1.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
2.
Representation theory of the Lorentz group
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The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space, in any relativistically invariant physical theory, these representations must enter in some fashion, physics itself must be made out of them. These have both historical importance in physics, as well as connections to more speculative present-day theories. The full theory of the representations of the Lie algebra of the Lorentz group is deduced using the general framework of the representation theory of semisimple Lie algebras. The finite-dimensional representations of the connected component SO+ of the full Lorentz group O are obtained by employing the Lie correspondence and the matrix exponential. The full finite-dimensional representation theory of the covering group SL of SO+ is obtained. The representatives of time reversal and space inversion are given in space inversion and time reversal, the general properties of the representations are outlined. Action on function spaces is considered, with the action on spherical harmonics, the infinite-dimensional case of irreducible unitary representations is classified and realized for the principal series and the complementary series. Finally, the Plancherel formula for SL is given, notable contributors are physicist E. P. The non-technical introduction contains some material for readers not familiar with representation theory. The Lie algebra basis and other adopted conventions are given in conventions, the present purpose is to illustrate the role of representation theory of groups in mathematics and in physics. Rigor and detail take the seat, as the main objective is to fix the notion of finite-dimensional and infinite-dimensional representations of the Lorentz group. The reader familiar with these concepts should skip by, many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. The representation theory also provides the ground for the concept of spin. The theory enters into general relativity in the sense that in small regions of spacetime. But these are also of mathematical interest and of direct physical relevance in other roles than that of a mere restriction. There were speculative theories, consistent with relativity and quantum mechanics, modern speculative theories potentially have similar ingredients per below. From the point of view that the goal of mathematics is to classify and characterize, but in association with the Bargmann–Wigner programme, there are yet unresolved purely mathematical problems, linked to the infinite-dimensional unitary representations
3.
E8 (mathematics)
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The E8 algebra is the largest and most complicated of these exceptional cases. Wilhelm Killing discovered the complex Lie algebra E8 during his classification of simple compact Lie algebras, though he did not prove its existence, Cartan determined that a complex simple Lie algebra of type E8 admits three real forms. Each of them rise to a simple Lie group of dimension 248. Chevalley introduced algebraic groups and Lie algebras of type E8 over other fields, for example, the Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is 8, therefore, the vectors of the root system are in eight-dimensional Euclidean space, they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations in the group, has order 21435527 =696729600. There is a Lie algebra Ek for every integer k ≥3, there is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimension 248. The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496 and this is simply connected, has maximal compact subgroup the compact form of E8, and has an outer automorphism group of order 2 generated by complex conjugation. The split form, EVIII, which has maximal compact subgroup Spin/, EIX, which has maximal compact subgroup E7×SU/, fundamental group of order 2 and has trivial outer automorphism group. For a complete list of forms of simple Lie algebras. Over finite fields, the Lang–Steinberg theorem implies that H1=0, meaning that E8 has no twisted forms, the characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. There are two non-isomorphic irreducible representations of dimension 8634368000, the fundamental representations are those with dimensions 3875,6696000,6899079264,146325270,2450240,30380,248 and 147250. The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations with the irreducible representations. These matrices were computed after four years of collaboration by a group of 18 mathematicians and computer scientists, led by Jeffrey Adams, the most difficult case is the split real form of E8, where the largest matrix is of size 453060×453060. The Lusztig–Vogan polynomials for all other simple groups have been known for some time. The announcement of the result in March 2007 received extraordinary attention from the media, the representations of the E8 groups over finite fields are given by Deligne–Lusztig theory. One can construct the E8 group as the group of the corresponding e8 Lie algebra. This algebra has a 120-dimensional subalgebra so generated by Jij as well as 128 new generators Qa that transform as a Weyl–Majorana spinor of spin and it is then possible to check that the Jacobi identity is satisfied
4.
Lorentz group
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In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz, in general relativity physics is that of special relativity in small enough regions of spacetime. The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime, Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is a subgroup of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is called the homogeneous Lorentz group while the Poincaré group is sometimes called the inhomogeneous Lorentz group. Lorentz transformations are examples of linear transformations, general isometries of Minkowski spacetime are affine transformations. Mathematically, the Lorentz group may be described as the orthogonal group O. This quadratic form is, when put on matrix form, interpreted in physics as the tensor of Minkowski spacetime. The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected, the four connected components are not simply connected, but rather doubly connected. The identity component of the Lorentz group is itself a group, and is called the restricted Lorentz group. The restricted Lorentz group consists of those Lorentz transformations that preserve the orientation of space, the restricted Lorentz group has often been presented through a facility of biquaternion algebra. The restricted Lorentz group arises in other ways in pure mathematics, for example, it arises as the point symmetry group of a certain ordinary differential equation. This fact also has physical significance, because it is a Lie group, the Lorentz group O is both a group and admits a topological description as a smooth manifold. As a manifold, it has four connected components, intuitively, this means that it consists of four topologically separated pieces. The subgroup of transformations is often denoted O+. Those that preserve orientation are called proper, and as linear transformations they have determinant +1, the subgroup of proper Lorentz transformations is denoted SO. The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, and is denoted by SO+. The set of the four connected components can be given a structure as the quotient group O/SO+
5.
Dynkin diagram
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In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled. The multiple edges are, within certain constraints, directed, the main interest in Dynkin diagrams are as a means to classify semisimple Lie algebras over algebraically closed fields. This gives rise to Weyl groups, i. e. to many finite reflection groups, Dynkin diagrams may also arise in other contexts. The term Dynkin diagram can be ambiguous, in this article, Dynkin diagram means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named. The fundamental interest in Dynkin diagrams is that they classify semisimple Lie algebras over algebraically closed fields, one classifies such Lie algebras via their root system, which can be represented by a Dynkin diagram. One then classifies Dynkin diagrams according to the constraints they must satisfy, the central classification is that a simple Lie algebra has a root system, to which is associated an Dynkin diagram, all three of these may be referred to as Bn, for instance. The unoriented Dynkin diagram is a form of Coxeter diagram, and corresponds to the Weyl group, thus Bn may refer to the unoriented diagram, the Weyl group, or the abstract Coxeter group. Note that while the Weyl group is isomorphic to the Coxeter group. Beware also that while Dynkin diagram notation is standardized, Coxeter diagram and group notation is varied and sometimes agrees with Dynkin diagram notation, lastly, sometimes associated objects are referred to by the same notation, though this cannot always be done regularly. Examples include, The root lattice generated by the root system and this is naturally defined, but not one-to-one – for example, A2 and G2 both generate the hexagonal lattice. An associated polytope – for example Gosset 421 polytope may be referred to as the E8 polytope, as its vertices are derived from the E8 root system, an associated quadratic form or manifold – for example, the E8 manifold has intersection form given by the E8 lattice. These latter notations are used for objects associated with exceptional diagrams – objects associated to the regular diagrams instead have traditional names. However, n does not equal the dimension of the module of the Lie algebra – the index on the Dynkin diagram should not be confused with the index on the Lie algebra. For example, B4 corresponds to s o 2 ⋅4 +1 = s o 9, which acts on 9-dimensional space. The simply laced Dynkin diagrams, those with no multiple edges classify many further mathematical objects, for example, the symbol A2 may refer to, The Dynkin diagram with 2 connected nodes, which may also be interpreted as a Coxeter diagram. The root system with 2 simple roots at a 2 π /3 angle, the Lie algebra s l 2 +1 = s l 3 of rank 2. The Weyl group of symmetries of the roots, isomorphic to the symmetric group S3, the abstract Coxeter group, presented by generators and relations, ⟨ r 1, r 2 ∣2 =2 =3 =1 ⟩. Dynkin diagrams must satisfy certain constraints, these are essentially those satisfied by finite Coxeter–Dynkin diagrams, Dynkin diagrams are closely related to Coxeter diagrams of finite Coxeter groups, and the terminology is often conflated
6.
Root system
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In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in graph theory. As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right and these vectors span the whole space. If you consider the line perpendicular to any root, say β, then the reflection of R2 in that line sends any other root, say α, moreover, the root to which it is sent equals α + nβ, where n is an integer. These six vectors satisfy the definition, and therefore they form a root system. Let V be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by, in this context, a root system that also satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2, then they call root systems satisfying condition 2 reduced, in this article, all root systems are assumed to be reduced and crystallographic. In view of property 3, the integrality condition is equivalent to stating that β, Note that the operator ⟨ ⋅, ⋅ ⟩, Φ × Φ → Z defined by property 4 is not an inner product. It is not necessarily symmetric and is only in the first argument. The rank of a root system Φ is the dimension of V, two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems A2, B2, and G2 pictured to the right, is said to be irreducible. Two root systems and are called if there is an invertible linear transformation E1 → E2 which sends Φ1 to Φ2 such that for each pair of roots. The group of isometries of V generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ, as it acts faithfully on the finite set Φ, the Weyl group is always finite. The root lattice of a root system Φ is the Z-submodule of V generated by Φ, there is only one root system of rank 1, consisting of two nonzero vectors. This root system is called A1, in rank 2 there are four possibilities, corresponding to σ α = β + n α, where n =0,1,2,3. Whenever Φ is a system in V, and U is a subspace of V spanned by Ψ = Φ ∩ U. Thus, the exhaustive list of four systems of rank 2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank
7.
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
8.
Hermann Weyl
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Hermann Klaus Hugo Weyl, ForMemRS was a German mathematician, theoretical physicist and philosopher. His research has had significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the century. Weyl published technical and some works on space, time, matter, philosophy, logic, symmetry. He was one of the first to conceive of combining general relativity with the laws of electromagnetism, while no mathematician of his generation aspired to the universalism of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, Weyl was born in Elmshorn, a small town near Hamburg, in Germany, and attended the gymnasium Christianeum in Altona. From 1904 to 1908 he studied mathematics and physics in both Göttingen and Munich and his doctorate was awarded at the University of Göttingen under the supervision of David Hilbert whom he greatly admired. In September 1913 in Göttingen, Weyl married Friederike Bertha Helene Joseph who went by the name Helene, Helene was a daughter of Dr. Bruno Joseph, a physician who held the position of Sanitätsrat in Ribnitz-Damgarten, Germany. Helene was a philosopher and also a translator of Spanish literature into German and it was through Helenes close connection with Husserl that Hermann became familiar with Husserls thought. Hermann and Helene had two sons, Fritz Joachim Weyl and Michael Weyl, both of whom were born in Zürich, Switzerland, Helene died in Princeton, New Jersey on September 5,1948. A memorial service in her honor was held in Princeton on September 9,1948, speakers at her memorial service included her son Fritz Joachim Weyl and mathematicians Oswald Veblen and Richard Courant. In 1950 Hermann married sculptress Ellen Bär, who was the widow of professor Richard Josef Bär of Zürich, einstein had a lasting influence on Weyl, who became fascinated by mathematical physics. In 1921 Weyl met Erwin Schrödinger, a theoretical physicist who at the time was a professor at the University of Zürich and they were to become close friends over time. Weyl left Zürich in 1930 to become Hilberts successor at Göttingen, leaving when the Nazis assumed power in 1933, particularly as his wife was Jewish. He had been offered one of the first faculty positions at the new Institute for Advanced Study in Princeton, New Jersey, as the political situation in Germany grew worse, he changed his mind and accepted when offered the position again. He remained there until his retirement in 1951, together with his second wife Ellen, he spent his time in Princeton and Zürich, and died from a heart attack on December 8,1955 while living in Zürich. Hermann Weyl was cremated in Zurich on December 12,1955 and his cremains remained in private hands until 1999, at which time they were interred in an outdoor columbarium vault in the Princeton Cemetery, located at 29 Greenview Avenue, Princeton, New Jersey. The remains of Hermanns son Michael Weyl are interred next to Hermanns ashes in the same columbarium vault in the Princeton Cemetery
9.
Representation of a Lie group
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In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a tool in their study being the use of the corresponding infinitesimal representations of Lie algebras. The physics literature sometimes passes over the distinction between Lie groups and Lie algebras, let us first discuss representations acting on finite-dimensional complex vector spaces. A representation of a Lie group G on a complex vector space V is a smooth group homomorphism Ψ. For n-dimensional V, the group of V is identified with a subset of the complex square matrices of order n. The automorphism group of V is given the structure of a manifold using this identification. The condition that Ψ is smooth, in the definition above, if a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into general linear group GL. This is known as a matrix representation, a representation of a Lie group G on a vector space V is a smooth group homomorphism G→Aut from G to the automorphism group of V. If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into general linear group GL and this is known as a matrix representation. Two representations of G on vector spaces V, W are equivalent if they have the same matrix representations with respect to some choices of bases for V and W. On the Lie algebra level, there is a linear mapping from the Lie algebra of G to End preserving the Lie bracket. See representation of Lie algebras for the Lie algebra theory, if the homomorphism is in fact a monomorphism, the representation is said to be faithful. A unitary representation is defined in the way, except that G maps to unitary matrices. If G is a compact Lie group, every representation is equivalent to a unitary one. This definition can handle representations on infinite-dimensional Hilbert spaces, such representations can be found in e. g. quantum mechanics, but also in Fourier analysis as shown in the following example. Let G=R, and let the complex Hilbert space V be L2 and we define the representation Ψ, R → B by Ψ → f. See also Wigners classification for representations of the Poincaré group, if G is a semisimple group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. The irreducibles are indexed by highest weight, the allowable highest weights satisfy a suitable positivity condition, the characters of the irreducible representations are given by the Weyl character formula
10.
Exponential map (Riemannian geometry)
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In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold M to M itself. The Riemannian metric determines a canonical affine connection, and the map of the Riemannian manifold is given by the exponential map of this connection. Let M be a manifold and p a point of M. An affine connection on M allows one to define the notion of a geodesic through the point p, let v ∈ TpM be a tangent vector to the manifold at p. Then there is a unique geodesic γv satisfying γv = p with initial tangent vector γ′v = v, the corresponding exponential map is defined by expp = γv. In general, the map is only locally defined, that is, it only takes a small neighborhood of the origin at TpM. This is because it relies on the theorem of existence and uniqueness for ordinary differential equations which is local in nature, an affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle. Intuitively speaking, the map takes a given tangent vector to the manifold, runs along the geodesic starting at that point and goes in that direction. Since v corresponds to the velocity vector of the geodesic, the distance traveled will be dependent on that. We can also reparametrize geodesics to be speed, so equivalently we can define expp = β where β is the unit-speed geodesic going in the direction of v. The Hopf–Rinow theorem asserts that it is possible to define the map on the whole tangent space if. In particular, compact manifolds are geodesically complete, however even if expp is defined on the whole tangent space, it will in general not be a global diffeomorphism. The radius of the largest ball about the origin in TpM that can be mapped diffeomorphically via expp is called the injectivity radius of M at p. The cut locus of the map is, roughly speaking. This means, in particular, that the sphere of a small ball about the origin in TpM is orthogonal to the geodesics in M determined by those vectors. This motivates the definition of normal coordinates on a Riemannian manifold. Via the exponential map, it now can be defined as the Gaussian curvature of a surface through p determined by the image under expp of a 2-dimensional subspace of TpM. In general, Lie groups do not have a bi-invariant metric, take the example that gives the honest exponential map