electron model and made it better able to explain the heat capacity. Two years later, Bloch used quantum mechanics to describe the motion of an electron in a periodic lattice. Magnetism as a property of matter has been known in China since 4000 BC, Pierre Curie studied the dependence of magnetization on temperature and discovered the Curie point phase transition in ferromagnetic materials. In 1906, Pierre Weiss introduced the concept of magnetic domains to explain the properties of ferromagnets
10.
Orthogonal group
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Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, an orthogonal matrix is a real matrix whose inverse equals its transpose. An important subgroup of O is the orthogonal group, denoted SO. This group is called the rotation group, because, in dimensions 2 and 3. In low dimension, these groups have been studied, see SO, SO and SO. This is a subgroup of the linear group GL given by O = where QT is the transpose of Q and I is the identity matrix. This article mainly discusses the groups of quadratic forms that may be expressed over some bases as the dot product, over the reals. Over the reals, for any quadratic form, there is a basis. Thus the orthogonal group depends only on the numbers of 1 and of −1, and is denoted O, for details, see indefinite orthogonal group. The derived subgroup Ω of O is an often studied object because, the Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form. The determinant of any orthogonal matrix is either 1 or −1, the orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O known as the special orthogonal group SO, consisting of all proper rotations. By analogy with GL–SL, the group is sometimes called the general orthogonal group and denoted GO. The term rotation group can be used to either the special or general orthogonal group. When this distinction is to be emphasized, the groups may be denoted O and O, reserving n for the dimension of the space. The letters p or r are also used, indicating the rank of the corresponding Lie algebra, in odd dimension the corresponding Lie algebra is s o, while in even dimension the Lie algebra is s o. In two dimensions, O is the group of all rotations about the origin and all reflections along a line through the origin, SO is the group of all rotations about the origin. These groups are related, SO is a subgroup of O of index 2. More generally, in any number of dimensions an even number of reflections gives a rotation, therefore, the rotations define a subgroup of O, but the reflections do not define a subgroup. A reflection through the origin may be generated as a combination of one reflection along each of the axes, the reflection through the origin is not a reflection in the usual sense in even dimensions, but rather a rotation
11.
Quartic interaction
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This article refers to a type of self-interaction in scalar field theory, a topic in quantum field theory. Other types of quartic interactions may be found under the topic of four-fermion interactions, a classical free scalar field φ satisfies the Klein–Gordon equation. If a scalar field is denoted φ, an interaction is represented by adding a potential term λ4. The coupling constant λ is dimensionless in 4-dimensional spacetime and this article uses the metric signature for Minkowski space. The Lagrangian for a scalar field with a quartic interaction is L =12 −14. This Lagrangian has a global Z2 symmetry mapping φ to − φ, with N real scalar fields, we can have a φ4 model with a global SO symmetry given by the Lagrangian L =12 −14 λ2, a =1. Expanding the complex field in real and imaginary parts shows that it is equivalent to the SO model of real scalar fields. In all of the models above, the coupling constant λ must be positive, since, otherwise, the potential would be unbounded below, also, the Feynman path integral discussed below would be ill-defined. In 4 dimensions, ϕ4 theories have a Landau pole and this means that without a cut-off on the high-energy scale, renormalization would render the theory trivial. The Feynman diagram expansion may be obtained also from the Feynman path integral formulation. All of these Greens functions may be obtained by expanding the exponential in Jφ in the generating function Z = ∫ D ϕ e i ∫ d 4 x = Z ∑ n =0 ∞1 n, a Wick rotation may be applied to make time imaginary. Changing the signature to then gives a φ4 statistical mechanics integral over a 4-dimensional Euclidean space, the standard trick to evaluate this functional integral is to write it as a product of exponential factors, schematically, Z ~ = ∫ D ϕ ~ ∏ p. The second two exponential factors can be expanded as power series, and the combinatorics of this expansion can be represented graphically, each vertex is represented by a factor -λ. At a given order λk, all diagrams with n external lines, each internal line is represented by a factor 1/, where q is the momentum flowing through that line. Any unconstrained momenta are integrated over all values, the result is divided by a symmetry factor, which is the number of ways the lines and vertices of the graph can be rearranged without changing its connectivity. Do not include graphs containing vacuum bubbles, connected subgraphs with no external lines, the last rule takes into account the effect of dividing by Z ~. The integrals over unconstrained momenta, called loop integrals, in the Feynman graphs typically diverge, a renormalization scale must be introduced in the process, and the coupling constant and mass become dependent upon it. It is this dependence that leads to the Landau pole mentioned earlier, alternatively, if the cutoff is allowed to go to infinity, the Landau pole can be avoided only if the renormalized coupling runs to zero, rendering the theory trivial
12.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
13.
Index notation
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In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used according to the subject. It is frequently helpful in mathematics to refer to the elements of an array using subscripts, the subscripts can be integers or variables. The array takes the form of tensors in general, since these can be treated as multi-dimensional arrays, special cases are vectors and matrices. The following is only an introduction to the concept, index notation is used in more detail in mathematics, see the main article for further details. For example, given the vector, a = then some entries are a 1 =10, the notation can be applied to vectors in mathematics and physics. The following vector equation a + b = c can also be written in terms of the elements of the vector and this expression represents a set of equations, one for each index. If the vectors each have n elements, meaning i =1,2, the notation ij should not be confused with i multiplied by j, it is read as i - j. For example, if A = then some entries are a 11 =9, a 12 =8, for indices larger than 9, the comma-based notation may be superior. Matrix equations are written similarly to vector equations, such as A + B = C in terms of the elements of the matrices A i j + B i j = C i j for all values of i and j. Again this expression represents a set of equations, one for each index, if the matrices each have m rows and n columns, meaning i =1,2. m and j =1,2. n, then there are mn equations. The notation allows a clear generalization to multi-dimensional arrays of elements, for example, A i 1 i 2 ⋯ + B i 1 i 2 ⋯ = C i 1 i 2 ⋯ representing a set of many equations. In tensor analysis, superscripts are used instead of subscripts to distinguish covariant from contravariant entities, see covariance and contravariance of vectors, in several programming languages, index notation is a way of addressing elements of an array. In general, the address of the ith element of an array with base address b and element size s is b+is. In the C programming language, we can write the above as * or base, coincidentally, since pointer addition is commutative, this allows for obscure expressions such as 3 which is equivalent to base. Things become more interesting when we consider arrays with more than one index, for example, a two-dimensional table. e. When the first method is used, the programmer decides how the elements of the array are laid out in the computers memory, the second method is used when the number of elements in each row is the same and known at the time the program is written. The programmer declares the array to have, say, three columns by writing e. g. elementtype tablename, one then refers to a particular element of the array by writing tablename
14.
Coupling constant
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In physics, a coupling constant or gauge coupling parameter is a number that determines the strength of the force exerted in an interaction. Usually, the Lagrangian or the Hamiltonian of a system describing an interaction can be separated into a kinetic part, the coupling constant determines the strength of the interaction part with respect to the kinetic part, or between two sectors of the interaction part. For example, the charge of a particle is a coupling constant that characterizes an interaction with two charge-carrying fields and one photon field. Since photons carry electromagnetism, this coupling constant determines how strongly electrons feel such a force and has its value fixed by experiment, a coupling constant plays an important role in dynamics. For example, one sets up hierarchies of approximation based on the importance of various coupling constants. In the motion of a lump of magnetized iron, the magnetic forces may be more important than the gravitational forces because of the relative magnitudes of the coupling constants. However, in classical mechanics one usually makes these decisions directly by comparing forces, coupling constants arise naturally in a quantum field theory. A special role is played in relativistic quantum theories by coupling constants that are dimensionless and this constant is proportional to the square of the coupling strength of the charge of an electron to the electromagnetic field. In a non-Abelian gauge theory, the gauge coupling parameter, g, in another widely used convention, G is rescaled so that the coefficient of the kinetic term is 1/4 and g appears in the covariant derivative. This should be understood to be similar to a version of the elementary charge defined as e ε0 ℏ c =4 π α ≈0.30282212. In a quantum theory with a dimensionless coupling constant g. In this case it is described by an expansion in powers of g. If the coupling constant is of one or larger, the theory is said to be strongly coupled. An example of the latter is the theory of strong interactions. In such a case non-perturbative methods have to be used to investigate the theory, one can probe a quantum field theory at short times or distances by changing the wavelength or momentum, k, of the probe one uses. With a high frequency probe, one sees virtual particles taking part in every process and this apparent violation of the conservation of energy can be understood heuristically by examining the uncertainty relation Δ E Δ t ≥ ℏ2, which allows such violations at short times. The previous remark only applies to formulations of quantum field theory, in particular. In other formulations, the event is described by virtual particles going off the mass shell
15.
Feynman diagram
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In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic particles. The scheme is named after its inventor, American physicist Richard Feynman, the interaction of sub-atomic particles can be complex and difficult to understand intuitively. Feynman diagrams give a simple visualization of what would otherwise be a rather arcane, while the diagrams are applied primarily to quantum field theory, they can also be used in other fields, such as solid-state theory. Feynman used Ernst Stueckelbergs interpretation of the positron as if it were an electron moving backward in time, thus, antiparticles are represented as moving backward along the time axis in Feynman diagrams. The calculation of probability amplitudes in theoretical particle physics requires the use of rather large and these integrals do, however, have a regular structure, and may be represented graphically as Feynman diagrams. A Feynman diagram is a contribution of a class of particle paths. Within the canonical formulation of field theory, a Feynman diagram represents a term in the Wicks expansion of the perturbative S-matrix. The transition amplitude is given as the matrix element of the S-matrix between the initial and the final states of the quantum system. The amplitude for scattering is the sum of each possible interaction history over all possible intermediate particle states, the number of times the interaction Hamiltonian acts is the order of the perturbation expansion, and the time-dependent perturbation theory for fields is known as the Dyson series. When the intermediate states at times are energy eigenstates the series is called old-fashioned perturbation theory. The Feynman diagrams are much easier to track of than old-fashioned terms. Each Feynman diagram is the sum of exponentially many old-fashioned terms, in a non-relativistic theory, there are no antiparticles and there is no doubling, so each Feynman diagram includes only one term. Feynman gave a prescription for calculating the amplitude for any given diagram from a field theory Lagrangian—the Feynman rules, in addition to their value as a mathematical tool, Feynman diagrams provide deep physical insight into the nature of particle interactions. Particles interact in every way available, in fact, intermediate virtual particles are allowed to propagate faster than light, the probability of each final state is then obtained by summing over all such possibilities. This is closely tied to the integral formulation of quantum mechanics. After renormalization, calculations using Feynman diagrams match experimental results with very high accuracy, Feynman diagram and path integral methods are also used in statistical mechanics and can even be applied to classical mechanics. Murray Gell-Mann always referred to Feynman diagrams as Stueckelberg diagrams, after a Swiss physicist, Ernst Stueckelberg, Feynman had to lobby hard for the diagrams, which confused the establishment physicists trained in equations and graphs. In quantum field theories the Feynman diagrams are obtained from Lagrangian by Feynman rules, dimensional regularization writes a Feynman integral as an integral depending on the spacetime dimension d and spacetime points
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Gluon
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In lay terms, they glue quarks together, forming protons and neutrons. In technical terms, gluons are vector gauge bosons that mediate interactions of quarks in quantum chromodynamics. Gluons themselves carry the charge of the strong interaction. This is unlike the photon, which mediates the electromagnetic interaction, gluons therefore participate in the strong interaction in addition to mediating it, making QCD significantly harder to analyze than QED. The gluon is a boson, like the photon, it has a spin of 1. In quantum field theory, unbroken gauge invariance requires that gauge bosons have zero mass, the gluon has negative intrinsic parity. Unlike the single photon of QED or the three W and Z bosons of the interaction, there are eight independent types of gluon in QCD. This may be difficult to understand intuitively, quarks carry three types of color charge, antiquarks carry three types of anticolor. Gluons may be thought of as carrying both color and anticolor, but to understand how they are combined, it is necessary to consider the mathematics of color charge in more detail. A relevant illustration in the case at hand would be a gluon with a state described by. This is read as red–antiblue plus blue–antired, the color singlet state is, /3. In words, if one could measure the color of the state, there would be equal probabilities of it being red-antired, blue-antiblue, there are eight remaining independent color states, which correspond to the eight types or eight colors of gluons. Because states can be mixed together as discussed above, there are ways of presenting these states. One commonly used list is, These are equivalent to the Gell-Mann matrices, there are many other possible choices, but all are mathematically equivalent, at least equally complex, and give the same physical results. Technically, QCD is a theory with SU gauge symmetry. Quarks are introduced as spinors in Nf flavors, each in the representation of the color gauge group. The gluons are vectors in the adjoint representation of color SU, for a general gauge group, the number of force-carriers is always equal to the dimension of the adjoint representation. For the simple case of SU, the dimension of this representation is N2 −1, in terms of group theory, the assertion that there are no color singlet gluons is simply the statement that quantum chromodynamics has an SU rather than a U symmetry
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Quark
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A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, due to a phenomenon known as color confinement, quarks are never directly observed or found in isolation, they can be found only within hadrons, such as baryons and mesons. For this reason, much of what is known about quarks has been drawn from observations of the hadrons themselves, Quarks have various intrinsic properties, including electric charge, mass, color charge, and spin. There are six types of quarks, known as flavors, up, down, strange, charm, top, up and down quarks have the lowest masses of all quarks. The heavier quarks rapidly change into up and down quarks through a process of particle decay, the transformation from a higher mass state to a lower mass state. Because of this, up and down quarks are generally stable and the most common in the universe, whereas strange, charm, bottom, and top quarks can only be produced in high energy collisions. For every quark flavor there is a type of antiparticle, known as an antiquark. The quark model was proposed by physicists Murray Gell-Mann and George Zweig in 1964. Accelerator experiments have provided evidence for all six flavors, the top quark was the last to be discovered at Fermilab in 1995. The Standard Model is the theoretical framework describing all the known elementary particles. This model contains six flavors of quarks, named up, down, strange, charm, bottom, antiparticles of quarks are called antiquarks, and are denoted by a bar over the symbol for the corresponding quark, such as u for an up antiquark. As with antimatter in general, antiquarks have the mass, mean lifetime, and spin as their respective quarks. Quarks are spin- 1⁄2 particles, implying that they are fermions according to the spin-statistics theorem and they are subject to the Pauli exclusion principle, which states that no two identical fermions can simultaneously occupy the same quantum state. This is in contrast to bosons, any number of which can be in the same state, unlike leptons, quarks possess color charge, which causes them to engage in the strong interaction. The resulting attraction between different quarks causes the formation of composite particles known as hadrons, there are two families of hadrons, baryons, with three valence quarks, and mesons, with a valence quark and an antiquark. The most common baryons are the proton and the neutron, the blocks of the atomic nucleus. A great number of hadrons are known, most of them differentiated by their quark content, the existence of exotic hadrons with more valence quarks, such as tetraquarks and pentaquarks, has been conjectured but not proven. However, on 13 July 2015, the LHCb collaboration at CERN reported results consistent with pentaquark states, elementary fermions are grouped into three generations, each comprising two leptons and two quarks
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Weyl semimetal
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Weyl fermions are massless chiral fermions that play an important role in quantum field theory and the standard model. They are considered a block for fermions in quantum field theory. For example, one-half of a charged Dirac fermion of a definite chirality is a Weyl fermion and they have not been observed as a fundamental particle in nature. Weyl fermions may be realized as emergent quasiparticles in a condensed matter system. This was first predicted by C, herring in the context of electronic band structures of solid state systems such as electronic crystals. An electronic Weyl fermion is not only charged but stable at room temperature there is no such superfluid or liquid state known. A Weyl semimetal is a solid state crystal whose low energy excitations are Weyl fermions that carry electrical charge even at room temperatures, a Weyl semimetal enables realization of Weyl fermions in electronic systems. It is a topologically nontrivial phase of matter, together with Helium-3 A superfluid phase, the Weyl fermions at zero energy correspond to points of bulk band degeneracy, the Weyl nodes, that are separated in momentum space. Weyl fermions have distinct chiralities, either left handed or right handed, hence the Weyl fermion quasiparticles in a Weyl semimetal possess a high degree of mobility. Due to the topology, a Weyl semimetal is expected to demonstrate Fermi arc electron states on its surface. These arcs are discontinuous or disjoint segments of a two dimensional Fermi contour, which are terminated onto the projections of the Weyl fermion nodes on the surface, a 2012 theoretical investigation of superfluid Helium-3, suggested Fermi arcs in neutral superfluids. This discovery was built upon previous theoretical predictions proposed in November 2014, Weyl points were also observed in non-electronic systems such as photonic crystals and Helium-3 superfluid quasiparticle spectrum. This article is about Weyl semimetals, TaAs is the first discovered Weyl semimetal. Large-size, high-quality TaAs single crystals can be obtained by vapor transport method using iodine as the transport agent. TaAs crystallizes in a tetragonal unit cell with lattice constants a =3.44 Å and c =11.64 Å. Ta and As atoms are six coordinated to each other and it should be noted that this structure lacks a horizontal mirror plane and thus inversion symmetry, which is essential to realize Weyl semimetal. TaAs single crystals have shiny facets, which can be divided into three groups, the two truncated surfaces are, the trapezoid or isosceles triangular surfaces are, and the rectangular ones, TaAs belongs to point group 4mm, the equivalent and planes should form a ditetragonal appearance. The observed morphology can be vary of degenerated cases of the ideal form, the Weyl fermions in the bulk and the Fermi arcs on the surfaces of Weyl semimetals are of interest in physics and materials technology
19.
Gerard 't Hooft
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Gerardus t Hooft is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G. Veltman for elucidating the structure of electroweak interactions. His work concentrates on gauge theory, black holes, quantum gravity and his contributions to physics include a proof that gauge theories are renormalizable, dimensional regularization, and the holographic principle. He is married to Albertha Schik and has two daughters, Saskia and Ellen, Saskia has translated one of her fathers popular speculative books Planetenbiljart into English. The books English title is Playing with Planets and was launched in Singapore in November 2008, Gerard t Hooft was born in Den Helder on July 5,1946, but grew up in The Hague, the seat of government of the Netherlands. He was the child of a family of three. He comes from a family of scholars and his grandmother was a sister of Nobel prize laureate Frits Zernike, and was married to Pieter Nicolaas van Kampen, who was a well-known professor of zoology at Leiden University. Following his familys footsteps, he showed interest in science at an early age, when his primary school teacher asked him what he wanted to be when he grew up, he boldly declared, a man who knows everything. After primary school Gerard attended the Dalton Lyceum, a school that applied the ideas of the Dalton Plan and he easily passed his science and mathematics courses, but struggled with his language courses. Nonetheless, he passed his classes in English, French, German, classical Greek, at the age of sixteen he earned a silver medal in the second Dutch Math Olympiad. After Gerard t Hooft passed his school exams in 1964. He opted for Utrecht instead of the much closer Leiden, because his uncle was a professor there and he wanted to attend his lectures. Because he was so focused on science, his father insisted that he join the Utrechtsch Studenten Corps, in the course of his studies he decided he wanted to go into what he perceived as the heart of theoretical physics, elementary particles. The resolution of the problem was completely unknown at the time, in 1969, t Hooft started on his PhD with Martinus Veltman as his advisor. He would work on the same subject Veltman was working on, in 1971 his first paper was published. In it he showed how to renormalize massless Yang–Mills fields, and was able to derive relations between amplitudes, which would be generalized by Andrei Slavnov and John C, taylor, and become known as the Slavnov–Taylor identities. The world took notice, but Veltman was excited because he saw that the problem he had been working on was solved. A period of collaboration followed in which they developed the technique of dimensional regularization
