There are so many times that something leaves you stumped. I was recently reading this paper and the derivation of the Maxwell's Equations from just Newton's Second Law and The Uncertainty Principle really intrigued me. Although they just derived the Bianchi set, yet with slight tweakings with re...

A notion encountered in field theory on non-globally hyperbolic manifolds is the notion of a spacetime being causally benign with respect to some field $\phi$, which is defined thusly : A neighbourhood $U$ is causally regular if $\bar U$ has a neighbourhood $U_1$ such that for every solution $...

Chemical potential is the difference in free energy to add a particle to the system. Consider two systems (1,2) with a contact with the reservoir at temperature $T$, then total free energy is \begin{equation} G = G_{1}+G_{2} \end{equation} But since the total number of particles is conserved \beg...

In a recent Math Stack Exchange question I asked about the function $$f(z)=\sum_{n=0}^\infty z^{2^n},$$ and was informed of its status is a canonical example of a lacunary series with natural boundary at $|z|=1$. A phenomenon observed by the accepted answer was that this function has a multitude ...

Hi! I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at every $2^n$-th root of unity for every $n$, I feel I'm missing some intuition into what exactly is g...

You wouldn't think it, from how easy it is to pose this question, but it is ridiculously nontrivial. As it happens, it is entirely impossible to find the position-basis matrix elements of this propagator. So far you've done good, and the identification $$ U\left( t_{2},t_{1}\right) =e^{\frac {-...

A notion encountered in field theory on non-globally hyperbolic manifolds is the notion of a spacetime being causally benign with respect to some field $\phi$, which is defined thusly : A neighbourhood $U$ is causally regular if $\bar U$ has a neighbourhood $U_1$ such that for every solution $...