Usually in probability theory, for a random variable whose value is in $\mathbb{R}$, we talk about its cumulative distribution function $F(x)$ and then its density $f(x)$, in good enough cases $F'(x)=f(x)$.
That's the setup I'm familiar with, so I got annoyed when physicists talk about unnormalized "densities". E.g. if the probabilistic density of the position of a particle on $\mathbb{R}$ is equal to 1 everywhere, that means it is equally likely to appear anywhere. More generally you can imagine them talking about a non-negative function $f(x)$ being the density of something with the density $f(x)$ is not integrable on $\mathbb{R}$ but locally integrable, i.e. $\int_{[a,b]}f(x)dx$ make sense for $a\leq b, a, b\in\mathbb{R}$. As $\int_{\mathbb{R}}f(x)dx$ is undefined ($=\infty$), one cannot divide by it to normalize.
Is there a mathematical way to make sense of such statements? There is one I have in mind, namely, one can talk about the density of some random variable $X$ up to a scalar multiple, such that for any intervals $[a,b]\subset [c,d]$ we can express the conditional probability as a quotient of integrals:
$$P(X\in[a,b] \big| X\in[c,d])=\dfrac{\int_{[a,b]}f(x)dx}{\int_{[c,d]}f(x)dx}.$$
I only know very basic probability theory so I don't know if this makes sense. Am I allowed to interpret unnormalized probabilistic density functions this way? Is this what physicists mean? Or are there any other interpretations? Do I have to worry about something else when thinking about things in this way?
Best Answer
The most coherent interpretation I know of what the physicists are doing in the real line example you mention is that they consider implicitely, for every positive $t$, a bona fide random variable $X_t$ with density $f_t$ where $f_t(x)=c_t^{-1}f(x)\,\mathbf 1_{[-t,t]}(x)$ and $c_t$ is the integral of $f$ on $[-t,t]$, and that they are convinced that the objects $X_t$ somehow converge when $t\to+\infty$ to... one does not know what kind of object exactly.
It happens that in most of the situations where (good) physicists are doing so, the properties of $X_t$ (of interest to them) somehow stabilize when $t\to+\infty$. Hence, although there is no random variable $X$ such that $X_t\to X$, nevertheless $X_t\approx X_s$ for every $s$ and $t$ large enough and this is all that is needed to proceed. So, in the end, there is (most often a pretty good amount of) reason in (them physicists') madness, although it may not be quite the brand of reason mathematicians are using.