星空浩淼
发表文章数: 1743
武功等级: 九阳神功
(第五重)
内力值: 617/617
我的重整化哲学
（不是班门弄斧，而是想通过在此暴露个人的认识，让其中种种谬解，能有人指出，令我有所提高，从而不枉此贴:-)）
数学家由于必须局限于严密，从而有所羁绊，在有些时候限制了新理论的出现和发展。而物理学家则有更大的自由度，更随心所欲——因为对他们而言，实验才是最终的检验标准，只要与实际相符，理论上哪怕死缠乱打也没关系。因此在有些时候，物理学家可以走在数学家前面而不需要先征得数学家们的同意，提前采用了一些“非法”手段。然后数学家们再醒过来，转而去挑刺，但最后他们总是惊讶于物理学家们的直觉：他们居然是对的！尽管或许欠缺严密，但根本上是对的。最终数学家们所需要做的就是对数学上的严密性进行修补。
首先来自于物理学家之手的Dirac δ函数，它的严密数学理论——广义函数论，是早已出来了的，此时Dirac δ函数只作为一个特例。我觉得重整化的严密数学理论早晚也会出来。尽管从数学上看重整化似乎很别扭，但它的正确性无容置疑。象“跑动耦合常数”、“重整化效应”等为重整化注入了直接的物理内容。换句话说，它远远不止是一种数学技巧，而且也直接体现物理本质、描述物理内容本身。
重整化群理论的出现，能带给我们关于对重整化的进一步理解。如今“重整化”这个东东的含义已经广义化了，在凝聚态物理中，在非线性物理中可以见到“重整化”的身影，在那里，你会感觉其含义似乎跟你原来理解的有些不一样，那或许是因为你原来的理解比较狭义。广义化意义上的“重整化”，其含义里面提取了重整化方法的本质。然而这样一来，其物理内容在不同场合可能会有所不同。我这个帖子里专指量子场论中的重整化。
我过去学习采用的教材，为了“完善了解各种重整化方法”，把历史上先后出现的重整化方法分别采用来对付不同的过程，所以那里可以看到无穷大的量一会儿被当作无穷小的量来对付，例如1/（1－L)=1+L，其中L是无穷大的量。有时候，说“在取无穷大极限之前，先把它看作小量”这种近似耍赖的话，总之给人感觉是：尽管最后结果是对的，但中间过程似乎歪打正着。
也许重整化方法那里包含有类似于利用发散级数（渐近级数）来干活一样的本质原理，从而理解它需要打破传统的思维惯性。我觉得那里应该有一些迷人的数学和物理原理。尽管在工具论的意义上一般人都不难掌握重整化，但在思想认识上，就不一定。也许还有隐藏在重整化背后的东西有待我们进一步去揭示，而且这些东西有利于我们进一步从物理和数学的角度上认识这个世界；也许正是这些东西，有利于我们对于引力量子化奥秘的认识，让我们变被动为主动地看待引力理论的不可重整性。
——也许引力量子化的主要困难，与其说是来源于它的不可重整性，不如说是来源于我们对于重整化本身的认识不够所造成的。总之，也许重整化的背后隐藏着有待我们去认识的玄机。
唯有与时间赛跑，方可维持一息尚存
| 发表时间：2005-04-06, 04:57:05 |
 作者资料
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萍踪浪迹
发表文章数: 1983
武功等级: 深不可测
内力值: 645/645
Re: 我的重整化哲学
Dirac δ函数的产生比Schwartz的广义函数论来得早.而且刚出来时很受数学家怀疑,但后来却完全找到了合法的数学基础
但是,重正化未必能够寻到更好的数学基础了,我个人觉得.
有些事情可能永远都无法到达要求,比如三体运动,不可能有解析解,再不满也只能用微分方程定性理论和数值解来寻求一些特殊的答案.
重整化和δ函数的事例不尽相同.δ函数只要定义一个弱收敛序列的极限就可以搞定,所以毕竟是数学家的事情.而重整化是要建立在可观测量的基础上进行的,对高阶散射的计算导致的无穷大只能从物理意义上解决.真空极化使裸电荷无法观测到.正是由于无法观测到裸电荷,物理学家才把重正化电荷当作物理上真实的电荷.
因此,数学意义上的重正化可能很难建立.
重整化肯定是不能完全令人满意的,就算在现在也是如此.
但它已经够伟大了,吹毛求疵的观点当然不足取.
不过为重正化找一个数学基础是一个很美好的愿望.希望有人能够实现
量子引力的困难远不止于重正化.
关于量子引力,恐怕不是这个回帖所能论述的~
重正化群不仅在物理上有重大意义,而且在天体力学的一些具体而重要的计算中也非常重要
物理几何是一家
共同携手走天涯
-----------------Chern
| 发表时间：2005-04-06, 06:27:09 |
作者资料
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sage
发表文章数: 1125
武功等级: 天山六阳掌
(第六重)
内力值: 535/535
Re: 我的重整化哲学
我过去学习采用的教材，为了“完善了解各种重整化方法”，把历史上先后出现的重整化方法分别采用来对付不同的过程，所以那里可以看到无穷大的量一会儿被当作无穷小的量来对付，例如1/（1－L)=1+L，其中L是无穷大的量。有时候，说“在取无穷大极限之前，先把它看作小量”这种近似耍赖的话，总之给人感觉是：尽管最后结果是对的，但中间过程似乎歪打正着。
I disagree there is anything not rigorous here. This might be the historical way of people doing things, and unfortunate, this is the way many old or bad (some old and bad) textbook do renormalization. This is not the correct point of view. Things are much clearer, even just in the BPH scheme.
could you please give a specific example where you think it is not rigorous?
I claim, unlike things such as the definition of path integral, renormalization by itself is mathematically rigorous. Please give me an example where it is not.
I remember Lu Changhai had similar claims in his article (during early days of his career). I just did not see an example.
| 发表时间：2005-04-06, 14:12:34 |
作者资料
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sage
发表文章数: 1125
武功等级: 天山六阳掌
(第六重)
内力值: 535/535
Re: 我的重整化哲学
真空极化使裸电荷无法观测到.正是由于无法观测到裸电荷,物理学家才把重正化电荷当作物理上真实的电荷.
因此,数学意义上的重正化可能很难建立.
I do not understand what you mean by renormalization in mathematical sense.
Bare charge does not exist as a physical parameter. The physical quantity is the parameter as a function of scale.
In physics, one only worries about things that can be observed.
I could use a physics theory to calculation all kinds of unphysical things. For example, you could calculate 'time' along space-like trajectories. It will give crazy answers. However, it does not mean there is anything wrong with the theory. It is just one is applying it in a wrong way.
| 发表时间：2005-04-06, 14:19:19 |
作者资料
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卢昌海
发表文章数: 1617
武功等级: 北冥神功
(第一重)
内力值: 602/602
Re: 我的重整化哲学
:: I remember Lu Changhai had similar claims in his article
:: (during early days of his career). I just did not see an example.
Good memory! :)
Indeed, that was what troubled me 13 years ago when I first encountered renormalization (as was mentioned in my old diaries). I don't remember clearly which textbook I read at that time. Maybe Hu2 Yao2 Guang1's. That is a fictitious issue due to the badly written textbook I came across. Summing up a chain of one-loop diagrams can easily fix that specific issue. One may still wonder about whether the expansion of 1/(1-L) convergent or not. In QED, for all physically meaningful cut-off (even if it is as high as Planck scale), it is convergent. I may write some articles update my description about renormalization when I have time.
宠辱不惊，看庭前花开花落
去留无意，望天空云卷云舒
| 发表时间：2005-04-06, 20:00:43 |
作者资料
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sage
发表文章数: 1125
武功等级: 天山六阳掌
(第六重)
内力值: 535/535
Re: 我的重整化哲学
——也许引力量子化的主要困难，与其说是来源于它的不可重整性，不如说是来源于我们对于重整化本身的认识不够所造成的。总之，也许重整化的背后隐藏着有待我们去认识的玄机。
This is a very good place to talk about effective field theory.
Before electroweak theory, there is fermi theory of weak interactions. Notice that there is in pinciple nothing wrong with Fermi theory. One has to only take care in apply it in its valid domain. When we are dealing with energy scales much lower than the electroweak scale, Fermi theory is perfectly OK. All the effect of a more fundamental theory (with W bosons) come in as non-renormalizable operators as a power expansion of (1/M_W)^2... Therefore, although the theory is non-renormalizable, it is perfecty fine in the low energy region. Then, when we want to make predictions around th electroweak scale, all those operators as powers of (1/M_W)^2 stop being a valid expansion. Therefore, we need to match onto some other theory. The new theory, is of course the Standard Model of electroweak interactions, with W, Z bosons.
Let me emphasize that Standard Model in principle is not renormalizable either. We at least know one higher scale where different physics will set in, the Planck scale. Therefore, in the Standard Model, there should at least by non-renormalizable operators like 1/M_P^2. Therefore, it is not renormalizable in the same sense as the Fermi theory.
We still consider the Standard Model a perfect success, even though it is not renormalizable. The reason is the same as why Fermi theory is a good theory. Standard Model, as a effective field theory, perfectly describes the physics at scales much lower than the Planck scale.
Now, why do we care about quantum gravity? the motivation is actually not very strong. The main reason is that we are curious. There are also problems like black-hole entropy which might always be blamed on quantum gravity.
Anyway, following our logic of effective field theory, quantum gravity is only necessary when we asking questions about the physics at Planck scale.
Why do quantum gravity, if such a theory is indeed found, has to be renormalizable? It does not have to be! It could be just like fermi theory, or the Standard Model, works as a effective field theory, with some higher scale enter as non-renormalizable operators.
On the other hand, probably because of our ignorance, we sometimes hope that quantum gravity is the final theory. It is a theory of everything. It is valid everywhere. However, if this is the case, making a renormalizable quantum gravity is not enough. It must be FINITE. There should not things like ultraviolet cutoff and renormalization. It should straight give the answer.
Renormalization is only necessary when we deal with effective field theories. In this sense, renormalization is only necessary when the theory is not renormalizable.
| 发表时间：2005-04-06, 21:18:44 |
作者资料
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卢昌海
发表文章数: 1617
武功等级: 北冥神功
(第一重)
内力值: 602/602
Re: 我的重整化哲学
:: Let me emphasize that Standard Model in principle is not renormalizable
:: either. We at least know one higher scale where different physics will
:: set in, the Planck scale. Therefore, in the Standard Model, there should
:: at least by non-renormalizable operators like 1/M_P^2. Therefore, it is
:: not renormalizable in the same sense as the Fermi theory.
hmm, this probably shouldn't be called "non-renormalizable". Standard model is certainly not a final theory, and will certainly break down at certain energy scale. But this is NOT equivalent to say that standard model is not renormalizable. Renormalizability is a well-defined concept in QFT (maybe not in the mathematically strict sense, but in the sense of physics), and standard model meets all the criteria. The difference between standard model and a non-renormalizable theory such as the Fermi theory is: Non-renormalizable theoy will break down BY ITSELF above certain energy scale (because infinite number of new parameters will emerge as energy goes up), while standard model can be applied to arbitrary energy scale without introducing new parameters (this, however, does NOT mean that the predictions of the standard model will always be correct).
Suppose we know nothing about gravity (therefore have no concept about Planck scale) and have no concept of anything (such as SUSY) that is not included in standard model, we would not know the limitation of standard model until we reach the point when it no longer matches experiments. But for Fermi theory and other non-renormalizable theory, we will still know it's limitation purely WITHIN the framework of QFT.
It is true that we can consider standard model as a low energy effective theory of a more general theory, in which we will probably see terms - IN ADDITION TO standard model terms - like 1/M_P^2, but the standard model piece is still renormalizable.
宠辱不惊，看庭前花开花落
去留无意，望天空云卷云舒
| 发表时间：2005-04-06, 22:34:54 |
作者资料
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walk_f