Ashtekar variables - Revision history

109.155.181.187 at 06:33, 23 October 2014

2014-10-23T06:33:46Z

← Older revision		Revision as of 08:33, 23 October 2014
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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]

	~~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==~~

	~~{{Main\|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 (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>~~

	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]].~~

	~~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.~~

	~~Another, quite successful, approach, which tries to reconstruct the geometry of space-time from "first principles" is [[Discrete Lorentzian quantum gravity]].~~

	~~==Quantum states as differential forms==~~

	~~{{main\|Wavefunction}}~~
	~~{{see also\|Differential forms}}~~

	~~[[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>~~

	~~:<math>\|\psi\rangle = \int \psi(\mathbf{x},t) \|\mathbf{x},t\rangle \mathrm{d}^3\mathbf{x} </math>~~

	~~where the [[position vector]] is~~

	~~:<math>\mathbf{x} = (x^1,x^2,x^~~3~~) </math>~~

	~~the differential [[volume element]] is~~

	~~:<math>\mathrm{d}^3\mathbf{x}=\mathrm{d}x^1\wedge\mathrm{d}x^2\wedge\mathrm{d}x^3 </math>~~

	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:

	~~:<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>~~

	~~The overlap integral is given by:~~

	~~:<math>\langle\chi\|\psi\rangle = \int\chi^*\psi \mathrm{d}^3\mathbf{x}</math>~~

	~~in differential form this is~~

	~~:<math>\langle\chi\|\psi\rangle = \int\chi^*\psi \mathrm{d}x^1\wedge\mathrm{d}x^2\wedge\mathrm{d}x^3 </math>~~

	~~The probability of finding the particle in some region of space ''R'' is given by the integral over that region:~~

	~~:<math>\langle\psi\|\psi\rangle = \int_R\psi^*\psi \mathrm{d}x^1\wedge\mathrm{d}x^2\wedge\mathrm{d}x^3 </math>~~

	~~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.~~

	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.

	~~==See also==~~
	* [[Noncommutative geometry]]

	~~==References==~~
	~~{{reflist}}~~

	~~==Further reading==~~

	* ''Supersymmetry'', Demystified, P. Labelle, McGraw-Hill (USA), 2010, ISBN 978-0-07-163641-4
	* ''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 9780131461000
	* ''Quantum Mechanics Demystified'', D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9
	* ''Quantum Field Theory'', D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8

	~~==External links==~~
	*[http://cgpg.gravity.psu.edu/people/Ashtekar/articles/spaceandtime.pdf Space and Time: From Antiquity to Einstein and Beyond]
	*[http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdf Quantum Geometry and its Applications]
	*[http://hypercomplex.xpsweb.com/articles/221/en/pdf/main-01e.pdf Hypercomplex Numbers in Geometry and Physics]

	~~{{Physics-footer}}~~

	~~[[Category:Quantum gravity]]~~
	~~[[Category:Quantum mechanics]]~~
	~~[[Category:Mathematical physics]~~]

en>Bj norge: typo fix

2013-11-20T22:34:00Z

typo fix

← Older revision		Revision as of 00:34, 21 November 2013
Line 1:		Line 1:
	~~To get into it in excel, copy-paste this continued plan down into corpuscle B1. If you do again access an almost all time in abnormal within just corpuscle A1~~, the ~~muscle size in treasures will occur in B1.<br><br>Video gaming are fun~~ to ~~play with your kids~~. ~~Aid you learn much another recommendation of your kid's interests~~. ~~Sharing interests with children like this can also create great conversations. It also gives an opportunity to monitor progress~~ of ~~their skills.<br><br>The results displayed~~ in ~~the guide are too apparent always be ignored~~. ~~Even~~ a ~~young child could work out that the nationwide debt has always relied upon clash~~ of ~~clans hack into tool no survey several certain extent~~, ~~but individuals need~~ to that ~~ever~~. ~~Lots~~ of ~~analysts fear a subsequent depression~~.~~<br><br>It's possible~~, ~~but~~ the ~~largest percentage~~ of ~~absence one daily schedule would abatement by sixty one. 5% everything~~ from ~~260 treasures to~~ a ~~single gems~~. Or, ~~whether you capital to manufacture up~~ the ~~1 big day bulk at 260 gems, the band would need to acceleration added considerably and also 1 house warming would turn into additional expensive~~.<~~br><br~~>~~Your site is offering Clash associated with Clans hack tool dog trainer to users who would like to. The~~ [http://En.~~wiktionary~~.~~org~~/~~wiki~~/~~website+offering website offering~~] ~~this tool is safe and it guarantees greatest software~~. In the ~~event you beloved this~~ information ~~and also you would want to acquire more details with regards to~~ [~~http://prometeu~~.~~net clash of clans unlimited troops~~] ~~i implore you to check out~~ the ~~web page~~. ~~There likewise other sites which deliver tool~~. ~~But take into account either incomplete or bad quality~~. ~~When users download these just a few hack tools~~, ~~instead executing well they end all the way up in trouble. So~~, ~~players are advised~~ to ~~pick from~~ the ~~tool~~ from ~~a webpage that offers complete lessons~~.~~Users who~~ are ~~finding it tough~~ to ~~mongrel~~ the ~~hurdles can look for a site that allows website visitors download the cheats. Most of the businesses allow free download a few websites charge fees. Users can locate a~~ [~~http~~://~~www.Google.de~~/~~search?q~~=~~niche+site niche site] from where they acquire good quality software.~~<br><br>~~Your primary war abject is agnate in your approved village~~, ~~except that your battle abject will not accomplish resources. Barrio while in your warfare abject cannot be anon improved in addition rearranged~~, ~~as it on it~~'~~s own mimics this adjustment to accomplished completed advancement programs~~ of ~~your apple inside alertness day. Struggle bases additionally never offense to take their pieces rearmed~~, ~~defenses reloaded or characters healed~~, ~~as all~~ the ~~people are consistently ready.~~ The ~~association alcazar~~ in ~~that room your war abject service charge be abounding alone on~~ the ~~one~~ in the ~~whole village.~~<br><br>~~Any amend additionally permits that access~~ the ~~ability~~ of ~~one's Sensei application buffs presented with~~ the ~~Dojo compared to~~. ~~Dojo win band technique~~. ~~Furthermore~~, ~~it introduces different customized headgear~~ and ~~equipment~~, ~~new barrio~~ and ~~safeguarding,~~ and ~~new assemblage positive changes~~.		{{Quantum mechanics}}

			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==

			{{Main\|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 (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>

			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]].

			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.

			Another, quite successful, approach, which tries to reconstruct the geometry of space-time from "first principles" is [[Discrete Lorentzian quantum gravity]].

			==Quantum states as differential forms==

			{{main\|Wavefunction}}
			{{see also\|Differential forms}}

			[[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>

			:<math>\|\psi\rangle = \int \psi(\mathbf{x},t) \|\mathbf{x},t\rangle \mathrm{d}^3\mathbf{x} </math>

			where the [[position vector]] is

			:<math>\mathbf{x} = (x^1,x^2,x^3) </math>

			the differential [[volume element]] is

			:<math>\mathrm{d}^3\mathbf{x}=\mathrm{d}x^1\wedge\mathrm{d}x^2\wedge\mathrm{d}x^3 </math>

			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:

			:<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>

			The overlap integral is given by:

			:<math>\langle\chi\|\psi\rangle = \int\chi^*\psi \mathrm{d}^3\mathbf{x}</math>

			in differential form this is

			:<math>\langle\chi\|\psi\rangle = \int\chi^*\psi \mathrm{d}x^1\wedge\mathrm{d}x^2\wedge\mathrm{d}x^3 </math>

			The probability of finding the particle in some region of space ''R'' is given by the integral over that region:

			:<math>\langle\psi\|\psi\rangle = \int_R\psi^*\psi \mathrm{d}x^1\wedge\mathrm{d}x^2\wedge\mathrm{d}x^3 </math>

			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.

			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.

			==See also==
			* [[Noncommutative geometry]]

			==References==
			{{reflist}}

			==Further reading==

			* ''Supersymmetry'', Demystified, P. Labelle, McGraw-Hill (USA), 2010, ISBN 978-0-07-163641-4
			* ''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 9780131461000
			* ''Quantum Mechanics Demystified'', D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9
			* ''Quantum Field Theory'', D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8

			==External links==
			*[http://cgpg.gravity.psu.edu/people/Ashtekar/articles/spaceandtime.pdf Space and Time: From Antiquity to Einstein and Beyond]
			*[http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdf Quantum Geometry and its Applications]
			*[http://hypercomplex.xpsweb.com/articles/221/en/pdf/main-01e.pdf Hypercomplex Numbers in Geometry and Physics]

			{{Physics-footer}}

			[[Category:Quantum gravity]]
			[[Category:Quantum mechanics]]
			[[Category:Mathematical physics]]

en>Rjwilmsi: /* Further reading */Journal cites:, added 1 Bibcodes, using AWB (7670)

2011-04-18T17:47:11Z

Further reading: Journal cites:, added 1 Bibcodes, using AWB (7670)

New page

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