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
