The main difference I know between bosons and fermions is that you can quantize a boson field with commutator [ab - ba] and a fermion field with anticommutators [ab + ba]. This was the main theme of my thesis in 1966. The part related to the boson field was published in Il Nuovo Cimento in 1967 (G. Bisiacchi et al; I was the al}. The second part was never published but I found some of its ideas in an Internet paper by the American physicist Mike Guidry called "Fermion Lie algebras and Microscopic Theories of Nuclear Structure". Of course he did not know my thesis but I sent him a copy of it with an English translation of my second part. I then lost my Internet connection and do not know his reaction. But if you do not publish (which is a costly matter) you lose all priority.
Tullio
I found a great little book, originally published in 1945, that teaches the mathematical basis of general relativity and touches on the main point of this thread - defining gravity. I highly recommend this book which is written as a Socratic dialog with plenty of really cool drawings.
The Einstein Theory of Relativity: A Trip to the Fourth Dimenion, by Lillian R. Lieber.
Congrats Tullio for being the al in et al. I'd always wondered who that was!!
Alas ( per Google etc ) it seems Il Nuovo Cimento was either discontinued or superceded in 1971. As you say, publish or perish.
G'day Dan! Thanks, I'll go out and order that one. Always looking for a fresh approach.
Quote:
.... about 20 introductory books later ....
I know the feeling. I've had a lifelong ( well, since my teens ) fascination with maths. Especially calculus. So I have Pre Calculus, Introductory Calculus, Calculus of One Variable, Calculus of Several Variables, Calculus To Read On The Train, Intermediate Calculus, Deleted Neighbourhoods Are Really OK, Advanced Calculus, Limits : Why Bother ?, Postgraduate Calculus, Introductory Real Analysis, Frontier Points - How Far Would You Go? , Introductory Complex Analysis, How To Close A Difficult Open Set, The Low Down On The Supremum, Real Analysis on Weekdays, The Not So Mean Value Theorem, Complex Analysis For A Long Weekend, Schaum's Outline of Differential Equations, Making Room For Accumulation Points, Continuity For Remote Students, Solvable Integro-Differential Equations ( total of five pages, including the title, preface, contents and index ), My Life As a Differential : The View from the Slope, A Quick Recipe For Lime Soup, Grow Your Own Compact Metric Space At Home, Calculus for Physicists Or Why You Can Cross Multiply by dx, Integration - Yes, Life Is a B**ch, What Newton and Leibniz Really Thought Of Each Other and my all time favourite You Have To Get A Computer to Numerically Solve It Anyway .... :-) :-)
Cheers, Mike.
( edit ) So Penrose's The Road To Reality is for me the pinnacle. Mind you the any of Schaum's series ( McGraw Hill ) are superb for hacking through the forest. You have to practice you see, to really get it, and their guides have long ( but graded ) problem sections with worked solutions. If you're really serious about a future in maths then you couldn't go ( far ) wrong with a bunch of pens ( I prefer 'felt' tipped ~ 0.3mm and various colors ), a stack of writing/graph pads, and a Schaum's book to plow through!! :-)
( edit ) Do not worry about your difficulties in mathematics, I assure you that mine are greater - Albert Einstein
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
I'm a big fan of Road to Reality as well. It is a nice book to see how the pieces fit together. Can't go wrong with Penrose.
If you're a fan of Calculus then you MUST have "div grad curl and all that" by h. m. Schey. This is a classic and has a lot of physics in it too. Just a small paperback, it is something you can stick in your bag for a long train ride. I read it on the plane to get that warm and fuzzy feeling that maths always brings me. Enjoy.
I'm a big fan of Road to Reality as well. It is a nice book to see how the pieces fit together. Can't go wrong with Penrose.
If you're a fan of Calculus then you MUST have "div grad curl and all that" by h. m. Schey. This is a classic and has a lot of physics in it too. Just a small paperback, it is something you can stick in your bag for a long train ride. I read it on the plane to get that warm and fuzzy feeling that maths always brings me. Enjoy.
Cool. I'll look that one up too. Good old Amazon! => Indeed one reviewer for that states :
Quote:
It's been over two decades since I first studied vector calculus from my old textbook on electromagnetic fields and waves (Lorrain and Corson, Freeman, 1970).
Another nifty pocket size one for the road ( train/plane/bus/hovercraft/interstellar-shuttle/matter-transfer-cabinet ) is Metric Spaces by E.T. Copson. It's one of the less brutal from the Cambridge Tracts in Mathematics, which you generally either read by the fireside or throw in to keep yourself warm .... :-)
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
Mike, evidently Il Nuovo Cimento (nuovo because the old was the Proceedings of the Accademia del Cimento in Galileo's times) suffered from the onslaught of on-line publishing. My Nuovo Cimento article was in the B Series, dedicated to relativity and mathematical methods in physics. Unfortunately my coauthor Giordano Bisiacchi, who was my thesis adviser, died in a car accident in 1972 and that put a stop to my publishing. Asides from some chapters in Mondadori and Isedi Encyclopedias, which count very little in academia, my biggest contribution was the Physics and Astronomy part written in English for a report "Current trends in basic research" written under a contract with UNESCO in 1972. It was to be a part of a book by Italian biologist Adriano Buzzati-Traverso, brother of writer and painter Dino Buzzati, who unfortunately delayed the printing of the book. When printed in 1978, some parts of the report were obsolescent or obsolete, and the "Nature" staff, which had competed with us of Mondadori Scientific Publishing staff for the contract, were quick to point it out. Publish or perish, as you say. Cheers.
Tullio
I've had a further look at the cross-product thingy, plus the imaginary time co-ordinate from The Road To Reality ( TRTR ). There's a deep topological theorem that places restrictions on the behaviour of quaternions etc .... giving the best ( most realistic ) results for 'spinors'. These model particles with spin BUT with an imaginary co-ordinate(s) that do for a 180 degree rotation what i does over 90 degrees. This implies that a 4PI rotation about an axis brings you back to the original state, so is sort of an identity transform.
This almost immediately reminded me of a neutron study done in the late 1970's by some chaps I knew at the University of Melbourne Physics Department ( on sabbatical to the neutron reactor facility in France ). They didn't unload anything ground breaking but fairly directly and cleverly showed the interference issue with neutrons ( as fermions ).
Quote:
Recall that spin-1/2 particles have to be turned around twice, at 360 degrees per turn, to come back 'the same'. Generally if a particle is spin-X then 360/X is the required least number of degrees to turn ( X non-zero ). A photon is spin-1 thus if, say, you have one coming towards you with a certain polarity ( the electric/magnetic field orientation ) and then you circle around it once ( 360 degrees ) that polarity looks unchanged. However a ( hypothetical ? ) graviton is spin-2, and this gives rise to two independent directions of polarization 45 degrees apart. Spin-0 by the way is like a perfectly smooth featureless sphere, you can't decide upon any special orientation regardless of your direction of view. There's considerable subtlety with the spinor construction, but I see that it is essential to get it correct - quantum mechanical calculations have no hope of coming out right otherwise.
Beware of the two uses of the word 'spin'. The first ( that we are referring to above ) is the total spin vector, which we can't actually measure per se but models 'as if'. The second is the component of that vector that we measure along some direction - while then remaining clueless about the components in the other two perpendicular directions.
Penrose doesn't like imaginary time, because it gives illusion of a 'proper' Euclidean metric.
Now I think I've found Tullio's effort : "SU n,1 Representation for the Harmonic Oscillator", G.Bisiacchi and T.Chersi. Il Nuovo Cimento B Series 10 ( 1967 ), vol. 51, issue 1 pages 195-198 ???? Which unfortunately SpringerLink won't allow prying from their fingers for less than $40 USD. :-(
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
I still have a printed copy of it. If you PM me a post address I might photocopy it and send it to you. We had a good number of requests for it, one from the Academy of Sciences of Kazakhstan. Cheers.
Tullio
Thanks! Actually if you PM me your address I'll send you a stamped self addressed post-pack type pre-paid priority etc thingy to return it in! I'd be delighted to read your efforts with T.Chersi at al :-) :-)
Another view on spin zero. The 360/X doesn't agree with the pattern of the others, as if you take X > 0 but tending to zero as an infinitesimal - that should give an unbounded ( tending to infinite ) value for the number of degrees to rotate and get the 'same thing'. Now if you take the converse case, the spin X tending to infinity hence 360/X goes to zero - all infinitesimal rotations are now giving the 'same' thing, which is more realistic for spin zero. So that suggests a duality b/w zero and infinity ( or unbounded and infinitesimal ), which is key real estate for the mapping from complex plane to complex plane via the Riemann sphere ( inside |z| = 1 goes to outside |z| = 1 and vice versa. ). The extended complex plane that is, with one way of the mapping you project from the 'south' pole at (0, 0, -1) : hitting the 'northern' hemisphere for |z| 1. The real line becomes 'painted' to a great circle through both poles and the pure real numbers at (-1, 0, 0 ) and (+1, 0, 0 ). The inverse mapping is done mutatis mutandis with projection from the 'north' pole at (0, 0, +1). The other nifty bit is the implicit complex conjugation ie. i-i, so one is not only going to the other 'side' ( inner outer ) of the circle at |z| = 1 via the mapping but there is an orientation reversal of PI or 180 degrees. Can you see how these themes just keep knocking on the door ??? :-)
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
An interesting take on Erik Verlinde Idea (Gravity is just a consequence of thermodynamics)
I have read a little bit. I like to read a little and digest. That way I don't get to fat :-)
The main difference I know
)
The main difference I know between bosons and fermions is that you can quantize a boson field with commutator [ab - ba] and a fermion field with anticommutators [ab + ba]. This was the main theme of my thesis in 1966. The part related to the boson field was published in Il Nuovo Cimento in 1967 (G. Bisiacchi et al; I was the al}. The second part was never published but I found some of its ideas in an Internet paper by the American physicist Mike Guidry called "Fermion Lie algebras and Microscopic Theories of Nuclear Structure". Of course he did not know my thesis but I sent him a copy of it with an English translation of my second part. I then lost my Internet connection and do not know his reaction. But if you do not publish (which is a costly matter) you lose all priority.
Tullio
I found a great little book,
)
I found a great little book, originally published in 1945, that teaches the mathematical basis of general relativity and touches on the main point of this thread - defining gravity. I highly recommend this book which is written as a Socratic dialog with plenty of really cool drawings.
The Einstein Theory of Relativity: A Trip to the Fourth Dimenion, by Lillian R. Lieber.
http://astore.amazon.com/amuletdevelopmco/detail/1589880447
After several false-starts, and about 20 introductory books later, I can now say general relativity is mine thanks to this book. Tenors rule!
Daniel
The Physics Groupie
Congrats Tullio for being the
)
Congrats Tullio for being the al in et al. I'd always wondered who that was!!
Alas ( per Google etc ) it seems Il Nuovo Cimento was either discontinued or superceded in 1971. As you say, publish or perish.
G'day Dan! Thanks, I'll go out and order that one. Always looking for a fresh approach.
I know the feeling. I've had a lifelong ( well, since my teens ) fascination with maths. Especially calculus. So I have Pre Calculus, Introductory Calculus, Calculus of One Variable, Calculus of Several Variables, Calculus To Read On The Train, Intermediate Calculus, Deleted Neighbourhoods Are Really OK, Advanced Calculus, Limits : Why Bother ?, Postgraduate Calculus, Introductory Real Analysis, Frontier Points - How Far Would You Go? , Introductory Complex Analysis, How To Close A Difficult Open Set, The Low Down On The Supremum, Real Analysis on Weekdays, The Not So Mean Value Theorem, Complex Analysis For A Long Weekend, Schaum's Outline of Differential Equations, Making Room For Accumulation Points, Continuity For Remote Students, Solvable Integro-Differential Equations ( total of five pages, including the title, preface, contents and index ), My Life As a Differential : The View from the Slope, A Quick Recipe For Lime Soup, Grow Your Own Compact Metric Space At Home, Calculus for Physicists Or Why You Can Cross Multiply by dx, Integration - Yes, Life Is a B**ch, What Newton and Leibniz Really Thought Of Each Other and my all time favourite You Have To Get A Computer to Numerically Solve It Anyway .... :-) :-)
Cheers, Mike.
( edit ) So Penrose's The Road To Reality is for me the pinnacle. Mind you the any of Schaum's series ( McGraw Hill ) are superb for hacking through the forest. You have to practice you see, to really get it, and their guides have long ( but graded ) problem sections with worked solutions. If you're really serious about a future in maths then you couldn't go ( far ) wrong with a bunch of pens ( I prefer 'felt' tipped ~ 0.3mm and various colors ), a stack of writing/graph pads, and a Schaum's book to plow through!! :-)
( edit ) Do not worry about your difficulties in mathematics, I assure you that mine are greater - Albert Einstein
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
Hi Mike, And G'Day to you
)
Hi Mike,
And G'Day to you too.
I'm a big fan of Road to Reality as well. It is a nice book to see how the pieces fit together. Can't go wrong with Penrose.
If you're a fan of Calculus then you MUST have "div grad curl and all that" by h. m. Schey. This is a classic and has a lot of physics in it too. Just a small paperback, it is something you can stick in your bag for a long train ride. I read it on the plane to get that warm and fuzzy feeling that maths always brings me. Enjoy.
Daniel
The Physics Groupie
RE: Hi Mike, And G'Day to
)
Cool. I'll look that one up too. Good old Amazon! => Indeed one reviewer for that states :
That book, Electromagnetic Fields and Waves, is precisely where I first touched the subject.
Another nifty pocket size one for the road ( train/plane/bus/hovercraft/interstellar-shuttle/matter-transfer-cabinet ) is Metric Spaces by E.T. Copson. It's one of the less brutal from the Cambridge Tracts in Mathematics, which you generally either read by the fireside or throw in to keep yourself warm .... :-)
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
Mike, evidently Il Nuovo
)
Mike, evidently Il Nuovo Cimento (nuovo because the old was the Proceedings of the Accademia del Cimento in Galileo's times) suffered from the onslaught of on-line publishing. My Nuovo Cimento article was in the B Series, dedicated to relativity and mathematical methods in physics. Unfortunately my coauthor Giordano Bisiacchi, who was my thesis adviser, died in a car accident in 1972 and that put a stop to my publishing. Asides from some chapters in Mondadori and Isedi Encyclopedias, which count very little in academia, my biggest contribution was the Physics and Astronomy part written in English for a report "Current trends in basic research" written under a contract with UNESCO in 1972. It was to be a part of a book by Italian biologist Adriano Buzzati-Traverso, brother of writer and painter Dino Buzzati, who unfortunately delayed the printing of the book. When printed in 1978, some parts of the report were obsolescent or obsolete, and the "Nature" staff, which had competed with us of Mondadori Scientific Publishing staff for the contract, were quick to point it out. Publish or perish, as you say. Cheers.
Tullio
I've had a further look at
)
I've had a further look at the cross-product thingy, plus the imaginary time co-ordinate from The Road To Reality ( TRTR ). There's a deep topological theorem that places restrictions on the behaviour of quaternions etc .... giving the best ( most realistic ) results for 'spinors'. These model particles with spin BUT with an imaginary co-ordinate(s) that do for a 180 degree rotation what i does over 90 degrees. This implies that a 4PI rotation about an axis brings you back to the original state, so is sort of an identity transform.
This almost immediately reminded me of a neutron study done in the late 1970's by some chaps I knew at the University of Melbourne Physics Department ( on sabbatical to the neutron reactor facility in France ). They didn't unload anything ground breaking but fairly directly and cleverly showed the interference issue with neutrons ( as fermions ).
Penrose doesn't like imaginary time, because it gives illusion of a 'proper' Euclidean metric.
Now I think I've found Tullio's effort : "SU n,1 Representation for the Harmonic Oscillator", G.Bisiacchi and T.Chersi. Il Nuovo Cimento B Series 10 ( 1967 ), vol. 51, issue 1 pages 195-198 ???? Which unfortunately SpringerLink won't allow prying from their fingers for less than $40 USD. :-(
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
I still have a printed copy
)
I still have a printed copy of it. If you PM me a post address I might photocopy it and send it to you. We had a good number of requests for it, one from the Academy of Sciences of Kazakhstan. Cheers.
Tullio
Thanks! Actually if you PM me
)
Thanks! Actually if you PM me your address I'll send you a stamped self addressed post-pack type pre-paid priority etc thingy to return it in! I'd be delighted to read your efforts with T.Chersi at al :-) :-)
Another view on spin zero. The 360/X doesn't agree with the pattern of the others, as if you take X > 0 but tending to zero as an infinitesimal - that should give an unbounded ( tending to infinite ) value for the number of degrees to rotate and get the 'same thing'. Now if you take the converse case, the spin X tending to infinity hence 360/X goes to zero - all infinitesimal rotations are now giving the 'same' thing, which is more realistic for spin zero. So that suggests a duality b/w zero and infinity ( or unbounded and infinitesimal ), which is key real estate for the mapping from complex plane to complex plane via the Riemann sphere ( inside |z| = 1 goes to outside |z| = 1 and vice versa. ). The extended complex plane that is, with one way of the mapping you project from the 'south' pole at (0, 0, -1) : hitting the 'northern' hemisphere for |z| 1. The real line becomes 'painted' to a great circle through both poles and the pure real numbers at (-1, 0, 0 ) and (+1, 0, 0 ). The inverse mapping is done mutatis mutandis with projection from the 'north' pole at (0, 0, +1). The other nifty bit is the implicit complex conjugation ie. i -i, so one is not only going to the other 'side' ( inner outer ) of the circle at |z| = 1 via the mapping but there is an orientation reversal of PI or 180 degrees. Can you see how these themes just keep knocking on the door ??? :-)
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
An interesting take on Erik
)
An interesting take on Erik Verlinde Idea (Gravity is just a consequence of thermodynamics)
I have read a little bit. I like to read a little and digest. That way I don't get to fat :-)
Entropic Gravity Predestrians
There are some who can live without wild things and some who cannot. - Aldo Leopold