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Orientation Entanglement
In mathematics and physics, the notion of orientation entanglement is sometimes used to develop intuition relating to the geometry of spinors or alternatively as a concrete realization of the failure of the special orthogonal groups to be simply connected. Elementary description Spatial vectors alone are not sufficient to describe fully the properties of rotations in space. Consider the following example. A coffee cup is suspended in a room by a pair of elastic rubber bands fixed to the walls of the room. The cup is rotated by its handle through a full twist of 360°, so that the handle is brought all the way around the central vertical axis of the cup and back to its original position. Note that after this rotation, the cup has been returned to its original orientation, but that its orientation with respect to the walls is ''twisted''. In other words, if we lower the coffee cup to the floor of the room, the two bands will coil around each other in one full twist of a double ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin One-Half (Slow)
Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning Spinning is an ancient textile arts, textile art in which fibre crop, plant, animal fibre, animal or synthetic fibre, synthetic fibres are drawn out and twisted together to form yarn. For thousands of years, fibre was spun by hand using simple ... * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally biased portrayal of something Spin, spinning or spinnin may also refer to: Physics and mathematics * Spin, the rotation of an object around a central axis * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin group, a particular double cover of the special orthogonal group SO(''n'') * Spin tensor, a tensor quantity for describing spinning motion in special relativity and general relativity * Spin (aerodynamics), autorotation of an aerodynamically stalle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Unitary Group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group , consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on \mathbb^n. It is itself a subgroup of the general linear group, \operatorname(n) \subset \operatorname(n) \subset \operatorname(n, \mathbb ). The groups find wide application in the Standard Model of particle physics, especially in the electroweak interaction and in quantum chromodynamics. The groups are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with n qubits and thus 2^n basis states. (Alternatively, the more genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Feynman Lectures On Physics
''The Feynman Lectures on Physics'' is a physics textbook based on some lectures by Richard Feynman, a Nobel laureate who has sometimes been called "The Great Explainer". The lectures were presented before undergraduate students at the California Institute of Technology (Caltech), during 1961–1963. The book's co-authors are Feynman, Robert B. Leighton, and Matthew Sands. ''The Feynman Lectures on Physics'' is perhaps the most popular physics book ever written. More than 1.5 million English-language copies have been sold; probably even more copies have been sold in a dozen foreign-language editions. A 2013 review in ''Nature'' described the book as having "simplicity, beauty, unity ... presented with enthusiasm and insight". Description The textbook comprises three volumes. The first volume focuses on mechanics, radiation, and heat, including relativistic effects. The second volume covers mainly electromagnetism and matter. The third volume covers quantum mechanics; for exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangloids
Tangloids is a mathematical game for two players created by Piet Hein to model the calculus of spinors. A description of the game appeared in the book ''"Martin Gardner's New Mathematical Diversions from Scientific American"'' by Martin Gardner from 1996 in a section on the mathematics of braiding.M. Gardner''Sphere Packing, Lewis Carroll, and Reversi: Martin Gardner's New Mathematical Diversions'', Cambridge University Press, September, 2009, Two flat blocks of wood each pierced with three small holes are joined with three parallel strings. Each player holds one of the blocks of wood. The first player holds one block of wood still, while the other player rotates the other block of wood for two full revolutions. The plane of rotation is perpendicular to the strings when not tangled. The strings now overlap each other. Then the first player tries to untangle the strings without rotating either piece of wood. Only translations (moving the pieces without rotating) are allowed. Aft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plate Trick
In mathematics and physics, the plate trick, also known as Dirac's string trick, the belt trick, or the Balinese cup trick, is any of several demonstrations of the idea that rotating an object with strings attached to it by 360 degrees does not return the system to its original state, while a second rotation of 360 degrees, a total rotation of 720 degrees, does. Mathematically, it is a demonstration of the theorem that SU(2) (which double-covers SO(3)) is simply connected. To say that SU(2) double-covers SO(3) essentially means that the unit quaternions represent the group of rotations twice over. A detailed, intuitive, yet semi-formal articulation can be found in the article on tangloids. Demonstrations Resting a small plate flat on the palm, it is possible to perform two rotations of one's hand while keeping the plate upright. After the first rotation of the hand, the arm will be twisted, but after the second rotation it will end in the original position. To do this, the hand m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an ''n''-sphere. Definition In coordinates, a 3-sphere with center and radius is the set of all points in real, 4-dimensional space () such that :\sum_^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 + ( x_1 - C_1 )^2 + ( x_2 - C_2 )^2+ ( x_3 - C_3 )^2 = r^2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted : :S^3 = \left\. It is often convenient to regard as the space with 2 complex dimensions () or the quaternions (). The unit 3-sphere is then given by :S^3 = \left\ or :S^3 = \left\. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word ''homeomorphism'' comes from the Greek words '' ὅμοιος'' (''homoios'') = similar or same and '' μορφή'' (''morphē'') = shape or form, introduced to mathematics by Henri Poincaré in 1895. Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Change Of Basis
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector on one basis is, in general, different from the coordinate vector that represents on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis. Such a conversion results from the ''change-of-basis formula'' which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis. Using matrices, this formula can be written :\mathbf x_\mathrm = A \,\mathbf x_\mathrm, where "old" and "new" refer respectively to the firstly defined basis and the other basis, \mathbf x_\mathrm and \mathbf x_\mathrm are the colu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although note that in quantum mechanics, A^\ast typically means the complex conjugate only, and not the conjugate transpose. Alternative characterizations Hermit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace-free
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Covering Group
In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. A frequently occurring case is a double covering group, a topological double cover in which ''H'' has index 2 in ''G''; examples include the spin groups, pin groups, and metaplectic groups. Roughly explained, saying that for example the metaplectic group Mp2''n'' is a ''double cover'' of the symplectic group Sp2''n'' means that there are always two elements in the metaplectic group representing one element in the symplectic group. Properties Let ''G'' be a covering group of ''H''. The kernel ''K'' of the covering homomorphism is just the fiber over the identity in ''H'' and is a discrete normal subgroup of ''G''. The kernel ''K'' is closed in ''G'' if and only if ''G'' is Hausdorff (and if and only if ''H'' is Hausdorff). Going in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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