Seminar on the Dirac equation

Next week Tadeusz Kulczycki and I start a seminar on the Dirac equation. The full title is Mathematical models in relativistic quantum physics, but actually we plan to focus on just one particular model. We will roughly follow (parts of) The Dirac equation book by B. Thaller, aiming at understanding the hydrogen atom model. From time to time I will post my notes on this blog. Hopefully this will become a mathematician-friendly introduction to the Dirac equation.

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π/2 conjecture

The second conjecture in this blog is related to my recent work on Lévy processes.

Suppose that x and y are positive reals, and \psi is an increasing function. Then, whenever the integral
\displaystyle\int_0^\infty\frac{(x+y)t^2}{(x^2+t^2)(y^2+t^2)}\exp\Biggl(\frac{1}{\pi}\int_0^\infty\biggl(\frac{x}{x^2+s^2}+\frac{y}{y^2+s^2}\biggr)\log\frac{\psi'(t^2)(s^2-t^2)}{\psi(s^2)-\psi(t^2)}ds\Biggr)dt
makes sense, it is equal to \pi/2.

Although the statement is quite elementary, I fail to find any elementary proof. Even for simple \psi, except \psi(\xi) = \xi and perhaps \psi(\xi) = \sqrt{\xi}, the conjecture seems to be highly non-trivial.

I showed this conjecture to several people. Let me cite here two comments: ‘It is the worst formula for \pi/2 I have ever seen!’, and ‘Come on, this must be either elementary or false’.

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Heronian triangles conjecture

Let my first blog post be about my six-year-old conjecture:

Every integer Heronian triangle is congruent to a lattice triangle.

A triangle is said to be an integer Heronian triangle, if it has integer side lengths and integer area. A triangle is said to be a lattice triangle, if all its vertices have integer coordinates.

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