My question is motivated by 2D CFT where one works in "complex coordinates". The question is the following: Suppose I am in 2D flat Euclidean space, i.e.  $$\mathrm{d}s^2 = \mathrm{d}x^2 + \mathrm{d}y^2$$  One of the crucial things here is that $x$ and $y$ are independent. Now it well known that the plane can also be described by one complex number. The point here is that it is ONE complex number, so once I write down:  $$z = x+iy$$  I have everything to describe the complex plane. And yet in every book, they then define  $$\bar z = x-iy$$  so the complex conjugate and write down the metric:  $$\mathrm{d}s^2 = \mathrm{d}z \mathrm{d}\bar z$$  Of course, the norm in the complex plane is given by $z\bar z$ but what I don't understand is that the complex conjugate is not an independent variable, so how does the above metric make sense. Of course, often in the books, they at some point mention that they complexify and essentially consider $\mathbb{C}^2$ in which case $z$ and $\bar z$ are to be considered as independent, then the metric makes more sense. But if you don't complexify, how am I supposed to understand the said metric?  Edit: It seems that the standard answer is that you consider the complexified $\mathbb{C}^2$, but that wasn't my question: If I don't want to go to $\mathbb{C}^2$, and just consider $\mathbb{R}^2$ as $\mathbb{C}$, what would the metric be?
The Setup: Let's say we want to study a Euclidean $\mathrm{CFT}_2$ on $\mathbb R^2$ with coordinates $\sigma^1$ and $\sigma^2$ and metric   $ds^2 = (d\sigma^1)^2+(d\sigma^2)^2$.    It seems to me that in the usual discussion (e.g. di Francesco, Ginsparg, Polchinski), one proceeds to consider an analytic continuation of the CFT to $\mathbb C^2$ with coordinates $z^1, z^2$ and complex metric  $ds^2 = (dz^1)^2+(dz^2)^2$  and then, one performs the coordinate transformation $z = z^1+iz^2$ and $\bar z = z^1-iz^2$.  In this way the coordinates $z$ and $\bar z$ can be considered "independent" because they are coordinates on a complex two-dimensional manifold.  Also, in these coordinates the metric becomes  $ds^2 = dz\,d\bar z$  and it becomes clear that conformal mappings consist of mappings: $(z, \bar z)\to (f(z), g(\bar z))$.  My confusion is this: Since our original theory was on $\mathbb R^2$, books say that when we do calculations, we should consider the physical theory as living on the copy of $\mathbb R^2$ embedded in $\mathbb C^2$ given by the condition $\bar z = z^*$.  But consider the mapping $(z, \bar z)\to (z^2, \bar z)$.  This is a conformal mapping on $\mathbb C^2$, but it does not map the surface $\bar z = z^*$ to itself; for example the point $(z, \bar z)=(2,2)$ gets mapped to the point $(z^2, \bar z) =(4,2)$ and $2$ is clearly not equal to $4^*$.  In particular, it seems to me that analytic continuation to a CFT on $\mathbb C^2$ enlarges the set of mappings one can have, so what relevance does it really have to the original CFT on $\mathbb R^2$?
Subsample of a random sample: random sample? What is the origin of BrEng ‘bird’ meaning “young woman”? Bird:         (Brit.) a girl or young woman, esp one's girlfriend ()      According to  bird:         "maiden, young girl," c.1300, confused with   (q.v.), but felt by later writers as a figurative use of  (n.1). Modern slang meaning "young woman" is from 1915, and probably arose independently of the older word.      also :         , bird was used for ‘girl’, but this was probably owing to confusion with another similar middle English word, burde, which also meant ‘young woman’. The usage crops up from time to time in later centuries, clearly as an independent metaphorical application, but there does not really seem to be an unbroken chain of occurrences leading up to the sudden explosion in the use of bird for ‘young woman’ in the 20th century.   (www.dictionarycentral.com)      Modern usage of bird meaning girl appears to be unrelated to the old Middle Ages meaning. Etymonline dates it back precisely to 1915.    What is the modern origin of bird meaning young woman?  In what context did they start to use it at the beginning of the 20th century? How to check if a column exists in a SQL Server table? Could I have a hint for testing the convergence of the following series please?  $$\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$$  I am very appreciative for your help. When I post an answer to a brand new question I usually find that someone else has submitted that same answer just moments before. (, usually ;-) )  What I've started to do lately is (I think) the right thing.   If I notice that another answer predates mine, and that I add no value over it, I immediately delete my own answer, and upvote the other guy.  I recommend that you too do the same. Otherwise the message will be diluted.  Similarly, if after posting your "answer" you realise that you are in fact wrong (and you've just misread the question) -- you can actually delete your answer.  Is this reasonable?  (The reason why I'm writing this down is because it feels completely counter-intuitive to delete your own stuff. But it really saves everyone time and effort) What was the joke that Geordi told Data in TNG: "Encounter at Farpoint"? states:     Apparently at some point during this mission, Geordi La Forge told Data a joke, which the humorless android did not "get" until years later, when he activated his emotion chip during the events of Star Trek: Generations.    Is it known what the joke is? Background notebook template with spiral Install on, install in, install to When I say "programs to install on a new PC" it sounds alright to me, but I'm not sure if it's the correct usage. Which one of the following should I use?   Programs to install on a new PC Programs to install in a new PC Programs to install to a new PC How to increase the width and height of the Rich Text area field in Visual force Page? Output from ls has newlines but displays on a single line. Why? How can I assign the output of a command to a shell variable? I want to assign the result of an expression (i.e., the output from a command) to a variable and then manipulate it – for example, concatenate it with a string, then echo it.  Here's what I've got: #!/bin/bash cd ~/Desktop; thefile= ls -t -U | grep -m 1 &quot;Screen Shot&quot;; echo &quot;Most recent screenshot is: &quot;$thefile;  But that outputs: Screen Shot 2011-07-03 at 1.55.43 PM.png Most recent screenshot is:   So, it looks like that isn't getting assigned to $thefile, and is being printed as it's executed. What am I missing? "going to come down" vs "going to have to come down" What is the difference between the following sentences?        This fence is going to come down.   This fence is going to have to come down.      I have doubt that when you are saying fence is going to have to come down, that is enough to say it right? but why I hear in movies saying fence is going to have to come down, is there a difference between them? I have a query which selects a max value over all the values of in a datetime column (dtimesent).  select dtimesent, max(ResponseTime) AS maxtime FROM allrequests  where Request like '%Requestsa%' group by dtimesent   I need to get the value of the request colmn for each max response time in each group by. Is there any easy way to do this?  There can be multiple rows per dtimesent value so I can't just include the request column in the group by,  Many thanks, How to calculate the signal-to-noise ratio (SNR) in an image? I am working with an image X, I apply the "adaptive median filter" in it and I get the image Y. I'd like to measure the SNR in both in order to evaluate the quantity of noise deleted.   I know the formula to calculate the SNR is:   SNR = Psignal / Pnoise  but I don't know how to get these values from both images I have. I was thinking in substract the image X from Y and get the noise value. But I am not sure about the method. I am studying General Relativity and some basic QFTs. It bothers me a lot that different books use different metric signatures, i.e. $(-+++)$ and $(+---).$ Can anyone tell me the advantages of using both of them, or why some people prefer the former while some prefer the latter? I would like to know which one should I while tackling different problems. The Equation $11^x + 13^x + 17^x =19^x  $  Has    No Real Roots Only One Real Roots Exactly Two Real Roots  More than Two Real Roots    What I have done is   The function $f(x)=11^x + 13^x + 17^x -19^x  $ is strictly increasing and being always positive it have no roots won't instead of using' wouldn't' is possible? I have asked many questions about "would" both in ELL and ELU. But I still have a question about the use of "would". I have recently asked a question which @Jay also answered. He used "wouldn't" in the following sentences: 1. Here, you wouldn't say, "May I ask you what is your name?" Though we break this rule for questions sometimes 2. Well, someone might say that in informal speech, but you wouldn't write it or use it in formal speech. 3. "May I ask you what's your name?" is awkward; a fluent speaker wouldn't say that.  I had asked  question in ELL and I have received answers for it. But there is one thing that I'm still confused about.  Do the meanings of the sentences above change if we use 'won't' instead of using 'wouldn't' ? can we use 'won't' in the place of 'wouldn't' without changing the meanings of the sentences above? We all know you can't do the following because of ConcurrentModificationException:  for (Object i : l) {     if (condition(i)) {         l.remove(i);     } }   But this apparently works sometimes, but not always. Here's some specific code:  public static void main(String[] args) {     Collection&lt;Integer&gt; l = new ArrayList&lt;&gt;();      for (int i = 0; i &lt; 10; ++i) {         l.add(4);         l.add(5);         l.add(6);     }      for (int i : l) {         if (i == 5) {             l.remove(i);         }     }      System.out.println(l); }   This, of course, results in:  Exception in thread "main" java.util.ConcurrentModificationException   Even though multiple threads aren't doing it. Anyway.  What's the best solution to this problem? How can I remove an item from the collection in a loop without throwing this exception?  I'm also using an arbitrary Collection here, not necessarily an ArrayList, so you can't rely on get. Restore the state of std::cout after manipulating it How to check if a String contains another String in a case insensitive manner in Java?