Christian's course on Lie algebras advanced and so did the problem sheets. Here is another exercise that nearly by accident ended up in the problem set: You are supposed to show that all unitary irreducible representations of SU(2) are finite dimensional.
OK, we physicists know how to do this: In terms of a Cartan basis (H,E,F) we take an eigenvector of H and than act on it a couple of times with E to show that if this does not terminate we end up with negative norm vectors. Easy.
But this was too easy. If we have to consider infinite dimensional representations one should be careful with functional analysis. It could be that H is only essentially self-adjoint (like the momentum operator) and thus does not have eigenvectors because it only has a continious spectrum. Can you repair the above prove in this case or show that it cannot happen or find a counter example?
Thursday, November 18, 2004
Wednesday, November 17, 2004
Theories of Everything
Everybody is talking about the landscape so I should probably also throw in a couple of words. It seems, the lesson from recent progress in understanding string compactifications with N=1 susy is that some people’s hopes for a unique vacuum with four macroscopic directions and SU(3)xSU(2)xU(1) gauge group and matter in the appropriate chiral representations was not fulfilled. It seems, there is not even a small number of N=1 vacuua but rather a number where even the logarithm is larger than some people’s ability to count.
This is quite disappointing for people who were hoping that string theory as the theory of everything would once and for all answer all questions about space-time dimensionality, matter content, masses, coupling constants, mixings etc. There are even people who argue with this new insight string theory lacks any predictability at all.
What I would like to argue is that both these views, the optimistic hope for a unique vacuum and the claim of predictive emptiness of string theory, have unrealistic expectations of a theory of everything.
We now have some centuries experience with more and more unifying and more and more fundamental theories (starting from Newton’s unification of gravity with celestical motions) that we should have learnt the lesson how these things work: There are always two theories, an old one and a new one, or an engeneering and a physical theory if you allow me this pun. The old one often actually consists of several independant subtheories (like the theory that "apples keep falling down" and "planet move on cubic sections") and the new theory in some sense "explains" the old theories.Like there is a simple, local force law corresponding to differential equations that happen to have cubic sections as their solutions.
Another example would be the theory of atomic spectra: Lacking any quantum mechnical model (not even Bohr's theory) one could have just a big book that contains the frequencies of the spectra of all elements (and possibly molecules). That is a theory of the type engeneers like to have: They have big tables with lots and lots of numbers describing for example all possible gears that are on the market with their mechnical properties.
Another example of such a theory is the particle data book. This contains many many numbers that describe out understanding of the standard model. Well, here I am cheating a bit because we believe that most of the numbers (especially all the branchings and all the baryonic data could in principle be computed from QCD etc), but there are still the first couple of pages. You get the idea. Now one might have hoped that once one has a sufficient understanding of string theory in situations with low or even no susy, there would be a unique ground state and similarly, one could have computed all those numbers (like mass of the electron, weak mixing angle, the ud matrix element of the CKM matrix) from string theory. Now the landscape seems to end these hopes.
But why should we have stopped with those microscopic numbers? If there would have been a unique state of string theory, we should be able to compute many more properties of this state (also known as our univers): Why not compute the mass of the earth. Or the number of water molecules in the Atlantic ocean. Or the length of my left index finger. If there is only one state (=one universe), all these numbers should at least in principle be computable from
.
Obviously, this is nonsense and nobody would have believed this in the first place. It might have been a possible scenario but nobody should have been so overoptimistic to expect a unique state of string thoery.
The real difference between the old and the new theory is that what used to be 'external parameters' in the old one (for example parameters of its equations of motion) are properties of the solution/initial conditions of the more fundamental one. This is like Maxewells equations: For all we know, they are the fundamental theory of electromagnetic radiation but nobody would expect to compute from them which song Radio 1 is playing at the moment. The song is just encoded in the initial and boundary conditions for a specific solution to dF=0, d*F=j.
I think, one should (have) expected a similar situation for string theory (or better M-Theory): As a theory of everything it should not have dimensionless parameters in its formulation, but still, there are many solutions to its equations of motion and at least some of the properties of our world (including maybe some of the standard model parameters) are simply properties of a specific solution.
But on the other hand, this does not mean that anything goes and (again in principle) string theory will never make any prediction: Simply take scattering at very very large energies. Those that correspond to distance scales much smaller than our compactification at hand: Than I would expect to see an excitation spectrum typical of strings with the tower of states etc. OK, this might not yet happen at LHC but I am talking in prinicple.
Furthermore, it might be that over all solutions to the string equations of motion not all but only a submanifold (or even only a discretuum) of the macroscopic parameter space is swept out: There are relations. It might be that the mass of the proton is always 612 times the mass of the electron times the number of generations. Who knows? That would clearly be a prediction. It's just at this stage we are not yet powerful enough to make these kinds of predictions.
Even in quantum mechanics, may properties of the emission spectra one is computing with Schrödinger's equation depend on intial and boudary contions: One has to specify for example the charge of the nucleus and the number of electrons before one can predict the energy spectrum. There is no unique solution to Schrödinger's equation but still, QM predicts for example Rydberg's law that you can write the energies as a constant times (1/m -1/n) for small integers m and n. Similar predictions be be possible in string theory even if there are many vacuua.
This is quite disappointing for people who were hoping that string theory as the theory of everything would once and for all answer all questions about space-time dimensionality, matter content, masses, coupling constants, mixings etc. There are even people who argue with this new insight string theory lacks any predictability at all.
What I would like to argue is that both these views, the optimistic hope for a unique vacuum and the claim of predictive emptiness of string theory, have unrealistic expectations of a theory of everything.
We now have some centuries experience with more and more unifying and more and more fundamental theories (starting from Newton’s unification of gravity with celestical motions) that we should have learnt the lesson how these things work: There are always two theories, an old one and a new one, or an engeneering and a physical theory if you allow me this pun. The old one often actually consists of several independant subtheories (like the theory that "apples keep falling down" and "planet move on cubic sections") and the new theory in some sense "explains" the old theories.Like there is a simple, local force law corresponding to differential equations that happen to have cubic sections as their solutions.
Another example would be the theory of atomic spectra: Lacking any quantum mechnical model (not even Bohr's theory) one could have just a big book that contains the frequencies of the spectra of all elements (and possibly molecules). That is a theory of the type engeneers like to have: They have big tables with lots and lots of numbers describing for example all possible gears that are on the market with their mechnical properties.
Another example of such a theory is the particle data book. This contains many many numbers that describe out understanding of the standard model. Well, here I am cheating a bit because we believe that most of the numbers (especially all the branchings and all the baryonic data could in principle be computed from QCD etc), but there are still the first couple of pages. You get the idea. Now one might have hoped that once one has a sufficient understanding of string theory in situations with low or even no susy, there would be a unique ground state and similarly, one could have computed all those numbers (like mass of the electron, weak mixing angle, the ud matrix element of the CKM matrix) from string theory. Now the landscape seems to end these hopes.
But why should we have stopped with those microscopic numbers? If there would have been a unique state of string theory, we should be able to compute many more properties of this state (also known as our univers): Why not compute the mass of the earth. Or the number of water molecules in the Atlantic ocean. Or the length of my left index finger. If there is only one state (=one universe), all these numbers should at least in principle be computable from
.
Obviously, this is nonsense and nobody would have believed this in the first place. It might have been a possible scenario but nobody should have been so overoptimistic to expect a unique state of string thoery.
The real difference between the old and the new theory is that what used to be 'external parameters' in the old one (for example parameters of its equations of motion) are properties of the solution/initial conditions of the more fundamental one. This is like Maxewells equations: For all we know, they are the fundamental theory of electromagnetic radiation but nobody would expect to compute from them which song Radio 1 is playing at the moment. The song is just encoded in the initial and boundary conditions for a specific solution to dF=0, d*F=j.
I think, one should (have) expected a similar situation for string theory (or better M-Theory): As a theory of everything it should not have dimensionless parameters in its formulation, but still, there are many solutions to its equations of motion and at least some of the properties of our world (including maybe some of the standard model parameters) are simply properties of a specific solution.
But on the other hand, this does not mean that anything goes and (again in principle) string theory will never make any prediction: Simply take scattering at very very large energies. Those that correspond to distance scales much smaller than our compactification at hand: Than I would expect to see an excitation spectrum typical of strings with the tower of states etc. OK, this might not yet happen at LHC but I am talking in prinicple.
Furthermore, it might be that over all solutions to the string equations of motion not all but only a submanifold (or even only a discretuum) of the macroscopic parameter space is swept out: There are relations. It might be that the mass of the proton is always 612 times the mass of the electron times the number of generations. Who knows? That would clearly be a prediction. It's just at this stage we are not yet powerful enough to make these kinds of predictions.
Even in quantum mechanics, may properties of the emission spectra one is computing with Schrödinger's equation depend on intial and boudary contions: One has to specify for example the charge of the nucleus and the number of electrons before one can predict the energy spectrum. There is no unique solution to Schrödinger's equation but still, QM predicts for example Rydberg's law that you can write the energies as a constant times (1/m -1/n) for small integers m and n. Similar predictions be be possible in string theory even if there are many vacuua.
Connected!!!
I arrived in the future! I got my wireless connection working. Now I am sitting in the silent work area of the IUB library with a laptop that has neither power nor network cable connection and still I can post nonsense to my blog. Wow.
Thursday, November 11, 2004
St. Martin
Being brought up with a protestant background, I usually don't care about saints and their anniversary days. But Martin (November 11th) has always been a bit of an exception, maybe because he was the patron of Martin Luther, maybe because my grandma had a liking for him (she wasn't so strict in these respects) or maybe as there are a couple of nice customs associated with this day.
Today, on my way to work, I heard a radio progam on Deutschlandfunk, the nation wide public radio, from which I learnt some factoids that I would like to share with the blogsphere.
The historic Martin was born as a roman (the name relates to Mars, the roman god of war) in the fourth century and started out as an officer in the roman army but later turned christian an became a archbishop in Tour, France. As such he of course was also a political figure and after his death in 397 was the first saint that did not die as a martyr.
His day is Martini, November 11th, which later became the last day of the peasant fiscal year: On this day, the contracts of farm labourers and servants ended and taxes, feud and leasehold had to be paid. It is also the beginning of a fourty day lent period and the way these things work was celebrated with a feast. This is the origin of the tradition to have a roast goose on Martini and for childern to have processions with hand lanterns. In some regions, this is also the day on which this year's wine is drunk for the first time.
Finally, 11/11 is the first day of the "fifth season", the carneval period that culminates in the days before the spring lent in February. But at least this custom is too Catholic for me to notice...
Today, on my way to work, I heard a radio progam on Deutschlandfunk, the nation wide public radio, from which I learnt some factoids that I would like to share with the blogsphere.
The historic Martin was born as a roman (the name relates to Mars, the roman god of war) in the fourth century and started out as an officer in the roman army but later turned christian an became a archbishop in Tour, France. As such he of course was also a political figure and after his death in 397 was the first saint that did not die as a martyr.
His day is Martini, November 11th, which later became the last day of the peasant fiscal year: On this day, the contracts of farm labourers and servants ended and taxes, feud and leasehold had to be paid. It is also the beginning of a fourty day lent period and the way these things work was celebrated with a feast. This is the origin of the tradition to have a roast goose on Martini and for childern to have processions with hand lanterns. In some regions, this is also the day on which this year's wine is drunk for the first time.
Finally, 11/11 is the first day of the "fifth season", the carneval period that culminates in the days before the spring lent in February. But at least this custom is too Catholic for me to notice...
Wednesday, November 03, 2004
New Dawn
OK, if no really big surprises will happen, George W. will be US president for four more years. And appearantly, this time he is even elected. Fine, if that's what you Americans want, you should have it.
TV says that values rather than issues decided this election. I just find it sad that there are so many people that have views on christianity, tolerance, social issues, ecology and international relations that on the long run I expect to turn against all of us. He will be your president, not mine, it's just that at least some of the time we have to share the same world.
Sean Caroll has written an entry on how to move on (with a long term perspective) in the Preposterous Universe that I support 100%!
TV says that values rather than issues decided this election. I just find it sad that there are so many people that have views on christianity, tolerance, social issues, ecology and international relations that on the long run I expect to turn against all of us. He will be your president, not mine, it's just that at least some of the time we have to share the same world.
Sean Caroll has written an entry on how to move on (with a long term perspective) in the Preposterous Universe that I support 100%!
Monday, November 01, 2004
Bold
Over at his reference frame, Lubos has been so bold to comment on all the papers of today. This is interesting to read for the ones he knows the background of and as amusing in the case of the others.
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