A question having the answer "Nobody has a clue really". 🤣
https://www.bbc.co.uk/videos/cp8yyj5j2j8o
https://www.bbc.co.uk/videos/cp8yyj5j2j8o
I have decided to follow the mighty Dirac's ideas about the Symbolic Method being superior to the method of Coordinates and Representations embodied in Schrodinger's Wave Mechanics and Heisenberg's Matrix Mechanics.
I am unsure which camp the Segal-Bargman space falls into. But surely, after Dirac, there is little more to be said?
As far as I have got so far. 312 pages to go. Somebody has to do the heavy lifting. 😎
That is at least the third time you have copied that picture into this thread!
The answer to the post above that is that "dark matter" is a famous Irish drink
which is universally applauded.
The answer to the post above that is that "dark matter" is a famous Irish drink
which is universally applauded.
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I looked into what lies behind this "symbolic method": https://pressbooks.pub/simplydirac/chapter/quantum-wizard/
Here are some edited extracts:
The version of quantum mechanics that Dirac developed in 1926 was known as “q-number algebra”.
q-numbers have different mathematical properties to ordinary numbers and don't obey the law of commutation (3 × 2 is equal to 2 × 3).
When Dirac established his algebraic theory, he thought in terms of pictures and diagrams. “I prefer the relationships which I can visualize in geometric terms” he noted.
His famous textbook, 'The Principles of Quantum Mechanics', is based on the symbolic method.
Just be thankful it is not the Tesseract!
Note that 'dark matter' is best consumed from a Feynman pint glass:
I was in my habitual place at the Southsea Theatre of the Seas tonight, and fell into conversation with an intelligent young woman who was brave enough to go there alone at night to take pictures on her 'phone of the Planets and Moon. She studies "Humanities" at Portsmouth University, which I had a vague notion is everything except useful Science and Mathematics:
There we finally spotted the winged messenger Mercury below Venus in the haze and just above the clouds:
NO WAY was it as bright as Sirius, which we could see, so I don't know who made that magnitude -1.5 stuff up.
We moved onto Mars, and she alertly also noticed Castor and Pollux in Gemini the Twins forming what I identified as an isosceles triangle:
Good little scientific brain lurking in that head of hers! We had to part, and it was such sweet sorrow, because. and there is no nice way to say this, women with brains are a rarity in Portsmouth.
I had to attempt an effort at the muddy bandstand, and it could have been better, I shone a torch on the top of it and got this with some computer trickery:
But I think I got the exposure wrong. Oh well,. I am fed up with going out in the cold, and ricking my back clambering about muddy slopes. Quoth the Raven: Nevermore! 🙁
There we finally spotted the winged messenger Mercury below Venus in the haze and just above the clouds:
NO WAY was it as bright as Sirius, which we could see, so I don't know who made that magnitude -1.5 stuff up.
We moved onto Mars, and she alertly also noticed Castor and Pollux in Gemini the Twins forming what I identified as an isosceles triangle:
Good little scientific brain lurking in that head of hers! We had to part, and it was such sweet sorrow, because. and there is no nice way to say this, women with brains are a rarity in Portsmouth.
I had to attempt an effort at the muddy bandstand, and it could have been better, I shone a torch on the top of it and got this with some computer trickery:
But I think I got the exposure wrong. Oh well,. I am fed up with going out in the cold, and ricking my back clambering about muddy slopes. Quoth the Raven: Nevermore! 🙁
Now you will have more time to read Dirac's "Principles" book and explain the intricacies of "q-number algebra" to us!
P.S. One would be forgiven for thinking that the "q" stands for "quantum", but Dirac himself said it stood for "queer".
P.P.S. Good luck with the mathematics, which does start off looking very "queer" to me...
Any q-number can be expressed in the form:
...and then gets complicated!
I am always keen to learn more about Dirac. The purest soul in Physics.
https://physicsworld.com/a/paul-dirac-the-purest-soul-in-physics/
Here photographed with Werner Heisenberg in Chicago in 1929:
His first paper was something essentially linking algebraic numbers with transcendentals in Physics I believe. An interesting topic of comparing the order of infinities.
In his later years he was fascinated with measuring the position of the planets to mere metres, believing there was something odd with time, maybe two dimensions.
https://www.americanscientist.org/article/an-interlude-with-dirac
He had a very practical engineering education, a good draughtsman, and was thus quite geometrical in approach.
Feynman, by contrast was later interested in the Bethe Ansatz, being a way to calculate larger assemblies of quantum particles, which was a weakness in Q.E.D.
It is my back that is complaining this week. Next week it may be a bellyache, after that a cough. This is about old age, one fears. No cure from Doctors. But I have an astronomical plan for the Spring:
There is my pet hate, Corona Borealis with no nova at all, but hark, what is that extraordinary cluster of Galaxies in Virgo? I think I know.
https://theskylive.com/
Every day a school day. 🙂
https://physicsworld.com/a/paul-dirac-the-purest-soul-in-physics/
Here photographed with Werner Heisenberg in Chicago in 1929:
His first paper was something essentially linking algebraic numbers with transcendentals in Physics I believe. An interesting topic of comparing the order of infinities.
In his later years he was fascinated with measuring the position of the planets to mere metres, believing there was something odd with time, maybe two dimensions.
https://www.americanscientist.org/article/an-interlude-with-dirac
He had a very practical engineering education, a good draughtsman, and was thus quite geometrical in approach.
Feynman, by contrast was later interested in the Bethe Ansatz, being a way to calculate larger assemblies of quantum particles, which was a weakness in Q.E.D.
It is my back that is complaining this week. Next week it may be a bellyache, after that a cough. This is about old age, one fears. No cure from Doctors. But I have an astronomical plan for the Spring:
There is my pet hate, Corona Borealis with no nova at all, but hark, what is that extraordinary cluster of Galaxies in Virgo? I think I know.
https://theskylive.com/
Every day a school day. 🙂
I read that the Virgo Cluster is the farthest 'object' to have a physical connection with our galaxy.
Image centered on our galaxy
https://lovethenightsky.com/virgo-galaxy-cluster-complete-guide/
Image centered on our galaxy
https://lovethenightsky.com/virgo-galaxy-cluster-complete-guide/
I suspect these shall be faint objects well beyond my capabilities! A disappointment, just like Dirac's monopoles....
Just got a shot of Corona and the head of the Serpent, in a murky sky, being nagged by the thought it could go off any second, and it is my chance of everlasting fame:
BUSTED again! I shall struggle off to bed, hoping for no backaches, perchance to dream. 😴
Just got a shot of Corona and the head of the Serpent, in a murky sky, being nagged by the thought it could go off any second, and it is my chance of everlasting fame:
BUSTED again! I shall struggle off to bed, hoping for no backaches, perchance to dream. 😴
Amongst much busyness today, I have installed this fetching picture as my wallpaper:
You can see that I am very busy indeed, with many ideas on the go awaiting the correct picture!
To wit, I have been considering P.A.M. Dirac's "c" and "q" numbers. I do not find them in "The Principles of Quantum Mechanics" so suppose he is merely dividing his numbers into two distinct sets for convenience.
This is the sort of thing Mathematicians enjoy doing a lot. Consider the Set of Algebraic Numbers, being those that can be easily calculated with polynomial methods and whatnot This covers nearly every number you can think of. Worryingly, there is an infinitely larger Set called the "Transcendental Numbers" which are rarely seen but familiar like "e" and "Pi" as in Euler's Formula, which has assumed "Treasure" status in complex analysis:
David Hilbert, who could have claimed General Relativity as his own, wanted to know more about Transcendental Numbers, and set this as his 7th. important problem for Mathematicians to solve:
I hadn't thought of it as a familiar Isosceles Triangle problem, as with Mars, Castor and Pollux the other night, but I am not a genius.
Long story short, the Gelfond-Schneider Theorem nailed it in 1934. And it usefully led to much interesting further transcendental numbers:
This I read about in Portsmouth University library, just after Wiles had solved the Fermat problem with interesting new consequences.
I expect I am right in thinking that "i" is algebraic, after all, i^2 = -1, but had never considered i^i !
To give an Astronomical Analogy of all this, I find this statement very beautiful:
And so should you. 🙂
You can see that I am very busy indeed, with many ideas on the go awaiting the correct picture!
To wit, I have been considering P.A.M. Dirac's "c" and "q" numbers. I do not find them in "The Principles of Quantum Mechanics" so suppose he is merely dividing his numbers into two distinct sets for convenience.
This is the sort of thing Mathematicians enjoy doing a lot. Consider the Set of Algebraic Numbers, being those that can be easily calculated with polynomial methods and whatnot This covers nearly every number you can think of. Worryingly, there is an infinitely larger Set called the "Transcendental Numbers" which are rarely seen but familiar like "e" and "Pi" as in Euler's Formula, which has assumed "Treasure" status in complex analysis:
David Hilbert, who could have claimed General Relativity as his own, wanted to know more about Transcendental Numbers, and set this as his 7th. important problem for Mathematicians to solve:
I hadn't thought of it as a familiar Isosceles Triangle problem, as with Mars, Castor and Pollux the other night, but I am not a genius.
Long story short, the Gelfond-Schneider Theorem nailed it in 1934. And it usefully led to much interesting further transcendental numbers:
This I read about in Portsmouth University library, just after Wiles had solved the Fermat problem with interesting new consequences.
I expect I am right in thinking that "i" is algebraic, after all, i^2 = -1, but had never considered i^i !
To give an Astronomical Analogy of all this, I find this statement very beautiful:
And so should you. 🙂
After such mysterious mathematics, I'm in need of some transcendental meditation.
I shall meditate on the stars in the firmament above. And so should you.
I have meditated upon the matter. With mixed results in MY PATH TO ENLIGHTENMENT... 🙁
The Mathematical Stars of the Algebraic Numbers seem VERY DIFFERENT from ALL THE HEAVENLY GLORY! 🤔
https://en.wikipedia.org/wiki/Algebraic_number
I decided, therefore to meditate upon "c" and "q" numbers in Quantum Mechanics, and this is very ancient wisdom around 1926.
The shortest path is usually the quickest, so I, lazily, eventually just skipped to the lengthy conclusion of your massive tome:
https://publishing.cdlib.org/ucpres...560&toc.depth=1&toc.id=d0e19422&brand=ucpress
Nope. Not a clue what that is about! And nor do you, I suspect! 🤣
Ah, The Crop! By their fruits shall we know them.
Now this seems familiar. Just add Lorentz Covariance to the Time Independent Schrodinger equation, which I helpfully elucidated earlier. And we have the Time Dependent Schrodinger Equation!
And there we must leave it, until next time. Things to do, people to meet, and all that. 😎
The Mathematical Stars of the Algebraic Numbers seem VERY DIFFERENT from ALL THE HEAVENLY GLORY! 🤔
https://en.wikipedia.org/wiki/Algebraic_number
I decided, therefore to meditate upon "c" and "q" numbers in Quantum Mechanics, and this is very ancient wisdom around 1926.
The shortest path is usually the quickest, so I, lazily, eventually just skipped to the lengthy conclusion of your massive tome:
https://publishing.cdlib.org/ucpres...560&toc.depth=1&toc.id=d0e19422&brand=ucpress
Nope. Not a clue what that is about! And nor do you, I suspect! 🤣
Ah, The Crop! By their fruits shall we know them.
Now this seems familiar. Just add Lorentz Covariance to the Time Independent Schrodinger equation, which I helpfully elucidated earlier. And we have the Time Dependent Schrodinger Equation!
And there we must leave it, until next time. Things to do, people to meet, and all that. 😎
It's enough for me to know that Dirac showed both Heisenberg's matrix mechanics and Schrödinger's wave mechanics to be inherent in his algebraic interpretation of quantum mechanics.
I shall simply ignore those dratted q-numbers as they do no more than give me a queer feeling!
It must be said that all three physicists were the Cream of the Crop!
Refreshed by a post-lunch nap, ALL FALLS INTO PLACE! 🙂
This job of understanding Quantum Mechanics for the ordinary person might seem a Sisyphean Task:
But I think it is simply one step after another.
Now we can do the Klein-Gordon equation:
Looks simple enough. Relativity, isn't it?
Feynman used that one with his 5 transformation matrices to do his Chirality stuff.
But the Dirac Equation is considered the "Seed of Modern Quantum Mechanics":
Something to do with making it First Order too, which was considered "Dirac's Coup".
Finding the square root of the simple differential operator, which is a bit weird if you ask me.
What does this MEAN? Apparently the method extends into Antimatter, Quantum Chromodynamics and the Higgs Particle.
The honest view amongst modern teachers of this stuff, is there are better books to learn this from than Dirac's one.
After learning it, you can read Dirac and see the brilliance. At which point, we mere toilers in the vinyard, can, for now, consider the matter closed. 😎
This job of understanding Quantum Mechanics for the ordinary person might seem a Sisyphean Task:
But I think it is simply one step after another.
Now we can do the Klein-Gordon equation:
Looks simple enough. Relativity, isn't it?
Feynman used that one with his 5 transformation matrices to do his Chirality stuff.
But the Dirac Equation is considered the "Seed of Modern Quantum Mechanics":
Something to do with making it First Order too, which was considered "Dirac's Coup".
Finding the square root of the simple differential operator, which is a bit weird if you ask me.
What does this MEAN? Apparently the method extends into Antimatter, Quantum Chromodynamics and the Higgs Particle.
The honest view amongst modern teachers of this stuff, is there are better books to learn this from than Dirac's one.
After learning it, you can read Dirac and see the brilliance. At which point, we mere toilers in the vinyard, can, for now, consider the matter closed. 😎
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Your expanding collection of incomprehensible extracts certainly smacks of eternal punishment!
I thought I summed it all up as simply as I could. After all, Einstein, on his arrival in New York, was asked to explain Relativity for the readers of a local American newspaper in one paragraph. A scoop, like.
"No," he said, "It would take 3 days!"
The modern Sisyphus:
Does this explain what generates gravity? 🙂
"No," he said, "It would take 3 days!"
The modern Sisyphus:
Does this explain what generates gravity? 🙂
Private Moon landings are suddenly the latest thing!
Firefly Aerospace's Blue Ghost lander, which touched down on March 2, has sent back this picture of its own shadow stretching across the lunar surface and pointing towards a distant planet Earth in the background:
Meanwhile, the private iSpace Resilience Moon lander has a target touchdown date of June 5 and has sent back this image of our Blue Marble:
Firefly Aerospace's Blue Ghost lander, which touched down on March 2, has sent back this picture of its own shadow stretching across the lunar surface and pointing towards a distant planet Earth in the background:
Meanwhile, the private iSpace Resilience Moon lander has a target touchdown date of June 5 and has sent back this image of our Blue Marble:
And there's another! (which I'll add separately here as I may have run out of my image size allowance in the post above.)
The private Intuitive Machines' Athena lander has reached lunar orbit ahead of a touchdown on March 6.
It took this selfie with the Earth in the background while on its way to the Moon:
This is actually Intuitive Machines' second lunar lander. You may remember that the first one tipped over 30 degrees on landing back in February 2024, although the mission was deemed a success.
The private Intuitive Machines' Athena lander has reached lunar orbit ahead of a touchdown on March 6.
It took this selfie with the Earth in the background while on its way to the Moon:
This is actually Intuitive Machines' second lunar lander. You may remember that the first one tipped over 30 degrees on landing back in February 2024, although the mission was deemed a success.
Call me world-weary, but I find very little interest in the Moon whatsoever.
It is a grey dreary place consisting of dark grey rock and irritating grey dust that gets in your lungs and everywhere, and almost nothing else.
I might find some morbid fascination in watching Moon rockets explode on the launch pad, but can think of no use for the Moon at all except as a base for huge and useful Space telescopes, where its lack of atmosphere becomes an asset.
This is why the public lost interest in the whole thing in the 1970s.
It is a grey dreary place consisting of dark grey rock and irritating grey dust that gets in your lungs and everywhere, and almost nothing else.
I might find some morbid fascination in watching Moon rockets explode on the launch pad, but can think of no use for the Moon at all except as a base for huge and useful Space telescopes, where its lack of atmosphere becomes an asset.
This is why the public lost interest in the whole thing in the 1970s.
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