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Effective Field Theory Q&A

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> 我并不是否认有效理论下对重整化不那么必要的讨论。

On the contrary, renormalization is absolutely necessary in an
effective field theory. (see below)


> 对QCD这样的理论感觉到现在为止 我
> 们的确还没找到一个比较好的有效理论，手征微扰论很漂亮(零阶时候) 但是
>  适用的能区实在太小
> 能描述的物理粒子也太少太少。有效理论很强大 但不是我们的目标）

Effective theory is by definition only
applicable in a range of scales. This is the best we have so
far. Every field theory we use to describe nature these days are
effective theories. They are not in principle different from chiral
perturbation theory. Below the the scale of chiral symmetry breaking,
chiral perturbation theory is a beautiful effective field theory. It
is the only method we have to gain analytical control of physics at
those scales.


> 个人感觉：可重整性并不是仅仅说：有一个紫外的物理的cutoff，在一定误差内，
> 理论就自洽了。它意味着的重正化不变. 1： 重整化不仅仅是处理紫外发散的。
> . 重整化不变性 有 或者没有 是物理的结论.
> . 重整化不变性 意味着我们不仅能由低能标的Green函数出发得到的高能标下
> 的Green函数 也能反过来。
> 它本身包含的物理意思 感觉跟发散没关系。即使没发散，
> 具有这种不变性的话 就可以做重整化。


This is exactly the statement from an effective field theory point of
view. Again, let me repeat. The theory is defined at some high energy
scale with couplings (or not so correctly, 'bare' couplings). However,
we want to deal have a theory which effectively and conveniently
describes the low energy experiment. Therefore, we integrate out high
momentum scales. This process changes (or renormalizes) the couplings
away from their bare values.

This is what happens when we do renormalization.

At the same time, I could also in principle do the calculation with
bare couplings. The result should be the same as using the
renormalized couplings at a lower scale. This is exactly the statement
of renormalization invariance that leads to Callan-Symanzik
equations.

> －在维数正规化里面 好像没看到需要紫外的cutoff

This is not true, Dimensional regularization is exactly a regulator
which regulates the untra-violet divergenece. (See Georgi's review
http://schwinger.harvard.edu/%7Egeorgi/index.htm)

> 2： 红外发散跟紫外发散具有类似的问题
> 重整化不变性意味着我们可以把微扰论中遇到的 所有紫外发散仍到 几个常数里面去。
> 个人感觉：
> 对QCD而言 夸克层次上振幅里面的红外发散(抵消不了的，并且在高能标时候 由
> 于渐近 自由 这个振幅具有物理意义) 类似于重正化(实际上 我是看不出物理思
> 想跟重整化有什么本质区别) 我们期望物理的夸克束缚态的描述能统一的吸收掉
> 所有这些红外发散
> (这个"能统一吸收"也肯定由某种不变性保证的 )－－－在这个做法里面
> 看不出涉及有效理论自然提供的cutoff所起到的作用......

It is technically similar but in principle different. the IR
divergence is the effect of defining a asymptotic states in a IR
non-trivial theory (either QCD or QED). The physical amplitude is
indeed depends on the IR cutoff (in the easier to understand case of
QED, it is determined by experimentally how well we can resolve a low
energy photon away from an electron. In QCD, this has something to do
with dressing up partons with gluons).

Ultraviolet cutoff of effective field theory of course does not say
anything about its IR divergence. It should not.

The statement of effective field is that such a theory captures the
essence of low energy physics after integrating out high energy
physics.

> 从高能标理论到低能有效理论 跟 重整化做的不完全是一个事情
> 前者不一定需要圈图等等的计算,也不涉及重整化点不变(这个说的不好:( 意思
> 比较模糊)
> 感觉仅仅是做一个match的事情(路径积分的形式说法:积掉高能自由度)
> (按照某个power counting 大家相互计算到特定阶数,然后match,
> 从场论到量子力学做的事情也许比较说明这点,前者可以一圈一圈的算完了跟后者中的
> 单个势能相match,后者(有效理论)不存在再重整化问题 )



Integrating out high energy scales has two classes of effects: 1)
generating higher dimensional operators suppressed by high scale. This
is done by matching. 2) usually more importantly, it generates
corrections to the couplings of low energy theory. This is exactly the
same effect as renormalization. It could not be anything else. Higher
energy scale does not contribute except in the loops and
non-renormalizable operators. It is therefore conceptually,
techinically, qualitatively and quantitatively identical to
renormaliztion.

> 我的感觉是 从高能标理论到低能有效理论 跟 重整化做的不完全是一个事情
> 前者不一定需要圈图等等的计算,也不涉及重整化点不变(这个说的不好:( 意思
> 比较模糊)
> 感觉仅仅是做一个match的事情(路径积分的形式说法:积掉高能自由度)
> (按照某个power counting 大家相互计算到特定阶数,然后match,
> 从场论到量子力学做的事情也许比较说明这点,前者可以一圈一圈的算完了跟后者中的
> 单个势能相match,后者(有效理论)不存在再重整化问题 )

Dimensional regularization is exactly a regulator
which regulates the untra-violet divergenece. (See Georgi's review
http://schwinger.harvard.edu/%7Egeorgi/index.htm)
--

二零零五年六月十五日