Gregor Wentzel and the path integral

Provoked by the previous blog post about the path integrals, Frank S. sent me two fascinating 1924 papers by Gregor Wentzel which "almost" defined the Feynman's path integral definition of quantum mechanics but two decades earlier (and a year before Heisenberg started quantum mechanics as we know it). If you find English somewhat more comprehensible than German, like I do, you may read a 1998 review of these papers by Salvatore Antoci and Dierck-E. Liebscher

Wentzel's Path Integrals (PDF may be downloaded there)

which probably contains more than what we will actually ever need, unless we plan to write a detailed book about the history of science. First, Wentzel may sound like a forgotten name. But on this blog, you find it in two previous texts. He had something to do with dispersion and Kramers etc. After all, Wentzel's name starts with a W, like WKB does, and indeed, he is also the first guy remembered in the WKB acronym. But how many people know what (J)WKB stands for now?

Just a Czech comment. Wentzel sounds very German because it's spelled in a German way. But it's just a Germanization of the Czech, Slavic name Václav. Why? Because the original form of the name, used some 1000 years ago, was Věnceslav (like Víceslav) which really means "more fame" or "more famous".

Ironically enough, "Boleslav" had the same meaning ("bole" is still a Russian word for "more"). Boleslav was the brother who murdered saint Václav. They were brothers but their first names had the same meaning. English adopted the old Czech "Věnceslav" as its (Good King) Wenceslaus, German shortened it to Wenzl or Wentzel, and modern Czech shortened it to Václav (or, colloquially, Vašek or Venca, the latter is probably a Bohemization of the German name Wentzel; note that Venclovský was the first Czech who swam over the English Channel while David Vencl – the surname is just a Bohemian respelled Wentzel – recently improved the world record in swimming under the ice to 81 meters, a great Czech dude)...

Feynman's path integral & superpositions of histories

What happens in between is "everything", Feynman diagrams are another basis of the "space of histories"

In January, I was asked on Quora What happens in between the initial and final states in quantum mechanics. Well, everything happens and nothing happens or it's meaningless to say what happens. A basic principle of quantum mechanics is that the results of the measurements of observables \(L\), i.e. statements \(L=\lambda_i\) obtained by measurements, are the only facts. Quantum mechanics is a set of rules to predict or derive or explain some facts (or their high probability) from other facts. A theory in classical physics could be given this description but also a "more specific one", a classical model of something, but the more specific one has been impossible after the discovery of quantum mechanics.

Feynman's path integral approach to quantum mechanics is a very explicit way to see "in what sense" everything happens. Assuming some initial and final states (in a position-like basis, to simplify the questions about the bases of initial and final Hilbert spaces), the probability amplitude for a transition (i.e. the complex number that determines the matrix element of a unitary evolution matrix) may be calculated as\[ {\mathcal A}_{{\rm initial}\to {\rm final}} = \int{\mathcal D}x(t)\,\exp(iS/\hbar) \] where the infinite-dimensional integral goes over all trajectories \(x(t)\) connecting the points \(x(t_{\rm initial}) = x_{\rm initial}\) and \(x(t_{\rm final}) = x_{\rm final}\). If you consider generic initial wave functions, you must pre-add two additional integrals over \(x_{\rm initial}\) and \(x_{\rm final}\), the weights \(\psi_{\rm initial}(x_{\rm initial})\) and \(\psi_{\rm final}(x_{\rm final})\), and then the integral becomes the integral over literally all functions \(x(t)\) defined on the interval of time between the initial and final time (be sure that while I admire \(\rm\TeX\), it's faster for me to write these mathematically heavy sentences in words than in mathematical symbols!).

Triangles in a row

My friend Prof Vlasta Dlab (Ottawa-Bzí) sent me his solution (which will appear in a Czech MathPhys Journal) to the following puzzle (well, its generalization from \(4\) to \(n\) triangles):



He was deeply dissatisfied with a solution of this puzzle on YouTube (posted in January 2021) because a part of that derivation is a blind application of a geometric formula (a sine formula for triangular areas) which doesn't really reveal the simplicity of the situation and its independence of angles, precise shapes... I completely agree with Vlasta. While the YouTube solution looks short, it uses an excessively heavy and irrelevant tool, and the person using these methods would probably waste too much extra time if he were supposed to solve a nearly isomorphic problem.

A good mathematician should look what matters and what doesn't matter. And an important point about this calculation of areas is that the angles don't matter at all! Correspondingly, you shouldn't be using the Pythagorean theorem, a sine formula for triangular areas, Heron's formula, and similar tools that deal with angles and, almost equivalently, with ratios of lengths of line intervals that go in different directions. In a good, conceptual solution, the "square roots of three" and similar things shouldn't appear at all (while they appear in the YouTube video).

SJW awokening harms mathematics more than communism did

At Twitter, PartTimeRacer has informed me about the essay (at "Common Sense With Bari Weiss", a brand that hasn't built name recognition with me so far)

There Is No Such Thing as "White" Math

by Sergiu Klainerman, a Romania-born Jewish male mathematics professor at Princeton. (Here we do try to maximize the information that readers might find important, and given the sociological focus of these discussions, rightfully so.) I have mentioned his name four months ago.



Bucharest

Klainerman is introduced as someone who left the Soviet bloc and who could do wonderful things in the U.S., partly because mathematics is a great equalizer. The solid method to look for the right and (to eliminate) wrong answers is universal across cultures, the beef of mathematics doesn't depend on the white race or any other group of humans in any way. Well, Klainerman is still PC in the sense that he likes to "assume" that mathematics skills are universal across cultures etc.

Do dogs and cats depend on eating plants?

A better, more strategy-oriented, and more quantitative "life sciences" curriculum for kids should prevent people from becoming scientifically illiterate vegan or Luddite ideologues

Last night, aside from many other interesting things ;-), I "learned" from an amateur Prague-based public intellectual that it is great to feed dogs and cats with a balanced diet rich in fruits and vegetables. Also, it turned out that animals in Nature never fight and if an animal ever dies, it is only an animal that was going to die very soon, too. The predators are very decent and avoid any physical violence. It was entertaining to some extent but I just don't have the stomach to stomach similar things that come from an adult for hours (unlike cows that have four stomachs and can stomach everything, as we will see).



Two weeks ago, I argued that a proper and captivating education of history for kids should be filled with the basic capitalist financial literacy. Now, I think that that the kids' biology classes should be full of adventurous and emotionally relevant National-Geographic-style stories about animals, something that the kids become involved in, and something that should ultimately teach the logic of Darwin's theory, evolution, and natural selection to the kids.

In fact, these two proposals are completely analogous – in some sense, they are almost the same thing. Capitalism is a counterpart of the "struggle, competition, and cooperation between animals" that is brought to the world of investment and corporations. The reasons why capitalism and natural selection lead to improved well-being of the whole and progress are fundamentally the same.

Edwin's understanding of QM: Part I, physics vs mathematics

Today I have listened to a large number of interviews, especially on the brilliant Free Universum Radio hosted by the smart opera singer Ms Martina Kociánová. But I also saw a 10-day-old quantum mechanical video with Edwin Steiner.

Millions of Czechs will strongly violate the draconian restrictions

It's often said that 40% of Czechs totally oppose most of the non-cosmetic Covid-19 restrictions and my reading of the in-real-life interactions as well as the Internet discussions agrees with this estimate (yes, of course, I am inside this 40% group). And yes, of course I think that this natural support for "ordinary freedoms needed to live ordinary lives at the individual level" (something that even all the foreign overlords respected at almost all times) is also the main reason why Czechia leads the world in the number of "Covid-19 cases" per million people – and I am proud about it. The number of Covid-19 cases really is a great approximation for a moral virtue in a nation. After the announcements last night, the anger in the discussions is intense and omnipresent.

Even though the Czech Parliament has refused to extend even the mild state of emergency that has existed so far, the Czech government has decided to introduce a new one with much more draconian measures from Monday, March 1st. So the face masks are mandatory outside and on the workplace; cloth masks aren't enough; two face masks or a respirator are needed in the public transportation (and...) and the grocery stores, almost all other shops are ordered to be shut down (exceptions are scheduled to drop by 50%, flower shops remain the most surprising exception, they're claimed to be open to remember the Covid victims but only 1-2 persons die in a district each day and what about the remaining thousands of flowers?).

But the restrictions on the movement seem most shocking and they almost resemble the Australian insanities. Every movement to another district (Czechia is divided to 76 districts plus Prague that will act as one district, good for them) must be justified by some incredible paperwork, a receipt from the employer or the doctor or something like that. So any "casual" movement in between the districts (like that of the people who have a romantic relationship in the adjacent municipality; or even the visit to parents or kids who live elsewhere) is claimed by the government to be banned from Monday (millions of people actively say that this is an inhuman restriction). One can walk a dog but only 500 meters from his home, and only during the day. Very shockingly, people who will be located on their cottage on Sunday night must stay there (or at least in the same district) for three weeks (the cottage will legally become their home for these purposes)!

Prague's municipal court endorses a freshman, orders in-person learning to be restarted

Every court must have its opinions about what is legal and constitutional

As "citizen" has pointed out, the Municipal Court in Prague decided against a high school where teachers were getting a salary while not fulfilling their duty to teach a student in person (which is what teachers do at most schools not only in Czechia now, as you know). The freshman T.N. has sued the school and won. The essence of the verdict is the judges' opinion that the state of emergency that was started on February 15th (last Monday) is the same as the previous one and whose extension was vetoed by the lower chamber of the Parliament on February 10th.



Gymnázium Na Zatlance, the high school that lost a lawsuit and was ordered to restore the proper education

Because there is no legally valid state of emergency now, the school has no valid justification for not fulfilling its pedagogical obligations.

Now, the prime minister has used a sleight-of-hand by emphasizing that a "fundamentally new" state of emergency started on November 15th, after the regional governors asked the government to extend the state of emergency (a discussion with zero legal implications) despite the lawmakers' veto (according to the prime minister, the state of emergency was "new" and responding to "new conditions" but this is in no conflict with his assertion that nothing changed about the restrictions between February 14th and 15th). The prime minister wasn't too clear why the state of emergency with the same cause and the same justification would be "new". But he must have believed that simply by saying it was "new", the courts would consider it "new": maybe it is because you never enter the same river twice. To say the least, he wanted 10.7 million Czech citizens to believe a story like that and to behave as if the state of emergency had been lawfully extended. According to the law, the government has the right to declare a "new" state of emergency at any moment. The government has to inform the Parliament and the new state of emergency may be later terminated by the Parliament.

Yellen, Lagarde vs Bitcoin

Two weeks ago, the president of the European Central Bank (ECB) Christine Lagarde said that Bitcoin was unlikely to be held by central banks, it is used for funny business. At about the same time, Elon Musk added the word "Bitcoin" to his Twitter profile. The Bitcoin price added some 15% in hours as a result – as millions of dimwits interpreted the comment as an order to BUY BUY BUY.

In the process of merging the two most notorious one-trillion bubbles of the contemporary crazy financial world, the uncompetitive carmaker Tesla bought $1.5 billion worth of Bitcoin and said that it would accept it as compensation. The supporters of Tesla and Bitcoin bubbles are highly overlapping sets of hopeless idiots. Thinking positively about one of these bubbles is quite a sign of the person's stupidity – but celebrating the hybridization of these two bubbles (in which all the risks amplify each other and the absence of any beef is illuminated from both sides) is probably yet another level. Of course, Tesla hasn't ever produced any real profit from cars so the Tesla profit from the appreciation of BTC surpassed all the conceivable profits of the carmakers that have something to do with cars. ;-) Imagine how fudged up the people have to be if they see nothing wrong about such "money" and such "carmakers". Do you really prefer a world where "producers" actually make their billions by manipulating millions of simpletons by ordering them BUY BUY BUY instead of producing useful, competitive products?

If your IQ happens to be below 70 but you were capable of learning to read, anyway, let me remind you that the fair Bitcoin price has always been and still is $0.00. Among Bitcoin cultists, this number $0.00 isn't applied too accurately, they are paying about $50,000 (and the peak was $58,354 a day or two days ago). In early 2019, the price at which the cult was trading the $0.00 Bitcoin was $3,100 (a local low). The 16-fold increase in these 2 years basically means that the number of dimwits who have been persuaded that it is clever to invest into this bubble grew by a factor of 16. Each of them evaluates the Bitcoin price inaccurately and the inaccuracy per person grew from the $3,100 discrepancy to $50,000, by the same factor of 16. That's why we could say that the total amount of stupidity contained in the Bitcoin cult actually grew with the second power of 16, by a factor of 256. I think that the people in this cult do believe that by collecting a sufficient number of dimwits and inflating the bubble to a huge enough size, they change the truth and $0.00 will be equal to $50,000 or something (just like when Bill Gates says that 2+2=4 is just a racist invention and other answers may be made more correct when minority SJWs are fighting hard enough). Sorry, idiots, it won't. The Bitcoin is still worth $0.00 and you are just a larger pile of more hardcore morons than what you were before.

The Gates fund "math is racist" junk

The number of such shocking news is absolutely terrifying these days. But this one was particularly disappointing minutes ago:

Rantz: Bill and Melinda Gates Foundation bankrolls ‘math is racist’ lunacy

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...more at Google News...

Far left activists are spreading the "A Pathway to Equitable Math Instruction" for children who are 12-14 years old and this absolute garbage is being funded by Melinda and Bill Gates. You can find relatively recent blog posts (pre-Covid posts, however) where I was still expressing a hope that Bill Gates was far wiser than other billionaires who run similar charities and "charities". I've liked him as a "Microsoft fan" when I was an undergrad and my sentiments got even more favorable when I learned tht he coded the BASIC for my Commodore 64 which I have spent a lot of time with as a boy. But should these things be enough to give thumbs up to a man?

Kosovo PM elect wants Kosovo to be annexed by Albania

Kosovo was the cradle of the Serbian statehood. In the recent century or so, Albanians over-migrated to this administrative region of Serbia (which was a part of Serbia even in the times of Yugoslavia) and over-copulated. The migration and the higher natality rate made them 90% of the population by now – well, it's been the case for a decade or two.



In 1939, when Adolf Hitler turned Czechia into the Protectorate of the Third Reich and bragged that he would be the greatest German at Valhalla ever, Benito Mussolini felt like the inferior member of the Axis and invaded Albania which was turned into a Protectorate of Italy. Well, he still remained inferior because Albania has always been heavily inferior to Czechia but let's not be distracted by such details LOL. This Italian-controlled Albania was supposed to expand to new territories including Kosovo but Mussolini and Hitler lost the war, Hitler wisely commited suicide, and victorious Serbia took control of Kosovo again.

In the 1990s, I was rather neutral about the Yugoslav wars. I had mixed feelings about almost everything. It was partly because like millions of Czechs, I still found some uncritical attitude to the West natural and the West wanted to punish Serbia. Poor Milošević – who was really nothing else than a Serbian clone of Joe Biden or something socialist like that – was promoted as the ultimate Devil. But I felt something really terrible was taking place. These days, and for a decade, I have no doubts that the Serbia were (and are) basically the good guys and the Kosovo separatists were the worst ones among the Serbia's foes. They were also responsible for drug sales, organ trafficking and other things.

Irreversibility of measurements is straightforward

For some reasons, the recent 24 hours brought an explosion of the confused talk about the irreversibility of quantum mechanics (and especially the measurements in it).

In this experiment, artist/director Ondrej Zunka give us a glimpse of a bifurcating canine, exploring the many-worlds interpretation of quantum mechanics. Source: https://t.co/pY0BlljQKi pic.twitter.com/f0Nhlj6lEA

— Cliff Pickover (@pickover) February 16, 2021

Well, people have posted lots of other crazy stuff about quantum mechanics. For example, the dog with many phase-shifted legs above, one created by the Czech 3D artist in London, Mr Ondřej Zunka, is surely totally cool. But the idea that this animation has anything to do with the way how our quantum mechanical Universe works is dumb beyond imagination. Quantum mechanics doesn't add many new legs to the dogs. Quantum mechanics just makes the uncertainty about the existing legs more unavoidable and adds some new causal relationships between the properties and their uncertainties.

American patriots, Second Amendment fans should celebrate the purchase of Colt by CZG

First, read some news about the purchase.

I must start with the proud disclaimer that I hold some stocks in CZG, the Czech Armory (Česká zbrojovka Group), a position that I doubled on Thursday.



Here is a monologue by a man who thinks that the operation (which was officially announced as completed on Thursday night, some stamps from regulators are waiting) is good news. So first, Colt is a legend, especially in Czechia.

Czech state of emergency should be over from Monday

On Thursday, I spent at least 5 hours by listening to the Czech Parliament (the lower house) which discussed the government request to extend the state of emergency again (the current one has been in place since late September). The state of emergency isn't quite the same thing as a lockdown; but they are closely related because the state of emergency is a legal framework that allows the government to do potentially harmful things that it simply cannot do otherwise, at least not without very special ad hoc justifications. They include the closure of all non-essential shops in an area. Our non-essential shops are closed now (although they were opened in December for weeks) and even some supermarkets are not allowed to sell non-essential goods (I couldn't clearly see the difference, however, the rules must be confusing).

Communists whose votes are needed to keep the minority government (coalition of ANO i.e. the billionaire PM Babiš and the social democrats) in power already had doubts about the extensions of the state of emergency in the past. They often succeeded in reducing the duration of the extension but it didn't matter because the state of emergency was later extended again, anyway. Now they helped to cancel it which finally could matter. Only ANO and social democrats want to continue. Other lawmakers have adopted the prevailing view of the Czech public that it is just too heavy a weapon that doesn't really bring much of an improvement in anything. The government hasn't been able to refine its policies so that the people would feel that the policies help more than they harm.

PandaX-II seems to slightly strengthen XENON1T's hints of dark matter particles

The recent Chinese 3D remake of the legendary cute Czech Little Mole features a new, bigger friend of the Central and Eastern Europe, I mean of the little mole, namely the People's Republic of China, I mean a big fat panda. ;-) This was the first episode among many.

This blog post is very brief because I don't think it's a good investment of time to write full-blown texts. But as you know, the hints of any new particle behind the Standard Model are extremely weak or almost non-existent.

Last June, Elena Aprile and her XENON1T folks (yes, one ton of xenon in the detector) saw some

Excess Electronic Recoil Events in XENON1T

(not to be confused with nuclear recoil events that they also like to observe) which represented the excess around 3.0-3.5 sigma above the background.