Noether’s theorem

In connection with classical mechanics and its formulation in terms of a least action principle there exists a remarkable theorem, the so called Noether theorem. It states:

For every continuous symmetry of a physical system, there exists a conserved quantity.

The reverse of this statement is also true as we will see later.

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.”