问一下重整化的问题 [文章类型: 原创]

如果理论本身不能重整化.人为的在拉氏量增加几项来抵消单圈双圈的发散项.这样的理论结果在低能条件下可信么?如果可信,在经典理论中又应该用原来的拉氏量,还是修改的拉氏量呢?

发表时间: 2007-09-27, 20:52:49 个人资料

sage



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Re: 问一下重整化的问题 [文章类型: 原创]

如果理论本身不能重整化.人为的在拉氏量增加几项来抵消单圈双圈的发散项.这样的理论结果在低能条件下可信么?

Yes, with some qualifications.



如果可信,在经典理论中又应该用原来的拉氏量,还是修改的拉氏量呢?


You should use the modified one.

It is probably useful if you say more precisely what Lagrangian you are using and which problem you are trying to solve.

发表时间: 2007-09-28, 20:34:56

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谢谢sage兄的回答,很早就想回了,可是内力恢复太满了.平常来这少了不知有这个地方,要不然第一次就发这了.

"You should use the modified one"
我也认同你的观点.但是如果理论是某个低能条件下非常精确的理论呢?比如爱因斯坦引力理论,将它按闵氏度规展加扰动展开后(作用量有无穷项),如果第3或第四阶需要增加新顶点来重正化(当然我没算但某一阶一定会有).那么原作用量就应该比较可信.

我也认同你的观点.但是如果理论是某个低能条件下非常精确的理论呢?比如爱因斯坦引力理论,将它按闵氏度规展加扰动展开后(作用量有无穷项),如果第3或第四阶需要增加新顶点来重正化(当然我没算但某一阶一定会有).那么原作用量就应该比较可信.

GR is a good example of your question. After including quantum corrections, you will induce terms in the Lagrangian of the form

a{[energy scale(or momentum)]/M_P}^{some power}

It is always a valid expansion as long as the energy/momentum/curvature is less than Planck scale. You will also notice that these terms become less and less important at low energy, the so called irrelevant operators.

However, the coefficient a, just like M_P in the leading order term, is not predicted by the perturbation theory. It must be fixed by some experiment, a very precise one in this case. This is another way of saying one must specify renormalization conditions, which is typically fixed by some observables.

When you say \sqrt{g}R is a very precise low energy theory, what you really should have said is that it is precise up to corrections due to those higher order terms, which is still outside of our experimental capability.

If some of them are within the experimental sensitivity, in principle, they should all be included, with coefficients measured by the experiments.

GR is a good example of your question. After including quantum corrections, you will induce terms in the Lagrangian of the form

a{[energy scale(or momentum)]/M_P}^{some power}
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引进的项应该是包含引力微扰的高阶导数的高次项,在费曼图中的体现就是上式是么?
如果这样对于高频(或空间导数大)的弱场不再是低能理论了是么(或者在这种情况下就不再是弱场)?因此在低能的条件下这些项作用很小,现有的GR理论就可以比较精确描述引力了.

对于所有不可重正理论,如果我们有它的低能条件下的比较精确的理论,我们能不能通过引入新的参数将它的适用能标范围提高呢?除去技术上困难,能不能象这样得到任意能标适用的理论呢?

引进的项应该是包含引力微扰的高阶导数的高次项,在费曼图中的体现就是上式是么?

Yes.

如果这样对于高频(或空间导数大)的弱场不再是低能理论了是么(或者在这种情况下就不再是弱场)?因此在低能的条件下这些项作用很小,现有的GR理论就可以比较精确描述引力了.

If the derivatives, or curvature, is comparable to Plack scale, you must include a lot of the higher order terms. This is certainly not a weak field. By the way, in reality, it is very hard to get such strong fields except in early universe or close to the singularity of a blackhole.


对于所有不可重正理论,如果我们有它的低能条件下的比较精确的理论,我们能不能通过引入新的参数将它的适用能标范围提高呢? 除去技术上困难,能不能象这样得到任意能标适用的理论呢?

In principle, if an effective theory is with a cut-off scale at Lambda, it means that we will have terms like

(non-renormalizable operator)/ Lambda^{power}

Given certain precision you want to achieve in your calculation, the closer you are to the cut-off scale, the more higher order terms you should include, all of them you will have to fix with some experimental observables. At the cut-off, You will have to include infinite number of terms, which means such a description is useless since you have lost predictive power.

If the derivatives, or curvature, is comparable to Plack scale, you must include a lot of the higher order terms. This is certainly not a weak field. By the way, in reality, it is very hard to get such strong fields except in early universe or close to the singularity of a blackhole.
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是不是指前面的M_P是planck质量么.

是不是指前面的M_P是planck质量么.

Yes.

谢谢sage兄的回答,看来还有很长的路要走