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Quantum Spacetime
In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some variables that are usually continuous may become discrete. Often only such discrete variables are called "quantized"; usage varies. The idea of quantum spacetime was proposed in the early days of quantum theory by Heisenberg and Ivanenko as a way to eliminate infinities from quantum field theory. The germ of the idea passed from Heisenberg to Rudolf Peierls, who noted that electrons in a magnetic field can be regarded as moving in a quantum spacetime, and to Robert Oppenheimer, who carried it to Hartland Snyder, who published the first concrete example. Snyder's Lie algebra was made simple by C. N. Yang in the same year. Overview Physical spacetime is a quantum spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to 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 (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncommutative Geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some generalized sense). A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions. An approach giving deep insight about noncommutative spaces is through operator algebras (i.e. algebras of bounded linear operators on a Hilbert space). Perhaps one of the typical examples of a noncommutative space is the " noncommutative tori", which played a key role in the early development of this field in 1980s and lead to noncommutativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variable Speed Of Light
A variable speed of light (VSL) is a feature of a family of hypotheses stating that the speed of light may in some way not be constant, for example, that it varies in space or time, or depending on frequency. Accepted classical theories of physics, and in particular general relativity, predict a constant speed of light in any local frame of reference and in some situations these predict apparent variations of the speed of light depending on frame of reference, but this article does not refer to this as a variable speed of light. Various alternative theories of gravitation and cosmology, many of them non-mainstream, incorporate variations in the local speed of light. Attempts to incorporate a variable speed of light into physics were made by Robert Dicke in 1957, and by several researchers starting from the late 1980s. VSL should not be confused with faster than light theories, its dependence on a medium's refractive index or its measurement in a remote observer's frame of ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubly Special Relativity
Doubly special relativity (DSR) – also called deformed special relativity or, by some, extra-special relativity – is a modified theory of special relativity in which there is not only an observer-independent maximum velocity (the speed of light), but also, an observer-independent maximum energy scale (the Planck energy) and/or a minimum length scale (the Planck length). This contrasts with other Lorentz-violating theories, such as the Standard-Model Extension, where Lorentz invariance is instead broken by the presence of a preferred frame. The main motivation for this theory is that the Planck energy should be the scale where as yet unknown quantum gravity effects become important and, due to invariance of physical laws, this scale should remain fixed in all inertial frames. History First attempts to modify special relativity by introducing an observer-independent length were made by Pavlopoulos (1967), who estimated this length at about . In the context of quantum gravity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-abelian Group
In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ''b'' ≠ ''b'' ∗ ''a''. This class of groups contrasts with the abelian groups. (In an abelian group, all pairs of group elements commute). Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order). Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shahn Majid
Shahn Majid (born 1960 in Patna, Bihar, India) is an English pure mathematician and theoretical physicist, trained at Cambridge University and Harvard University and, since 2001, a Professor of Mathematics at the School of Mathematical Sciences, Queen Mary, University of London. Majid is best known for his pioneering work on quantum groups where he introduced one of the two main known classes of these objects and worked on all aspects of their theory. His 1995 textbook ''Foundations of Quantum Group Theory'' is a standard text still used by researchers today. He also pioneered a quantum groups approach to noncommutative geometry and the use of such methods as a route to quantum gravity, leading in 1994 to the first model with testable predictions of quantum spacetime. He is also known for a range of results in algebra and category theory, notably for his theory of braided Hopf algebras and for a new view of the octonions. Although many regard Majid as a pure mathematician, his m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''simple'' objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context. For example, if ''G'' is a finite group (mathematics), group, then a nontrivial finite-dimensional group representation, representation ''V'' over a field (mathematics), field is said to be ''simple'' if the only subrepresentations it contains are either or ''V'' (these are also called irreducible representations). Now Maschke's theorem says that any finite-dimensional representation of a finite group is a direct sum of simple representations (provided the characteristic (field), characteristic of the base field does not divide the order of the group). So in the case of finite group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Differential Calculus
In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra A over a field k means the specification of a space of differential forms over the algebra. The algebra A here is regarded as a coordinate ring but it is important that it may be noncommutative and hence not an actual algebra of coordinate functions on any actual space, so this represents a point of view replacing the specification of a differentiable structure for an actual space. In ordinary differential geometry one can multiply differential 1-forms by functions from the left and from the right, and there exists an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following: 1. An A-A-bimodule \Omega^1 over A, i.e. one can multiply elements of \Omega^1 by elements of A in an associative way: : a(\omega b)=(a\omega)b,\ \forall a,b\in A,\ \omega\in\Omega^1. 2. A linear map :A\to\Omega^1 obeying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our understanding of the most basic fundamentals of physics. Overview A Minkowski spacetime isometry has the property that the interval between events is left invariant. For example, if everything were postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stop-watch you carried with you would be the same. Or if everything were shifted five kilometres to the west, or turned 60 degrees to the right, you would also see no change in the interval. It turns out that the proper length of an object is also unaffected by such a shift. A time or space reversal (a reflection) is also an isometry of this group. In Minkowski space (i.e. ignoring the effec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
