Turgon

1. 4天前
2019-01-18 18:38:50

@baishuxu 其实按照ST的设定，只要你warp field消除掉就能立刻停下来吧……

那还能不能活着就另说了。。。ENT里边进warp2那次不就很惨嘛

2. 5天前
2019-01-17 14:19:27

@baishuxu 额……一般这种科幻电影设定中，带warp drive的飞船都会有long-range sensor和emergency stop吧……而且也确实有那种突然出现一个障碍物然后来不及反应的情节……（具体我一下也想不起来了）

long-range sensor的原理比较迷。。。（话说emergency stop不就是drop out of warp嘛，感觉他们哪怕warp9都可以直接跳出

3. 6天前
2019-01-16 22:53:04

所以一般科幻作品不都会出现意外嘛（不然故事怎么进行下去

歪个楼，前段时间刷arxiv看到篇文章给了下FTL的review：1812.10350

4. 上周
2019-01-14 15:07:25
Turgon 更新于 粒子的空间局域性

Cluster decomposition principle...或者翻译成集团分解原理，簇分解原理之类的（不是很清楚怎么翻译），这部分内容我感觉你也有点基础，看Weinberg3，4章比我讲的强多了。另外类似的空间关联性质也有些量可以描述，原则上说你都可以自己造一个量测量。对撞机物理里边大概有这种量，但是我不是很熟（因为他们造测量量很happy。。。），splitting function，thurst之类的大概间接有点联系。啊还有jet broadening

5. 2019-01-13 19:55:27

One Loop Corrections

In last section we derived that
$$T\phi(x)\phi(0)=\phi^2(0)+\cdots$$
Now that's assume there're loop corrections to the coefficient of $\phi^2$. We're not sure whether the correction is finite or not, by experience it's more likely to be divergent. To our prior knowledge, $T\phi(x)\phi(0)$ gives finite result when $x$ is not zero (all divergence will be suppressed by the exponential factor). Thus we can define a renormalized operator $[\phi^2]$ and construct
$$T\phi(x)\phi(0)=C(x)[\phi^2(0)]+\cdots.$$
This way, $C(x)$ is also well-defined. We already know the leading order of $C$, so
$$C(x)=1+\frac{g}{16\pi}c(x)$$
The factor is purely set for convenience.

Consider the following diagram (I hope I didn't mess up with some signs or factors, nor J. Collins):

[attachment:5c436480b4809]

$$\frac{i}{p_1^2-m^2}\frac{i}{p_2^2-m^2}ig\int\frac{\dd^4q}{(2\pi)^4}\frac{e^{-iq\cdot x}}{(q^2-m^2)((q-p_1-p_2)^2-m^2)}$$

T.B.D

6. 2019-01-13 19:38:49
Turgon 更新于 粒子的空间局域性

@蹑履思登 [quote=47328:@Turgon]
emmm...我以为我已经说得很明白了？
A particle is localized at some specific space region.
我想知道的是，对于一个具体的物理态（姑且只考虑单粒子态好了），如何描述这个粒子……怎么个local法。应该要有一个普遍的判断标准、有一个明确的数学量来描述——例如QM里坐标波函数模方表示粒子出现在该坐标处的概率密度，这就是一个明确的判断标准，判断粒子的空间局域性质。
场有坐标label，但场算符又不是描述物理状态的态矢量，这能说明什么？场算符的负频部分φ^(-)(x)作用在真空态|0>上能表示一个在x点的正粒子吗？假设能，毕竟关联函数就是这样写的。但即使这样设定——对于一个一般的态，它的坐标空间方面的性质如何？没有坐标算符，没有坐标本征态。一个一般的粒子态，如

∫d^3p[f(p)|p>]

|p>是单粒子动量本征态（直接归一成δ函数好了，不采用Lorentz不变的归一化），f(p)是一个已知的、正常的动量波函数，如何求这个态……局域在哪一空间区域？别告诉我说没有描述“粒子空间局域性”的数学量，激光往前一照，发出的光子总不会跟右边一兆光年外的电子来个Compton散射吧？现实中的粒子可以局域在一定的空间区域里，那如何从态矢量上读出这些相关信息？
真的求求各位大佬了……这应该是一个很明确、很容易的问题，大佬们好心就直接告诉我好了，别耍我了。

这么说吧，你讨论单个粒子出现在某个位置的概率有什么意义，反正是自由场。。。考虑粒子的locality应该看它的相互作用，相互作用Weinberg已经说的很清楚了，cluster decomposition principle，这是个假设，说的是激光往前一照，发出的光子不会跟右边一兆光年外的电子来个Compton散射（划掉），说的其实是“experiments that are sufficiently separated in space have unrelated results”。这是散射理论能成立的前提，不然反正你实验都不对，100w光年外无数干扰能影响你的实验，鬼知道理论怎么造。。。实现上说S矩阵元就是这么造的，相互作用只在很小的一个尺度进行，拉到无穷远的初末态可以看成自由态，粒子被认为是波包（就像2楼说的那样），在碰撞前几个波包互不关联，碰撞视为波包的叠加。（顺便一提，我之前说两点关联函数是开玩笑，那不是物理可观测量。）

7. 2019-01-13 18:28:19

The topic of this thread is about Operator Product Expansion (OPE) and related stuffs. I'll briefly introduce OPE with a simple $\phi^4$ theory, emphasize the importance of large momentum expansion and take 4-fermion theory as an example to prove this point, maybe talk a little about complications in CFT, applications in curved spacetime (quantization, etc.) or DIS and QCD sum rule, even quasi PDF (if I'm ever gonna master them, that is). Nowadays people tend to use EFT in place of OPE, but somehow I believe there're subtle diffenerces.

History of OPE
Well, Kim Wilson invented it in 1969, shortly afterwards Zimmermann proved it mathematically in a lecture in 1970. HEP physicists adopted it to prove the factorization theorem of DIS experiment, and it was a triumph. Then it became famous. There're a lot of applications ever since, but I won't make a list here.

What is OPE?
What is OPE? In short, it's Laurent expansion. We can have a general expression as following:
$$T\phi(x)\prod_i\phi(x_i)\approx\sum_iC_i(x_i^\mu)[\mathcal{O}_i(x^\mu)]$$
where $[\mathcal{O}_i(x)]$ is renormalized local/composite operator and $C_i$ is often called Wilson coefficient.

$\phi^4$ example and tree level contributions
Let's take $\phi^4$ theory (3+1 dimension) as an example:
$$\mathcal{L}=(\partial_\mu\phi)^2/2-m^2\phi^2/2-g\phi^4/24+\text{counterterms}.$$
We can start with OPE btw. two operators and put them into a 4-point Green function so we can actually do some calculation:
$$\langle 0|T\phi(x)\phi(0)\tilde\phi(p_1)\tilde\phi(p_2)|0\rangle\approx\sum_iC_i(x^\mu)\langle 0|[\mathcal{O}_i(0)\tilde\phi(p_1)\tilde\phi(p_2)]|0\rangle$$
and we can define its momentum space counterpart
$$\newcommand{\dd}{\text{d}}T\tilde\phi(q)\phi(0)=\int\dd^4 xe^{iq\cdot x}T\phi(x)\phi(0)\approx\sum_i\tilde C_i(q^\mu)[\mathcal{O}_i(0)].$$
(We use tilde to denote momentum space quantities and square bracket to denote renormalized quantities. )

Now we can compute the tree diagrams of the Green function:

[attachment:5c3b12283a035]

we have
$$\frac{i}{p_1^2-m^2}\frac{i}{p_2^2-m^2}\big[e^{-ip_1\cdot x}+e^{-ip_2\cdot x}\big]$$
Expand it around $x=0$ we have
$$\frac{i}{p_1^2-m^2}\frac{i}{p_2^2-m^2}[2-i(p_1+p_2)\cdot x-(p_1\cdot x)^2/2-(p_2\cdot x)^2/2+\cdots]$$
These are exactly the OPE relation in coordinate space with lowest order Wilson coefficients. This is normally how we derive the basic forms of $\mathcal{O}_i$s.

The OPE relation is just
$$T\phi(x)\phi(0)=\phi^2(0)+\frac{1}{2}x^\mu\partial\mu\phi^2+\frac{1}{2}x^\mu x^\nu\phi\partial\mu\partial\nu\phi+\cdots$$
where all $\phi$ fields appear in the r.h.s. are evaluated at $x=0$. The result is finite, so there's no need to involve renormalization label.

It's easy to prove that this relation is an operator relation thus independent of external states. One may start with a 6-point Green function ($\phi(x)\phi(0)$ and 4 external legs) and check if this relation is still valid.

From the look of it, OPE is trivially Taylor expansion. However, one must note that we haven't compute loop corrections yet. So for classical scenarios, Taylor expansion might be enough, but there're quantum fluctuations we must take into account.

8. 2019-01-12 22:55:54
Turgon 更新于 粒子的空间局域性

不明白你想问什么，你是想说cluster decomposition principle有没有数学形式的表达？还是场论本身如何区分local和nonlocal理论？如果是后者的话本身场是有坐标的label啊。还是说你是想找到一个具体的物理量对应到坐标？这个我就不太清楚了，但是你应该可以在坐标空间算一个两点关联函数（大雾）。

9. 2019-01-12 22:29:52

能搞清楚微分作用的对象大概确实是件好事吧。。。当初学规范场的时候看着一堆不知道往哪作用的算符超气。。。最后还是得自己从头推。。。。但是这种注释略繁琐

10. 2月前
2018-10-28 03:45:32

@DTSIo 抱歉我没看明白，你是怎么把原来那个积分变成两个简单积分的卷积的？