The Stormy Onset of Group Theory in the New Quantum Mechanics

When I first started studying quantum mechanics I read quite a bit about the remarkable history of the subject, especially about the brief period from 1925-27 when the subject grew dramatically out of the incoherent ideas of the old quantum theory to the full quantum mechanical formalism that is still taught today. This was the work of a small group of physicists: especially Heisenberg, Born and Jordan in Göttingen, Schrödinger in Zurich, Dirac in Cambridge, and Pauli in Hamburg. Recently I’ve been reading again about some of this history, but paying attention especially to the interactions of mathematics and physics during these years. An excellent very recent article that covers some of this is by Luisa Bonolis, entitled “From the Rise of the Group Concept to the Stormy Onset of Group Theory in the New Quantum Mechanics”. (It seems that this link is inaccessible unless you’re at a university site that has a subscription. The article should also be available at most physics research libraries as vol 27, numbers 4-5 of the 2004 issue of Rivista del Nuovo Cimento.)

I’ve written a bit about this history before, especially about the mathematician Hermann Weyl’s role, but quite a few other mathematicians were closely involved, including Hilbert, von Neumann, Emmy Noether, and van der Waerden. Much of the interaction between mathematicians and physicists took place at Göttingen, where Hilbert was the leading mathematical figure, and Weyl was sometimes a visitor, with both of them lecturing on quantum mechanics. This period was very much a high point of the interaction of mathematics and physics, interactions of a sort that were not seen again until the 1980s. Heisenberg and his collaborators learned about matrices from Hilbert and the other mathematicians at Göttingen, and Weyl was responsible for educating physicists about group representation theory and turning it into an important tool in quantum mechanics.

The Bonolis article has some amusing quotes from physicists who were having trouble absorbing what the mathematicians were telling them. Heisenberg wrote to Jordan “Now the learned Göttingen mathematicians talk so much about Hermitian matrices, but I do not even know what a matrix is,” and to Pauli “Göttingen is divided into two camps, those who, like Hilbert (or also Weyl, in a letter to Jordan), talk about the great success which has been scored by the introduction of matrix calculus into physics; the others, like Franck, who say that one will never be able to understand matrices.” Pauli was scornful about this new, unphysical, mathematical formalism of matrices, drawing a testy response from Heisenberg: “When you reproach us that we are such big donkeys that we have never produced anything new physically, it well may be true. But then, you are also an equally big jackass because you have not accomplished it either.”

Immediately after having to get used to matrices, physicists were confronted by Weyl with high-powered group representation theory, which they found even harder to understand than matrices. Famously, Pauli referred to the group theory that mathematicians were talking about as the “Gruppenpest”, but the late twenties saw a very fruitful exchange of ideas between mathematicians and physicists around this topic. Weyl’s proof of the Peter-Weyl theorem and von Neumann’s work on representation theory grew out of quantum mechanics, and the Brauer-Weyl theory of spinor representations was inspired by Dirac’s work on the Dirac equation.

It’s also interesting to note how in the years just preceding this period, much interaction between math and physics had grown out of general relativity. Noether’s work on what is now known as the Noether theorem came about because she was asked questions by Einstein and Hilbert who were trying to sort out conservation laws in GR. Weyl took up representation theory as a result of his work on the symmetries of the curvature tensor.

An amusing story I hadn’t heard before that is in the Bonolis article was one told by Edward Condon about Hilbert. He claims that when Born and Heisenberg went to Hilbert to get help with matrices, he told them that “the only times that he had ever had anything to do with matrices was when they came up as a sort of by-product of the eigenvalues of the boundary-value problem of a differential equation. So if you look for the differential equation which has these matrices you can probably do more with that. They had thought it was a goofy idea and that Hilbert did not know what he was talking about. So he was having a lot of fun pointing out to them that they could have discovered Schrödinger’s wave mechanics six months earlier if they had paid a little more attention to him.”

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65 Responses to The Stormy Onset of Group Theory in the New Quantum Mechanics

  1. Juan R. says:

    No problem Peter,

    but by a question of education and consistency, please use always the same phylosophy for stoping/erasing any no relevant comments.

    It is my belief (and of others) that your meaning of “irrelevant” is rather flexible, specially when you open a topic on a “stringy theme” and people here finalize it attacking to Lubos Motl. I still wonder that you maintain several of those comments and personal attacks intact.

    Of course, this is your blog and I respect your decision. No problem by my part, this was only a comment.

  2. Chris W. says:

    The paper that is the topic of this post has been downloaded and may be found here:

    (..via It’s Equal But It’s Different)

    Regarding my previous comment (partly in response to one of Ben’s), see Section 6. – Einstein vs. mathematicians: Minkowski and the special theory of relativity (p. 20-27).

  3. Peter Woit says:

    No, I was up in Boston at another conference, which I’ll write about soon.

    Matti and Juan,

    Please stop using this weblog as a discussion forum for your own ideas that have nothing to do with the topics here. I’ll delete any further comments of this kind.

  4. Anonymous says:

    Sorry, Peter to change the topic.
    A question I have is, did you attend the
    workshop on string cosmology at Columbia organized
    on Friday 13th which Lubos Motl discusses in his blog. Maybe you can report on it if you did

  5. Juan R. says:

    Thanks Matti,

    It has been a pleasure read your post. Still i may apologize because i do have studied TGD, but when I have some time free i will do.

    I also work with a two-time formalism, therefore some times i talk about a 4+1D formalism. Humm, interesting! What is the status of the “geometric time appearing in field equations of physics” in your formalism?

    I think that with apparently breaking of the second law in self asemmbly you mean integral decreasing of disorder or perhaps or on a diferential rate sense (negative production of entropy).

    I found time ago that spacetime cannot be represented by usual differentiable manifolds (bye bye Calabi-Yaus), but still I cannot claim for fractal-like behavior, since i do know if a fractal description would be exact or only an approximation valid in certain regimes. I simply are not sure.

    Really interesting, in my approach the failure of scattering theory is associated to density of matter. In particle physics with small effective densities, S-matrix work perfectly. for condensed matter situations, all the formalism breaks down and one works with phenomenological issues ad hoc.

    The failure of usual relativistic description of bound states is more complex in my approach

    Yes i also found that reductionsim fail, in fact it is a proven fact (usually ignored by physicists and by all string theorists) that string theory does not verify the equation for complexity level.

    In my approach nature has a hierarchical structure and no one level is in deep more important that other. Upper levels are not totally reduced to simple lower levels, in fact there is information that is not contained in lower levels, e.g. particle. That is the failure of particle physics to explain upper structures for example biomolecules.

    A priori my theory contain MOND approach like a limiting case for the explaining of anomalous acceleration and the missing matter problem.

  6. To Juan R.:

    Thank you for interesting comments.

    The most obvious apparent violations of second law relate to self assembly and behavior of phase conjugate light.

    In my own theoretical framework they are apparent violations so that here we agree. I feel it necessary to distinguish between two times: the geometric time appearing in field equations of physics and the experienced time whose basic unit is quantum jump and which corresponds in average sense to some increment of geometric time which is however proportional to hbar so that there is a hierarcy for the geometric average durations of quantum jump (moment of consciousness) just like there is hierarchy of material systems: elementary particles, hadrons, nuclei, atoms…

    The differences between these times are obvious: consider only reversibility (irreversibility) of geometric (experienced) time. With respect to the experienced time second law holds still true but since TGD predicts that both positive and negative arrows of geometric time (positive and negative sign of conserved inertial energy, two possible manners to select the fermionic Fock state in second quantization), processes such as self assembly for which controlling process proceeds backwards in the geometric time, apparently break the second law.

    Concerning the “more quantal” issue. Increase of hbar means essentially fractal scaling: quantum coherence lengths and times are scaled up. For scattering cross sections in perturbative regime the effect is perhaps somewhat surprisingly just the opposite since higher order corrections come in powers of alpha= g^2/4*pi*hbar, which is reduced. For bound state energies which cannot be understood perturbatively the situation is different: in the case of hydrogen effects is simple scaling by 1/hbar^2 proportionality of binding energy. An interseting hypothesis is that hbar increases when the perturbative series for S-matrix fails.

    Macrostructure indeed affects microstructure and reductionism fails: this is one of the main implications of TGD. This is already implied by what I call topological quantization: space-time surface has a many-sheeted structure with sheets having outer boundary (magnetic flux tubes, “topological light rays”, etc..) identifiable as quantum coherence regions and forming a length and time scale hierarchy. Quantum classical correspondence together with the fact that these regions can have arbitrarily large but finite size suggests a generalization of quantum theory and dynamical and quantized hbar provides it.

    Concerning strange unobserved matter: I am believer in TGD based variants of string like objects identifiable as magnetic flux tubes. Simplest of them are cosmic strings, 4-D surfaces X^2xY^2, where X^2 is string orbit in M^4 and Y^2 holomorphically imbedded 2-manifold of CP_2. A cosmic string traversing through the nucleus of galaxy in a direction transverse to the galactic plane (naturally assignable to the galactic jet) creates a Newtonian 1/rho potential, which explains the constant velocity spectrum of stars: no dark matter elsewhere would be the minimum option. I do not know how closely this relates to your explanation. Actually TGD allows to identify galactic black hole as a highly convoluted cosmic string. Galaxies would be pearls in a cosmic necklace.

    I think that the limits of classical physics are encountered when one tries to understand intentional action and the coherent behavior of the matter in living organisms.

    Matti Pitkanen

  7. Juan R. says:

    For Matti.

    Some time ago I revised some of supposed violations of the second law in quantum regimes. The famous San Diego conference. Finally i discovered that all revised claims of violation of the second law were based in obvious misunderstanding of thermodynanmics and/or errors.

    I also “showed” that formation of structures is compatible with a new generalized version of the second law for mesoscopic regimes.

    Therefore, since that i known several of usual methods in laser physics, i doubt that any group can find real violation of the second law in laser phenomena.

  8. Juan R. says:


    I am not completely sure now of that the increase of hbar in Delta x*Delta p=about hbar–>hbar_s would make the system more quantal, just more uncertainty in coupled observables. I don’t think that can be exclusively represented like more quantum character in all situations. Remember classical statistical mechanics, asumed to be classical but with Delta x*Delta p different from zero.

    Therein my emphasis in that perhaps you are working with some like alpa·h, with alpha a system parameter, instead of with variable h.

    Could variation of your h explain cosmological redsift like the effect of travel of light for different phases of universe? Or am i wrong?

    I don’t know the details of TGD and therefore I cannot do any serious comment still. However, I think that there is no real dark matter in the universe (this is another argument against ST and supposed dark matter explained from “cosmostrings”).

    My explaining of galaxies and cluster “dark matter” like a discrepancy in standard gravity appears to be supported by experimental data. In fact, i can derive the well-known (1/r) behavior without invoking to strange unobserved matter.

    What is your opinion?

    “The possibility of several values of hbar would allow interaction between widely different time and length scales.”

    It appears a fractal like behavior. Does macrostructure affects to microstructure in your TGD? If yes, this may be a violation of typical reductionism of particle physics.

    Living matter is really interesting, still i found no sufficient time for doing research in that. Now i am working in gravitation and cosmology.

    My point is that living matter is characterized by “long-range correlations”, but i don’t call that a “gigantic quantum structure”, since formulas for understanding living phenomena are really classical ones, e.g. chemical kinetics.

    Of course, perhaps i am wrong, but i don’t know any macro-quantum effect violating classical laws usually applied, with success, in biology.

    Cannot the movement of your hand be modelled with chemical kinetics (muscle) + transport theory (electrons, ions, etc.) + EM + classical mechanics (skeleton)?

  9. Anonymous says:

    In reply to Ben:

    I’m a mathematician (though unlike some – most? – people here, I like to think that math and physics are essentially the same; like two different sides of the same thing). What you describe/ask (abstract commutation relations vs. concrete PDSs) happens very often in mathematics. For example, consider finite groups; every finite group is a subgroup of some symmetric group (that is, all permutations of a set), and indeed that was the way people thinked of groups in the 19th century. Only in the 20th century started people think about abstract groups and their representation. Similarly, in the 19th century, continuous groups mostly meant groups of transformations of vector spaces; manifolds meant submanifolds of euclidean spaces (and again, every smooth manifold can be imbedded in a large euclidean space), etc. So why consider abstract objects instead “concrete” ones?
    First, it is not always true that all abstract objects can be realized as “concrete” ones (like subobjects of some model object); and, even more significantly, the abstract viewpoint turned out to be very fruitful. For example, you can make a difference between intrinsic and extrinsic properties of an object: which are properties of the abstract object itself and which are consequences of the particular realization of it.

  10. To Juan R.:

    The increase of hbar in Delta x*Delta p=about hbar–>hbar_s would make the system more quantal.

    One can imagine that a system consisting of ordinary elementary particles can make a transition to a large hbar phase without an appreciable change in four momenta of particles but with an increase in quantal size hbar/m defined by the Compton length. Macroscopic quantum phase is a natural outcome due to the quantum overlap of particles.

    For instance, suppose that hbar_s/hbar= about 2^11 (a preferred value for hbar_s for certain reasons). Ordinary IR photon with energy of 1.24 eV corresponds to wavelength of one micrometer whereas “dark photon” would correspond to a microwave wavelength of 5 millimeters.

    There are good arguments (in TGD Universe) for believing that dark matter particles form analogs of Bose Einstein condensates and emit coherently BE condensates of dark photons behaving very much like laser beams and decaying to ordinary photons with wavelength shorter by a factor 2^(-11) in our example (decoherence).

    The possibility of several values of hbar would allow interaction between widely different time and length scales. This kind of interactions characterize living matter. Consider only how my intentional action to raise my hand eventually boils down to *coherently* occurring interactions in molecular and atomic length and time scales.

    Matti Pitkanen

  11. Juan R. says:


    You claim for modifications of the Planck.

    What is the interpretation of deltaE = hw in your theory for large systems?

    Are you perhaps really claiming for the substitution

    h –> alpha·h

    in formulas, with alpha a parameter instead of asuming that h is not a real constant?

  12. To Chris:

    My purpose is not to propose any ad hoc modifications of quantum theory. The value of hbar remains free in quantum theory: this is a fact. Second fact is that TGD leads to a well-educated guess for the spectrum of allowed values of hbar based on mathematics associated with so called hyper-finite type II_1 factor of von Neumann algebras (see the chapter at my homepage). This appears naturally as the Clifford algebra of infinite-dimensional spaces (now the space of 3-surfaces in certain 8-D imbedding space) and are partially characterized by the requirement that infinite-dimensional unit matrix has unit trace (might be relevant for the finiteness of the theory). A considerable generalization of the structure of quantum theory is involved.

    As far as energy conservation is involved, I am extreme conservative. TGD was born from the requirement that inertial energy is strictly defined and conserved: in general relativity this is of course not the case and has led to numerous difficulties discussed also in this blog. Non-trivial representations of Diff^4 required by the identification of momenta as Diff^4 generators have central extension and lead to Diff anomaly. The problem finds an elegant resolution if Poincare symmetries correspond to those of imbedding space rather than space-time surface.

    To blame that free energy people have never heard of energy conservation and second law is of course cheap rhetoric. The anomalous effects involved with free energy effects and cold fusion (to name only few of them) represent the borderline of our knowledge, and it could be very rewarding for theoretical physicists to come down from the academic heights, and start to think more what refinements of the basic concepts are required by these anomalies if they are indeed real. There are also strange effects apparently breaking second law associated with phase conjugate laser beams. Self assembly in living systems is also very interesting in this respect. Of course, the conceptual problems of general relativity alone should provide enough food for original thought.

    Matti Pitkanen

  13. Anonymous says:

    To Ben:

    Remembering the basics: QM algebra mumbo jumbo describes discrete values of observables not just a time-continuous evolution of probability waves ( in the Born interpretation ). The discrete values of an observable are nothing but the eigenvalues of a Hamiltonian acting as a hermitian operator on a Hilbert space which is spanned by orthogonal vectors representing solutions of the Schroedinger equation ( the eigenvectors of the Hamiltonian ). If hermitian operators of different observables cannot be simultanously diagonalized, measurements on the observable can not be measured accurately the same time. The Schroedinger-equation connects the Hamiltonian with continous time evolution and Hilbertspace/Matrix formalism connects the same Hamiltonian with observable spectra. Each element of the mathematical formalism has a clear physical interpretation.

    To Matti:

    If You want to vary physics on demand of some unobserved speculative effect, why not dispute energy conservation or entropy law ( which makes people depressive )? The “free-energy” crowd that creates and sells perpetuum-mobiles would be grateful.

  14. Chris Oakley says:

    Hi Ben,

    I think that the reason for wanting to base QM/QFT on commutation relations is simply that you have got to start somewhere. But I agree that it is a mistake. After all, it does not work for quantum fields other than spin zero, and even if one accepts mathematically meaningless constructs such as the differencing of divergent integrals, “quantization” of GR always fails. One might also add that the premise of quantization if nonsensical: we start with a classical theory and then “quantize” it – are we really saying that the classical theory is the fundamental thing here rather than just some kind of limit?

  15. Thanks for Ben, the engineer, for very thoughtful comments. The canonical commutation relations for observables are often taken as a more or less sacred thing although there are of course generalizations. Symplectic geometry phase space gives a good justification for the general form of commutation relations apart from the value of hbar in the case of quantum mechanics. Bose (Fermi) statistics is behind the commutation (anticommutation) relations in quantum field theory, and in two-dimensional case braiding statistics leads to more general commutation relations.

    What puzzles me is why the value of Planck constant is taken as sacred (and that also I took it as sacred for so long) so that it disappears from quantum physics formulas totally by the choice hbar=1,c=1. Dynamical, possibly quantized, Planck constant able to have large values, would be well-come to anyone attempting to understand living matter as a macroscopic quantum system since Compton lengths, etc. would be scaled up. A phase in which protons have atomic Compton lengths would be very different from ordinary condensed matter and might allow to understand some claimed anomalies such as cold fusion.

    Variations of hbar do not lead to any dramatic effects in scattering of free particles if the classical cross sections representing hbar=0 limit are non-vanishing so that only higher order perturbative effects are affected (in fact reduced, since gauge coupling strengths are proportional to 1/hbar). Situation is different for processes like photon-photon scattering for which classical cross section vanishes. Also the spectrum of binding energies for say hydrogen atom scaling like 1/hbar^2 would be strongly affected.

    One could play with the thought that the value of hbar must be such that classical bound states make sense also quantally. In the case of gravitational bound states of masses larger than Planck mass this would have rather interesting consequences. Planck constant would become gigantic and the black hole formation as a gravitational counterpart of infrared catastrophe for hydrogen atom would be prevented by the formation of quantum gravitational bound states. This line of thinking would mean a bottom-up approach to quantum gravity starting from gravitational wave mechanics (of dark matter perhaps) instead of not so successful top-down approach provided by M-theory.

    For these and many other reasons I see the possible effects related to the dynamical hbar as worth of studying. More ponderings about this at my blog site.

    Matti Pitkanen

  16. Ben the Engineer says:

    Chris W said: “In quantum field theory and general relativity we seem to have left this simple starting point (and its closely related predecessors) way behind, and modern mathematics has been absolutely essential in doing so. Have we lost something as well? ”

    Hi, I’m Ben the (not terribly mathematical) engineer, and thanks to those who answered my post about Feynman vs. Schwinger. They’re both great of course!

    Expanding on this discussion about abstract math in modern physics, please indulge me in a sweeping and naive question about quantum physics. Also, please forgive the length of this post, but this is the first (or rather second) time I have dared participate on a physics list and I wish to unload something that has been bothering me for a long time. The basic question is:

    Why are the commutation relations so sacred?

    This may seem like a strange question, but what I mean is, Why is a certain formal manipulation of symbols taken as sacred dogma by ALL physicists? Yes, I know it works in QED and QCD to upteen decimal places and all that, but why do the superstring and other quantum gravity people simply transplant it without any questions? Every other crazy idea can be entertained, but this is sacred. Why? At least, that’s the impression I get. Perhaps it is this blind faith in the commutation relations which is the reason that gravity and the other forces have not yet been unified. Perhaps they simply don’t work for gravity. Perhaps they don’t even work for the other forces in the regimes explored by unification theory.

    If I could elaborate a bit, my question might make a bit more sense. As an EE (electrical engineer), I have no problem with wave equations. In fact, I love them. So naturally, I take a Schrodinger view of things, which I know is the simplest quantum approach. Hence, to me, the commutation relations simply express a straightforward relationship between partial differential operators. I know that the modern view is that the latter are only a *representation* and that the truth, following Dirac and others, is to be found in the abstract approach of a mere algebraic relationship between operators, which have themselves become quite abstract. (How ironic that Dirac started out as an engineer!)

    Well, this clashes with a strong philosophical prejudice of mine, namely, that I want to be able to *visualize* everything. After all, whatever is happening down there is happening in space and time, so it must involve regions of space distorting and evolving in some way, and this means that it can be imagined in principle. Never mind what the regions are filled with, if anything at all. The point is that there must be regions in which whatever is happening is happening. (By ‘regions’ I am thinking primarily of boundaries in spacetime, though I realize that according to GR the spacetime itself may be distorting and evolving.)

    So the bottom line is that I want to be able to visualize what is happening, even if there are profound ontological issues with *what* is being visualized. The idea of a probability wave may seem esoteric in some sense, but even engineers can feel happy with the complex exponentials or Bessel functions or Hermite polynomials or whatever that one gets when one solves the Schrodinger equation. At least we can *draw* them.

    (Note that even Maxwell’s theory can seem a bit ‘metaphysical’ in that one may ask *what* is waving. This led to the idea of the ether and its subsequent repudiation. Yet we all feel comfortable with waves, whether classical or quantum, because we can at least imagine them, even if they have some ghostly ontological aspect.)

    Things get even more mysterious for me when the commutation relations are applied to the electromagnetic field. How can something that was invented for particles be simply transplanted to the electromagnetic field? You may say that the EM field is ‘made up of’ particles called photons, but there is a difference with, say, the particle in a potential well for which Schrodinger derived his equation. The difference is simply that the ‘classical’ entity with which we begin is, in one case, a particle and, in the other, a wave. How can the same algebraic gimmick (if I might say so) simply apply in both cases? At least, why don’t physicists spend more time wondering about this? Every QED and QFT textbook I have seen simply ‘postulates’ the commutation relations for any pair of ‘conjugate’ variables, according to Lagrangian theory. I don’t deny that this somehow works in a number of cases, but the fact that it does should be profoundly puzzling to physicists. After all, not only is there the particle/wave difference in the classical starting point, but in one case we have a particle in an *external* potential and in the other case we have a *free* plane wave NOT in a potential. Yet the physicists just wave their magic wand and justify this magic by the pretext that it happens to work, as if that were a sufficient excuse! 🙂

    Then when it comes to strings or the Planck regime, the questions and puzzles simply increase by orders of magnitude. I think I’ve said enough and you can see where I’m going. I’d like to dispense with algebraic abstractions like commutation relations and get back to good old partial differential equations describing waves that I can visualize evolving in space. If you argue that the commutation relations are in fact always equivalent to PDEs, then why not go to the PDEs directly? And the way I have seen the CRs applied in modern physics papers, in what seems like a mechanical and robotic fashion, makes me wonder if all contact with PDEs (and hence visualizability) has simply been lost, and THIS may be the problem. It may have all degenerated into quantum algebraic mumbo jumbo. Is this something any of you care about? Thanks

  17. Cy Cantrell says:


    Interesting summary of Bonolis’ article… I’ll try to get a paper

    The story about the conversation between Born, Heisenberg and Hilbert, in which Hilbert pointed out the connection between matrices and eigenvalue problems of partial differential equations, and Born/Heisenberg went away thinking that Hilbert was a very strange old man, is part of the lore of physics. I first heard it from Kurt Gottfried in a course on quantum mechanics at Harvard.

  18. D R Lunsford says:

    Hey! Here’s Ed Witten photographed with incontrovertible evidence of strings.


  19. Juan R. says:

    I sorry i had a problem with the axial flux-collector and with the catalyzer of supersimmetry. Now!!

    O (closed)

    S (open)

    8 (cluster of two closed strings)

    They look quiet at ambient temperature, but really are vibrating. Moreover, I am working in new glasses for seeing it in 10D and in full technicolor.

    P.D: I forgot say that is just a math laboratory.

  20. Juan R. says:

    I have already sinthetized superstring and clusters in my laboratory. I post it below.

    [b]O[/b] (closed)

    [b]S[/b] (open)

    [b]8[/b] (cluster of two closed strings)

  21. Quantoken says:

    Thomas Larson:

    Yeah I saw Lubos meantioning that paper, too. I did not comment because I thought that’s just a joke that some solid state physicists were trying to ridicule their counter part colleagues in super string research.

    Certainly if super string simply does not exist, no one can make an experimental device to produce them. But that’s not even the point.

    We know, super symmetry, if it exists, must exist at an energy scale very high, much higher than the energy accessible by today’s accelerators, i.e., above TeV, which corresponds above 1×10^16 degrees temperature.

    On the other hand, the Bose-Einstein condensation of heavy metal atoms, as we know it, involves energy scale extremely weak, only at super cold temperature, sub mili degree absolute temperature, can those fermions condense into bosons. Therefore, any interaction withou the slightest amount of energy could easily thaw it out of Bose-Einstein condensation. Ther super symmetry interactions, certainly, is 1×10^20 times higher than that’s required energy level to destroy the Bose-Einstein condensation. So such an experimental setting could not have produced anything even if super string theory is all correct.


  22. Thomas Larsson says:

    Note the standard caveats with if’s and would’s:

    “If their idea can be put into practice, it would allow aspects of string theory to be explored in an experiment for the first time.”

    If arch-angles are produced at the LHC, it would allow aspects of Christianity to be explored in an experiment for the first time.

  23. D R Lunsford says:

    Hey, back to strings! In the lab!


  24. Juan R. says:


    It appears obvious that when we improve our understanding of nature, it is necessary more and more mental power. In some sense you can explain GR, QFT or ST for public in usual, “cotidiane”, terms, but the real understanding is only achieved from the mathematical formulation of those ideas and that math is each time more difficult and abstract.

    I could offer you good examples of these topics but since Peter probably would erase my post, you would see my recent work in epsilon-calculus.
    Recently, a mathematical research has done some good comments in my ideas on calculus and criticism of Connes, on an recent post (on GR) in a well-known forum. I think that you can find me easily.

    It is rather probable that one day nobody can understand new mathematical formalism. In fact, that fatal epoque has already arrived to pure math, like some of us probably know.

    In general, people has no idea of QM, GR, Strings or others. Since that was a teacher of young people, and some of my friends continue to be. I can say you that people has no best understanding of Newton gravity now that 100 years ago.

  25. Quantoken says:

    Some one said:

    “This means that real physics is increasing abstract and simplying physics for the man on the onminbus is now impossible.”

    I completely disagree. The whole point of science is try to de-mystify nature and try to simplify and explain nature in ways we human can understand. If science grows more and more complicated and abstract and is detached from the comprehension ability of the people, eventually it will grow to the point that no one in the world understand science any more. Then science is no longer relevant to the development of human society if no one could understand it. I do not think that is the trend.

    Back in stone age, nobody even knows how much is 1+1. Back in Newton’s time, no one even realize the existence of gravity. But today even an average person knows a little bit of weird stuff of 10 dimentional super strings. And any weird field you publish your paper, there are at least a couple thousand people around the world who would like to read and can understand your stuff.

    Science is becoming more accessible to people, not less accessible!

    Imagine what will happen if the opposite is true. Einstein’s GR was once said to be understood by only 12 people. Now a theory twice complicated and twice as advanced would probaly be understood by 6 people. Continue on and pretty soon the comprehensibility is reduced to zero. And the knowledge will be accessible to not a single living human being any more, zero.

    Is that possible? Not at all. As long as science is still part of human culture, it will always remain accessible to at least a portion of the population, and this portion could only grow, not shrink. The information age makes it more likely that a vast majority of population will become very familiar with at least some partucular areas of science.


  26. Juan R. says:

    Like an expert in chemical questions, i can sure you that chemistry newer was reduced to/ explained by physics (either inside or outside of QM, SM, and ST).

    I known the usual (very wrong) belief of that all of chemistry is already known (e.g. popular claims by Weinberg, Witten, etc). But they are so correct like Newtonian physicists claiming for an understanding of chemical reactions or 19th century physicists claiming for ultimate models of chemical bond based in classical electrodynamics.

    For me it is so arrogant the claim by physicists of that all of chemistry is known like the claim by string theorist of that ST is the TOE. Somewhat like Peter considers useful to explain that ST is not a TOE (in fact one cannot predict anything) I consider good to present the current status of chemistry like an autonomous science.

  27. Selrach says:

    Hi Chris W,

    Reading your post made me think about my
    favourite issue with 21st century Physics.

    What is to replace the Hierarchy?

    What I mean is that Science upto now has explained the world thus: Geology then Biology then Chemistry then Physics then Mathematics.

    However, with Quantum Mechanics, this clear delineation has broken down with superposition effects and with String Theory with mirror symmetries (proven as mathematics!).

    So called paradoxes are paradoxes if you try to ‘simplify’ the problem at a higher Hierarchy level .

    This means that real physics is increasing abstract and simplying physics for the man on the onminbus is now impossible.

    An amateur mathematician.

  28. robert says:

    It’s good to see Schwinger’s rehabilitation in a couple of the posts following on from the ‘can mathmos do physics and vice versa?’ debate. His papers are models of clear exposition, whether they address the quantisation of gravity or the generation of synchrotron radiation, and, where possible, make direct numerical contact with physical reality. As is evident when reading his ‘Classical Electrodynamics’ text, he felt that the maths should emerge from the physics, rather than the other way round. This is particularly evident in his amazing ‘On Angular Momentum’ paper, which brings us neatly back to the Gruppenpest question.

    Whatever; it is sadly true that those of us raised on Morse and Feshbach find it rather hard to penetrate the hep-th arxiv these days.

  29. Thomas Larsson says:

    Re Einstein’s disregard for mathematics. While developing GR, AE did some absolutely marvellous inventions in mathematical technology, namely the summation convention and the idea to put contravariant indices upstairs (though I’m unsure whether this is really due to AE). In my experience, many mathematicians still seem unconfortable with these inventions, 90 years afterwards.

  30. Steve M says:

    I recently had a look at old papers of Schwinger’s from Physical Review in the 50s. They are an absolute calculational tour de force. He states what he is going to do and derive, then does the stunning calculation in complete detail. No excuses, no handwaving, no waffling, no speculation and no wishful thinking. The stark contrast with many of the modern arxiv papers kind of hits you.

  31. Chris W. says:

    Ben said, “By the way, even though Einstein developed respect for mathematics after laboring over General Relativity, I did read somewhere that later in life he complained that the mathematicians had so transformed (or veiled) his theory that he no longer understood it!”

    Actually, I’m quite sure that he said this early in his career, in response to Minkowski’s explicitly geometrical formulation of special relativity. Notwithstanding his initial misgivings, Minkowski’s formulation set the stage (in part) for posing the questions that led to general relativity, ie, as Einstein reconsidered his relativistic theory of space-time measurement in the context of the equivalence principle and what was understood about gravitation prior to 1915.


    This is how new mathematical ideas can be so helpful; they can make it possible to state clearly and objectively notions and assertions that experience, physical intuition, ordinary language, and pre-existing mathematics can only motivate and sketch in an incomplete and sometimes contradictory fashion. Nonetheless the crude initial formulations are essential when new and fundamental problems are being confronted. They make it possible to sensibly discuss the problem of what new mathematics is needed for addressing physical questions, and why. My feeling about much recent work is that it relies on formal precision and rigor as a comforting and professionally rewarding refuge—a way of avoiding the crucial, difficult (and usually somewhat ill-posed) questions while still demonstrating admirable technical mastery and (perhaps) making contributions of substantial value to mathematicians and mathematical physicists.

    In this context, consider Feynman’s famous remark:

    If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis that

    All things are made of atoms—little particles that that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.

    In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.

    In quantum field theory and general relativity we seem to have left this simple starting point (and its closely related predecessors) way behind, and modern mathematics has been absolutely essential in doing so. Have we lost something as well? That is, have we obstructed a path to a lucid reconsideration of this metaphysical starting point? It seems to me [see comments] that the fundamental issues raised in attempting to develop a theory of quantum gravity make this question more important* than it has been in a very long time.

    * In his recent lecture at Perimeter Institute Leslie Ballentine emphasizes what I believe to be a closely related point:

    Einstein’s “God” talk was purely metaphorical, like “Mother Nature”, or “Father Time”. His objection to indeterminism was more serious, but has unfortunately been over-emphasized. [John Stachel has argued that it has also been badly misunderstood. -CW]
    His most powerful criticism of the “Copenhagen” interpretation did not involve determinism/indeterminism.
    Rather, it concerned realism (or ontology).

    For some time now it has not been clear what physics at the most fundamental level is about. What it is about has certainly been clear enough to do much valuable research, up to a point, but I strongly suspect that in certain areas we have reached that point, and must now find a deeper, more lucid, and more unified answer to this basic question.

  32. D R Lunsford says:

    RE Klein – he at once understood the light cone, in fact he invented the theory of “light cones” insofar as projective geometry needs a quadric to become a metric geometry. How wonderful for Klein, in my opinion the Newton of mathematics, to see his program so gloriously realized.


  33. D R Lunsford says:

    Peter – you should check out Schwinger’s book on electrodynamics, based on lecture notes and mostly reviewed by Schwinger himself. This book is one of my prized possessions. You really get a clear idea of how Schwinger did physics, and in this context it’s just amazingly to the point.

    Sorry for the long url.


  34. Peter says:

    Actually I don’t think either Schwinger or Feynman had much use for mathematicians, although Feynman’s style was more intuitive and pictorial and Schwinger’s more formal.

    I’ve always found Schwinger hard to read, whether or not you know the subject he is writing about, but Feynman can also be hard if you are just learning the subject. His lectures on physics for undergraduates famously baffled most of the students. I’ve heard that when they both presented their work on QED, people found Schwinger more understandable, since he presented coherent derivations, whereas Feynman seemed to be engaging in repeated leaps of logic. On the other hand, I’ve also heard about these early calculations that “Feynman made it look like anyone could do it, Schwinger that only he could have done it”.

  35. Ben says:

    Hi. I’m an engineer by profession with an interest in physics, but my mathematical knowledge is not deep. Speaking of the overuse of math in physics, I have an impression regarding Schwinger and Feynman, and I am wondering if it is correct. Schwinger was supposed to be one of the most virtuosic mathematical physicists of his day, allegedly capable of calculations nobody else could do or perhaps even understand. Feynman, on the other hand, seems to have taken a relatively ‘intuitive’ approach to QED, which dispensed with heavy formalisms in favor of the famous Feynman diagrams. Yet, who had the greater impact on physics? It seems quite clear to me that it was Feynman. I suspect that this comparison is almost a cliche among physicists. I did read somewhere that Feynman once referred disparagingly to ‘fancy schmancy differential geometry’. By the way, even though Einstein developed respect for mathematics after laboring over General Relativity, I did read somewhere that later in life he complained that the mathematicians had so transformed (or veiled) his theory that he no longer understood it!

  36. Peter Woit says:

    To expand on the previous comment:

    Klein’s point of view on geometry as being about Lie groups and Riemann’s point of view as it being about metrics and curvature were unified by Cartan, who really was the first one to come up with the modern view of geometry, in which the connection on a principal G-bundle plays the central role.

    Hilbert was definitely the major figure in mathematics in the early part of the century, since by this time Klein wasn’t very active (he retired in 1913). Klein was the one who built up the great school of mathematics at Gottingen, but by the time quantum mechanics came around, he was dead and Hilbert was the leading figure there. Hilbert’s influence is due partly to the fact that he worked in an amazingly wide array of mathematical areas, with geometry only one of many.

    By 1925, Hilbert was getting old (63). It is Weyl who was really at the height of his powers during those years, and had the most influence on physics during that period.

  37. Walt Pohl says:

    Selrach: Klein’s Erlanger programme was done in by changes in mathematical fashion more than anything else. Riemannian geometry simultaneously encompassed Klein’s geometries and the classical differential geometry of curves and surfaces, so it has drawn most of the attention. The Erlanger programme just became a special case of highly-symmetric Riemannian manifolds.

    DRL: Within mathematics itself, Hilbert is a more influential figure than either Klein or Weyl — probably the most influential figure on mathematics in the twentieth century.

  38. D R Lunsford says:


    In my opinion, there are Klein, Weyl, and a lot of second stringers 🙂


  39. D R Lunsford says:


    Your comment RE Lie is extremely interesting and exactly to the point.


  40. Thomas Larsson says:

    DRL, I don’t think that one should regard Hilbert’s remark as Einstein-bashing, but rather as an observation that mathematical skills and physical intuition are very different things. Of the two, the latter is more important. Your math can be improved if needed, but without an understanding of the physics you don’t know which math is needed.

  41. Selrach says:

    Hi drl,

    I am not sure that you can compare Klein and Hilbert in that way!

    Both were mathematicians who created grand overarching philosophies which have proved influential but flawed. Klein’s geometry is group theory (Erlanger program) was shown to be inadequate, when Peano/Weierstrass discovered continous curves which have no derivatives anywhere!

    Hilbert’s program was of course undone famously by K Godel.

    My belief is that they are both great flawed mathematicians!

    An amateur mathematician

  42. Thomas Larsson says:

    In my experience, the mathematical knowledge of string theorists is rather lopsided. Many of them are strong in algebraic geometry (which I am not), but it is unclear to what extent algebraic geometry is really needed in physics. Real physics, i.e. GR and SM, can be understood without even knowing about manifolds and bundles. This is obvious, since these theories were discovered before fiber bundles became popular in physics around 1980. If we want to calculate some QCD amplitude, does it really help to know that the gauge potential is a connection on some SU(3) bundle? Some problems, like the Dirac monopole and the Aharonov-Bohm effect, can perhaps be understood better in modern language, but I doubt that neither Dirac, Aharonov nor Bohm knew about it.

    A good illustration of the strange selection of well-known math is given by simple Lie algebras of vector fields over the complex numbers. The Cartan-Killing classification in the finite-dimensional case (A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2) is common knowledge, but Cartan’s classifictation of the infinite-dimensional ones (W_n, S_n, H_n, K_n) is not. Which is strange, because these algebras were well known to Sophus Lie himself, and they play a much more prominent role in physics than do the finite-dimensional ones (except A_1 and A_2).

  43. Juan R. says:

    Famous Feynmann criticism to math-oriented physicists continue to be true.

    Some of my colleagues focus on mathematical research and believe that are doing physics when only are providing new mathematical views on old physical problems.

    1º) Physics, after math.

    Moreover, I believe that in some decades we will find limitations on the use of math for modelling nature.

    When i read the GR manual from Wald, I wonder that only after several chapters one begins to see some of physics. What do you think?

  44. Kay says:

    “Einstein … often spoke against abusive use of mathematiccs in physics. Physics, he would say, is essentially a concrete and intuitive science”. “I don’t believe in mathematics”, Einstein is reported to have affirmed before 1910.

    Einstein re-articulates here an old and somehow paradoxal intellectual relationship towards the use of language which can be dated back to Platon. It is very funny to read this citation of Einstein because remember that his enemies in Germany dispraised his physics as jewish and abstract. They bashed Einstein and all modern physics with exactly the argument that physics should be concrete and intuitive. The key-term in these philosophical polemics was the concept of “Anschauung” ( in orig. german ). “Anschauung” cannot be simply translated into “view” or “concept” but it includes a sensual and a contemplative aspect of “watching the true shapes of the ideas” and is not a mere technical reasoning. Remember also that Einstein was an opponent of Bohr/Heisenberg style of positivism which led the classical world of “Anschauung” completely behind and transformed physics into an interface-language for holding a conversation(!) with nature ( later it was Prigogine/Spengers who insisted in this “dialog with nature” ). This controversy was not less popular in mathematics and the most prominent proponent of positivism/formalism was no one else than David Hilbert.

    The role of language in the mindset of “Anschauung” was that of a service. So it was mathematics to the physics-community of that time. Language had no different role than fixing vagueness and making ideas communicable and testable but it had no function of it’s own. The modern mindest is far away from the platonic ideal of “Anschauung” but it comes close to that of investigating language and it’s effects. It’s more Kabbalah than nature mystics.


  45. “I don’t believe in mathematics”, Einstein is reported to have affirmed before 1910.
    – p 3.

    True, but he changed his mind later.

  46. D R Lunsford says:

    Wow, Einstein bashing even from Hilbert. Jealous I suppose. Of course it’s ridiculous. I read the papers, so I don’t give a fig about Hilbert’s lofty opinion of himself. Notice that one never finds Klein making such comments, and Hilbert was no Klein.


  47. Thomas Larsson says:

    Some quotes from physics/0504179 :

    “Einstein … often spoke against abusive use of mathematiccs in physics. Physics, he would say, is essentially a concrete and intuitive science”. “I don’t believe in mathematics”, Einstein is reported to have affirmed before 1910.
    – p 3.

    “Every boy in the streets of Goettingen understands more about four-dimensional geometry than Einstein. Yet, … Einstein did the work and not the mathematicians”
    – attributed to Hilbert, p 11.

  48. JC says:


    I got the impression the more “hardcore” mathematically inclined string folks and other theorists seem to be fond of using a lot of “new math”. Papers written by folks of this sort seem to pop up frequently in various journals like J. Math. Phys. or Comm. Math. Phys., and other lesser known journals specializing in “mathematical physics”.

    Some of these guys seem to be quite far away and disconnected from experimental data.

  49. Chris W. says:

    On the genesis and significance of Noether’s work in connection with general relativity, see this review by Nina Byer (UCLA, 1999).

    From Hermann Weyl’s 1935 memorial to her:

    “A stormy time of struggle like this one we spent in Göttingen in the summer of 1933 draws people closely together; thus I have a vivid recollection of these months. Emmy Noether – her courage, her frankness, her unconcern about her own fate, her conciliatory spirit – was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace.”

    (For more, see this page.)

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