有没有物理问题与Yang-Mill方程（SU(2)或SU(3)丛）的经典解有关？

what classical solution you are referring to?

also, are you talking about pure YM?

其实我也说不清楚，我对物理的了解挺少。其实我是想在Yang-Mill方程方面有没有类似于Maxwell的电磁场理论那样的理论，我现在看到的大部分和gauge theory相关的都是量子化场论，是否有像经典电磁场那样的宏观现象用到Yang-Mill理论来解释。这里可以不是自由场的例子，因为实际的物理现象都是相互作用的理论。

其实我也说不清楚，我对物理的了解挺少。其实我是想在Yang-Mill方程方面有没有类似于Maxwell的电磁场理论那样的理论，我现在看到的大部分和gauge theory相关的都是量子化场论，是否有像经典电磁场那样的宏观现象用到Yang-Mill理论来解释。这里可以不是自由场的例子，因为实际的物理现象都是相互作用的理论。
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All other gauge interactions we know does not have macroscopic configurations since they are short-ranged (either because they are higgsed or they are confining). Therefore, It is hard to have macroscopic configurations where a lot of field quanta condense (such as in the case of E&M).

There are, however, field configurations which cannot be interpreted as single exictations. In this sense, they are classical. Examples of these are instantons.

do you mean instantons can be observed physically? I am not sure the instantons you said is same as the one we talk in mathematics, whose energy reachs the absolute minimum rather than the critical point of Yang-Mill functional.

我的问题可能太过于泛泛了很难回答，我用google搜索了一下，才知道原来instanton这个概念不只在Yang-Mill理论中有，虽然我只关心Yang-Mill理论，但还是可以了解一下物理上一般的instanton是什么意思。如果sage兄知道有什么好的读物，别忘了告诉我，谢谢。

在数学上Atiyah等在70年代构造出S4上的所有instanton,这个构造过程叫做ADHM construction. 这里面的数学有点不好读。Atiyah的学生Donaldson在此基础上发展了4维流形的Donaldson theory。后来Donaldson曾经试图将这个理论推广到高维，现在还没有完全成功。当然物理学家不会关心这样的推广，甚至不会关心Donaldson theory本身。附带说一下，Witten曾说明Donaldson theory实际上是topological quantum field theory.

oh, instantons in mathematics are always absolute minimum rather than critical points of Yang-Mill functional? in the pure Y-M case they are, but i think in physics we mean critical points of the action.

SU(2) weak force is higgsed. SU(3) strong force is confined. when you run the renormalization group you find they dont really have obvious macroscopic effect.

i think there is no observation of instantons as of now. personally, i seriously wonder if they really exist.

Donaldson : Seiberg-Witten as Electric field : Monopole, though i dunno the details.

High dimensional donaldson theory? in 2n-d? 4n-d? but 4 is just so special where funny things happen between those small groups.

i think there is no observation of instantons as of now.
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That's not true. We have seen their effect already in QCD vaccuum.



personally, i seriously wonder if they really exist.
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I don't know what is the basis of your wonder. Unless our understanding of field theory is seriously wrong, they must exist.

场论中如果不用instanton或者说求解经过ward rotation后得到的Euclidean形式的场方程，那就很难得到哪些诸如tunnel effect等现象的定量计算，还是说路径积分也能解这类问题，而instanton只是一种语言？ 说明一下，我不能很好的理解说为什么用instanton来解释tunnel effect，这两者有什么区别和联系？我所能理解的程度的量子力学上纯粹用波动观点来看tunnel effect，是不是说instanton是二次量子化后的解释？

原谅我一个不懂物理的在胡说八道，不过我真想知道答案。谢谢。

"场论中如果不用instanton或者说求解经过ward rotation后得到的Euclidean形式的场方程，那就很难得到哪些诸如tunnel effect等现象的定量计算，还是说路径积分也能解这类问题，而instanton只是一种语言？"
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这里指的是Wick rotation吧? 用不是那么讲究严格性的观念来看 只要这个积分的pole
的结构我们清楚 Euclidean形式的方程总是可以等价到Mink空间的形式吧
不用欧式空间形式中的instanton解来做这个积分 估计很难吧


"说明一下，我不能很好的理解说为什么用instanton来解释tunnel effect，这两者有什么区别和联系？我所能理解的程度的量子力学上纯粹用波动观点来看tunnel effect，是不是说instanton是二次量子化后的解释？"
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纯YM 场里面提到的tunnel effect 应该是指不同真空间的相互跃迁吧
如果没有instanton存在的话 因为是对全空间积分 相当于势是无穷大 这个跃迁是0
量子力学里面的tunnel effect相应的势不是无穷大 穿过的概率不是0
就有可观察的tunnel effect
instanton只是一个单纯的经典解吧 和二次量子化没关系吧
只是这个经典解到底有多大的的量子效应 可能还并没有完全弄的很清楚?
(不知道理解的对不)
changhai 做过拓扑场论吧 呵呵 给我们大家来普及普及

Dear sage, could you give some materials on the instanton effect in QCD vaccuum? thank you!

Dear sage, could you give some materials on the instanton effect in QCD vaccuum? thank you!
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The best intro about instanton I have found so far is by Sidney Coleman

``The use of instantons"

you can find it in his book

``Aspects of Symmetry"

Is it just an approximated model for calculation? I find there are some other models for QCD vacuum.