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| {{Quantum mechanics}}
| | My name is Ola (47 years old) and my hobbies are Baseball and Skateboarding.<br><br>my blog post ... [http://www.denrrou.com/%E6%BF%80%E5%AE%89-%E3%83%A1%E3%83%B3%E3%82%BA-%E3%83%95%E3%82%A1%E3%83%83%E3%82%B7%E3%83%A7%E3%83%B3-%E6%96%B0%E5%88%B0%E7%9D%80/%E6%A4%9C%E6%9F%BB%E5%90%88%E6%A0%BC-3-4-tee-%E3%82%A2%E3%82%A6%E3%83%88%E3%83%AC%E3%83%83%E3%83%88 メンズ ファッション 3 4 tee] |
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| 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.
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| ==Quantum gravity==
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| {{Main|quantum gravity}}
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| 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 (string theory)|mirror symmetry]], [[topology]]-changing transitions, minimal possible distance scale, and other effects that challenge our usual geometrical intuition. More technically, quantum geometry refers to the shape of the spacetime manifold as seen by [[D-branes]] which includes the quantum corrections to the [[metric tensor]], such as the worldsheet [[instanton]]s. For example, the quantum volume of a cycle is computed from the mass of a [[Membrane (M-theory)|brane]] wrapped on this cycle. As another example, a distance between two quantum mechanics particles can be expressed in terms of the [[Lukaszyk–Karmowski metric]].<ref>[http://www.springerlink.com/content/y4fbdb0m0r12701p A new concept of probability metric and its applications in approximation of scattered data sets], Łukaszyk Szymon, Computational Mechanics Volume 33, Number 4, 299–304, Springer-Verlag 2003 {{doi|10.1007/s00466-003-0532-2}}</ref>
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| In an alternative approach to quantum gravity called [[loop quantum gravity]] (LQG), the phrase "quantum geometry" usually refers to the [[Scientific formalism|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 [[observable]]s, such as the area, have a [[discrete spectrum]]. It has also been shown that the loop quantum geometry is [[non-commutative geometry|non-commutative]].
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| 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.
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| Another, quite successful, approach, which tries to reconstruct the geometry of space-time from "first principles" is [[Discrete Lorentzian quantum gravity]].
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| ==Quantum states as differential forms==
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| {{main|Wavefunction}}
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| {{see also|Differential forms}}
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| [[Differential forms]] are used to express [[quantum states]], using the [[wedge product]]:<ref>''The Road to Reality'', Roger Penrose, Vintage books, 2007, ISBN 0-679-77631-1</ref>
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| :<math>|\psi\rangle = \int \psi(\mathbf{x},t) |\mathbf{x},t\rangle \mathrm{d}^3\mathbf{x} </math>
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| where the [[position vector]] is
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| :<math>\mathbf{x} = (x^1,x^2,x^3) </math>
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| the differential [[volume element]] is
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| :<math>\mathrm{d}^3\mathbf{x}=\mathrm{d}x^1\wedge\mathrm{d}x^2\wedge\mathrm{d}x^3 </math>
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| and ''x''<sup>1</sup>, ''x''<sup>2</sup>, ''x''<sup>3</sup> are an arbitrary set of coordinates, the upper [[Index notation|indices]] indicate [[Covariance and contravariance of vectors|contravariance]], lower indices indicate [[Covariance and contravariance of vectors|covariance]], so explicitly the quantum state in differential form is:
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| :<math>|\psi\rangle = \int \psi(x^1,x^2,x^3,t) |x^1,x^2,x^3,t\rangle \mathrm{d}x^1\wedge\mathrm{d}x^2\wedge\mathrm{d}x^3 </math>
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| The overlap integral is given by:
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| :<math>\langle\chi|\psi\rangle = \int\chi^*\psi \mathrm{d}^3\mathbf{x}</math>
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| in differential form this is
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| :<math>\langle\chi|\psi\rangle = \int\chi^*\psi \mathrm{d}x^1\wedge\mathrm{d}x^2\wedge\mathrm{d}x^3 </math>
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| The probability of finding the particle in some region of space ''R'' is given by the integral over that region:
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| :<math>\langle\psi|\psi\rangle = \int_R\psi^*\psi \mathrm{d}x^1\wedge\mathrm{d}x^2\wedge\mathrm{d}x^3 </math>
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| provided the wave function is [[Wave function|normalized]]. When ''R'' is all of 3d position space, the integral must be 1 if the particle exists.
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| Differential forms are an approach for describing the geometry of [[Mathematical curves|curves]] and [[surfaces]] in a coordinate independent way. In [[quantum mechanics]], idealized situations occur in rectangular [[Cartesian coordinates]], such as the [[potential well]], [[particle in a box]], [[quantum harmonic oscillator]], and more realistic approximations in [[spherical polar coordinates]] such as [[electrons]] in [[atoms]] and [[molecules]]. For generality, a formalism which can be used in any coordinate system is useful.
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| ==See also==
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| * [[Noncommutative geometry]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * ''Supersymmetry'', Demystified, P. Labelle, McGraw-Hill (USA), 2010, ISBN 978-0-07-163641-4
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| * ''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 9780131461000
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| * ''Quantum Mechanics Demystified'', D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9
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| * ''Quantum Field Theory'', D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
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| ==External links==
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| *[http://cgpg.gravity.psu.edu/people/Ashtekar/articles/spaceandtime.pdf Space and Time: From Antiquity to Einstein and Beyond]
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| *[http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdf Quantum Geometry and its Applications]
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| *[http://hypercomplex.xpsweb.com/articles/221/en/pdf/main-01e.pdf Hypercomplex Numbers in Geometry and Physics]
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| {{Physics-footer}}
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| [[Category:Quantum gravity]]
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| [[Category:Quantum mechanics]]
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| [[Category:Mathematical physics]]
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My name is Ola (47 years old) and my hobbies are Baseball and Skateboarding.
my blog post ... メンズ ファッション 3 4 tee