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Supernumber
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra of a complex vector space. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed. Informal discussion Grassmann numbers are generated by anti-commuting elements or objects. The idea of anti-commuting objects arises in multiple areas of mathematics: they are typically seen in differential geometry, where the differential forms are anti-commuting. Differential forms are normally defined in terms of derivatives on a manifold; however, one can contemplate the situation where one "forgets" or "ignores" the existence of any underlying manifold, and "forgets" or "ignores" that the forms were define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Dual numbers can be added component-wise, and multiplied by the formula : (a+b\varepsilon)(c+d\varepsilon) = ac + (ad+bc)\varepsilon, which follows from the property and the fact that multiplication is a bilinear operation. The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements. History Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as , where is the angle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector v in V. The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol \wedge and the fact that the product of two elements of V is "outside" V. The wedge product of k vectors v_1 \wedge v_2 \wedge \dots \wedge v_k is called a ''blade (geometry), blade of degree k'' or ''k-blade''. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude (mathematics), magnitude of a bivector, -blade v\wedge w is the area of the parallelogram defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the Parallelepiped#Parallelotope, parallelotope defined by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics, known as physical mathematics. Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world. Classical mechanics Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Group
In mathematics the spin group, denoted Spin(''n''), page 15 is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathbb_2 \to \operatorname(n) \to \operatorname(n) \to 1. The group multiplication law on the double cover is given by lifting the multiplication on \operatorname(n). As a Lie group, Spin(''n'') therefore shares its dimension, , and its Lie algebra with the special orthogonal group. For , Spin(''n'') is simply connected and so coincides with the universal cover of SO(''n''). The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −. Spin(''n'') can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(''n''). A distinct article discusses the spin representations. Use for physics models The spin group is use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current standard model of particle physics is based on QFT. History Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory—quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inabili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac, and were later developed, most notably, by Pascual Jordan and Vladimir Fock. In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory. Quantum many- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ladder Operator
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising and lowering operators are commonly known as the creation and annihilation operators, respectively. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum. Terminology There is a relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field theory which lies in representation theory. The creation operator ''a''''i''† increments the number of particles in state ''i'', while the corresponding annihilation operator ''ai'' decrements the number of particles in state ''i''. This clearly satisfies the requirements of the above definition of a ladder operator: the incrementin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter), psi, respectively). Wave functions are complex number, complex-valued. For example, a wave function might assign a complex number to each point in a region of space. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density function, probability density of measurement in quantum mechanics, measuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called ''normalization''. Since the wave function is complex-valued, only its relative phase and relative magnitud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Spinor
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions. None of the elementary particles in the Standard Model are Weyl fermions. Previous to the confirmation of the neutrino oscillations, it was considered possible that the neutrino might be a Weyl fermion (it is now expected to be either a Dirac or a Majorana fermion). In condensed matter physics, some materials can display quasiparticles that behave as Weyl fermions, leading to the notion of Weyl semimetals. Mathematically, any Dirac fermion can be decomposed as two Weyl fermions of opposite chirality coupled by the mass term. History The Dirac equation was published in 1928 by Paul Dirac, and was first used to model spin-1/2 particles in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford (1845–1879). The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''. Introduction and basic properties A Clifford algebra is a unital associative algebra that contains and is generated by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
