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Wheeler–DeWitt Equation
The Wheeler–DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity. In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called 'problem of time'. More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergman–Komar "group" (which ''is'' the diffeomorphism group on-shell). Quantum gravity All defined and understood descriptions of string/M-theory deal with fixed asymptotic conditions on the background spacetime. At infinity, the "right" choice of the time coordinate "t" is determined (because the space-time is asymptotic to some fixed space-time) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Review
''Physical Review'' is a peer-reviewed scientific journal established in 1893 by Edward Nichols. It publishes original research as well as scientific and literature reviews on all aspects of physics. It is published by the American Physical Society (APS). The journal is in its third series, and is split in several sub-journals each covering a particular field of physics. It has a sister journal, ''Physical Review Letters'', which publishes shorter articles of broader interest. History ''Physical Review'' commenced publication in July 1893, organized by Cornell University professor Edward Nichols and helped by the new president of Cornell, J. Gould Schurman. The journal was managed and edited at Cornell in upstate New York from 1893 to 1913 by Nichols, Ernest Merritt, and Frederick Bedell. The 33 volumes published during this time constitute ''Physical Review Series I''. The American Physical Society (APS), founded in 1899, took over its publication in 1913 and star ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of the real coordinate space R''n'' into the cosets ''x'' + R''p'' of the standardly embedded subspace R''p''. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class ''Cr''), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class ''Cr'' it is usually understood that ''r'' ≥ 1 (otherwise, ''C''0 is a topological foliation). The number ''p'' (the dimension of the leaves) is called the dimension of the foliation and is called its codimension. In some papers on general relativity by mathematical physicists, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Momentum Constraint
In theoretical physics, it is often important to study theories with the diffeomorphism symmetry such as general relativity. These theories are invariant under arbitrary coordinate transformations. Equations of motion are generally derived from the requirement that the action is stationary. There are special variations that are equivalent to spatial diffeomorphisms. The invariance of the action under these variations implies non-dynamical equations of motion i.e. constraints. These equations must be satisfied or, at least, they must annihilate the physical states in a quantum version of the theory. \infty See also * Wheeler–DeWitt equation The Wheeler–DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and general r ... Quantum gravity Diffeomorphisms {{relativity-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar (mathematics), mathematical number (dimensionless) or a scalar (physics), scalar physical quantity (with unit of measurement, units). In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory. Definition Mathematically, a scalar field on a Region (mathematical analysis), region ''U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clock Operator
A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and the year. Devices operating on several physical processes have been used over the millennia. Some predecessors to the modern clock may be considered as "clocks" that are based on movement in nature: A sundial shows the time by displaying the position of a shadow on a flat surface. There is a range of duration timers, a well-known example being the hourglass. Water clocks, along with the sundials, are possibly the oldest time-measuring instruments. A major advance occurred with the invention of the verge escapement, which made possible the first mechanical clocks around 1300 in Europe, which kept time with oscillating timekeepers like balance wheels., pp. 103–104., p. 31. Traditionally, in horology, the term ''clock'' was used for a striki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decoherence
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics. If a quantum system were perfectly isolated, it would maintain coherence indefinitely, but it would be impossible to manipulate or investigate it. If it is not perfectly isolated, for example during a measurement, coherence is shared with the environment and appears to be lost with time; a process called quantum decoherence. As a result of this process, quantum behavior is apparently lost, just as e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by t ... [...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] |
Functional (mathematics)
In mathematics, a functional (as a noun) is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). * In linear algebra, it is synonymous with linear forms, which are linear mapping from a vector space V into its Field (mathematics), field of scalars (that is, an element of the dual space V^*) "Let ''E'' be a free module over a commutative ring ''A''. We view ''A'' as a free module of rank 1 over itself. By the dual module ''E''∨ of ''E'' we shall mean the module Hom(''E'', ''A''). Its elements will be called functionals. Thus a functional on ''E'' is an ''A''-linear map ''f'' : ''E'' → ''A''." * In functional analysis and related fields, it refers more generally to a mapping from a space X into the field of Real numbers, real or complex numbers. "A numerical function ''f''(''x'') defined on a normed linear space ''R'' will be called a ''functional''. A functional ''f''(''x'') is said to be ''linear'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Class Constraint
A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanishing of all the constraints). To calculate the first class constraint, one assumes that there are no second class constraints, or that they have been calculated previously, and their Dirac brackets generated. First and second class constraints were introduced by as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate. The terminology of first and second class constraints is confusingly similar to that of primary and secondary constraints, reflecting the manner in which these are generated. These divisions are independent: both first and second class constraints can be either primary or secondary, so this gives altogether four different classes of constraints. Poisson brackets Consider a Po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Wavefunction
The universal wavefunction (or wave function), introduced by Hugh Everett in his PhD thesis ''The Theory of the Universal Wave Function,'' informs a core concept in the relative state interpretationHugh Everett, Relative State Formulation of Quantum Mechanics, ''Reviews of Modern Physics'' vol 29, (1957) pp 454–462. An abridged summary of ''The Theory of the Universal Wavefunction'' or many-worlds interpretation of quantum mechanics. It later received investigation from James Hartle and Stephen Hawking in which they derived a specific solution to the Wheeler–deWitt equation to explain the initial conditions of the Big Bang cosmology. Everett's thesis introduction reads: The universal wave function is the wavefunction or quantum state of the totality of existence, regarded as the "basic physical entity" or "the fundamental entity, obeying at all times a deterministic wave equation."Everett 956973, "Theory of the Universal Wavefunction", chapter 6 (e) Everett's response to Str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hartle–Hawking State
The Hartle–Hawking state is a proposal in theoretical physics concerning the state of the Universe prior to the Planck epoch. It is named after James Hartle and Stephen Hawking. Hartle and Hawking suggest that if we could travel backwards in time towards the beginning of the Universe, we would note that quite near what might otherwise have been the beginning, time gives way to space such that at first there is only space and no time. According to the Hartle–Hawking proposal, the Universe has no origin as we would understand it: the Universe was a singularity in both space and time, pre- Big Bang. However, Hawking does state "...the universe has not existed forever. Rather, the universe, and time itself, had a beginning in the Big Bang, about 15 billion years ago.", but that the Hartle-Hawking model is not the steady state Universe of Hoyle; it simply has no initial boundaries in time or space. Technical explanation The Hartle–Hawking state is the wave function of the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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