What does "mathematically well defined" quantum field theory mean? Reading Wald's book (page 380, end of the first paragraph of section 14.1) while the author is giving an overall discussion of quantum field theories you can read     However, for the more interesting case of a theory with interactions (i.e., for a field or fields satisfying nonlinear equations) one is led unavoidably to consider the products of field operators at the same spacetime point. Such quantities have no natural mathematical meaning, and consequently, with the exception of some simple models in lower spacetime dimensions, there are at the present no known examples of quantum field theories of physically reasonable interacting fields which are mathematically well defined.   What does Wald exactly mean by mathematically well defined, which are the problems, for example in QED that make it ill defined?
What does it mean for a QFT to not be well-defined? It is usually said that QED, for instance, is not a well-defined QFT. It has to be embedded or completed in order to make it consistent. Most of these arguments amount to using the renormalization group to extrapolate the coupling up to a huge scale, where it formally becomes infinite.  First of all, it seems bizarre to me to use the renormalization group to increase resolution (go to higher energies). The smearing in the RG procedure is irreversible, and the RG doesn't know about the non-universal parts of the theory which should be important at higher energies.   I have read in a note by Weinberg that Kallen was able to show that QED was sick using the spectral representation of the propagator. Essentially, he evaluated a small part of the spectral density and showed that it violated the inequalities for the field strength $Z$ that are imposed by unitarity + Lorentz invariance, but I do not have the reference.  So what I would like to know is :   What does it take to actually establish that a QFT is not well-defined?   In other words, if I write down some arbitrary Lagrangian, what test should I do to determine if the theory actually exists. Are asymptotically free theories the only well-defined ones?   My initial thought was: We would like to obtain QED (or some other ill-defined theory) using the RG on a deformation of some UV CFT. Perhaps there is no CFT with the correct operator content so that we obtain QED from the RG flow. However, I don't really know how to make that precise.   Any help is appreciated.
What does the bracketed capital letter indicate in the revised version below? Is it essential, per the rules of proper punctuation, to do so? Are the ellipsis points (i.e., the spacing of the dots) spot on in that sentence to indicate the omission of one or more sentences?  Original quotation  "It need hardly be said that shortness is a merit in words. There are often reasons why shortness is not possible; much less often there are occasions when length, not shortness, is desirable. But it is a general truth that the short words are not only handier to use, but more possible in effect; extra syllables reduce, not increase, vigor."  Revised quotation  "It need hardly be said that shortness is a merit in words. . . . [S]hort words are not only handier to use, but more possible in effect; extra syllables reduce, not increase, vigor."  Thank you. This is a question from the free Harvard online abstract algebra .  I'm posting my solutions here to get some feedback on them.  For a fuller explanation, see  post.  This problem is from assignment $5$.     Prove directly that distinct cosets do not overlap.   Let $H$ be a subgroup of $G$ and let $a$, $b$, and $c$ be elements of $G$ such that $b\not\in aH$ and $c$ is in both $aH$ and $bH$.  Then there are elements $h$ and $h^\prime$ in $H$ such that $c=ah$ and $c=bh^\prime$.  So $ah=bh^\prime$ and $b=ahh^{\prime -1}$.  But $hh^{\prime -1}\in H$ so $b\in aH$.  This contradicts our original assumption. Therefore, there can be no element in more than one distinct coset.  Again, I welcome any critique of my reasoning and/or my style as well as alternative solutions to the problem.  Thanks. Let $\mathbb F$ be a field and $\mathbb  K $ be an extension field of $\mathbb F$  such that $\mathbb K$ is algebraically closed.   Let $\mathbb L$ be the field of all elements of $\mathbb K$ which are algebraic over $\mathbb F$. Then $\mathbb L_{|\mathbb F}$ is an algebraic extension.   My question is : Is $\mathbb L$ algebraically closed ?  I am trying to prove the existence of algebraic closure, so please don't assume that every field has an algebraic closure. I attended a business meeting in the Midwest and one of our clients took our management team and a few other business partners to dinner at an awesome Mexican restaurant. When the appetizers were served I commented that it was a wonderful presentation and that everything looked very fresh, including the avocado slices.   Speaking to the woman seated across from me I said that even though I use a good amount of lemon juice, whenever I prepare avocados they still turned dark pretty quickly. This happens whether they are sliced, used in guacamole, etc. She responded that the secret is to use lime juice. She proceeded to tell me that she was born and raised in Central America and that it is customary there to use fresh limes or juice rather than lemon.  I have tried it and lime seems to work much better than lemon, so it would stand to reason that they are more acidic. However, I never thought there was that much difference. So, on to the questions.  Are limes more acidic than lemons? Are there any other differences in their properties that could make a difference? Is there an (easy) way to create or personalize .bst files? What does UNDEF and notype() mean when seeing the C++ library information using dumpbin? I want to derive a closed formula for the integral $$I_n= \int \frac {dx}{\prod_{r=0}^n (x+r)}$$  On writing out first few terms we get   For $n=0$, $$I_0=\ln \vert x\vert+C$$ For $n=1$, $$I_1=\ln \vert x\vert-\ln \vert x+1\vert+C$$ For $n=2$ $$I_2=\frac {1}{2!}\left(\sum_{r=0}^2 (-1)^r\binom {2}{r} \ln \vert x+r\vert\right)+C$$ For $n=3$ $$I_3=\frac {1}{3!}\left(\sum_{r=0}^3 (-1)^r\binom {3}{r} \ln \vert x+r\vert\right)+C$$  Hence for generalized $n$ we have $$I_n=\frac {1}{n!}\left(\sum_{r=0}^n (-1)^r\binom {n}{r} (\ln \vert x+r\vert)\right)+C$$  Now this is just an observation but I want to prove that it is correct. I have tried lot of methods but not useful. Partial fractions would have been most useful but would go out on tedious task which is nearly impossible.  Also integration by parts won't help nor any trig substitution. So any ideas are welcome.  And ya,  this is not a homework question,  it's a question which I just saw in a integral challenge paper. Global Inbox -> Global RSS feed for all my questions across all the sites? Encrypting as3 flash .swf Xml configuration versus Annotation based configuration In a few large projects i have been working on lately it seems to become increasingly important to choose one or the other (XML or Annotation). As projects grow, consistency is very important for maintainability.   My questions are: what are the advantages of XML-based configuration over Annotation-based configuration and what are the advantages of Annotation-based configuration over XML-based configuration? Two git repositories in one directory? Is it possible to have 2 git repositories in one directory?  I'd think not, but thought I'd ask.  Basically, I'd like to check in my home directory config files (e.g. .emacs) which should be common across all of the machines I work on, but have a second repository for local files (e.g. .emacs.local), which contains machine-specific configurations.  The only way I can think of to do that is to have the local config in a subdirectory and ignore that subdirectory from the main git repository.  Any other ideas? I cant use lhopital's rule here is my attempt:  Limit of $(\frac{n^{2}-2n+1}{n^{2}-4n+2})^{n}$ =$e^{\lim_{n\to \infty} n\times \ln(\frac{1 -2-n +\frac{1}{n^{2}}}{1 -\frac{4}{n} +\frac{2}{n^{2}}})}$ I got stuck here I have now enough reputation points on Stack Overflow to edit every post directly. But sometimes (especially if this is about a field where I'm not savvy, or when I try to edit a question title without having understood the question itself) I might be not totally sure about whether my edit really is an improvement.  For now, I have these options:   Do the edit anyway, hope that someone else will catch it if it is wrong Don't do the edit, and hope that someone else will see this post and edit it in the right way Let a comment and hope that someone will see it (either after editing or instead of it). Flag the post for moderator attention (but will the moderators know it better?) Search the right chat room to find someone who knows more about the topic.   All these are not really optimal, I think.  I would like to have a check box     [_] Submit this edit for approval   which, when checked, would submit the edit into the usual edit approval queue. Then I would get at least one or two other pairs of eyes looking over my edit, and maybe even improving it.  (I don't really care about the +2 reputation for the edit - I think it should be without any reputation difference here.)  There's an  for nearly giving everyone the ability to suggest edits that the post owner can approve, but I'm simply interested in being able to insert them into the general edit queue where anyone can approve them.  Actually, I could do this right now by having a low-reputation sock puppet (or a totally anonymous edit), but I think this is not the intended use for sock puppets, is it? Also, I would like to be able to get feedback for my edit. .gitignore for Visual Studio Projects and Solutions How to strike through obliquely, e.g. to indicate cancellation? How can I strike through an expression (e.g. "2x") obliquely, e.g. from bottom left to top right, so as to indicate cancellation? With the \sout command from the ulem package, I can only strike horizontally. Why doesn't a fly fall off the wall? grep strange behaviour with single letter words I am removing stop words from a text, roughly using this   I have the following  $ cat file file types extensions  $ cat stopwords i file types   grep -vwFf stopwords file  I am expecting the result: extensions  but I get the ( I think incorrect)  file extensions   It is as if the word file has been skipped in the stopwords file. Now here's the cool bit: if I modify the stopwords file, by changing the single word/letter i on the first line, to any other ascii letter apart from f, i, l, e, then the same grep command gives me a different and correct result of extensions.  What is going on here and how do I fix it?  I'm using grep (BSD grep) 2.5.1-FreeBSD on a Mac OSX GNU bash, version 4.4.12(1) Python Script returns unintended "None" after execution of a function Is it possible to set the "urgent" flag in Gmail for outgoing mail? I'd like to be able to flag time-sensitive emails as such. Accidently deleted boot option on Dell laptop I have a Dell laptop in which I have dual booted Ubuntu with Windows 10. Usually the laptop loads the Grub menu first.  Recently I had some problems with Windows Bitlocker: every time I open Windows I am prompted to enter the Bitlocker password. In trying to solve this, I accidentaly clicked Ubuntu in the "Delete Boot" option in the boot sequence. Now the laptop boots directly to Windows without first loading the Grub menu.  Luckily, when I restarted the system and clicked "add boot option" I found the efi folder in which the Ubuntu folder is present. In Ubuntu there is fwgrupx64.efi, grubx64.efi, grub.cfg, shimx64.efi, mmx64.efi, and BOOTX64.CSV. On clicking grubx64.efi it shows "Boot name not found".  How can I recover the Grub boot menu?