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I'm having problems in an integral of many variables in which one of them leads to a Dirac delta and a Principal value.

I have to solve

\begin{equation} Int = \int dp dp' dk dk' dq f[p,p'] f[k,k'] \delta (p-p'-q)\delta (k-k'+q) \int_0^\infty e^{-i (g[k] -g[k'] - h[q])t} dt, \end{equation}

in which $f[p,p']$ denotes a function of $p$ and $p'$, for instance. I know that \begin{equation} \int_0 ^\infty e^{-i (g[k] -g[k'] - h[q])t} dt = \pi \delta (g[k] -g[k'] - h[q]) + i PV \frac{1}{(g[k] -g[k'] - h[q])}, \end{equation} in which PV denotes the Cauchy Principal Value. The ''delta part'' is ok, the PV part no. I know that PV(1/x), for instance, requires a test function $f[x]$ to be computed. So, PV(1/x) of some function would be \begin{equation} PV\int_{-\infty}^\infty \frac{f[x]}{x} dx. \end{equation}

The problem here is that I have many variables, and as soon as I will eventually integrate over all variables, I really do not know in relation of what I have to integrate in the resulting PV.

So far, I thought I could do \begin{equation} \int dp dp' dk dk' f[p,p'] f[k,k'] PV \int_{-\infty}^{\infty} dq \delta (p-p'-q) \frac{\delta (k-k'+q)}{(g[k] -g[k'] - h[q])} . \end{equation} Applying one of these delta ''functions'' I would obtain \begin{equation}\int dp dp' dk dk' f[p,p'] f[k,k']\frac{\delta (p-p'+k-k')}{(g[k] -g[k'] - h[k'-k])} \end{equation} and finally \begin{equation} \int dp dp' dk \frac{f[p,p'] f[k, k +p -p']}{(g[k] -g[k+p-p'] - h[p-p'])}. \end{equation}

Is it correct? For me it is a bit strange, because it seems like I am cheating choosing the integral to be in $q$ since I have another integral in $q$ to do. Moreover, in any variable I would choose, they are all dependent of o $p$, $p'$ and so on. What kind of detail I lost?

Maybe you will need to know that $g[k]$ and $h[q]$ only obtain non-negative values.

Thank you very much.

Wellington Ribeiro
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  • $\newcommand\R{\mathbb{R}}$ Sorry, I didn't worked to most of this, but the PV is no different for higher dimensions. Take for instance $x\in \R^n$, then $PV(\frac{x_i}{|x|^{n+1}})$ makes sense for test functions f: $\R^n \to \R$. Also notice that similar to $\delta$ the $PV$ is also a distribution. – hal4math Sep 24 '19 at 23:17
  • Yes, PV is also a distribution. But the problem is that I do not know the action of PV in these cases. It is also important to say that I will be very grateful if someone recommend a bibliography in which I can search for details. I does not need to be very rigorous. I just want to understand better the action of a PV. – Wellington Ribeiro Sep 24 '19 at 23:34
  • Mh, but this will highly depend about which PV you are talking about. I was also a bit incorrect, because $PV$ is as it self of course meaningless, it only becomes meaning when plugging in a function. And not for every function PV becomes meaningful and a distribution. So understand that action in general, is not possible I think, not like $\delta$ in any case. – hal4math Sep 24 '19 at 23:38
  • Yes, exactly. PV needs a function in order to have meaning. But, allow me to be a bit more simple. – Wellington Ribeiro Sep 24 '19 at 23:47
  • Allow me to be a bit more simple. Consider the following integral \beging{equation} \int_{-\infty}^{\infty} dx \int_{-\infty}^{\infty} f[x,y] dy \int_0^{\infty} e^{i t (x-y)} dt \end{equation} The last integral leads \begin{equation} \int_0^{\infty} e^{i t (x-y)} dt = \pi \delta(x-y) -i PV (1/x-y). \end{equation} Ok, but in what variable I must integrate? – Wellington Ribeiro Sep 24 '19 at 23:54
  • Both $x$ and $y$. Now this is quite tricky I feel. Also I am quite confused about the fact that the $\delta$ takes a real number whereas I think it should take a point $(x,y)$. – hal4math Sep 25 '19 at 00:02
  • The delta part of this ''confusing'' integral will be ok, in my problem (see the definition of $Int$). Thanks to the many integrals, after applying the deltas I will obtain $\int dp dk f[p,p] f[k,k]$, and it is ok. But it is really confusing what I have to do with the PV. – Wellington Ribeiro Sep 25 '19 at 00:13
  • Okay, I really don't see why this should be true. Btw. I assume this is some quantum mechanics? Did you try post it in the physics part of this side? – hal4math Sep 25 '19 at 00:18
  • Yes, it is. I did not try because it is a pure application of some mathematical techniques. Is it ok to post it in physics part? – Wellington Ribeiro Sep 25 '19 at 00:22
  • I really think so, sure. There is just this problem that this of course super abstract stuff, and natural you using quite of a bit different notation I am used to and also some surly useful abuse of notation that I think is common among physicists but not so much for mathematician at least for me it is hard to decipher. – hal4math Sep 25 '19 at 00:25
  • Ok. Thank you very much for the discussion and availability. – Wellington Ribeiro Sep 25 '19 at 00:28

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