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Quantum Geometry
In theoretical physics, quantum geometry is the set of mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at distance scales comparable to the Planck length. At these distances, quantum mechanics has a profound effect on physical phenomena. Quantum gravity Each theory of quantum gravity uses the term "quantum geometry" in a slightly different fashion. String theory, a leading candidate for a quantum theory of gravity, uses the term quantum geometry to describe exotic phenomena such as T-duality and other geometric dualities, mirror symmetry, topology-changing transitions, minimal possible distance scale, and other effects that challenge intuition. More technically, quantum geometry refers to the shape of a spacetime manifold as experienced by D-branes which includes quantum corrections to the metric tensor, such as the worldsheet instantons. For example, the quantum volume of a cycle is computed from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Curves
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline is €¦the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which €¦will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image of an interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves such as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi, respectively). The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance And Contravariance Of Vectors
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation, the role is sometimes swapped. In physics, a basis is sometimes thought of as a set of reference axes. A change of scale on the reference axes corresponds to a change of units in the problem. For instance, by changing scale from meters to centimeters (that is, ''dividing'' the scale of the reference axes by 100), the components of a measured velocity vector are ''multiplied'' by 100. A vector changes scale ''inversely'' to changes in scale to the reference axes, and consequently is called ''contravariant''. As a result, a vector often has units of distance or distance with other units (as, for example, velocity has units of distance divided by time). In contrast, a covector, also called a ''dual vector'', typically has units of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Notation
In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to the elements of a list, a vector, or a matrix, depending on whether one is writing a formal mathematical paper for publication, or when one is writing a computer program. In mathematics It is frequently helpful in mathematics to refer to the elements of an array using subscripts. The subscripts can be integers or variables. The array takes the form of tensors in general, since these can be treated as multi-dimensional arrays. Special (and more familiar) cases are vectors (1d arrays) and matrices (2d arrays). The following is only an introduction to the concept: index notation is used in more detail in mathematics (particularly in the representation and manipulation of tensor operations). See the main article for further details. One-d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume Element
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV = \rho(u_1,u_2,u_3)\,du_1\,du_2\,du_3 where the u_i are the coordinates, so that the volume of any set B can be computed by :\operatorname(B) = \int_B \rho(u_1,u_2,u_3)\,du_1\,du_2\,du_3. For example, in spherical coordinates dV = u_1^2\sin u_2\,du_1\,du_2\,du_3, and so \rho = u_1^2\sin u_2. The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume elements to be defined as a kind of measure on a manifold. On an ori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Position Vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s, it corresponds to the straight line segment from ''O'' to ''P''. In other words, it is the displacement or translation that maps the origin to ''P'': :\mathbf=\overrightarrow The term "position vector" is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus. Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces and affine spaces of any dimension.Keller, F. J, Gettys, W. E. et al. (1993), p 28–29 Relative position The relative position of a point ''Q'' with respect to point ''P'' is the Euclidean vector resulting from the subtraction of the two absolute position vectors (each with respect to the origin): :\Delta \mathbf=\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wedge Product
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converting a force applied to its blunt end into forces perpendicular (normal) to its inclined surfaces. The mechanical advantage of a wedge is given by the ratio of the length of its slope to its width..''McGraw-Hill Concise Encyclopedia of Science & Technology'', Third Ed., Sybil P. Parker, ed., McGraw-Hill, Inc., 1992, p. 2041. Although a short wedge with a wide angle may do a job faster, it requires more force than a long wedge with a narrow angle. The force is applied on a flat, broad surface. This energy is transported to the pointy, sharp end of the wedge, hence the force is transported. The wedge simply transports energy in the form of friction and collects it to the pointy end, consequently breaking the item. History Wedges have exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum States
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement in quantum mechanics, measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture distribution, mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray (quantum theory), ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrix, density matrices, which are Definiteness of a matrix, positive semidefinite operators that act on Hilbert spaces. Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Forms
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Lorentzian Quantum Gravity
Causal dynamical triangulation (abbreviated as CDT) theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, is an approach to quantum gravity that, like loop quantum gravity, is background independent. This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. There is evidence that at large scales CDT approximates the familiar 4-dimensional spacetime, but shows spacetime to be 2-dimensional near the Planck scale, and reveals a fractal structure on slices of constant time. These interesting results agree with the findings of Lauscher and Reuter, who use an approach called Quantum Einstein Gravity, and with other recent theoretical work. Introduction Near the Planck scale, the structure of spacetime itself is supposed to be constantly changing due to quantum fluctuations and topological fluctuations. CDT theory uses a triangulation process which varies dynamically and fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
