|
|
| Line 1: |
Line 1: |
| {{multiple issues|
| | Hi there. Allow me begin by introducing the writer, her title is Sophia. The preferred pastime for him and his kids is to perform lacross and he would never give it up. Credit authorising is how he makes money. Mississippi is the only location I've been residing in but I will have to move in a yr or two.<br><br>Here is my blog post: psychic chat online; [http://Www.Octionx.Sinfauganda.Co.ug/node/22469 visit the next internet site], |
| {{Technical|date=December 2012}}
| |
| {{Expert-subject|Physics|talk=Proof by Dynin?|date=March 2011}}
| |
| }}
| |
| | |
| In [[mathematical physics]], the '''Yang–Mills existence and mass gap problem''' is an [[open problem|unsolved problem]] and one of the seven [[Millennium Prize Problems]] defined by the [[Clay Mathematics Institute]] which has offered a prize of US$1,000,000 to the one who solves it.
| |
| | |
| The problem is phrased as follows: | |
| | |
| :'''Yang–Mills Existence and Mass Gap.''' Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on <math>\mathbb{R}^4</math> and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in {{Harvtxt|Streater|Wightman|1964}}, {{Harvtxt|Osterwalder|Schrader|1973}} and {{Harvtxt|Osterwalder|Schrader|1975}}.
| |
| | |
| In this statement, [[Yang–Mills theory]] is the (non-Abelian) [[quantum field theory]] underlying the [[Standard Model]] of [[particle physics]]; <math>\mathbb{R}^4</math> is [[Euclidean space|Euclidean 4-space]]; the [[mass gap]] Δ is the mass of the least massive particle predicted by the theory. Therefore, the winner must first prove that [[Yang–Mills theory]] exists and that it satisfies the standard of rigor that characterizes contemporary [[mathematical physics]], in particular [[constructive quantum field theory]], which is referenced in the papers 45 and 35 cited in the official problem description by Jaffe and Witten. The winner must then prove that the mass of the least massive particle of the force field predicted by the theory is strictly positive. For example, in the case of G=SU(3)—the strong nuclear interaction—the winner must prove that [[glueball]]s have a lower mass bound, and thus cannot be arbitrarily light.
| |
| | |
| {{Millennium Problems}}
| |
| | |
| ==Background==
| |
| {{Cquote|[...] one does not yet have a mathematically complete example of a [[quantum gauge theory]] in four-dimensional [[space-time]], nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so!|15px|15px|From the Clay Institute's official problem description by [[Arthur Jaffe]] and [[Edward Witten]].}}
| |
| | |
| Most known and nontrivial (i.e. interacting) [[quantum field theories]] in 4 dimensions are [[effective field theory|effective field theories]] with a [[cutoff (physics)|cutoff]] scale. Since the [[beta-function]] is positive for most models, it appears that most such models have a [[Landau pole]] as it is not at all clear whether or not they have nontrivial [[UV fixed point]]s. This means that if such a [[Quantum field theory|QFT]] is well-defined at all scales, as it has to be to satisfy the axioms of [[axiomatic quantum field theory]], it would have to be trivial (i.e. a [[free field theory]]).
| |
| | |
| [[Quantum Yang-Mills theory]] with a [[Non-abelian gauge theory|non-abelian]] [[gauge group]] and no quarks is an exception, because [[asymptotic freedom]] characterizes this theory, meaning that it has a trivial [[UV fixed point]]. Hence it is the simplest nontrivial constructive QFT in 4 dimensions. ([[Quantum chromodynamics|QCD]] is a more complicated theory because it involves [[quark]]s.)
| |
| | |
| It has already been well proven—at least at the level of rigor of [[theoretical physics]] but not that of [[mathematical physics]]—that the quantum Yang–Mills theory for a non-abelian [[Lie group]] exhibits a property known as [[color confinement|confinement]]. This property is covered in more detail in the relevant QCD articles ([[Quantum chromodynamics|QCD]], [[color confinement]], [[lattice gauge theory]], etc.), although not at the level of rigor of mathematical physics. A consequence of this property is that beyond a certain scale, known as the [[Coupling constant#QCD scale|QCD scale]] (more properly, the [[Color confinement|confinement scale]], as this theory is devoid of quarks), the color charges are connected by [[QCD string|chromodynamic flux tubes]] leading to a linear potential between the charges. Hence free color charge and free [[gluon]]s cannot exist. In the absence of confinement, we would expect to see massless gluons, but since they are confined, all we would see are color-neutral bound states of gluons, called [[glueball]]s. If glueballs exist, they are massive, which is why we expect a mass gap.
| |
| | |
| Results from [[lattice gauge theory]] have convinced many that quantum Yang–Mills theory for a non-abelian [[Lie group]] model exhibits confinement—as indicated, for example, by an area law for the falloff of the [[Vacuum expectation value|vacuum expectation value (VEV)]] of a [[Wilson loop]]. However, these methods and results are not mathematically rigorous and are extremely complex.
| |
| | |
| ==See also==
| |
| * [[Mass gap]]
| |
| * [[Quantum field theory]]
| |
| * [[Standard model]]
| |
| * [[Yang–Mills theory]]
| |
| * [[Constructive quantum field theory]]
| |
| | |
| ==References==
| |
| *[[Arthur Jaffe]] and [[Edward Witten]] "[http://www.claymath.org/sites/default/files/yangmills.pdf Quantum Yang-Mills theory.]" Official problem description.
| |
| *{{cite book |last1=Streater |first1=R. |last2=Wightman |first2=A. |title=PCT, Spin and Statistics and all That |publisher=W. A. Benjamin |year=1964 |ref=harv}}
| |
| *{{cite journal |last1=Osterwalder |first1=K. |last2=Schrader |first2=R. |title=Axioms for Euclidean Green’s functions |journal=[[Communications in Mathematical Physics|Comm. Math. Phys.]] |volume=31 |issue=2 |pages=83–112 |year=1973 |ref=harv |doi=10.1007/BF01645738 }}
| |
| *{{cite journal |last1=Osterwalder |first1=K. |last2=Schrader |first2=R. |title=Axioms for Euclidean Green’s functions II |journal=Comm. Math. Phys. |volume=42 |issue=3 |pages=281-305 |year=1975 |ref=harv |doi=10.1007/BF01608978 }}
| |
| | |
| ==External links==
| |
| * [http://www.claymath.org/millennium/Yang-Mills_Theory/ The Millennium Prize Problems: Yang–Mills and Mass Gap]
| |
| | |
| {{DEFAULTSORT:Yang-Mills Existence And Mass Gap}}
| |
| [[Category:Quantum chromodynamics]]
| |
| [[Category:Unsolved problems in mathematics]]
| |
| [[Category:Millennium Prize Problems]]
| |
Hi there. Allow me begin by introducing the writer, her title is Sophia. The preferred pastime for him and his kids is to perform lacross and he would never give it up. Credit authorising is how he makes money. Mississippi is the only location I've been residing in but I will have to move in a yr or two.
Here is my blog post: psychic chat online; visit the next internet site,