We all have been told when we were kids that constant terms in Lagrangians are irrelevant because when doing classical dynamics the Euler Lagrange equations involve derivatives of the Lagrangians so the constant terms just vanish.   This explanation is solid. I want the same in quantum field theory. For illustration I shall consider two real scalar fields with a mexican hat potential. Using dimensional regularization the action is  $$ I[\phi_1,\phi_2]=\int d^dx\bigg(\frac{1}{2}\partial_{\mu}\phi_1\partial^{\mu}\phi_1+\frac{1}{2}\partial_{\mu}\phi_2\partial^{\mu}\phi_2+\frac{1}{2}\mu^2(\phi_1^2+\phi_2^2)-\frac{1}{4}\lambda^2(\phi_1^2+\phi_2^2)^2\bigg) $$ where both $\mu$ and $\lambda$ are positive. The potential above is $$ V(|\phi|)=-\frac{1}{2}\mu^2|\phi|^2+\frac{1}{4}\lambda^2|\phi|^4 $$ has a minimum at $$ |\phi|=\frac{\mu}{\lambda} $$ I pick to expand around $$ \phi_1=\frac{\mu}{\lambda}\qquad\phi_2=0 $$ redefining the parameters $$ \eta=\phi_1-\frac{\mu}{\lambda}\qquad\xi=\phi_2 $$ and re-expanding the lagrangian $$ I[\phi_1=\frac{\mu}{\lambda}+\eta,\phi_2=\xi]\equiv I'[\eta,\xi]= \int d^dx\bigg(\frac{1}{2}\partial_{\mu}\eta\partial^{\mu}\eta-\mu^2\eta^2+\frac{1}{2}\partial_{\mu}\xi\partial^{\mu}\xi-\mu\lambda(\eta^3+\eta\xi^2)-\frac{\lambda^2}{4}(\eta^4+\xi^4+2\eta^2\xi^2)+\frac{\mu^4}{4\lambda^2}\bigg) $$ Now, this action goes in a generating functional. $$ \mathcal{Z}[J_{\eta},J_{\xi}]=\int[d\eta][d\xi]\exp(iI'[\eta,\xi]+i\int d^dx\,\bigg[J_{\eta}\eta+J_{\xi}\xi\bigg]) $$ I am concerned about the constant piece in the action $$ \Delta I=\int d^dx\,\frac{\mu^4}{4\lambda^2} $$ This integral above is infinite even in $d=4-2\epsilon$ dimensions! so, in  which sense can we ignore constant terms in the Lagrangian such as above in quantum field theory, and why?
Why can we always drop any constant term in a Lagrangian density in quantum field theory?  This issue is somehow related to the constant term being some kind of cosmological constant.  Can you please explain this issue?
Find all Quotient (or Factor) Groups of $D_4$ (Dihedral Group of order $8$) I need to be able to find all of the quotient groups for dihedral group 4 with $D_4 = \{ e,R,R^2,R^3,V,H,D,D'\}$. I know I have to start by finding the normal subgroups, which are $\{e,R^2\}$ $\{e,R,R^2,R^3\}$ $\{e,R^2,V,H\}$ $\{e,R^2,D,D'\}$. Then I need to find the sets defined by $G/H=\{ aH : a \in G\}$ with operation $aH  bH = abH$. I am stuck here; could someone please show me how to find all of the factor groups for D4? If every pair of congruence equations admits solutions, then the entire system admits solutions Let a system of three linear congruence equations in integers be given;      \begin{cases}x\equiv b_1\mod c_1\\ x\equiv b_2\mod c_2\\ x\equiv b_3\mod c_3\\ \end{cases}   with $c_1,c_2,c_3\in\mathbb Z_{+}$. Show that, if every pair of congruence equations admits solutions, then the entire system admits solutions.    Now, if every pairs admits solutions, then for example $b_1\equiv b_2\equiv a\mod \gcd(c_1,c_2)$ for some $a$ then;   $x-a\equiv b_1-a\mod c_1$  $x-a\equiv b_2-a\mod c_2$  replacing $x$ by $x':=\frac{x-y}{\gcd(c_1,c_2)}$, we obtain:  $x'\equiv b_1'\mod c_1'$  $x'\equiv b_2'\mod c_2'$  and since $\gcd(c_1,c_2)=1$ by Chinese Remainder Theorem we have a unique solution $\mod{c_1'c_2'}$, so $x\equiv (\gcd(c_1,c_2)x'+a)\mod{\operatorname{lcm(c_1',c_2')}}\quad(1)$  We can do it with another pair (take $2$nd and $3$rd, since we know that such a pair admits solutions;  $x\equiv(\gcd(c_2,c_3)x''+a')\mod{\operatorname{lcm(c_2',c_3')}}\quad(2)$  Now how can I show that for the pair $(1)$ and $(2)$ also solutions exist ? Find all functions $f$ such that $(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$ Can I tap a creature with summoning sickness to pay "tap an untapped creature you control" costs? Maximum and Minimum values for ints How to prove that a nilpotent operator has a basis representation that is strictly upper triangular? Proving question on sets I am unable to understand this question. I have to say whether its true of false and then prove it but I can't proceed with the question unless I understand it.     Let $S$ be the set of stars in our galaxy. There is exactly the same   number of subsets of stars in our galaxy as there are functions   $f:S\rightarrow\{\alpha,\beta\}$ How can a file be copied? Where are the Unity desktop wallpapers located? Please see the picture below. Where are the Unity desktop wallpapers located? Suppose that $X$ and $Y$ are chosen randomly and independently according to the uniform distribution from $(0,1)$. Define $$ Z=\frac{\max(X,Y)}{\min(X,Y)}.$$ Compute the probability distribution function of $Z$. Can anyone give me some hints on how to proceed? I can only note that $\mathbb{P}[Z\geq 1]=1$ and $$F_Z(t)= \mathbb{P}[Z \leq t]=\mathbb{P}[X\leq Y, Y\leq tX]+\mathbb{P}[Y \leq X, x\leq tY]$$ If $x+{1\over x} = r $ then what is $x^3+{1\over x^3}$? If $$x+{1\over x} = r $$ then what is $$x^3+{1\over x^3}$$  Options: $(a) 3,$ $(b) 3r,$ $(c)r,$ $(d) 0$ I asked a question that has been marked as a duplicate. I read a lot of similar questions and explanations in advance but I couldn't find a solution. In the question that should answer my question the asker used a different (not working) approach. There is even an answer that would work for my case however I'd rather consider it as a workaround than a solution since I have seen examples using my approach.  So I would like your opinion in that case. Would it be ok to repost my question while explaining why other answers didn't help me? I edited my question but I think it's not getting any attention anymore. Naturally I will remove the old question. Strange Bittorrent Log On My Server What does a dot after the file permission bits mean? I performed an ls -la on directory on my CentOS 6.4 server here and the permissions for a given file came out as:  -rwxr-xr-x.   I understand what -rwxr-xr-x means, what I don't understand is the . after the last attribute.  Can someone explain it to me? Is it harmful in any way? Can it be removed? Login screen resolution So I just installed Ubuntu 16.04 on my Dell XPS 13. When I get to the login screen it starts at 3200x1800 for a quick second, then I get a way lower resolution like 800x600 but as soon as I login it adjusts to 3200x1800. How would I get 3200x1800 permanently on my login screen? I didn't have this issue on 14.04. Let $t\in \mathbb R$. Prove that $\displaystyle{\sum_{n=1}^{\infty}n^{3}\sin\left(\frac{1}{n}\right)-n^{2}+t}$ converges iff $t=\frac{1}{6}$.  While the $(\Rightarrow)$ direction was fairly easy (as $a_n$ should converge to $0$, forcing $t=\frac{1}{6}$), i'm facing some hard time trying to prove the second direction, that is $\displaystyle{\sum_{n=1}^{\infty}(n^{3}\sin\left(\frac{1}{n}\right)-n^{2}+\frac{1}{6})}$ converges. I did notice that $a_n=a_{-n}$ and tried to write this series as a Leibnitz series, but had no luck since the sequence doesn't decrease. I also tried to prove absolute convergence but then again, no luck. Thanks. Router to router setup works fine with straight cable up to 2-3 meter it is not working for 45 meter why? I don't always answer myself, but when I do, I fail to ask a good question Sometimes I'll encounter problems while coding, or sometimes out of simple interest for a question, I'll google for an answer, and most of the time I find that answer is here on stackoverflow.com.   If I can't find an answer, or if it's not enough to satisfy my curiosity, sometimes I expend my own effort to solve those questions. And after I've solved it, I'm happy to make my contribution to the community, and I post them as the Q&amp;A style posts. For example:     It seems relatively easier to ask a question if I didn't already have a solution, and I can put what I have so far in the question body, such as:     The problem to me is, if I have my own solution to my own question, then most of what I have so far becomes part of the solution. It's likely I have failed in making my contribution by asking a question to which I have an answer, for example:     So I'm wondering if there are some guidelines or good practices for Q&amp;A style posts?   How do I do it better? The international Stack Overflow sites in Spanish, Portuguese, Russian, and Japanese seem to have a bug in their quick site descriptions (under their titles), for they all start with the word  For...  in English, instead of their respective languages As an example, the Spanish Stack Overflow site description is:  For programadores y profesionales de la informática  Obviously, this isn't something too big of a deal, but since they offer those sites in their respective languages, it would be nice if this issue were fixed. You can see this under &quot;More communities&quot; in the top bar.    You can also see this on the  page, where the descriptions begin with  Q&amp;A for... Render issue - Model doesn't appear to be on any layers and won't render Really odd on this... using Blender 2.75... I started modelling and lost the objects from the 3d view, but they were still showing in the object lists on the outlier... I was able to copy them and continued working... when I did a test render I discovered that only the original lost objects were rendering and nothing that I had modelled over the previous 2/3hrs was showing up...  I have no idea what I did but I created another 3d view and the original bits showed up along with the layer buttons which had been missing from the screen which I thought was just a new feature from  2.75...!  Picture to show the 2 screens, top one doesn't render but bottom one does    Any idea's what I've done?