30 | July | 2024 | Physics Town

A First Course in String Theory

Residues

The behavior for non-positive $\displaystyle{z}$ is more intricate. Euler’s integral does not converge for $\displaystyle{\Re (z)\leq 0}$ , but the function it defines in the positive complex half-plane has a unique analytic continuation to the negative half-plane. One way to find that analytic continuation is to use Euler’s integral for positive arguments and extend the domain to negative numbers by repeated application of the recurrence formula,

$\displaystyle{\Gamma (z)={\frac {\Gamma (z+n+1)}{z(z+1)\cdots (z+n)}}}$ ,

choosing $\displaystyle{n}$ such that $\displaystyle{z+n}$ is positive. The product in the denominator is zero when $\displaystyle{z}$ equals any of the integers $\displaystyle{0,-1,-2,\ldots}$ . Thus, the gamma function must be undefined at those points to avoid division by zero; it is a meromorphic function with simple poles at the non-positive integers.

— Wikipedia on Gamma function

2024.07.30 Tuesday ACHK

An important and unique property of quantum key distribution is the ability of the two communicating users to detect the presence of any third party trying to gain knowledge of the key. This results from a fundamental aspect of quantum mechanics: the process of measuring a quantum system in general disturbs the system. A third party trying to eavesdrop on the key must in some way measure it, thus introducing detectable anomalies.

— Wikipedia on Quantum key distribution

The common language of quantum mechanics is convenient but not accurate:

Eavesdropping would cause the collapse of the wave function, so Alice and Bob must be aware of it.

The accurate language:

A wave function encodes the probability distribution of various possible experimental outcomes of a system. In other words, the wave function is a property of the system (the experimental setup), encompassing the experimental operations, including measurements.

To eavesdrop, Eve has to add an extra detector to the system. Thus, the system is altered (replaced). So the probability distribution is no longer that of the original system. That is the meaning of “collapse of the wave function”.

— Me@2024-06-19 02:17:35 PM

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continues

Daily Archives: July 30, 2024

3.6 Analytic continuation for gamma function, 6

Quantum encryption