Vector measure - Revision history

178.190.94.37: Wrong statements corrected: Vector measures are not generalizations of measures that can take infinite values. Vector measures on set algebras generalize finite measures only. ASlateff

2015-01-02T15:09:42Z

Wrong statements corrected: Vector measures are not generalizations of measures that can take infinite values. Vector measures on set algebras generalize finite measures only. ASlateff

← Older revision		Revision as of 17:09, 2 January 2015
Line 1:		Line 1:
	~~Golda~~ is ~~what's created on my birth certification even though it is not the title on my birth certificate~~. ~~The favorite pastime for him~~ and his ~~kids is fashion and he'll be starting some thing else along with~~ it. ~~She functions as~~ a ~~journey agent but quickly~~ she~~'ll be on her personal. My wife~~ and ~~I live in Kentucky~~.<br><br>my ~~web blog; online psychic readings -~~ [http://~~help~~.~~ksu~~.~~edu.sa~~/~~node~~/~~65129 mouse click the next web page~~] -		The author is known as Araceli Gulledge. Alabama has always been his home and his family members enjoys it. My job is a messenger. What she enjoys performing is bottle tops gathering and she is trying to make it a occupation.<br><br>Feel free to visit my webpage [http://elearning.Qamtraining.net/UserProfile/tabid/42/userId/141840/language/en-US/Default.aspx elearning.Qamtraining.net]

en>Monkbot: /* Lyapunov's theorem */Fix CS1 deprecated date parameter errors (test)

2014-02-26T23:42:35Z

Lyapunov's theorem: Fix CS1 deprecated date parameter errors (test)

← Older revision		Revision as of 01:42, 27 February 2014
Line 1:		Line 1:
	~~{{About\|String duality\|other forms of duality\|Duality (disambiguation)}}~~		Golda is what's created on my birth certification even though it is not the title on my birth certificate. The favorite pastime for him and his kids is fashion and he'll be starting some thing else along with it. She functions as a journey agent but quickly she'll be on her personal. My wife and I live in Kentucky.<br><br>my web blog; online psychic readings - [http://help.ksu.edu.sa/node/65129 mouse click the next web page] -

	'''String duality''' is a class of [[symmetry in physics\|symmetries]] in [[physics]] that link different [[string theory\|string theories]], theories which assume that the fundamental building blocks of the [[universe]] are [[string (physics)\|string]]s instead of [[point particle]]s.

	~~Before the so-called "duality revolution" there were believed to be five distinct versions of string theory, plus the (unstable) bosonic and gluonic theories.~~

	~~{\|class="wikitable" mm_noconvert="TRUE"~~
	~~\|- bgcolor="#FFFFFF"~~
	~~! colspan="3" class="dark" \| String Theories~~
	\|-
	~~! class="dark" \| Type~~
	~~! class="dark" \| Spacetime dimensions<br>~~
	~~! class="dark" \| Details~~
	\|-
	~~! class="dark" \| Bosonic~~
	~~\| align="CENTER" class="dark" \| 26~~
	\| class="dark" \| Only [[boson]]s, no [[fermion]]s means only forces, no matter, with both open and closed strings; major flaw: a [[Particle physics\|particle]] with imaginary [[mass]], called the [[tachyon]], representing an instability in the theory.
	\|-
	~~! class="dark" \| I~~
	~~\| align="CENTER" class="dark" \| 10~~
	~~\| class="dark" \| [[Supersymmetry]] between forces and matter, with both closed strings and open strings, no [[tachyon]], group symmetry is [[special orthogonal group\|SO(32)]]~~
	\|-
	~~! class="dark" \| IIA~~
	~~\| align="CENTER" class="dark" \| 10~~
	~~\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings and open strings bound to [[D-brane]]s, no [[tachyon]], massless [[fermion]]s spin both ways (nonchiral)~~
	\|-
	~~! class="dark" \| IIB~~
	~~\| align="CENTER" class="dark" \| 10~~
	~~\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings and open strings bound to [[D-brane]]s, no [[tachyon]], massless [[fermion]]s only spin one way (chiral)~~
	\|-
	~~! class="dark" \| HO~~
	~~\| align="CENTER" class="dark" \| 10~~
	~~\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings only, no [[tachyon]], [[heterotic]], meaning right moving and left moving strings differ, group symmetry~~ is ~~[[special orthogonal group\|SO(32)]]~~
	\|-
	~~! class="dark" \| HE~~
	~~\| align="CENTER" class="dark" \| 10~~
	\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings only, no [[tachyon]], [[heterotic]], meaning right moving and left moving strings differ, group symmetry is [[E8 (mathematics)\|''E''<sub>8</sub>×''E''~~<sub>8</sub>]]~~
	\|}

	Note that in the type IIA and type IIB string theories closed strings are allowed to move everywhere throughout the ten-dimensional space-time (called the ''bulk''), while open strings have their ends attached to [[D-brane]]s~~, which are membranes of lower dimensionality (their dimension is odd - 1,3,5,7 or 9 - in type IIA and~~ even ~~- 0,2,4,6 or 8 - in type IIB, including the time direction).~~

	Before the 1990s, string theorists believed there were five distinct superstring theories: [[type I string\|type I]], [[type IIA string\|types IIA]] and [[type IIB string\|IIB]], and the two [[heterotic string]] theories ([[special orthogonal group\|SO(32)]] and [[E8 (mathematics)\|''E''<sub>8</sub>×''E''<sub>8</sub>]]). The thinking was that out of these five candidate theories, only one was the actual [[theory of everything]], and that theory was the theory whose low energy limit, with ten dimensions spacetime [[compactification (physics)\|compactified]] down to four, matched the physics observed in our world today. It is ~~now known that the five superstring theories are~~ not fundamental, but are instead different limits of a more fundamental theory, dubbed [[M-theory]]. These theories are related by transformations called dualities. If two theories are related by a duality transformation, each observable of the ~~first theory can be mapped in some way to the second theory to yield equivalent predictions~~. The two theories are then said to be dual to one another under that transformation. Put differently, the two theories are two mathematically different descriptions of the same phenomena. A simple example of a duality is the equivalence of [[particle physics]] upon replacing matter with antimatter; describing our universe in terms of anti-particles would yield identical predictions for ~~any possible experiment.~~

	String dualities often link quantities that appear to be separate: Large and small distance scales, strong and weak coupling strengths. These quantities have always marked very distinct limits of behavior of a physical system, in both [[classical field theory]] and ~~quantum [[particle physics]]. But strings can obscure the difference between large and small, strong and weak, and this~~ is ~~how these five very different theories end up being related.~~

	~~==T-duality==~~
	~~{{main\|T-duality}}~~
	~~Suppose we are in ten spacetime dimensions, which means we have nine space dimensions~~ and one time. Take one of those nine space dimensions and make it a circle of radius R, so that traveling in that direction for a distance L = 2πR takes you around the circle and brings you back to where you started. A particle traveling around this circle will have a quantized [[momentum]] around the circle, because its momentum is linked to its [[wavelength]] (see [[Wave-particle duality]]), and 2πR must be a multiple of that. In fact, the particle momentum around the circle - and the contribution to its energy - is of the form n/R (in [[standard units]], for an integer n), so that at large R there will be many more states compared to small R (for a given maximum energy). A string, in addition to traveling around the circle, may also wrap around it. ~~The number of times the string winds around the circle is called the [[winding number]], and that is also quantized (~~as it must be an integer). Winding around the circle requires energy, because the string must be stretched against its tension, so it contributes an amount of energy of the form <math>wR/L_{st}^2</math>, where <math>L_{st}</math> is a ~~constant called the~~ '~~'string length'' and w is the winding number (an integer). Now (for a given maximum energy) there will~~ be ~~many different states (with different momenta) at large R, but there will also be many different states (with different windings) at small R~~. ~~In fact, a theory with large R~~ and ~~a theory with small R are equivalent, where the role of momentum in the first is played by the winding~~ in ~~the second, and vice versa~~. ~~Mathematically, taking R to~~ <~~math~~>~~L_{st}^2/R~~<~~/math~~> ~~and switching n and w will yield the same equations. So exchanging momentum and winding modes of the string exchanges a large distance scale with a small distance scale.~~

	This type of duality is called [[T-duality]]. T-duality relates [[type IIA string\|type IIA]] superstring theory to [[type IIB string\|type IIB]] superstring theory. That means if we take type IIA and Type IIB theory and compactify them both on a circle (one with a large radius and the other with a small radius) then switching the momentum and winding modes, and switching the distance scale, changes one theory into the other. The same is also true for the two heterotic theories. T-duality also relates [[type I string\|type I]] superstring theory to both type IIA and type IIB superstring theories with certain boundary conditions (termed [[orientifold]]).

	Formally, the location of the string on the circle is described by two fields living on it, one which is left-moving and another which is right-moving. The movement of the string center (and hence its momentum) is related to the sum of the fields, while the string stretch (and hence its winding number) is related to their difference. T-duality can be formally described by taking the left-moving field to minus itself, so that the sum and the difference are interchanged, leading to switching of momentum and winding.

	~~==S-duality==~~
	~~{{main\|S-duality\|M-theory}}~~
	Every [[force]] has a [[coupling constant]], which is a measure of its strength, and determines the chances of one particle to emit or absorb another particle. For [[electromagnetism]], the coupling constant is proportional to the square of the [[electric charge]]. When physicists study the [[Quantum electrodynamics\|quantum behavior of electromagnetism]], they can't solve the whole theory exactly, because every particle may emit and absorb many other particles, which may also do the same, endlessly. So events of emission and absorption are considered as perturbations and are dealt with by a series of approximations, first assuming there is only one such event, then correcting the result for allowing two such events, etc. (this method is called [[Perturbation theory]]). This is a reasonable approximation only if the coupling constant is small, which is the case for electromagnetism. But if the coupling constant gets large, that method of calculation breaks down, and the little pieces become worthless as an approximation to the real physics.

	This also can happen in string theory. String theories have a coupling constant. But unlike in particle theories, the string coupling constant is not just a number, but depends on one of the [[oscillation]] modes of the string, called the [[dilaton]]. Exchanging the dilaton field with minus itself exchanges a very large coupling constant with a very small one. This symmetry is called [[S-duality]]. If two string theories are related by S-~~duality, then one theory with a strong coupling constant is the same as the other theory with weak coupling constant. The theory with strong coupling cannot be understood by means of~~ [[perturbation theory]], but the theory with weak coupling can. So if the two theories are related by S-duality, then we just need to understand the weak theory, and that is equivalent to understanding the strong theory.

	~~Superstring theories related by S-duality are~~: ~~[[type I string\|type I]] superstring theory with [[heterotic]] [[special orthogonal group\|SO(32)]] superstring theory, and [[type IIB string\|type IIB]] theory with itself~~.

	~~Furthermore, [[type IIA string\|type IIA]] theory in strong coupling behaves like an 11-dimensional theory, with the [[dilaton]] field playing the role of an eleventh dimension~~. ~~This 11-dimensional theory is known as [[M-theory]]~~.

	~~Unlike~~ the T-duality, however, S-duality has not been proven to even a physics level of rigor for any of the aforementioned cases. It remains, strictly speaking, a conjecture, although most string theorists believe in its validity.

	~~==See also==~~
	*[[Dilaton]]
	*[[M-theory]]
	*[[S-~~duality]]~~
	*[[String (physics)\|String]]
	*[[String theory]]
	*[[T-duality]]
	*[[U-duality]]

	~~==References==~~
	~~{{Unreferenced\|date=February 2007}}~~

	~~[[Category:String theory]]~~

en>Monkbot: /* Lyapunov's theorem */Fix CS1 deprecated date parameter errors

2014-01-31T13:44:16Z

Lyapunov's theorem: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 15:44, 31 January 2014
Line 1:		Line 1:
	~~The author~~'s ~~name~~ is ~~Christy~~. ~~Kentucky~~ is ~~exactly~~ where ~~I've usually been living~~. He is an ~~order clerk~~ and it'~~s something he truly enjoy~~. ~~Doing ballet~~ is ~~some thing she would~~ by ~~no means give up~~.<br><br>~~Here~~ is ~~my site~~ ... [~~https~~:~~//www~~-~~ocl~~.~~gist~~.ac.~~kr/work/xe/?document_srl~~=~~605236 online psychic reading~~]		{{About\|String duality\|other forms of duality\|Duality (disambiguation)}}

			'''String duality''' is a class of [[symmetry in physics\|symmetries]] in [[physics]] that link different [[string theory\|string theories]], theories which assume that the fundamental building blocks of the [[universe]] are [[string (physics)\|string]]s instead of [[point particle]]s.

			Before the so-called "duality revolution" there were believed to be five distinct versions of string theory, plus the (unstable) bosonic and gluonic theories.

			{\|class="wikitable" mm_noconvert="TRUE"
			\|- bgcolor="#FFFFFF"
			! colspan="3" class="dark" \| String Theories
			\|-
			! class="dark" \| Type
			! class="dark" \| Spacetime dimensions<br>
			! class="dark" \| Details
			\|-
			! class="dark" \| Bosonic
			\| align="CENTER" class="dark" \| 26
			\| class="dark" \| Only [[boson]]s, no [[fermion]]s means only forces, no matter, with both open and closed strings; major flaw: a [[Particle physics\|particle]] with imaginary [[mass]], called the [[tachyon]], representing an instability in the theory.
			\|-
			! class="dark" \| I
			\| align="CENTER" class="dark" \| 10
			\| class="dark" \| [[Supersymmetry]] between forces and matter, with both closed strings and open strings, no [[tachyon]], group symmetry is [[special orthogonal group\|SO(32)]]
			\|-
			! class="dark" \| IIA
			\| align="CENTER" class="dark" \| 10
			\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings and open strings bound to [[D-brane]]s, no [[tachyon]], massless [[fermion]]s spin both ways (nonchiral)
			\|-
			! class="dark" \| IIB
			\| align="CENTER" class="dark" \| 10
			\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings and open strings bound to [[D-brane]]s, no [[tachyon]], massless [[fermion]]s only spin one way (chiral)
			\|-
			! class="dark" \| HO
			\| align="CENTER" class="dark" \| 10
			\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings only, no [[tachyon]], [[heterotic]], meaning right moving and left moving strings differ, group symmetry is [[special orthogonal group\|SO(32)]]
			\|-
			! class="dark" \| HE
			\| align="CENTER" class="dark" \| 10
			\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings only, no [[tachyon]], [[heterotic]], meaning right moving and left moving strings differ, group symmetry is [[E8 (mathematics)\|''E''<sub>8</sub>×''E''<sub>8</sub>]]
			\|}

			Note that in the type IIA and type IIB string theories closed strings are allowed to move everywhere throughout the ten-dimensional space-time (called the ''bulk''), while open strings have their ends attached to [[D-brane]]s, which are membranes of lower dimensionality (their dimension is odd - 1,3,5,7 or 9 - in type IIA and even - 0,2,4,6 or 8 - in type IIB, including the time direction).

			Before the 1990s, string theorists believed there were five distinct superstring theories: [[type I string\|type I]], [[type IIA string\|types IIA]] and [[type IIB string\|IIB]], and the two [[heterotic string]] theories ([[special orthogonal group\|SO(32)]] and [[E8 (mathematics)\|''E''<sub>8</sub>×''E''<sub>8</sub>]]). The thinking was that out of these five candidate theories, only one was the actual [[theory of everything]], and that theory was the theory whose low energy limit, with ten dimensions spacetime [[compactification (physics)\|compactified]] down to four, matched the physics observed in our world today. It is now known that the five superstring theories are not fundamental, but are instead different limits of a more fundamental theory, dubbed [[M-theory]]. These theories are related by transformations called dualities. If two theories are related by a duality transformation, each observable of the first theory can be mapped in some way to the second theory to yield equivalent predictions. The two theories are then said to be dual to one another under that transformation. Put differently, the two theories are two mathematically different descriptions of the same phenomena. A simple example of a duality is the equivalence of [[particle physics]] upon replacing matter with antimatter; describing our universe in terms of anti-particles would yield identical predictions for any possible experiment.

			String dualities often link quantities that appear to be separate: Large and small distance scales, strong and weak coupling strengths. These quantities have always marked very distinct limits of behavior of a physical system, in both [[classical field theory]] and quantum [[particle physics]]. But strings can obscure the difference between large and small, strong and weak, and this is how these five very different theories end up being related.

			==T-duality==
			{{main\|T-duality}}
			Suppose we are in ten spacetime dimensions, which means we have nine space dimensions and one time. Take one of those nine space dimensions and make it a circle of radius R, so that traveling in that direction for a distance L = 2πR takes you around the circle and brings you back to where you started. A particle traveling around this circle will have a quantized [[momentum]] around the circle, because its momentum is linked to its [[wavelength]] (see [[Wave-particle duality]]), and 2πR must be a multiple of that. In fact, the particle momentum around the circle - and the contribution to its energy - is of the form n/R (in [[standard units]], for an integer n), so that at large R there will be many more states compared to small R (for a given maximum energy). A string, in addition to traveling around the circle, may also wrap around it. The number of times the string winds around the circle is called the [[winding number]], and that is also quantized (as it must be an integer). Winding around the circle requires energy, because the string must be stretched against its tension, so it contributes an amount of energy of the form <math>wR/L_{st}^2</math>, where <math>L_{st}</math> is a constant called the ''string length'' and w is the winding number (an integer). Now (for a given maximum energy) there will be many different states (with different momenta) at large R, but there will also be many different states (with different windings) at small R. In fact, a theory with large R and a theory with small R are equivalent, where the role of momentum in the first is played by the winding in the second, and vice versa. Mathematically, taking R to <math>L_{st}^2/R</math> and switching n and w will yield the same equations. So exchanging momentum and winding modes of the string exchanges a large distance scale with a small distance scale.

			This type of duality is called [[T-duality]]. T-duality relates [[type IIA string\|type IIA]] superstring theory to [[type IIB string\|type IIB]] superstring theory. That means if we take type IIA and Type IIB theory and compactify them both on a circle (one with a large radius and the other with a small radius) then switching the momentum and winding modes, and switching the distance scale, changes one theory into the other. The same is also true for the two heterotic theories. T-duality also relates [[type I string\|type I]] superstring theory to both type IIA and type IIB superstring theories with certain boundary conditions (termed [[orientifold]]).

			Formally, the location of the string on the circle is described by two fields living on it, one which is left-moving and another which is right-moving. The movement of the string center (and hence its momentum) is related to the sum of the fields, while the string stretch (and hence its winding number) is related to their difference. T-duality can be formally described by taking the left-moving field to minus itself, so that the sum and the difference are interchanged, leading to switching of momentum and winding.

			==S-duality==
			{{main\|S-duality\|M-theory}}
			Every [[force]] has a [[coupling constant]], which is a measure of its strength, and determines the chances of one particle to emit or absorb another particle. For [[electromagnetism]], the coupling constant is proportional to the square of the [[electric charge]]. When physicists study the [[Quantum electrodynamics\|quantum behavior of electromagnetism]], they can't solve the whole theory exactly, because every particle may emit and absorb many other particles, which may also do the same, endlessly. So events of emission and absorption are considered as perturbations and are dealt with by a series of approximations, first assuming there is only one such event, then correcting the result for allowing two such events, etc. (this method is called [[Perturbation theory]]). This is a reasonable approximation only if the coupling constant is small, which is the case for electromagnetism. But if the coupling constant gets large, that method of calculation breaks down, and the little pieces become worthless as an approximation to the real physics.

			This also can happen in string theory. String theories have a coupling constant. But unlike in particle theories, the string coupling constant is not just a number, but depends on one of the [[oscillation]] modes of the string, called the [[dilaton]]. Exchanging the dilaton field with minus itself exchanges a very large coupling constant with a very small one. This symmetry is called [[S-duality]]. If two string theories are related by S-duality, then one theory with a strong coupling constant is the same as the other theory with weak coupling constant. The theory with strong coupling cannot be understood by means of [[perturbation theory]], but the theory with weak coupling can. So if the two theories are related by S-duality, then we just need to understand the weak theory, and that is equivalent to understanding the strong theory.

			Superstring theories related by S-duality are: [[type I string\|type I]] superstring theory with [[heterotic]] [[special orthogonal group\|SO(32)]] superstring theory, and [[type IIB string\|type IIB]] theory with itself.

			Furthermore, [[type IIA string\|type IIA]] theory in strong coupling behaves like an 11-dimensional theory, with the [[dilaton]] field playing the role of an eleventh dimension. This 11-dimensional theory is known as [[M-theory]].

			Unlike the T-duality, however, S-duality has not been proven to even a physics level of rigor for any of the aforementioned cases. It remains, strictly speaking, a conjecture, although most string theorists believe in its validity.

			==See also==
			*[[Dilaton]]
			*[[M-theory]]
			*[[S-duality]]
			*[[String (physics)\|String]]
			*[[String theory]]
			*[[T-duality]]
			*[[U-duality]]

			==References==
			{{Unreferenced\|date=February 2007}}

			[[Category:String theory]]

en>Sodin: /* Books */ fix lk (cosmetic, since right now the old links are automatically redirected)

2012-08-06T02:28:45Z

Books: fix lk (cosmetic, since right now the old links are automatically redirected)

New page

The author's name is Christy. Kentucky is exactly where I've usually been living. He is an order clerk and it's something he truly enjoy. Doing ballet is some thing she would by no means give up.<br><br>Here is my site ... [https://www-ocl.gist.ac.kr/work/xe/?document_srl=605236 online psychic reading]

← Older revision		Revision as of 01:42, 27 February 2014
Line 1:		Line 1:
	~~{{About\|String duality\|other forms of duality\|Duality (disambiguation)}}~~		Golda is what's created on my birth certification even though it is not the title on my birth certificate. The favorite pastime for him and his kids is fashion and he'll be starting some thing else along with it. She functions as a journey agent but quickly she'll be on her personal. My wife and I live in Kentucky.<br><br>my web blog; online psychic readings - [http://help.ksu.edu.sa/node/65129 mouse click the next web page] -

	'''String duality''' is a class of [[symmetry in physics\|symmetries]] in [[physics]] that link different [[string theory\|string theories]], theories which assume that the fundamental building blocks of the [[universe]] are [[string (physics)\|string]]s instead of [[point particle]]s.

	~~Before the so-called "duality revolution" there were believed to be five distinct versions of string theory, plus the (unstable) bosonic and gluonic theories.~~

	~~{\|class="wikitable" mm_noconvert="TRUE"~~
	~~\|- bgcolor="#FFFFFF"~~
	~~! colspan="3" class="dark" \| String Theories~~
	\|-
	~~! class="dark" \| Type~~
	~~! class="dark" \| Spacetime dimensions<br>~~
	~~! class="dark" \| Details~~
	\|-
	~~! class="dark" \| Bosonic~~
	~~\| align="CENTER" class="dark" \| 26~~
	\| class="dark" \| Only [[boson]]s, no [[fermion]]s means only forces, no matter, with both open and closed strings; major flaw: a [[Particle physics\|particle]] with imaginary [[mass]], called the [[tachyon]], representing an instability in the theory.
	\|-
	~~! class="dark" \| I~~
	~~\| align="CENTER" class="dark" \| 10~~
	~~\| class="dark" \| [[Supersymmetry]] between forces and matter, with both closed strings and open strings, no [[tachyon]], group symmetry is [[special orthogonal group\|SO(32)]]~~
	\|-
	~~! class="dark" \| IIA~~
	~~\| align="CENTER" class="dark" \| 10~~
	~~\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings and open strings bound to [[D-brane]]s, no [[tachyon]], massless [[fermion]]s spin both ways (nonchiral)~~
	\|-
	~~! class="dark" \| IIB~~
	~~\| align="CENTER" class="dark" \| 10~~
	~~\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings and open strings bound to [[D-brane]]s, no [[tachyon]], massless [[fermion]]s only spin one way (chiral)~~
	\|-
	~~! class="dark" \| HO~~
	~~\| align="CENTER" class="dark" \| 10~~
	~~\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings only, no [[tachyon]], [[heterotic]], meaning right moving and left moving strings differ, group symmetry~~ is ~~[[special orthogonal group\|SO(32)]]~~
	\|-
	~~! class="dark" \| HE~~
	~~\| align="CENTER" class="dark" \| 10~~
	\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings only, no [[tachyon]], [[heterotic]], meaning right moving and left moving strings differ, group symmetry is [[E8 (mathematics)\|''E''<sub>8</sub>×''E''~~<sub>8</sub>]]~~
	\|}

	Note that in the type IIA and type IIB string theories closed strings are allowed to move everywhere throughout the ten-dimensional space-time (called the ''bulk''), while open strings have their ends attached to [[D-brane]]s~~, which are membranes of lower dimensionality (their dimension is odd - 1,3,5,7 or 9 - in type IIA and~~ even ~~- 0,2,4,6 or 8 - in type IIB, including the time direction).~~

	Before the 1990s, string theorists believed there were five distinct superstring theories: [[type I string\|type I]], [[type IIA string\|types IIA]] and [[type IIB string\|IIB]], and the two [[heterotic string]] theories ([[special orthogonal group\|SO(32)]] and [[E8 (mathematics)\|''E''<sub>8</sub>×''E''<sub>8</sub>]]). The thinking was that out of these five candidate theories, only one was the actual [[theory of everything]], and that theory was the theory whose low energy limit, with ten dimensions spacetime [[compactification (physics)\|compactified]] down to four, matched the physics observed in our world today. It is ~~now known that the five superstring theories are~~ not fundamental, but are instead different limits of a more fundamental theory, dubbed [[M-theory]]. These theories are related by transformations called dualities. If two theories are related by a duality transformation, each observable of the ~~first theory can be mapped in some way to the second theory to yield equivalent predictions~~. The two theories are then said to be dual to one another under that transformation. Put differently, the two theories are two mathematically different descriptions of the same phenomena. A simple example of a duality is the equivalence of [[particle physics]] upon replacing matter with antimatter; describing our universe in terms of anti-particles would yield identical predictions for ~~any possible experiment.~~

	String dualities often link quantities that appear to be separate: Large and small distance scales, strong and weak coupling strengths. These quantities have always marked very distinct limits of behavior of a physical system, in both [[classical field theory]] and ~~quantum [[particle physics]]. But strings can obscure the difference between large and small, strong and weak, and this~~ is ~~how these five very different theories end up being related.~~

	~~==T-duality==~~
	~~{{main\|T-duality}}~~
	~~Suppose we are in ten spacetime dimensions, which means we have nine space dimensions~~ and one time. Take one of those nine space dimensions and make it a circle of radius R, so that traveling in that direction for a distance L = 2πR takes you around the circle and brings you back to where you started. A particle traveling around this circle will have a quantized [[momentum]] around the circle, because its momentum is linked to its [[wavelength]] (see [[Wave-particle duality]]), and 2πR must be a multiple of that. In fact, the particle momentum around the circle - and the contribution to its energy - is of the form n/R (in [[standard units]], for an integer n), so that at large R there will be many more states compared to small R (for a given maximum energy). A string, in addition to traveling around the circle, may also wrap around it. ~~The number of times the string winds around the circle is called the [[winding number]], and that is also quantized (~~as it must be an integer). Winding around the circle requires energy, because the string must be stretched against its tension, so it contributes an amount of energy of the form <math>wR/L_{st}^2</math>, where <math>L_{st}</math> is a ~~constant called the~~ '~~'string length'' and w is the winding number (an integer). Now (for a given maximum energy) there will~~ be ~~many different states (with different momenta) at large R, but there will also be many different states (with different windings) at small R~~. ~~In fact, a theory with large R~~ and ~~a theory with small R are equivalent, where the role of momentum in the first is played by the winding~~ in ~~the second, and vice versa~~. ~~Mathematically, taking R to~~ <~~math~~>~~L_{st}^2/R~~<~~/math~~> ~~and switching n and w will yield the same equations. So exchanging momentum and winding modes of the string exchanges a large distance scale with a small distance scale.~~

	This type of duality is called [[T-duality]]. T-duality relates [[type IIA string\|type IIA]] superstring theory to [[type IIB string\|type IIB]] superstring theory. That means if we take type IIA and Type IIB theory and compactify them both on a circle (one with a large radius and the other with a small radius) then switching the momentum and winding modes, and switching the distance scale, changes one theory into the other. The same is also true for the two heterotic theories. T-duality also relates [[type I string\|type I]] superstring theory to both type IIA and type IIB superstring theories with certain boundary conditions (termed [[orientifold]]).

	Formally, the location of the string on the circle is described by two fields living on it, one which is left-moving and another which is right-moving. The movement of the string center (and hence its momentum) is related to the sum of the fields, while the string stretch (and hence its winding number) is related to their difference. T-duality can be formally described by taking the left-moving field to minus itself, so that the sum and the difference are interchanged, leading to switching of momentum and winding.

	~~==S-duality==~~
	~~{{main\|S-duality\|M-theory}}~~
	Every [[force]] has a [[coupling constant]], which is a measure of its strength, and determines the chances of one particle to emit or absorb another particle. For [[electromagnetism]], the coupling constant is proportional to the square of the [[electric charge]]. When physicists study the [[Quantum electrodynamics\|quantum behavior of electromagnetism]], they can't solve the whole theory exactly, because every particle may emit and absorb many other particles, which may also do the same, endlessly. So events of emission and absorption are considered as perturbations and are dealt with by a series of approximations, first assuming there is only one such event, then correcting the result for allowing two such events, etc. (this method is called [[Perturbation theory]]). This is a reasonable approximation only if the coupling constant is small, which is the case for electromagnetism. But if the coupling constant gets large, that method of calculation breaks down, and the little pieces become worthless as an approximation to the real physics.

	This also can happen in string theory. String theories have a coupling constant. But unlike in particle theories, the string coupling constant is not just a number, but depends on one of the [[oscillation]] modes of the string, called the [[dilaton]]. Exchanging the dilaton field with minus itself exchanges a very large coupling constant with a very small one. This symmetry is called [[S-duality]]. If two string theories are related by S-~~duality, then one theory with a strong coupling constant is the same as the other theory with weak coupling constant. The theory with strong coupling cannot be understood by means of~~ [[perturbation theory]], but the theory with weak coupling can. So if the two theories are related by S-duality, then we just need to understand the weak theory, and that is equivalent to understanding the strong theory.

	~~Superstring theories related by S-duality are~~: ~~[[type I string\|type I]] superstring theory with [[heterotic]] [[special orthogonal group\|SO(32)]] superstring theory, and [[type IIB string\|type IIB]] theory with itself~~.

	~~Furthermore, [[type IIA string\|type IIA]] theory in strong coupling behaves like an 11-dimensional theory, with the [[dilaton]] field playing the role of an eleventh dimension~~. ~~This 11-dimensional theory is known as [[M-theory]]~~.

	~~Unlike~~ the T-duality, however, S-duality has not been proven to even a physics level of rigor for any of the aforementioned cases. It remains, strictly speaking, a conjecture, although most string theorists believe in its validity.

	~~==See also==~~
	*[[Dilaton]]
	*[[M-theory]]
	*[[S-~~duality]]~~
	*[[String (physics)\|String]]
	*[[String theory]]
	*[[T-duality]]
	*[[U-duality]]

	~~==References==~~
	~~{{Unreferenced\|date=February 2007}}~~

	~~[[Category:String theory]]~~

← Older revision		Revision as of 15:44, 31 January 2014
Line 1:		Line 1:
	~~The author~~'s ~~name~~ is ~~Christy~~. ~~Kentucky~~ is ~~exactly~~ where ~~I've usually been living~~. He is an ~~order clerk~~ and it'~~s something he truly enjoy~~. ~~Doing ballet~~ is ~~some thing she would~~ by ~~no means give up~~.<br><br>~~Here~~ is ~~my site~~ ... [~~https~~:~~//www~~-~~ocl~~.~~gist~~.ac.~~kr/work/xe/?document_srl~~=~~605236 online psychic reading~~]		{{About\|String duality\|other forms of duality\|Duality (disambiguation)}}

			'''String duality''' is a class of [[symmetry in physics\|symmetries]] in [[physics]] that link different [[string theory\|string theories]], theories which assume that the fundamental building blocks of the [[universe]] are [[string (physics)\|string]]s instead of [[point particle]]s.

			Before the so-called "duality revolution" there were believed to be five distinct versions of string theory, plus the (unstable) bosonic and gluonic theories.

			{\|class="wikitable" mm_noconvert="TRUE"
			\|- bgcolor="#FFFFFF"
			! colspan="3" class="dark" \| String Theories
			\|-
			! class="dark" \| Type
			! class="dark" \| Spacetime dimensions<br>
			! class="dark" \| Details
			\|-
			! class="dark" \| Bosonic
			\| align="CENTER" class="dark" \| 26
			\| class="dark" \| Only [[boson]]s, no [[fermion]]s means only forces, no matter, with both open and closed strings; major flaw: a [[Particle physics\|particle]] with imaginary [[mass]], called the [[tachyon]], representing an instability in the theory.
			\|-
			! class="dark" \| I
			\| align="CENTER" class="dark" \| 10
			\| class="dark" \| [[Supersymmetry]] between forces and matter, with both closed strings and open strings, no [[tachyon]], group symmetry is [[special orthogonal group\|SO(32)]]
			\|-
			! class="dark" \| IIA
			\| align="CENTER" class="dark" \| 10
			\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings and open strings bound to [[D-brane]]s, no [[tachyon]], massless [[fermion]]s spin both ways (nonchiral)
			\|-
			! class="dark" \| IIB
			\| align="CENTER" class="dark" \| 10
			\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings and open strings bound to [[D-brane]]s, no [[tachyon]], massless [[fermion]]s only spin one way (chiral)
			\|-
			! class="dark" \| HO
			\| align="CENTER" class="dark" \| 10
			\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings only, no [[tachyon]], [[heterotic]], meaning right moving and left moving strings differ, group symmetry is [[special orthogonal group\|SO(32)]]
			\|-
			! class="dark" \| HE
			\| align="CENTER" class="dark" \| 10
			\| class="dark" \| [[Supersymmetry]] between forces and matter, with closed strings only, no [[tachyon]], [[heterotic]], meaning right moving and left moving strings differ, group symmetry is [[E8 (mathematics)\|''E''<sub>8</sub>×''E''<sub>8</sub>]]
			\|}

			Note that in the type IIA and type IIB string theories closed strings are allowed to move everywhere throughout the ten-dimensional space-time (called the ''bulk''), while open strings have their ends attached to [[D-brane]]s, which are membranes of lower dimensionality (their dimension is odd - 1,3,5,7 or 9 - in type IIA and even - 0,2,4,6 or 8 - in type IIB, including the time direction).

			Before the 1990s, string theorists believed there were five distinct superstring theories: [[type I string\|type I]], [[type IIA string\|types IIA]] and [[type IIB string\|IIB]], and the two [[heterotic string]] theories ([[special orthogonal group\|SO(32)]] and [[E8 (mathematics)\|''E''<sub>8</sub>×''E''<sub>8</sub>]]). The thinking was that out of these five candidate theories, only one was the actual [[theory of everything]], and that theory was the theory whose low energy limit, with ten dimensions spacetime [[compactification (physics)\|compactified]] down to four, matched the physics observed in our world today. It is now known that the five superstring theories are not fundamental, but are instead different limits of a more fundamental theory, dubbed [[M-theory]]. These theories are related by transformations called dualities. If two theories are related by a duality transformation, each observable of the first theory can be mapped in some way to the second theory to yield equivalent predictions. The two theories are then said to be dual to one another under that transformation. Put differently, the two theories are two mathematically different descriptions of the same phenomena. A simple example of a duality is the equivalence of [[particle physics]] upon replacing matter with antimatter; describing our universe in terms of anti-particles would yield identical predictions for any possible experiment.

			String dualities often link quantities that appear to be separate: Large and small distance scales, strong and weak coupling strengths. These quantities have always marked very distinct limits of behavior of a physical system, in both [[classical field theory]] and quantum [[particle physics]]. But strings can obscure the difference between large and small, strong and weak, and this is how these five very different theories end up being related.

			==T-duality==
			{{main\|T-duality}}
			Suppose we are in ten spacetime dimensions, which means we have nine space dimensions and one time. Take one of those nine space dimensions and make it a circle of radius R, so that traveling in that direction for a distance L = 2πR takes you around the circle and brings you back to where you started. A particle traveling around this circle will have a quantized [[momentum]] around the circle, because its momentum is linked to its [[wavelength]] (see [[Wave-particle duality]]), and 2πR must be a multiple of that. In fact, the particle momentum around the circle - and the contribution to its energy - is of the form n/R (in [[standard units]], for an integer n), so that at large R there will be many more states compared to small R (for a given maximum energy). A string, in addition to traveling around the circle, may also wrap around it. The number of times the string winds around the circle is called the [[winding number]], and that is also quantized (as it must be an integer). Winding around the circle requires energy, because the string must be stretched against its tension, so it contributes an amount of energy of the form <math>wR/L_{st}^2</math>, where <math>L_{st}</math> is a constant called the ''string length'' and w is the winding number (an integer). Now (for a given maximum energy) there will be many different states (with different momenta) at large R, but there will also be many different states (with different windings) at small R. In fact, a theory with large R and a theory with small R are equivalent, where the role of momentum in the first is played by the winding in the second, and vice versa. Mathematically, taking R to <math>L_{st}^2/R</math> and switching n and w will yield the same equations. So exchanging momentum and winding modes of the string exchanges a large distance scale with a small distance scale.

			This type of duality is called [[T-duality]]. T-duality relates [[type IIA string\|type IIA]] superstring theory to [[type IIB string\|type IIB]] superstring theory. That means if we take type IIA and Type IIB theory and compactify them both on a circle (one with a large radius and the other with a small radius) then switching the momentum and winding modes, and switching the distance scale, changes one theory into the other. The same is also true for the two heterotic theories. T-duality also relates [[type I string\|type I]] superstring theory to both type IIA and type IIB superstring theories with certain boundary conditions (termed [[orientifold]]).

			Formally, the location of the string on the circle is described by two fields living on it, one which is left-moving and another which is right-moving. The movement of the string center (and hence its momentum) is related to the sum of the fields, while the string stretch (and hence its winding number) is related to their difference. T-duality can be formally described by taking the left-moving field to minus itself, so that the sum and the difference are interchanged, leading to switching of momentum and winding.

			==S-duality==
			{{main\|S-duality\|M-theory}}
			Every [[force]] has a [[coupling constant]], which is a measure of its strength, and determines the chances of one particle to emit or absorb another particle. For [[electromagnetism]], the coupling constant is proportional to the square of the [[electric charge]]. When physicists study the [[Quantum electrodynamics\|quantum behavior of electromagnetism]], they can't solve the whole theory exactly, because every particle may emit and absorb many other particles, which may also do the same, endlessly. So events of emission and absorption are considered as perturbations and are dealt with by a series of approximations, first assuming there is only one such event, then correcting the result for allowing two such events, etc. (this method is called [[Perturbation theory]]). This is a reasonable approximation only if the coupling constant is small, which is the case for electromagnetism. But if the coupling constant gets large, that method of calculation breaks down, and the little pieces become worthless as an approximation to the real physics.

			This also can happen in string theory. String theories have a coupling constant. But unlike in particle theories, the string coupling constant is not just a number, but depends on one of the [[oscillation]] modes of the string, called the [[dilaton]]. Exchanging the dilaton field with minus itself exchanges a very large coupling constant with a very small one. This symmetry is called [[S-duality]]. If two string theories are related by S-duality, then one theory with a strong coupling constant is the same as the other theory with weak coupling constant. The theory with strong coupling cannot be understood by means of [[perturbation theory]], but the theory with weak coupling can. So if the two theories are related by S-duality, then we just need to understand the weak theory, and that is equivalent to understanding the strong theory.

			Superstring theories related by S-duality are: [[type I string\|type I]] superstring theory with [[heterotic]] [[special orthogonal group\|SO(32)]] superstring theory, and [[type IIB string\|type IIB]] theory with itself.

			Furthermore, [[type IIA string\|type IIA]] theory in strong coupling behaves like an 11-dimensional theory, with the [[dilaton]] field playing the role of an eleventh dimension. This 11-dimensional theory is known as [[M-theory]].

			Unlike the T-duality, however, S-duality has not been proven to even a physics level of rigor for any of the aforementioned cases. It remains, strictly speaking, a conjecture, although most string theorists believe in its validity.

			==See also==
			*[[Dilaton]]
			*[[M-theory]]
			*[[S-duality]]
			*[[String (physics)\|String]]
			*[[String theory]]
			*[[T-duality]]
			*[[U-duality]]

			==References==
			{{Unreferenced\|date=February 2007}}

			[[Category:String theory]]