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. Physical reasons have been given to believe that physical spacetime is a quantum spacetime. are already noncommutative, obey the Heisenberg uncertainty principle, and are continuous. Again, physical spacetime is expected to be quantum because physical coordinates are already slightly noncommutative. Both arguments are based on pure gravity and quantum theory, and they limit the measurement of time by the only time constant in pure quantum gravity, the Planck time. The Lie algebra should be semisimple (Yang, I. Bicrossproduct model spacetime
Quantum entanglement Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently – instead, a quantum state may be given for the system as a whole. Such phenomena were the subject of a 1935 paper by Albert Einstein, Boris Podolsky and Nathan Rosen, describing what came to be known as the EPR paradox, and several papers by Erwin Schrödinger shortly thereafter. Einstein and others considered such behavior to be impossible, as it violated the local realist view of causality (Einstein referred to it as "spooky action at a distance"), and argued that the accepted formulation of quantum mechanics must therefore be incomplete. History However, they did not coin the word entanglement, nor did they generalize the special properties of the state they considered. Concept Meaning of entanglement Apparent paradox The hidden variables theory
Proportionality (mathematics) y is directly proportional to x. In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant. The constant is called the coefficient of proportionality or proportionality constant. If one variable is always the product of the other and a constant, the two are said to be directly proportional. x and y are directly proportional if the ratio is constant.If the product of the two variables is always equal to a constant, the two are said to be inversely proportional. x and y are inversely proportional if the product is constant. If a linear function transforms 0, a and b into 0, c and d, and if the product a b c d is not zero, we say a and b are proportional to c and d. where no term is zero, is called a proportion. The two rectangles with stripes are similar, the ratios of their dimensions are horizontally written within the image. or In the proportion Other symbols include: Since
Coulomb's law Value of the constant The exact value of Coulomb's constant ke comes from three of the fundamental, invariant quantities that define free space in the SI system: the speed of light c0 , magnetic permeability μ0 , and electric permittivity ε0 , related by Maxwell as: Use of Coulomb's constant Coulomb's constant is used in many electric equations, although it is sometimes expressed as the following product of the vacuum permittivity constant: Some examples of use of Coulomb's constant are the following: Coulomb's law: Electric potential energy: Electric field: See also References Potential well A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is captured in the local minimum of a potential well. Therefore, a body may not proceed to the global minimum of potential energy, as it would naturally tend to due to entropy. Overview Energy may be released from a potential well if sufficient energy is added to the system such that the local maximum is surmounted. In quantum physics, potential energy may escape a potential well without added energy due to the probabilistic characteristics of quantum particles; in these cases a particle may be imagined to tunnel through the walls of a potential well. The graph of a 2D potential energy function is a potential energy surface that can be imagined as the Earth's surface in a landscape of hills and valleys. Quantum confinement See also
Uncertainty reigns over Heisenberg's measurement analogy A row has broken out among physicists over an analogy used by Werner Heisenberg in 1927 to make sense of his famous uncertainty principle. The analogy was largely forgotten as quantum theory became more sophisticated but has enjoyed a revival over the past decade. While several recent experiments suggest that the analogy is flawed, a team of physicists in the UK, Finland and Germany is now arguing that these experiments are not faithful to Heisenberg's original formulation. Heisenberg's uncertainty principle states that we cannot measure certain pairs of variables for a quantum object – position and momentum, say – both with arbitrary accuracy. When Heisenberg proposed the principle in 1927, he offered a simple physical picture to help it make intuitive sense. Not necessarily wrong Then in 1988 Masanao Ozawa at Nagoya University in Japan argued that Heisenberg's original relationship between error and disturbance does not represent a fundamental limit of uncertainty. Truer to Heisenberg?
Efimov state The Efimov effect is an effect in the quantum mechanics of Few-body systems predicted by the Russian theoretical physicist V. N. Efimov in 1970. Efimov’s effect refers to a scenario in which three identical bosons interact, with the prediction of an infinite series of excited three-body energy levels when a two-body state is exactly at the dissociation threshold. The unusual Efimov state has an infinite number of similar states. In 2005, for the first time the research group of Rudolf Grimm and Hanns-Christoph Nägerl from the Institute for Experimental Physics (University of Innsbruck, Austria) experimentally confirmed such a state in an ultracold gas of caesium atoms. The interest in the "universal phenomena" of cold atomic gases is still growing, especially because of the long awaited experimental results. The discipline of universality in cold atomic gases nearby the Efimov states are sometimes commonly referred to as "Efimov physics". References Jump up ^ В.И.
Linear function In mathematics, the term linear function refers to two different, although related, notions: As a polynomial function In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero). For a function of any finite number independent variables, the general formula is and the graph is a hyperplane of dimension k. A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. In this context, the other meaning (a linear map) may be referred to as a homogeneous linear function or a linear form. As a linear map In linear algebra, a linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication: Some authors use "linear function" only for linear maps that take values in the scalar field; these are also called linear functionals. See also
Gravitational constant The gravitational constant, approximately 6.67×10−11 N·(m/kg)2 and denoted by letter G, is an empirical physical constant involved in the calculation(s) of gravitational force between two bodies. It usually appears in Sir Isaac Newton's law of universal gravitation, and in Albert Einstein's theory of general relativity. It is also known as the universal gravitational constant, Newton's constant, and colloquially as Big G. It should not be confused with "little g" (g), which is the local gravitational field (equivalent to the free-fall acceleration), especially that at the Earth's surface. Laws and constants According to the law of universal gravitation, the attractive force (F) between two bodies is directly proportional to the product of their masses (m1 and m2), and inversely proportional to the square of the distance, r, (inverse-square law) between them: The constant of proportionality, G, is the gravitational constant. with relative standard uncertainty 1.2×10−4. . and
Zero-point energy Zero-point energy, also called quantum vacuum zero-point energy, is the lowest possible energy that a quantum mechanical physical system may have; it is the energy of its ground state. All quantum mechanical systems undergo fluctuations even in their ground state and have an associated zero-point energy, a consequence of their wave-like nature. The uncertainty principle requires every physical system to have a zero-point energy greater than the minimum of its classical potential well. This results in motion even at absolute zero. For example, liquid helium does not freeze under atmospheric pressure at any temperature because of its zero-point energy. History In 1900, Max Planck derived the formula for the energy of a single energy radiator, e.g., a vibrating atomic unit: where is Planck's constant, is the frequency, k is Boltzmann's constant, and T is the absolute temperature. According to this expression, an atomic system at absolute zero retains an energy of ½hν. Varieties .
Amplituhedron An amplituhedron is a geometric structure that enables simplified calculation of particle interactions in some quantum field theories. In planar N = 4 supersymmetric Yang–Mills theory, an amplituhedron is defined as a mathematical space known as the positive Grassmannian. The connection between the amplituhedron and scattering amplitudes is at present a conjecture that has passed many non-trivial checks, including an understanding of how locality and unitarity arise as consequences of positivity. Research has been led by Nima Arkani-Hamed. Description In the approach, the on-shell scattering process "tree" is described by a positive Grassmannian, a structure in algebraic geometry analogous to a convex polytope, that generalizes the idea of a simplex in projective space. A polytope is a kind of higher dimensional polyhedron, and the values being calculated are scattering amplitudes, and so the object is called an amplituhedron. Implications See also References
Personal and Historical Perspectives of Hans Bethe Polynomial The graph of a polynomial function of degree 3 Etymology According to the Oxford English Dictionary, polynomial succeeded the term binomial, and was made simply by replacing the Latin root bi- with the Greek poly-, which comes from the Greek word for many. Notation and terminology It is a common convention to use upper case letters for the indeterminates and the corresponding lower case letters for the variables (arguments) of the associated function. It may be confusing that a polynomial P in the indeterminate X may appear in the formulas either as P or as P(X). Normally, the name of the polynomial is P, not P(X). In particular, if a = X, then the definition of P(a) implies This equality allows writing "let P(X) be a polynomial" as a shorthand for "let P be a polynomial in the indeterminate X". Definition A polynomial in a single indeterminate can be written in the form where This can be expressed more concisely by using summation notation: For example: is a term. then is from