The Dirac equation (3): heuristic derivation

In Section 3 we introduced the Klein-Gordon equation \partial_t^2 \psi - \Delta \psi + m^2 \psi = 0 (formula (3.2)). Although it is a fundamental equation in relativistic quantum mechanics, it does not really fit the quantum world framework. The wave function  \psi(t) does not completely describe the state of a particle at time t. This is because the Klein-Gordon equation is a second order differential equation in time variable (unlike the Schrödinger equation). In this post we will discuss a way to take a ‘square-root’ of the Klein-Gordon equation.

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Numerical computation of the distribution of the supremum functional

Let X_t be a Lévy process with Lévy-Khintchin exponent \psi(\xi^2) for a complete Bernstein function \psi (some examples below). Let M_t = \sup_{s \in [0, t]} X_s be the supremum functional of X_t. Given some growth condition on X_t, a formula for \mathbf{P}(M_t < x) was given in my recent preprint written with Jacek Małecki and Michał Ryznar. (The same one where the \pi/2 conjecture is solved). In this blog post I would like to discuss a numerical scheme to calculate \mathbf{P}(M_t < x), which is based on this formula.

It is perhaps worth noting that if \tau_x denotes the smallest t such that X_t > x, then \mathbf{P}(\tau_x > t) = \mathbf{P}(M_t < x), so equally well this note could have a title Numerical computation of the distribution of the first passage time.

This post is not about the supremum functional or first passage times. It is about numerical approximation to a rather complicated object. No knowledge of Lévy processes, complete Bernstein functions and other theoretical objects is required to follow this post.

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π/2 conjecture solved

A few posts ago I stated the pi-over-two concecture. Recently, my friend and collegue Jacek Małecki proved it in full generality. (He did it while I was on my vacation in November — sometimes it is good to take a few weeks off!) This is one of the results of our joint preprint with Michał Ryznar, available here. The solution is quite elementary, but is somehow hidden in the rather technical paper. For this reason, I would like to describe it briefly in this note.

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The Dirac equation (2): heuristic derivation and basic properties of the Klein-Gordon equation

For more than a century now, we know that light presents some properties typical to particles (for example, in the photoelectric effect, explained in 1905 by Albert Einstein), and also matter can behave as waves (which was first conjectured by Louis de Broglie in 1924, and then observed for example in electron diffraction experiments). This phenomenon is known as wave-particle duality, and it suggests that the equations of motion for matter should at least resemble Maxwell’s equations for light. This is a starting point for our first approximation to relativistic quantum dynamics.  Continue reading

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Job in a city park

Since October 2010 I work in the Wrocław branch of the Institute of Mathematics of the Polish Academy of Sciences, where I have a two-year position. The institute is located in the heart of Szczytnicki Park, one of the largest city parks in Wrocław. Riding a bike through this magnificent area every morning and every afternoon always makes me smile. Just take a look at Google images results, and you will understand why. Let me share with you a few pictures of the institute. I took them in October.

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When a snowy weather comes back to Wrocław, I will upload some winter pictures as well.

Links to images: ***********

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The Dirac equation (1): introduction

Although it took me much more time than I expected, the introductory part of the notes on the Dirac equation is ready. I welcome all comments.

In order to properly understand the Dirac equation, one needs some background on the Lorentz transformation. In this post, we also discuss briefly some aspects of Maxwell’s equations, which will become important later when we couple the Dirac particle with electromagnetic field.

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Mathematica and hypergeometric functions

Recently I did some numerical experiments in Mathematica involving the hypergeometric function _2F_1. The results were clearly wrong (a positive-definite matrix having negative eigenvalues, for example), so I spent a couple of hours checking the code. Finally, I discovered a bug, but it was not in my program!

This post is not about the hypergeometric function, it is enough to know that it accepts four arguments, usually typed as _2F_1(a,b;c;x). For some parameters a, b, c it reduces to elementary functions, for example _2F_1(\frac{1}{2},\frac{1}{2};\frac{3}{2};-x) = \mathrm{arsinh}(\sqrt{x}) / \sqrt{x}. Of course, Wolfram Research products are aware of that (link). A sligth modification of the parameters of the hypergeometric function does not change the value of the function significantly, it is smooth in all four arguments. But according to Wolfram Alpha, _2F_1(\frac{1}{2},\frac{1}{2}-\frac{1}{1000};\frac{3}{2}-\frac{1}{1000};-x) is not even continuous! Click on an image to see the plot at the Wolfram Alpha site.

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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
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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