1.
Levi-Civita symbol
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It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the symbol, antisymmetric symbol, or alternating symbol. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis, ε i 1 i 2 … i n where each index i1, i2, …, there are nn indexed values of εi1i2…in, which can be arranged into an n-dimensional array. The key definitive property of the symbol is total antisymmetry in all the indices, when any two indices are interchanged, equal or not, the symbol is negated, ε … i p … i q … = − ε … i q … i p …. If any two indices are equal, the symbol is zero, the value ε12…n must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose ε12…n = +1, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal and this choice is used throughout this article. The values of the Levi-Civita symbol are independent of any metric tensor, also, the specific term symbol emphasizes that it is not a tensor because of how it transforms between coordinate systems, however it can be interpreted as a tensor density. The Levi-Civita symbol allows the determinant of a matrix. The three- and higher-dimensional Levi-Civita symbols are used more commonly, in three dimensions only, the cyclic permutations of are all even permutations, similarly the anticyclic permutations are all odd permutations. This means in 3d it is sufficient to take cyclic or anticyclic permutations of and easily obtain all the even or odd permutations. Analogous to 2-dimensional matrices, the values of the 3-dimensional Levi-Civita symbol can be arranged into a 3 ×3 ×3 array, the formula is valid for all index values, and for any n. However, computing the formula above naively is O in time complexity, a tensor whose components in an orthonormal basis are given by the Levi-Civita symbol is sometimes called a permutation tensor. It is actually a pseudotensor because under a transformation of Jacobian determinant −1. As the Levi-Civita symbol is a pseudotensor, the result of taking a product is a pseudovector. Under a general coordinate change, the components of the permutation tensor are multiplied by the Jacobian of the transformation matrix. This implies that in coordinate frames different from the one in which the tensor was defined, if the frame is orthonormal, the factor will be ±1 depending on whether the orientation of the frame is the same or not. In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual, thus, one could write ε i j … k = ε i j … k

2.
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

3.
BPST instanton
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In theoretical physics, the BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a solution to the equations of motion of SU Yang–Mills theory in Euclidean space-time. This hope was not realized, however, the BPST instanton is an essentially non-perturbative classical solution of the Yang–Mills field equations. It is found when minimizing the Yang–Mills SU Lagrangian density, L = −14 F μ ν a F μ ν a with Fμνa = ∂μAνa – ∂νAμa + gεabcAμbAνc the field strength. The instanton is a solution with finite action, so that Fμν must go to zero at space-time infinity, space-time infinity of our four-dimensional world is S3. The gauge group SU has exactly the structure, so the solutions with Aμ pure gauge at infinity are mappings from S3 onto itself. These mappings can be labelled by a number q, the Pontryagin index. Instantons have q =1 and thus correspond to gauge transformations which cannot be deformed to unity. The BPST solution is thus topologically stable and it can be shown that self-dual configurations obeying the relation Fμνa = ± ½ εμναβ Fαβa minimize the action. Solutions with a sign are called instantons, those with the minus sign are anti-instantons. The integer is called instanton number, explicitly the instanton solution is given by A μ a =2 g η μ ν a ν2 + ρ2 with zμ the center and ρ the scale of the instanton. ηaμν is the t Hooft symbol, η μ ν a = { ϵ a μ ν μ, ν =1,2,3 − δ a ν μ =4 δ a μ ν =40 μ = ν =4. For large x2, ρ becomes negligible and the gauge field approaches that of the gauge transformation. Indeed, the strength is,12 ϵ i j k F a j k = F a 0 i =4 ρ2 δ a i g 2. The BPST solution has many symmetries, translations and dilations transform a solution into other solutions. Coordinate inversion transforms an instanton of size ρ into an anti-instanton with size 1/ρ, rotations in Euclidean four-space and special conformal transformations leave the solution invariant. The classical action of an instanton equals S =8 π2 g 2, the expression for the BPST instanton given above is in the so-called regular Landau gauge. Another form exists, which is gauge-equivalent with the expression given above, in both these gauges, the expression satisfies ∂μAμ =0

4.
Instanton
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An instanton is a notion appearing in theoretical and mathematical physics. An instanton is a solution to equations of motion with a finite, non-zero action. More precisely, it is a solution to the equations of motion of the field theory on a Euclidean spacetime. In such quantum theories, solutions to the equations of motion may be thought of as points of the action. The critical points of the action may be local maxima of the action, local minima, for example, the classical path is the path that minimizes the action and is therefore a global minimum. Instantons are topologically nontrivial solutions of Yang–Mills equations that absolutely minimize the energy functional within their topological type, many methods developed in studying instantons have also been applied to monopoles. This is because magnetic monopoles arise as solutions of a reduction of the Yang–Mills equations. An instanton can be used to calculate the probability for a quantum mechanical particle tunneling through a potential barrier. One example of a system with an effect is a particle in a double-well potential. In contrast to a particle, there is non-vanishing probability that it crosses a region of potential energy higher than its own energy. One way to calculate this probability is by means of the semi-classical WKB approximation, the Schrödinger equation for the particle reads d 2 ψ d x 2 =2 m ℏ2 ψ. If the potential were constant, the solution would be a plane wave and this means that if the energy of the particle is smaller than the potential energy, one obtains an exponentially decreasing function. The associated tunneling amplitude is proportional to e −1 ℏ ∫ a b 2 m d x, alternatively, the use of path integrals allows an instanton interpretation and the same result can be obtained with this approach. In path integral formulation, the amplitude can be expressed as K = ⟨ x = a | e − i H t ℏ | x = b ⟩ = ∫ d e i S ℏ. The potential energy changes sign V → − V under the Wick rotation, another way to understand the concept of instantons is to consider the action in the path integral. We generally want to look for solutions to a Hamiltonian that minimize the action and we know that the classical solution is the minimum of this. However, if we choose a path that deviates slightly from the path it is possible that its action is infinite. It is possible in cases to find a solution that deviates from the classical path

5.
ArXiv
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In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14,1991, arXiv. org passed the half-million article milestone on October 3,2008, by 2014 the submission rate had grown to more than 8,000 per month. The arXiv was made possible by the low-bandwidth TeX file format, around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Additional modes of access were added, FTP in 1991, Gopher in 1992. The term e-print was quickly adopted to describe the articles and its original domain name was xxx. lanl. gov. Due to LANLs lack of interest in the rapidly expanding technology, in 1999 Ginsparg changed institutions to Cornell University and it is now hosted principally by Cornell, with 8 mirrors around the world. Its existence was one of the factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists regularly upload their papers to arXiv. org for worldwide access, Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv. The annual budget for arXiv is approximately $826,000 for 2013 to 2017, funded jointly by Cornell University Library, annual donations were envisaged to vary in size between $2,300 to $4,000, based on each institution’s usage. As of 14 January 2014,174 institutions have pledged support for the period 2013–2017 on this basis, in September 2011, Cornell University Library took overall administrative and financial responsibility for arXivs operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it was supposed to be a three-hour tour, however, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. The lists of moderators for many sections of the arXiv are publicly available, additionally, an endorsement system was introduced in 2004 as part of an effort to ensure content that is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, new authors from recognized academic institutions generally receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for allegedly restricting scientific inquiry, perelman appears content to forgo the traditional peer-reviewed journal process, stating, If anybody is interested in my way of solving the problem, its all there – let them go and read about it. The arXiv generally re-classifies these works, e. g. in General mathematics, papers can be submitted in any of several formats, including LaTeX, and PDF printed from a word processor other than TeX or LaTeX. The submission is rejected by the software if generating the final PDF file fails, if any image file is too large. ArXiv now allows one to store and modify an incomplete submission, the time stamp on the article is set when the submission is finalized