Ah, I see, thanks Tullio. Sounds like Testa Grigia is a great place to “catch some rays� :)
But I was wondering more from a theoretical standpoint: if there are only three generations of matter, might that mean that just more copious jets of the same particles in the Standard Model will be formed despite using ever higher accelerator energies? Is there not expected to be a limit, a point of equilibrium (as in 'quark soup'?), where new particles are being formed at the same rate as other particles are decaying? (With the result that adding more energy just shifts the point of equilibrium towards producing more of the same [Standard Model] particles...?)
I always thought that the real symmetry group of elementary particles is not SU(3) but SU(4). Take a four dimensional Riemaniann manifold with pseudo-euclidean metric and calculate its tangent space at a fixed point. You must differentiate any of a set of infinitely differentiable functions on it. The differential operators form a Lie algebra. Is it the algebra of SU(4)? I've been unable to prove this.
Tullio
I always thought that the real symmetry group of elementary particles is not SU(3) but SU(4).
I recall the post you made a while back about E8 and I've been trying to learn all the things required for understanding the nature of that, and also the well known SU(3)xSU(2)xU(1), and I admit I still have a ways to go.
What motivated me to ask about the three generations being a limit was actually studying the general equation for thrust from a rocket (because I didn't understand why the flow should ever become “choked� regardless of adding more pressure). I learned that some things can't be caused (or made to happen) simply by increasing the force, and that's when it dawned on me that it might not be possible to create a lepton heavier than a tau, or a quark heavier than the top. I guess I should have said “fundamental standpoint� instead of 'theoretical'...?
It's exciting to think that the LHC might produce evidence for exotic particles and hence also for one of the theories that go beyond the Standard Model. And I guess it makes sense that these (or at least the Higgs) would be observed before seeing a fourth generation of matter. Probably all related to how the gap in the masses arises? Also begs the question of how powerful an accelerator would be required to produce a 4th generation particle :)
Quote:
Take a four dimensional Riemannian manifold with pseudo-euclidean metric and calculate its tangent space at a fixed point. You must differentiate any of a set of infinitely differentiable functions on it. The differential operators form a Lie algebra. Is it the algebra of SU(4)? I've been unable to prove this.
Well, I've set a personal record for the number of open tabs in my browser trying to fully understand the question, and I don't yet. The task is to show that their respective Lie groups are isomorphic ?
Incidentally, I happened to stumble across this: 3 Generations: E6, E7, E8
Seems straightforward enough and I wish I knew enough to evaluate it, but I don't – yet. :)
I always thought that the real symmetry group of elementary particles is not SU(3) but SU(4).
I recall the post you made a while back about E8 and I've been trying to learn all the things required for understanding the nature of that, and also the well known SU(3)xSU(2)xU(1), and I admit I still have a ways to go.
What motivated me to ask about the three generations being a limit was actually studying the general equation for thrust from a rocket (because I didn't understand why the flow should ever become “choked� regardless of adding more pressure). I learned that some things can't be caused (or made to happen) simply by increasing the force, and that's when it dawned on me that it might not be possible to create a lepton heavier than a tau, or a quark heavier than the top. I guess I should have said “fundamental standpoint� instead of 'theoretical'...?
It's exciting to think that the LHC might produce evidence for exotic particles and hence also for one of the theories that go beyond the Standard Model. And I guess it makes sense that these (or at least the Higgs) would be observed before seeing a fourth generation of matter. Probably all related to how the gap in the masses arises? Also begs the question of how powerful an accelerator would be required to produce a 4th generation particle :)
Quote:
Take a four dimensional Riemannian manifold with pseudo-euclidean metric and calculate its tangent space at a fixed point. You must differentiate any of a set of infinitely differentiable functions on it. The differential operators form a Lie algebra. Is it the algebra of SU(4)? I've been unable to prove this.
Well, I've set a personal record for the number of open tabs in my browser trying to fully understand the question, and I don't yet. The task is to show that their respective Lie groups are isomorphic ?
Incidentally, I happened to stumble across this: 3 Generations: E6, E7, E8
Seems straightforward enough and I wish I knew enough to evaluate it, but I don't – yet. :)
I must admit that I dabbled with Lie groups and algebras many years ago while working toward my thesis. I even published a paper in "Il Nuovo Cimento" in 1967 whose title was "SU(n,1)representation for the harmonic oscillator". My coauthor was my thesis advisor, Giordano Bisiacchi, a young and brilliant theoretical physicist from Trieste University and International Center for Theoretical Physics (now called Abdus Salam Center, for the Nobel Prize winner Pakistani physicist). Then Giordano died in 1972 in a car crash. Even physicists die. But, really, what was published was only the result of the first half of my thesis, using commutators of the kind ab - ba, suitable for bosons. A second part, using anticommutators of the kind ab + ba (where a and b are Lie algebra operators) and suitable for fermions was never published and never will be. But in 1996, while surfing the Internet on a Pentium I, I discovered that an American physicist, Mike Guidry of the University of Tennessee had used the same algebras to describe a very important fermion system, the nucleus of uranium. His article, visible on Internet, is called "Fermion Lie Algebras and Microscopic Theories of Nuclear Matter". I contacted him and he asked me for a translated copy of my thesis, which I sent him together with the Nuovo Cimento article. Then my Internet connection fell and he sent an answer to my brother Franco, a mathematician at Trieste University (I was then in Milano), whose email address I had given Guidry as an alternative. My brother received an answer and, since I had no longer an Internet connection, I asked Franco to send me a hardcopy of the Guidry mail. But, being a mathematician, he was unable to use a printer and this ended the matter. Had Giordano not died in his Alfa Romeo Giulietta Sprint we might have preceded Guidry by about thirty years. Such is life. Cheers.
Tullio
That may not be so bad. Remember, Neither Regan, or either Bush were lawyers, nor, if I recall correctly was Blair or Thatcher, and look at the fine messes they have put the world in.
That may not be so bad. Remember, Neither Regan, or either Bush were lawyers, nor, if I recall correctly was Blair or Thatcher, and look at the fine messes they have put the world in.
Sorry Michael! I don't mean they are all bad. :-)
Actually, in Aussie politics it is certainly true that doctors have made shocking politicians ( and vice-versa ).
Sometimes one yearns for a 'man/woman in the street' to be elected, but then they no longer are 'in the street'. Reminds me of, I think, Robert Redford in The Candidate.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Actually Blair (Mr) was a failed lawyer, as is his wife, and Thatcher was a qualified Research Chemist (originally).
I must say Thatcher guided the country economically much better than Blair/Brown have done. The latter 2 are slowly unravelling their inheritence from Thatcher, and, at least, Thatcher had a vision of where she wanted the country to be during her first 10 years (unlike Blair/Brown who have no vision and only play the power game because they enjoy them)).
Needless to say the LHC will find the charming quark of all hamsters, the answer to all the conundrums facing astronomy and high energy Physisc at the moment.
Shih-Tzu are clever, cuddly, playful and rule!! Jack Russell are feisty!
RE: Ah, I see, thanks
)
I always thought that the real symmetry group of elementary particles is not SU(3) but SU(4). Take a four dimensional Riemaniann manifold with pseudo-euclidean metric and calculate its tangent space at a fixed point. You must differentiate any of a set of infinitely differentiable functions on it. The differential operators form a Lie algebra. Is it the algebra of SU(4)? I've been unable to prove this.
Tullio
RE: I always thought that
)
I recall the post you made a while back about E8 and I've been trying to learn all the things required for understanding the nature of that, and also the well known SU(3)xSU(2)xU(1), and I admit I still have a ways to go.
What motivated me to ask about the three generations being a limit was actually studying the general equation for thrust from a rocket (because I didn't understand why the flow should ever become “choked� regardless of adding more pressure). I learned that some things can't be caused (or made to happen) simply by increasing the force, and that's when it dawned on me that it might not be possible to create a lepton heavier than a tau, or a quark heavier than the top. I guess I should have said “fundamental standpoint� instead of 'theoretical'...?
It's exciting to think that the LHC might produce evidence for exotic particles and hence also for one of the theories that go beyond the Standard Model. And I guess it makes sense that these (or at least the Higgs) would be observed before seeing a fourth generation of matter. Probably all related to how the gap in the masses arises? Also begs the question of how powerful an accelerator would be required to produce a 4th generation particle :)
Well, I've set a personal record for the number of open tabs in my browser trying to fully understand the question, and I don't yet. The task is to show that their respective Lie groups are isomorphic ?
Incidentally, I happened to stumble across this: 3 Generations: E6, E7, E8
Seems straightforward enough and I wish I knew enough to evaluate it, but I don't – yet. :)
RE: RE: I always thought
)
I must admit that I dabbled with Lie groups and algebras many years ago while working toward my thesis. I even published a paper in "Il Nuovo Cimento" in 1967 whose title was "SU(n,1)representation for the harmonic oscillator". My coauthor was my thesis advisor, Giordano Bisiacchi, a young and brilliant theoretical physicist from Trieste University and International Center for Theoretical Physics (now called Abdus Salam Center, for the Nobel Prize winner Pakistani physicist). Then Giordano died in 1972 in a car crash. Even physicists die. But, really, what was published was only the result of the first half of my thesis, using commutators of the kind ab - ba, suitable for bosons. A second part, using anticommutators of the kind ab + ba (where a and b are Lie algebra operators) and suitable for fermions was never published and never will be. But in 1996, while surfing the Internet on a Pentium I, I discovered that an American physicist, Mike Guidry of the University of Tennessee had used the same algebras to describe a very important fermion system, the nucleus of uranium. His article, visible on Internet, is called "Fermion Lie Algebras and Microscopic Theories of Nuclear Matter". I contacted him and he asked me for a translated copy of my thesis, which I sent him together with the Nuovo Cimento article. Then my Internet connection fell and he sent an answer to my brother Franco, a mathematician at Trieste University (I was then in Milano), whose email address I had given Guidry as an alternative. My brother received an answer and, since I had no longer an Internet connection, I asked Franco to send me a hardcopy of the Guidry mail. But, being a mathematician, he was unable to use a printer and this ended the matter. Had Giordano not died in his Alfa Romeo Giulietta Sprint we might have preceded Guidry by about thirty years. Such is life. Cheers.
Tullio
RE: Or anti-hamsters.
)
That may not be so bad. Remember, Neither Regan, or either Bush were lawyers, nor, if I recall correctly was Blair or Thatcher, and look at the fine messes they have put the world in.
"We must be the change we wish to see."
Mahatma Gandhi
RE: That may not be so bad.
)
Sorry Michael! I don't mean they are all bad. :-)
Actually, in Aussie politics it is certainly true that doctors have made shocking politicians ( and vice-versa ).
Sometimes one yearns for a 'man/woman in the street' to be elected, but then they no longer are 'in the street'. Reminds me of, I think, Robert Redford in The Candidate.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Actually Blair (Mr) was a
)
Actually Blair (Mr) was a failed lawyer, as is his wife, and Thatcher was a qualified Research Chemist (originally).
I must say Thatcher guided the country economically much better than Blair/Brown have done. The latter 2 are slowly unravelling their inheritence from Thatcher, and, at least, Thatcher had a vision of where she wanted the country to be during her first 10 years (unlike Blair/Brown who have no vision and only play the power game because they enjoy them)).
Needless to say the LHC will find the charming quark of all hamsters, the answer to all the conundrums facing astronomy and high energy Physisc at the moment.
Shih-Tzu are clever, cuddly, playful and rule!! Jack Russell are feisty!
It was just a matter of
)
It was just a matter of time:
http://hasthelargehadroncolliderdestroyedtheworldyet.com/
;-)
RE: It was just a matter of
)
Hahahahhahahhahahah. That's priceless humor. Simple, yet effective. :D
Hadron Collider forced to
)
Hadron Collider forced to halt
RE: ...... a tonne of
)
ooooh, snuck out of the superconducting state. Expensive!
[spontaneous ditty][pre] Buy the Helium,
Cool the Helium,
Spill the Helium over ..... [/pre][\spontaneous ditty]
There's energy in that thar beam! :-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal