The Euler-Lagrange equation and the principle of least action

In classical, non-relativistic particle mechanics the action of a system is given by
$$ S[\vec{q}(t)] = \int_{t_1}^{t_2} L(\vec{q}, \dot{\vec{q}}, t) ~ dt $$
which is a functional $S: F^n \rightarrow \mathbb{R}$, i.e. it maps the $n$ functions $\vec{q}(t) = (q_1(t),\dots,q_n(t)) \in F^n$ of some function space $F$ onto the real numbers. Continue reading “The Euler-Lagrange equation and the principle of least action”

Bell’s theorem is wrong.

I revisit the orignal proof of Bell’s theorem and give a thorough discussion about the assumptions that have been made. This shows that Bell’s inequality is violated in general not only by quantum systems but also by simple cassical systems with correlation. To get a grasp on that topic I will start with the general notion of entanglement.

Entanglement isn’t about interaction or information transfer betweeen entangled particles.

Consider spin-entaglement of two spin-$\frac{1}{2}$ particles.
Let them be in a singulet state relative to an arbitrary axis (say z-axis):

$$ |\Psi \rangle = \frac{1}{\sqrt{2}} (\ |\uparrow_z, \downarrow_z \rangle – |\downarrow_z,\uparrow_z\rangle \ ) $$ Continue reading “Bell’s theorem is wrong.”

What are covariant and contravariant coordinates?

Usually when people first study special relativity or general relativity, they will get introduced to the concept of covariant and contravariant coordinates. They will learn that these two sets of coordinates are connected through the metric tensor $g_{ij}$ and that these coordinates are tensor themselves. As always in physics, this notion and the way to talk about it is obscured and the connection to a proper mathematical treatment is left out. This post is a short introdcution into covariant and contravariant coordinates in terms of linear algebra. Continue reading “What are covariant and contravariant coordinates?”