Lubos Motl
Dec27-04, 07:05 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Let me offer a text from my blog, attempting to provoke a discussion.\n\n=================================== ===================================\nhttp://motls.blogspot.com/2004/12/theories-are-increasingly-theoretical.html\n================================ ======================================\n\nThis text follows my discussions with Nima Arkani-Hamed and David Goss.\n\nSome people don\'t like the fact that the arguments in string theory are\nincreasingly theoretical in nature, and that our theories seem to give us\nless exactly calculable sharp predictions that are verified\nexperimentally.\n\nHowever: it\'s not just string theory: the whole particle physics has been\nbecoming increasingly theoretical and string theory just continues in the\nsame direction. What do I mean?\n\nQED, Electroweak theory, QCD: increasing groups, decreasing accuracy\n\nThe peak of the old-fashioned quantitative predictivity of very particular\nfacts in physics was QED which stands for Quantum Electrodynamics. You\nknow that people could have calculated its predictions already 50 years\nago, including the quantum loop corrections, even though they did not\nquite understand why their methods were working (The Renormalization\nGroup), and the most precise predictions - like the anomalous magnetic\nmoment of the electron - have been successfully tested with the accuracy\nof 13 decimal places!\n\nThen the physicists found the electroweak force that naturally predicted\nthe neutral currents, W bosons, the Z boson, and so forth. It is also a\nrelatively very predictive theory (Glashow, Salam, Weinberg) although its\npredictions were never tested as exactly as for QED. Nevertheless, all the\ncross sections and decay rates are measured rather precisely, the\nelectroweak scattering is "clean".\n\nAnother step: QCD\n\nThen you go to QCD which is now an accepted and "Nobelized" part of our\ntheoretical canon. QCD, in some sense, confirmed the things people\n"guessed" by other means, and one might criticize it using some very\nsimilar words as those often applied to string theory by its critics.\n\nYou know, QCD is claimed to be a theory of the strong force, but it talks\nabout the gluons, quarks, and especially their three colors, three\nconcepts that were never directly seen; and according to QCD no one will\never see either of them. Also, no one has been able to calculate the\nproperties of the proton, neutron, and nuclei - which used to be thought\nto be the objects from the strong interaction - from this theory too well.\nThe actual calculations often rely on some properties of the quark and\ngluon distribution functions, and the critics might say that these\nfunctions have never been really derived from QCD. Even if one accepts the\nexistence of quarks, they were not really invented by QCD: Gell-Mann\nreceived a Nobel prize for quarks in 1969, five years before QCD was\nproposed. The new quark flavors, such as the c-quark found in the J/psi\nparticle, were naturally predicted by the electroweak theory (the GIM\nmechanism from 1970), not by QCD. In this respect, QCD seems to have had\nno "striking" new predictions. So why do we say that QCD is a good theory\nof the strong interaction?\n\nThe low-energy properties of the hadrons have not been calculated\naccurately enough simply because QCD is a pretty difficult machine to\ncalculate with at low energies - but this difficulty is a fact of Nature.\nIn the same way, the vacuum structure in string theory is also rather\ncomplicated, which also seems to be a fact of Nature. At high energies,\nthe quarks are almost free (due to asymptotic freedom, which is really\nwhat our friends got their 2004 Nobel prize for). If the quarks are free,\nperturbation theory is great and one can easily and precisely calculate\nthe high-energy events. But for the effects important for the nuclear\nphysics, the interaction is strong - more or less by definition. QCD is a\nstrongly coupled theory at longer distances. The perturbation theory\nbreaks down and the nonlinear equations of QCD are just very difficult -\nsome progress can be obtained numerically using lattices and some other\ntools (the AdS/CFT correspondence has become the most powerful new tool).\n\nIn this sense, I believe that one could use nearly the same criticism not\nonly against string theory, but also QCD itself. However I feel that it\'s\nnot hard to realize that in the QCD case, it would be unreasonable. Not\nonly because of the Nobel prize!\n\nSo what does QCD predict that makes us sure that it\'s right? It predicts\nthe jets in the high energy collissions - "dressed" quarks and gluons. But\npeople qualitatively knew about these things experimentally already before\nQCD, so it was not a real prediction. They also knew about the\norganization of strongly interacting particles into families (with\ndifferent composition of quarks, depending on the particular member of the\nfamily - i.e. of the multiplet). So this was not a "real" prediction\neither. QCD was constructed to agree with the scaling laws - it was an\ninput and one of Gross\'s motivations - but it did not predict much\nafterwards, as long as one talks about some completely new, visible\neffects.\n\nThe advantage of QCD is claimed to be beauty - it is a nice SU(3)\nYang-Mills theory - and the pure QCD has no dimensionless parameters - the\nsame virtue as string theory: the original dimensionless coupling is\nconverted into a dimensionful scale by the dimensional transmutation.\nYang-Mills theory seems to be the unique way how to obtain asymptotic\nfreedom (vanishing of interactions at very high energies) from a quantum\nfield theory.\n\nDavid Gross likes to say that a theory without dimensionless parameters\n(QCD) can now explain all the "anthropic" mysteries from nuclear physics.\nNima Arkani-Hamed correctly points out that it\'s not quite correct because\nthe various "coincidences" relating the masses of the nucleons etc. depend\non all these small parameters like the quark bare masses. Well, I am not\nterribly happy to admit that Nima\'s objection is fair because his\nobjection is a small argument in favor of the anthropic thinking.\nNevertheless I must admit that Nima is right because he is. ;-)\n\nThe success of QCD is that it is really the only theory that explains the\ndata that had been known already before QCD was found - and it\'s able to\nput these data into a coherent framework. And it is a very beautiful\ntheory - it has nice symmetries and no dimensionless parameters in its\n"pure" version. These things were enough for the authors of QCD to know\nthat it was correct as early as in 1975.\n\nWe\'re saying the very same things about string theory. String theory is\nreally the only theory that can agree with the existing facts about\nquantum field theory but also with physics of general relativity i.e. with\ngravity. Of course, there is a difference between QCD and string theory is\nthat QCD has given us some new predictions that were unavailable for the\nprevious rules to understand the strong interactions, and these\npredictions are tested at the 1% accuracy, while string theory is still\nwaiting for the right experiments that will eliminate its critics. Let me\nbe more specific: the 1% accuracy was only achieved in the 1990s, twenty\nyears after the fathers of QCD knew that QCD was correct.\n\nNevertheless, you see that the character of our theories is evolving in a\nparticular direction - even if we study the evolution within the Standard\nModel itself. String theory is just one more step in this progression; it\ncertainly implies no "qualitative" change in our understanding what\nphysics theories are good for. We\'re marching towards more strongly\ncoupled - and more difficult to calculate - theories that may look\n"richer" but that are also increasingly more constrained, and we are using\nincreasingly complex mathematics - and the observations about the\nuniqueness of the consistent solutions of our problems - as our arguments.\nIt is happening simply because the naive, simple math that can be easily\ncalculated and compared with the experiments was already calculated a long\ntime ago.\n\nAs our theories become more mathematical and abstract - which is a\nnecessary process, as I tried to explain - the number of the people who\nactually understand the logic behind these new steps decreases. Not too\nmany "ordinary" people understand relativity; quantum mechanics is even\nmore difficult for most physics fans. Quantum field theory requires a\nspecial training, among other things, and in the case of string theory it\nis simply true that a PhD degree from theoretical physics is not a\nsufficient condition to understand the inevitability of its claims. I\nagree with the critics of string theory that a theoretical physics PhD\nshould be enough to understand string theory, but my ideas how to achieve\nthis goal are very different from theirs. ;-)\n\nAs our theories are becoming more mathematical, we are simultaneously\nrevising the concepts dramatically and we are finding new connections\nbetween the previous concepts, and their limitations that looked\nimpossible previously. The latter was happening in every revolution of\nphysics, including the quantum revolution.\n\nSo I don\'t really understand what is it exactly that makes so many people\nfeel so uneasy about string theory and why. Of course, I understand why\npeople may be frustrated that the progress is slow, but it\'s harder to see\nhow can string theory be blamed for it. Where we\'re going - in the\nperspective of a decade or so - is arguably the right way, and all\nphilosophical properties and trends of this progress agree with what has\nbeen proved fruitful in the past and recently.\n\nMuch of the recent progress, including the construction of QCD, was about\npushing "reductionism" as far as we can. We could not be satisfied with a\nlist of 200 strongly interacting "elementary" particles and their messy\ninteractions; people eventually convinced themselves that the right\nelementary particles are quarks (and gluons), although the hadrons remain\na good description at low energies. In a similar fashion, we cannot be\nsatisfied with the list of the elementary particles of the Standard Model\nplus the graviton, whose interactions furthermore don\'t work at the loop\nlevel, and this is why we are happy to reduce these concepts further to\nthe level of strings (and their non-perturbative friends) - because this\nreduction seems possible which is itself a shocking, nontrivial fact.\nAgain, the previous language of low-energy effective theories remains good\nat long distances.\n\nString theory marvellously has all the desired qualitative features and\nthe quantitative power to explain everything we know about the real world,\nand we know that the unification of quantum field theories with gravity is\na very difficult task and a generic proposed theory usually solves nothing\nat all, while string theory seems to solve a lot. This is why we "know"\nthat string theory is probably correct, even though it may take decades or\neven centuries to convince the critics. But the situation is qualitatively\nanalogous to QCD. The difference is that string theory is even more\ndependent on theoretical arguments than QCD, and it works with much higher\nenergy scales. But there is no qualitative phase transition in the\ndefinition of physics!\n\nWe may be unhappy about the particular developments in the last 1 year or\nperhaps even 5 years or something like that. But every time I see what the\nalternatives could be, it reassures me that we are on the right track. The\nalternatives usually want to return science at least 40 years into the\npast, and perhaps to the 19th century.\n\nIt\'s hard to convince anyone about the analogy if he or she does not feel\nit this way, but let me try anyway. There are creationists who reject\nevolution. Let\'s call them the 1860 crackpots. There are people who reject\nspecial relativity, right? Let\'s call them the 1905 crackpots. Some of\nthese insist on the luminiferous aether (even though some of them may call\nit spin foam). Then there are people that reject general relativity, the\n1916 crackpots, and quantum mechanics, the 1926 crackpots. Then there are\nthinkers who reject the (divergent) loop diagrams and their\nregularization; let\'s call them the 1949 crackpots, and who reject quarks,\nwho are called the 1973 speculative colleagues.\n\nAs I go towards the present, physics of these topics becomes increasingly\ndifficult, requires higher education, expertise - and I think that\nsomething remotely similar exists in any other sufficiently complex field\nof science, including e.g. number theory, too. Proving the Fermat Last\nTheorem is a pretty fancy thing that requires some new technology, does\nnot it?\n\nThe people who reject our understanding collected in the last 20 years\nthat string theory is the only way to exceed the limitations (and repair\nthe divergent behavior) of quantum field theory and classical GR - and who\nreject hundreds of the particular more detailed insights about string\ntheory and quantum field theory that we\'ve made and we will never unlearn\n- are, of course, not quite as clear crackpots as the previous categories\nbecause they only failed to follow (or decided to deny) the last 20 years\nand the questions studied by string theory are still "work in progress".\nBut ignoring these insights still seems as a pretty bad starting point for\nmaking contributions to physics - or trying to direct physics - in 2004.\n\nWhat I find more obvious is that the people who want to ignore string\ntheory actually want to neglect some older, well-established insights as\nwell - the renormalization group, semiclassical gravity (of Hawking), and\nothers - perhaps even perturbation theory or the S-matrix as the important\nconcepts in quantum relativistic physics.\n\nOne may ask why I feel so sure that string theory is most likely on the\nright track. It is a combination of both aspects: the impressive power of\nstring theory demonstrated in many contexts, but also the naive picture of\nphysics that the proponents of "alternatives" want to advocate. One must\nalways choose some principles when he or she tries to go beyond the known\nrealm. But the non-stringy people in physics just generally choose\nprinciples that look very simple-minded and obsolete. It\'s pretty hard to\nexplain non-technically and exactly why I almost always feel so certain\nabout it. I understand why the people feel that my certainty looks like\n"religion" - it would also look like religion to me if I did not know most\nof the things I know, or if they were not organized in my brain the same\nway.\n\nAether, hidden variables: repeating the errors forever\n\nBut it\'s like if you remember some error that you did 15 years ago, and\nyou later understood perfectly why it was silly and how your viewpoint on\nthe problem was uninformed and narrow-minded and 19th-century-like (or\nperhaps it was not you, just some other people around). Today, you may\nunderstand that all your confusion 15 years ago was unjustified, and that\nthere exists a completely meaningful and rigorous answer to all your\nquestions you had - and these answers are often different than you\nthought. Also, you may realize today that you used to neglect a huge\namount of important knowledge - you were just too ignorant about too many\nthings - which invalidates all your previous reasoning.\n\nAnd suddenly, 15 years later, someone comes with the same or even more\nunlikely approach and claims that it is an important idea that is meant to\nrevolutionize physics.\n\nLike those loop quantum gravity people. Most of them probably don\'t know\nthat Maxwell did not write just his equations; he constructed a few\ndiscrete models of aether. George FitzGerald even constructed working\nmodels of such an aether that produced the transverse electromagnetic\nwaves! And this model really worked. Such problems involving gears and\nwheels were what the 19th century physics was about. All this aether,\nsomething discrete that fills the vacuum, was exactly the trash that\nEinstein had to throw away, and this non-trivial act was one of the main\nreasons why Einstein was such a revolutionary. Of course, Einstein could\nhave done it because he was standing on the shoulders of giants, including\nHendrik Lorentz.\n\nAnd then 100 years later someone comes and proposes a new model of aether,\na discrete substrate filling the vacuum. Now it should explain gravity\ninstead of electromagnetism. A difference is that the "modern" models,\nunlike FitzGerald\'s model, quite obviously do not work and cannot give you\nthe right physics. No 21st century FitzGerald will be able to construct a\nmechanical model of a spin foam that behaves like general relativity -\nbecause it does not behave this way. These models cannot agree with\nspecial relativity because of the very same reasons as the 19th century\naether. Another difference is that it is not 1860, but 2004. The progress\nin science was not so terribly non-linear after all - and it is going in\nsome direction. There are just too many people who want to revert science\nand return us to the trees. In many cases, one can easily decide that\ncertain progress would be "negative".\n\nIn physics, we have learned something, and it is impossible to "unlearn"\nmost of these insights. There is a lot of recent insights that will stay\nwith us even if string theory will be proved irrelevant for the\nexperiments. But let\'s not be too pessimistic. String theory agrees with\nall the basic (and often also with the non-basic) discoveries and contains\nall the methods of the previous successful theories - quantum field\ntheory, general relativity, gauge theories, chiral fermions organized into\nfamilies, Higgs mechanism, confinement, relations between them,\nRenormalization Group effects, non-perturbative physics, the S-matrix.\nIt\'s the only known theory different from the old, incomplete framework of\nquantum field theory that can do everything good that the old theories\nwere able to do as well.\n\nThe self-described "competitors" just don\'t care about the actual physics\n- I really mean primarily experimental physics. They don\'t really care\nwhether their theory has something new to say about QCD, general\nrelativity, black holes, particle spectrum, scattering amplitudes - the\nphysical phenomena that really exist. They don\'t even care whether their\ntheory is consistent with the older insights. They prefer to extend some\nobsolete and narrow-minded dogmas - such as "the world is discrete" or\n"the vacuum must be made of something" - dogmas that have really nothing\nto do with the discoveries physics made in the last 200 years. Dogmas that\nhave been more or less falsified. And that makes a difference.\n\nSome people want physics to become "postmodern" and allow hundreds of\ndifferent trends that revive various old theories of aether,\nLorentz-FitzGerald contractions, hidden variables, and many other wrong\nthings from physics of the past that our heroes had to struggle with for\nso long before they saw the new light.\n\nI would really prefer if theoretical physics were interrupted completely\nrather than becoming a "diverse" arena of all these pseudoscientists who\nare rejecting random principles we learned - as well as the majority of\nthe actual data - and who keep on constructing toy models with very\nlimited ability to agree with anything we actually observe: interrupted\nphysics can continue in the future once people become more reasonable and\ncreative. On the other hand, a return to the proto-science or even\npseudo-science would effectively convert the culture of theoretical\nphysicists into the culture of intellectual monkeys once again.\n\nThe string theorists know what they\'re doing and how their theory fits all\nsuccessful - and experimentally verified - previous insights about Nature;\nothers don\'t. Our civilization certainly does not have enough resources to\npay for all conceivable proto-theories that are comparably attractive as\nloop quantum gravity - simply because the space of such\nnot-terribly-serious ideas off the track is virtually infinite.\n\nConcerning string theory: don\'t get me wrong: I am far from being certain\nthat we will have great new successes in the next 2 years, for example.\nAnd it\'s not clear in advance what the LHC will see. I am not even sure\nwhether the number of string theorists is already too high or still too\nsmall. But most of my statements are based on a comparison of string\ntheory with the alternatives, and in this respect, my feeling is that\nthere is no rational justification at this point why the alternatives\nshould "grow".\n_________________________________________ _____________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\nWebs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Let me offer a text from my blog, attempting to provoke a discussion.
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http://motls.blogspot.com/2004/12/theories-are-increasingly-theoretical.html
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This text follows my discussions with Nima Arkani-Hamed and David Goss.
Some people don't like the fact that the arguments in string theory are
increasingly theoretical in nature, and that our theories seem to give us
less exactly calculable sharp predictions that are verified
experimentally.
However: it's not just string theory: the whole particle physics has been
becoming increasingly theoretical and string theory just continues in the
same direction. What do I mean?
QED, Electroweak theory, QCD: increasing groups, decreasing accuracy
The peak of the old-fashioned quantitative predictivity of very particular
facts in physics was QED which stands for Quantum Electrodynamics. You
know that people could have calculated its predictions already 50 years
ago, including the quantum loop corrections, even though they did not
quite understand why their methods were working (The Renormalization
Group), and the most precise predictions - like the anomalous magnetic
moment of the electron - have been successfully tested with the accuracy
of 13 decimal places!
Then the physicists found the electroweak force that naturally predicted
the neutral currents, W bosons, the Z boson, and so forth. It is also a
relatively very predictive theory (Glashow, Salam, Weinberg) although its
predictions were never tested as exactly as for QED. Nevertheless, all the
cross sections and decay rates are measured rather precisely, the
electroweak scattering is "clean".
Another step: QCD
Then you go to QCD which is now an accepted and "Nobelized" part of our
theoretical canon. QCD, in some sense, confirmed the things people
"guessed" by other means, and one might criticize it using some very
similar words as those often applied to string theory by its critics.
You know, QCD is claimed to be a theory of the strong force, but it talks
about the gluons, quarks, and especially their three colors, three
concepts that were never directly seen; and according to QCD no one will
ever see either of them. Also, no one has been able to calculate the
properties of the proton, neutron, and nuclei - which used to be thought
to be the objects from the strong interaction - from this theory too well.
The actual calculations often rely on some properties of the quark and
gluon distribution functions, and the critics might say that these
functions have never been really derived from QCD. Even if one accepts the
existence of quarks, they were not really invented by QCD: Gell-Mann
received a Nobel prize for quarks in 1969, five years before QCD was
proposed. The new quark flavors, such as the c-quark found in the J/\psi
particle, were naturally predicted by the electroweak theory (the GIM
mechanism from 1970), not by QCD. In this respect, QCD seems to have had
no "striking" new predictions. So why do we say that QCD is a good theory
of the strong interaction?
The low-energy properties of the hadrons have not been calculated
accurately enough simply because QCD is a pretty difficult machine to
calculate with at low energies - but this difficulty is a fact of Nature.
In the same way, the vacuum structure in string theory is also rather
complicated, which also seems to be a fact of Nature. At high energies,
the quarks are almost free (due to asymptotic freedom, which is really
what our friends got their 2004 Nobel prize for). If the quarks are free,
perturbation theory is great and one can easily and precisely calculate
the high-energy events. But for the effects important for the nuclear
physics, the interaction is strong - more or less by definition. QCD is a
strongly coupled theory at longer distances. The perturbation theory
breaks down and the nonlinear equations of QCD are just very difficult -
some progress can be obtained numerically using lattices and some other
tools (the AdS/CFT correspondence has become the most powerful new tool).
In this sense, I believe that one could use nearly the same criticism not
only against string theory, but also QCD itself. However I feel that it's
not hard to realize that in the QCD case, it would be unreasonable. Not
only because of the Nobel prize!
So what does QCD predict that makes us sure that it's right? It predicts
the jets in the high energy collissions - "dressed" quarks and gluons. But
people qualitatively knew about these things experimentally already before
QCD, so it was not a real prediction. They also knew about the
organization of strongly interacting particles into families (with
different composition of quarks, depending on the particular member of the
family - i.e. of the multiplet). So this was not a "real" prediction
either. QCD was constructed to agree with the scaling laws - it was an
input and one of Gross's motivations - but it did not predict much
afterwards, as long as one talks about some completely new, visible
effects.
The advantage of QCD is claimed to be beauty - it is a nice SU(3)
Yang-Mills theory - and the pure QCD has no dimensionless parameters - the
same virtue as string theory: the original dimensionless coupling is
converted into a dimensionful scale by the dimensional transmutation.
Yang-Mills theory seems to be the unique way how to obtain asymptotic
freedom (vanishing of interactions at very high energies) from a quantum
field theory.
David Gross likes to say that a theory without dimensionless parameters
(QCD) can now explain all the "anthropic" mysteries from nuclear physics.
Nima Arkani-Hamed correctly points out that it's not quite correct because
the various "coincidences" relating the masses of the nucleons etc. depend
on all these small parameters like the quark bare masses. Well, I am not
terribly happy to admit that Nima's objection is fair because his
objection is a small argument in favor of the anthropic thinking.
Nevertheless I must admit that Nima is right because he is. ;-)
The success of QCD is that it is really the only theory that explains the
data that had been known already before QCD was found - and it's able to
put these data into a coherent framework. And it is a very beautiful
theory - it has nice symmetries and no dimensionless parameters in its
"pure" version. These things were enough for the authors of QCD to know
that it was correct as early as in 1975.
We're saying the very same things about string theory. String theory is
really the only theory that can agree with the existing facts about
quantum field theory but also with physics of general relativity i.e. with
gravity. Of course, there is a difference between QCD and string theory is
that QCD has given us some new predictions that were unavailable for the
previous rules to understand the strong interactions, and these
predictions are tested at the 1% accuracy, while string theory is still
waiting for the right experiments that will eliminate its critics. Let me
be more specific: the 1% accuracy was only achieved in the 1990s, twenty
years after the fathers of QCD knew that QCD was correct.
Nevertheless, you see that the character of our theories is evolving in a
particular direction - even if we study the evolution within the Standard
Model itself. String theory is just one more step in this progression; it
certainly implies no "qualitative" change in our understanding what
physics theories are good for. We're marching towards more strongly
coupled - and more difficult to calculate - theories that may look
"richer" but that are also increasingly more constrained, and we are using
increasingly complex mathematics - and the observations about the
uniqueness of the consistent solutions of our problems - as our arguments.
It is happening simply because the naive, simple math that can be easily
calculated and compared with the experiments was already calculated a long
time ago.
As our theories become more mathematical and abstract - which is a
necessary process, as I tried to explain - the number of the people who
actually understand the logic behind these new steps decreases. Not too
many "ordinary" people understand relativity; quantum mechanics is even
more difficult for most physics fans. Quantum field theory requires a
special training, among other things, and in the case of string theory it
is simply true that a PhD degree from theoretical physics is not a
sufficient condition to understand the inevitability of its claims. I
agree with the critics of string theory that a theoretical physics PhD
should be enough to understand string theory, but my ideas how to achieve
this goal are very different from theirs. ;-)
As our theories are becoming more mathematical, we are simultaneously
revising the concepts dramatically and we are finding new connections
between the previous concepts, and their limitations that looked
impossible previously. The latter was happening in every revolution of
physics, including the quantum revolution.
So I don't really understand what is it exactly that makes so many people
feel so uneasy about string theory and why. Of course, I understand why
people may be frustrated that the progress is slow, but it's harder to see
how can string theory be blamed for it. Where we're going - in the
perspective of a decade or so - is arguably the right way, and all
philosophical properties and trends of this progress agree with what has
been proved fruitful in the past and recently.
Much of the recent progress, including the construction of QCD, was about
pushing "reductionism" as far as we can. We could not be satisfied with a
list of 200 strongly interacting "elementary" particles and their messy
interactions; people eventually convinced themselves that the right
elementary particles are quarks (and gluons), although the hadrons remain
a good description at low energies. In a similar fashion, we cannot be
satisfied with the list of the elementary particles of the Standard Model
plus the graviton, whose interactions furthermore don't work at the loop
level, and this is why we are happy to reduce these concepts further to
the level of strings (and their non-perturbative friends) - because this
reduction seems possible which is itself a shocking, nontrivial fact.
Again, the previous language of low-energy effective theories remains good
at long distances.
String theory marvellously has all the desired qualitative features and
the quantitative power to explain everything we know about the real world,
and we know that the unification of quantum field theories with gravity is
a very difficult task and a generic proposed theory usually solves nothing
at all, while string theory seems to solve a lot. This is why we "know"
that string theory is probably correct, even though it may take decades or
even centuries to convince the critics. But the situation is qualitatively
analogous to QCD. The difference is that string theory is even more
dependent on theoretical arguments than QCD, and it works with much higher
energy scales. But there is no qualitative phase transition in the
definition of physics!
We may be unhappy about the particular developments in the last 1 year or
perhaps even 5 years or something like that. But every time I see what the
alternatives could be, it reassures me that we are on the right track. The
alternatives usually want to return science at least 40 years into the
past, and perhaps to the 19th century.
It's hard to convince anyone about the analogy if he or she does not feel
it this way, but let me try anyway. There are creationists who reject
evolution. Let's call them the 1860 crackpots. There are people who reject
special relativity, right? Let's call them the 1905 crackpots. Some of
these insist on the luminiferous aether (even though some of them may call
it spin foam). Then there are people that reject general relativity, the
1916 crackpots, and quantum mechanics, the 1926 crackpots. Then there are
thinkers who reject the (divergent) loop diagrams and their
regularization; let's call them the 1949 crackpots, and who reject quarks,
who are called the 1973 speculative colleagues.
As I go towards the present, physics of these topics becomes increasingly
difficult, requires higher education, expertise - and I think that
something remotely similar exists in any other sufficiently complex field
of science, including e.g. number theory, too. Proving the Fermat Last
Theorem is a pretty fancy thing that requires some new technology, does
not it?
The people who reject our understanding collected in the last 20 years
that string theory is the only way to exceed the limitations (and repair
the divergent behavior) of quantum field theory and classical GR - and who
reject hundreds of the particular more detailed insights about string
theory and quantum field theory that we've made and we will never unlearn
- are, of course, not quite as clear crackpots as the previous categories
because they only failed to follow (or decided to deny) the last 20 years
and the questions studied by string theory are still "work in progress".
But ignoring these insights still seems as a pretty bad starting point for
making contributions to physics - or trying to direct physics - in 2004.
What I find more obvious is that the people who want to ignore string
theory actually want to neglect some older, well-established insights as
well - the renormalization group, semiclassical gravity (of Hawking), and
others - perhaps even perturbation theory or the S-matrix as the important
concepts in quantum relativistic physics.
One may ask why I feel so sure that string theory is most likely on the
right track. It is a combination of both aspects: the impressive power of
string theory demonstrated in many contexts, but also the naive picture of
physics that the proponents of "alternatives" want to advocate. One must
always choose some principles when he or she tries to go beyond the known
realm. But the non-stringy people in physics just generally choose
principles that look very simple-minded and obsolete. It's pretty hard to
explain non-technically and exactly why I almost always feel so certain
about it. I understand why the people feel that my certainty looks like
"religion" - it would also look like religion to me if I did not know most
of the things I know, or if they were not organized in my brain the same
way.
Aether, hidden variables: repeating the errors forever
But it's like if you remember some error that you did 15 years ago, and
you later understood perfectly why it was silly and how your viewpoint on
the problem was uninformed and narrow-minded and 19th-century-like (or
perhaps it was not you, just some other people around). Today, you may
understand that all your confusion 15 years ago was unjustified, and that
there exists a completely meaningful and rigorous answer to all your
questions you had - and these answers are often different than you
thought. Also, you may realize today that you used to neglect a huge
amount of important knowledge - you were just too ignorant about too many
things - which invalidates all your previous reasoning.
And suddenly, 15 years later, someone comes with the same or even more
unlikely approach and claims that it is an important idea that is meant to
revolutionize physics.
Like those loop quantum gravity people. Most of them probably don't know
that Maxwell did not write just his equations; he constructed a few
discrete models of aether. George FitzGerald even constructed working
models of such an aether that produced the transverse electromagnetic
waves! And this model really worked. Such problems involving gears and
wheels were what the 19th century physics was about. All this aether,
something discrete that fills the vacuum, was exactly the trash that
Einstein had to throw away, and this non-trivial act was one of the main
reasons why Einstein was such a revolutionary. Of course, Einstein could
have done it because he was standing on the shoulders of giants, including
Hendrik Lorentz.
And then 100 years later someone comes and proposes a new model of aether,
a discrete substrate filling the vacuum. Now it should explain gravity
instead of electromagnetism. A difference is that the "modern" models,
unlike FitzGerald's model, quite obviously do not work and cannot give you
the right physics. No 21st century FitzGerald will be able to construct a
mechanical model of a spin foam that behaves like general relativity -
because it does not behave this way. These models cannot agree with
special relativity because of the very same reasons as the 19th century
aether. Another difference is that it is not 1860, but 2004. The progress
in science was not so terribly non-linear after all - and it is going in
some direction. There are just too many people who want to revert science
and return us to the trees. In many cases, one can easily decide that
certain progress would be "negative".
In physics, we have learned something, and it is impossible to "unlearn"
most of these insights. There is a lot of recent insights that will stay
with us even if string theory will be proved irrelevant for the
experiments. But let's not be too pessimistic. String theory agrees with
all the basic (and often also with the non-basic) discoveries and contains
all the methods of the previous successful theories - quantum field
theory, general relativity, gauge theories, chiral fermions organized into
families, Higgs mechanism, confinement, relations between them,
Renormalization Group effects, non-perturbative physics, the S-matrix.
It's the only known theory different from the old, incomplete framework of
quantum field theory that can do everything good that the old theories
were able to do as well.
The self-described "competitors" just don't care about the actual physics
- I really mean primarily experimental physics. They don't really care
whether their theory has something new to say about QCD, general
relativity, black holes, particle spectrum, scattering amplitudes - the
physical phenomena that really exist. They don't even care whether their
theory is consistent with the older insights. They prefer to extend some
obsolete and narrow-minded dogmas - such as "the world is discrete" or
"the vacuum must be made of something" - dogmas that have really nothing
to do with the discoveries physics made in the last 200 years. Dogmas that
have been more or less falsified. And that makes a difference.
Some people want physics to become "postmodern" and allow hundreds of
different trends that revive various old theories of aether,
Lorentz-FitzGerald contractions, hidden variables, and many other wrong
things from physics of the past that our heroes had to struggle with for
so long before they saw the new light.
I would really prefer if theoretical physics were interrupted completely
rather than becoming a "diverse" arena of all these pseudoscientists who
are rejecting random principles we learned - as well as the majority of
the actual data - and who keep on constructing toy models with very
limited ability to agree with anything we actually observe: interrupted
physics can continue in the future once people become more reasonable and
creative. On the other hand, a return to the proto-science or even
pseudo-science would effectively convert the culture of theoretical
physicists into the culture of intellectual monkeys once again.
The string theorists know what they're doing and how their theory fits all
successful - and experimentally verified - previous insights about Nature;
others don't. Our civilization certainly does not have enough resources to
pay for all conceivable proto-theories that are comparably attractive as
loop quantum gravity - simply because the space of such
not-terribly-serious ideas off the track is virtually infinite.
Concerning string theory: don't get me wrong: I am far from being certain
that we will have great new successes in the next 2 years, for example.
And it's not clear in advance what the LHC will see. I am not even sure
whether the number of string theorists is already too high or still too
small. But most of my statements are based on a comparison of string
theory with the alternatives, and in this respect, my feeling is that
there is no rational justification at this point why the alternatives
should "grow".
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
================================================== ====================
http://motls.blogspot.com/2004/12/theories-are-increasingly-theoretical.html
================================================== ====================
This text follows my discussions with Nima Arkani-Hamed and David Goss.
Some people don't like the fact that the arguments in string theory are
increasingly theoretical in nature, and that our theories seem to give us
less exactly calculable sharp predictions that are verified
experimentally.
However: it's not just string theory: the whole particle physics has been
becoming increasingly theoretical and string theory just continues in the
same direction. What do I mean?
QED, Electroweak theory, QCD: increasing groups, decreasing accuracy
The peak of the old-fashioned quantitative predictivity of very particular
facts in physics was QED which stands for Quantum Electrodynamics. You
know that people could have calculated its predictions already 50 years
ago, including the quantum loop corrections, even though they did not
quite understand why their methods were working (The Renormalization
Group), and the most precise predictions - like the anomalous magnetic
moment of the electron - have been successfully tested with the accuracy
of 13 decimal places!
Then the physicists found the electroweak force that naturally predicted
the neutral currents, W bosons, the Z boson, and so forth. It is also a
relatively very predictive theory (Glashow, Salam, Weinberg) although its
predictions were never tested as exactly as for QED. Nevertheless, all the
cross sections and decay rates are measured rather precisely, the
electroweak scattering is "clean".
Another step: QCD
Then you go to QCD which is now an accepted and "Nobelized" part of our
theoretical canon. QCD, in some sense, confirmed the things people
"guessed" by other means, and one might criticize it using some very
similar words as those often applied to string theory by its critics.
You know, QCD is claimed to be a theory of the strong force, but it talks
about the gluons, quarks, and especially their three colors, three
concepts that were never directly seen; and according to QCD no one will
ever see either of them. Also, no one has been able to calculate the
properties of the proton, neutron, and nuclei - which used to be thought
to be the objects from the strong interaction - from this theory too well.
The actual calculations often rely on some properties of the quark and
gluon distribution functions, and the critics might say that these
functions have never been really derived from QCD. Even if one accepts the
existence of quarks, they were not really invented by QCD: Gell-Mann
received a Nobel prize for quarks in 1969, five years before QCD was
proposed. The new quark flavors, such as the c-quark found in the J/\psi
particle, were naturally predicted by the electroweak theory (the GIM
mechanism from 1970), not by QCD. In this respect, QCD seems to have had
no "striking" new predictions. So why do we say that QCD is a good theory
of the strong interaction?
The low-energy properties of the hadrons have not been calculated
accurately enough simply because QCD is a pretty difficult machine to
calculate with at low energies - but this difficulty is a fact of Nature.
In the same way, the vacuum structure in string theory is also rather
complicated, which also seems to be a fact of Nature. At high energies,
the quarks are almost free (due to asymptotic freedom, which is really
what our friends got their 2004 Nobel prize for). If the quarks are free,
perturbation theory is great and one can easily and precisely calculate
the high-energy events. But for the effects important for the nuclear
physics, the interaction is strong - more or less by definition. QCD is a
strongly coupled theory at longer distances. The perturbation theory
breaks down and the nonlinear equations of QCD are just very difficult -
some progress can be obtained numerically using lattices and some other
tools (the AdS/CFT correspondence has become the most powerful new tool).
In this sense, I believe that one could use nearly the same criticism not
only against string theory, but also QCD itself. However I feel that it's
not hard to realize that in the QCD case, it would be unreasonable. Not
only because of the Nobel prize!
So what does QCD predict that makes us sure that it's right? It predicts
the jets in the high energy collissions - "dressed" quarks and gluons. But
people qualitatively knew about these things experimentally already before
QCD, so it was not a real prediction. They also knew about the
organization of strongly interacting particles into families (with
different composition of quarks, depending on the particular member of the
family - i.e. of the multiplet). So this was not a "real" prediction
either. QCD was constructed to agree with the scaling laws - it was an
input and one of Gross's motivations - but it did not predict much
afterwards, as long as one talks about some completely new, visible
effects.
The advantage of QCD is claimed to be beauty - it is a nice SU(3)
Yang-Mills theory - and the pure QCD has no dimensionless parameters - the
same virtue as string theory: the original dimensionless coupling is
converted into a dimensionful scale by the dimensional transmutation.
Yang-Mills theory seems to be the unique way how to obtain asymptotic
freedom (vanishing of interactions at very high energies) from a quantum
field theory.
David Gross likes to say that a theory without dimensionless parameters
(QCD) can now explain all the "anthropic" mysteries from nuclear physics.
Nima Arkani-Hamed correctly points out that it's not quite correct because
the various "coincidences" relating the masses of the nucleons etc. depend
on all these small parameters like the quark bare masses. Well, I am not
terribly happy to admit that Nima's objection is fair because his
objection is a small argument in favor of the anthropic thinking.
Nevertheless I must admit that Nima is right because he is. ;-)
The success of QCD is that it is really the only theory that explains the
data that had been known already before QCD was found - and it's able to
put these data into a coherent framework. And it is a very beautiful
theory - it has nice symmetries and no dimensionless parameters in its
"pure" version. These things were enough for the authors of QCD to know
that it was correct as early as in 1975.
We're saying the very same things about string theory. String theory is
really the only theory that can agree with the existing facts about
quantum field theory but also with physics of general relativity i.e. with
gravity. Of course, there is a difference between QCD and string theory is
that QCD has given us some new predictions that were unavailable for the
previous rules to understand the strong interactions, and these
predictions are tested at the 1% accuracy, while string theory is still
waiting for the right experiments that will eliminate its critics. Let me
be more specific: the 1% accuracy was only achieved in the 1990s, twenty
years after the fathers of QCD knew that QCD was correct.
Nevertheless, you see that the character of our theories is evolving in a
particular direction - even if we study the evolution within the Standard
Model itself. String theory is just one more step in this progression; it
certainly implies no "qualitative" change in our understanding what
physics theories are good for. We're marching towards more strongly
coupled - and more difficult to calculate - theories that may look
"richer" but that are also increasingly more constrained, and we are using
increasingly complex mathematics - and the observations about the
uniqueness of the consistent solutions of our problems - as our arguments.
It is happening simply because the naive, simple math that can be easily
calculated and compared with the experiments was already calculated a long
time ago.
As our theories become more mathematical and abstract - which is a
necessary process, as I tried to explain - the number of the people who
actually understand the logic behind these new steps decreases. Not too
many "ordinary" people understand relativity; quantum mechanics is even
more difficult for most physics fans. Quantum field theory requires a
special training, among other things, and in the case of string theory it
is simply true that a PhD degree from theoretical physics is not a
sufficient condition to understand the inevitability of its claims. I
agree with the critics of string theory that a theoretical physics PhD
should be enough to understand string theory, but my ideas how to achieve
this goal are very different from theirs. ;-)
As our theories are becoming more mathematical, we are simultaneously
revising the concepts dramatically and we are finding new connections
between the previous concepts, and their limitations that looked
impossible previously. The latter was happening in every revolution of
physics, including the quantum revolution.
So I don't really understand what is it exactly that makes so many people
feel so uneasy about string theory and why. Of course, I understand why
people may be frustrated that the progress is slow, but it's harder to see
how can string theory be blamed for it. Where we're going - in the
perspective of a decade or so - is arguably the right way, and all
philosophical properties and trends of this progress agree with what has
been proved fruitful in the past and recently.
Much of the recent progress, including the construction of QCD, was about
pushing "reductionism" as far as we can. We could not be satisfied with a
list of 200 strongly interacting "elementary" particles and their messy
interactions; people eventually convinced themselves that the right
elementary particles are quarks (and gluons), although the hadrons remain
a good description at low energies. In a similar fashion, we cannot be
satisfied with the list of the elementary particles of the Standard Model
plus the graviton, whose interactions furthermore don't work at the loop
level, and this is why we are happy to reduce these concepts further to
the level of strings (and their non-perturbative friends) - because this
reduction seems possible which is itself a shocking, nontrivial fact.
Again, the previous language of low-energy effective theories remains good
at long distances.
String theory marvellously has all the desired qualitative features and
the quantitative power to explain everything we know about the real world,
and we know that the unification of quantum field theories with gravity is
a very difficult task and a generic proposed theory usually solves nothing
at all, while string theory seems to solve a lot. This is why we "know"
that string theory is probably correct, even though it may take decades or
even centuries to convince the critics. But the situation is qualitatively
analogous to QCD. The difference is that string theory is even more
dependent on theoretical arguments than QCD, and it works with much higher
energy scales. But there is no qualitative phase transition in the
definition of physics!
We may be unhappy about the particular developments in the last 1 year or
perhaps even 5 years or something like that. But every time I see what the
alternatives could be, it reassures me that we are on the right track. The
alternatives usually want to return science at least 40 years into the
past, and perhaps to the 19th century.
It's hard to convince anyone about the analogy if he or she does not feel
it this way, but let me try anyway. There are creationists who reject
evolution. Let's call them the 1860 crackpots. There are people who reject
special relativity, right? Let's call them the 1905 crackpots. Some of
these insist on the luminiferous aether (even though some of them may call
it spin foam). Then there are people that reject general relativity, the
1916 crackpots, and quantum mechanics, the 1926 crackpots. Then there are
thinkers who reject the (divergent) loop diagrams and their
regularization; let's call them the 1949 crackpots, and who reject quarks,
who are called the 1973 speculative colleagues.
As I go towards the present, physics of these topics becomes increasingly
difficult, requires higher education, expertise - and I think that
something remotely similar exists in any other sufficiently complex field
of science, including e.g. number theory, too. Proving the Fermat Last
Theorem is a pretty fancy thing that requires some new technology, does
not it?
The people who reject our understanding collected in the last 20 years
that string theory is the only way to exceed the limitations (and repair
the divergent behavior) of quantum field theory and classical GR - and who
reject hundreds of the particular more detailed insights about string
theory and quantum field theory that we've made and we will never unlearn
- are, of course, not quite as clear crackpots as the previous categories
because they only failed to follow (or decided to deny) the last 20 years
and the questions studied by string theory are still "work in progress".
But ignoring these insights still seems as a pretty bad starting point for
making contributions to physics - or trying to direct physics - in 2004.
What I find more obvious is that the people who want to ignore string
theory actually want to neglect some older, well-established insights as
well - the renormalization group, semiclassical gravity (of Hawking), and
others - perhaps even perturbation theory or the S-matrix as the important
concepts in quantum relativistic physics.
One may ask why I feel so sure that string theory is most likely on the
right track. It is a combination of both aspects: the impressive power of
string theory demonstrated in many contexts, but also the naive picture of
physics that the proponents of "alternatives" want to advocate. One must
always choose some principles when he or she tries to go beyond the known
realm. But the non-stringy people in physics just generally choose
principles that look very simple-minded and obsolete. It's pretty hard to
explain non-technically and exactly why I almost always feel so certain
about it. I understand why the people feel that my certainty looks like
"religion" - it would also look like religion to me if I did not know most
of the things I know, or if they were not organized in my brain the same
way.
Aether, hidden variables: repeating the errors forever
But it's like if you remember some error that you did 15 years ago, and
you later understood perfectly why it was silly and how your viewpoint on
the problem was uninformed and narrow-minded and 19th-century-like (or
perhaps it was not you, just some other people around). Today, you may
understand that all your confusion 15 years ago was unjustified, and that
there exists a completely meaningful and rigorous answer to all your
questions you had - and these answers are often different than you
thought. Also, you may realize today that you used to neglect a huge
amount of important knowledge - you were just too ignorant about too many
things - which invalidates all your previous reasoning.
And suddenly, 15 years later, someone comes with the same or even more
unlikely approach and claims that it is an important idea that is meant to
revolutionize physics.
Like those loop quantum gravity people. Most of them probably don't know
that Maxwell did not write just his equations; he constructed a few
discrete models of aether. George FitzGerald even constructed working
models of such an aether that produced the transverse electromagnetic
waves! And this model really worked. Such problems involving gears and
wheels were what the 19th century physics was about. All this aether,
something discrete that fills the vacuum, was exactly the trash that
Einstein had to throw away, and this non-trivial act was one of the main
reasons why Einstein was such a revolutionary. Of course, Einstein could
have done it because he was standing on the shoulders of giants, including
Hendrik Lorentz.
And then 100 years later someone comes and proposes a new model of aether,
a discrete substrate filling the vacuum. Now it should explain gravity
instead of electromagnetism. A difference is that the "modern" models,
unlike FitzGerald's model, quite obviously do not work and cannot give you
the right physics. No 21st century FitzGerald will be able to construct a
mechanical model of a spin foam that behaves like general relativity -
because it does not behave this way. These models cannot agree with
special relativity because of the very same reasons as the 19th century
aether. Another difference is that it is not 1860, but 2004. The progress
in science was not so terribly non-linear after all - and it is going in
some direction. There are just too many people who want to revert science
and return us to the trees. In many cases, one can easily decide that
certain progress would be "negative".
In physics, we have learned something, and it is impossible to "unlearn"
most of these insights. There is a lot of recent insights that will stay
with us even if string theory will be proved irrelevant for the
experiments. But let's not be too pessimistic. String theory agrees with
all the basic (and often also with the non-basic) discoveries and contains
all the methods of the previous successful theories - quantum field
theory, general relativity, gauge theories, chiral fermions organized into
families, Higgs mechanism, confinement, relations between them,
Renormalization Group effects, non-perturbative physics, the S-matrix.
It's the only known theory different from the old, incomplete framework of
quantum field theory that can do everything good that the old theories
were able to do as well.
The self-described "competitors" just don't care about the actual physics
- I really mean primarily experimental physics. They don't really care
whether their theory has something new to say about QCD, general
relativity, black holes, particle spectrum, scattering amplitudes - the
physical phenomena that really exist. They don't even care whether their
theory is consistent with the older insights. They prefer to extend some
obsolete and narrow-minded dogmas - such as "the world is discrete" or
"the vacuum must be made of something" - dogmas that have really nothing
to do with the discoveries physics made in the last 200 years. Dogmas that
have been more or less falsified. And that makes a difference.
Some people want physics to become "postmodern" and allow hundreds of
different trends that revive various old theories of aether,
Lorentz-FitzGerald contractions, hidden variables, and many other wrong
things from physics of the past that our heroes had to struggle with for
so long before they saw the new light.
I would really prefer if theoretical physics were interrupted completely
rather than becoming a "diverse" arena of all these pseudoscientists who
are rejecting random principles we learned - as well as the majority of
the actual data - and who keep on constructing toy models with very
limited ability to agree with anything we actually observe: interrupted
physics can continue in the future once people become more reasonable and
creative. On the other hand, a return to the proto-science or even
pseudo-science would effectively convert the culture of theoretical
physicists into the culture of intellectual monkeys once again.
The string theorists know what they're doing and how their theory fits all
successful - and experimentally verified - previous insights about Nature;
others don't. Our civilization certainly does not have enough resources to
pay for all conceivable proto-theories that are comparably attractive as
loop quantum gravity - simply because the space of such
not-terribly-serious ideas off the track is virtually infinite.
Concerning string theory: don't get me wrong: I am far from being certain
that we will have great new successes in the next 2 years, for example.
And it's not clear in advance what the LHC will see. I am not even sure
whether the number of string theorists is already too high or still too
small. But most of my statements are based on a comparison of string
theory with the alternatives, and in this respect, my feeling is that
there is no rational justification at this point why the alternatives
should "grow".
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^