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,[1] describing what came to be known as the EPR paradox, and several papers by Erwin Schrödinger shortly thereafter.[2][3] 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"),[4] and argued that the accepted formulation of quantum mechanics must therefore be incomplete. History[edit] However, they did not coin the word entanglement, nor did they generalize the special properties of the state they considered. Concept[edit] Meaning of entanglement[edit] Apparent paradox[edit] The hidden variables theory[edit]
Quantum geometry In theoretical physics, quantum geometry is the set of new mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at very short distance scales (comparable to Planck length). At these distances, quantum mechanics has a profound effect on physics. Quantum gravity[edit] In an alternative approach to quantum gravity called loop quantum gravity (LQG), the phrase "quantum geometry" usually refers to the formalism within LQG where the observables that capture the information about the geometry are now well defined operators on a Hilbert space. In particular, certain physical observables, such as the area, have a discrete spectrum. It has also been shown that the loop quantum geometry is non-commutative[citation needed]. It is possible (but considered unlikely) that this strictly quantized understanding of geometry will be consistent with the quantum picture of geometry arising from string theory. See also[edit]
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. Edward Witten described the work as “very unexpected" and said that "it is difficult to guess what will happen or what the lessons will turn out to be Description[edit] Using twistor theory, BCFW recursion relations involved in the scattering process may be represented as a small number of twistor diagrams. Implications[edit] See also[edit] References[edit] Notes[edit] Bibliography[edit] External links[edit]
Density matrix Explicitly, suppose a quantum system may be found in state with probability p1, or it may be found in state with probability p2, or it may be found in state with probability p3, and so on. The density operator for this system is[1] By choosing a basis (which need not be orthogonal), one may resolve the density operator into the density matrix, whose elements are[1] For an operator (which describes an observable is given by[1] In words, the expectation value of A for the mixed state is the sum of the expectation values of A for each of the pure states Mixed states arise in situations where the experimenter does not know which particular states are being manipulated. Pure and mixed states[edit] In quantum mechanics, a quantum system is represented by a state vector (or ket) . is called a pure state. and a 50% chance that the state vector is . A mixed state is different from a quantum superposition. Example: Light polarization[edit] An example of pure and mixed states is light polarization. . and . . .
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 space-time, and to Robert Oppenheimer, who carried it to Hartland Snyder, who published the first concrete example.[1] Snyder's Lie algebra was made simple by C. The Lie algebra should be semisimple (Yang, I. for the spatial variables . .
Quantum gauge theory Older approaches to quantization for Abelian models use the Gupta-Bleuler formalism with a "semi-Hilbert space" with an indefinite sesquilinear form. However, it is much more elegant to just work with the quotient space of vector field configurations by gauge transformations. An alternative approach using lattice approximations is covered in (Wick rotated) lattice gauge theory. To establish the existence of the Yang-Mills theory and a mass gap is one of the seven Millennium Prize Problems of the Clay Mathematics Institute. Planck constant Plaque at the Humboldt University of Berlin: "Max Planck, discoverer of the elementary quantum of action h, taught in this building from 1889 to 1928." In 1905 the value (E), the energy of a charged atomic oscillator, was theoretically associated with the energy of the electromagnetic wave itself, representing the minimum amount of energy required to form an electromagnetic field (a "quantum"). Further investigation of quanta revealed behaviour associated with an independent unit ("particle") as opposed to an electromagnetic wave and was eventually given the term photon. The Planck relation now describes the energy of each photon in terms of the photon's frequency. This energy is extremely small in terms of ordinary experience. Since the frequency , wavelength λ, and speed of light c are related by λν = c, the Planck relation for a photon can also be expressed as The above equation leads to another relationship involving the Planck constant. Value[edit] Significance of the value[edit]
Mass gap In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle. Since exact energy eigenstates are infinitely spread out and are therefore usually excluded from a formal mathematical description, a stronger definition is that the mass gap is the greatest lower bound of the energy of any state which is orthogonal to the vacuum. Mathematical definitions[edit] For a given real field with being the lowest energy value in the spectrum of the Hamiltonian and thus the mass gap. with the constant being finite. Examples from classical theories[edit] An example of mass gap arising for massless theories, already at the classical level, can be seen in spontaneous breaking of symmetry or Higgs mechanism. This equation has the exact solution —where and being References[edit]