:: Michael Flohr ::

:: Research ::

I am a mathematical physicist. In physics, you have two breeds of scientists, the experimentalists and the theoreticians. The former conduct experiments which yield facts one may wonder about. The latter formulate Laws of Nature in the language of mathematics to explain these facts, so that you may marvel about them even more. The mathematical physicists are a special, rather extreme, group of theoreticians which attempt to find or identify mathematical structures, which may be suitable to formulate new theories and new Laws of Nature.
    Humanity always strove to understand Nature, to predict its phenomena. Since the earliest days that Humans looked towards the stars, they invented more and more elaborate versions of mathematics and astronomy to understand the movements of the stars. Mathematics as the language of Nature is since then an incredible success story. Theories are more or less consistent mathematical buildings which allow to predict phenomena of a certain kind. For example, Newtonian mechanics allows to predict the movements of the planets and moons to a certain degree of precision. Maxwell's theory of electro-magnetism allows to predict all phenomena which have to do with electric and magnetic fields, electric charges and currents, as long as no quantum effects play a role.
    All theories developed so far are in a sense effective theories. They have a limited range of validity in energy or scale. Nevertheless, certain entities in all these theories are considered to be fundamental or elementary, at least within the scope of the theory. For example, in quantum electro-dynamics, the electron is considered to be an elementary particle. However, it is believed that the electron will turn out to be not an elementary particle in a more fundamental theory. It is believed that more fundamental theories exist. It is not so important, whether this is the case. The important thing is that it is a Human imperative to search for such theories, to strive to understand Nature as deeply as possible. This desire actually is the largest and most effective source for technological progress and innovation in Human civilizations. This remains true even if one discards all cultural importance of fundamental research as trivial or unnecessary. However, I personally hold that giving up interest in fundamental research is equivalent with giving up to be Human. Fundamental research is one of the most important and precious things, not every single Human, but a society of Humans can and should do.
    And one more thing: As conscious beings, Humans suffer from fear, and fear is the greatest source of why Humans inflict pain, suffering and death upon other beings and other Humans. And there is one major cause for fear, as already Seneca pointed out a long time ago: “timendi causa est nescire  ::  ignorance is the cause of fear.”


:: Fundamental Theories ::

Our current picture of Nature is that there are four fundamental forces: gravity, electro-magnetism, and the weak and strong nuclear forces. All matter is made up out of electrons and neutrinos, both coming in three varieties, and quarks, which make up the nucleons. Three of the four fundamental forces, with the exception of gravity, can be described by quantum field theories, while gravity is very successfully described by a classical field theory called the theory of general relativity. The electro-weak unified quantum field theory of electro-magnetism and weak nuclear forces and the theory of general relativity constitute the most precise theories with the highest predictive power conceived so far by Humans.
    However, this building of physical theories, the so-called standard model, is certainly not a truly fundamental theory. Actually, it depends on quite a large number of parameters which can only be obtained from experiment. It cannot predict these parameters. A theory is the more fundamental the less external parameters it needs it cannot predict. The current state is in this respect quite unsatisfactory. Moreover, there are principle obstacles to unify gravity with the other three forces, as it seems impossible to formulate a quantum field theory of gravity which would allow us to understand gravity on its quantum scale, i.e. the Planck length of about 10-33 cm. Of course, one might ask whether we should expect that all fundamental forces could be unified into one Theory of Everything, whether there should be something like a quantum theory of gravity. The detailed answer is quite involved and lengthy. Here, it may suffice to say that yes, gravity should have a valid description on the quantum level, and yes, there is good reason to expect that all forces including gravity can be unified into one more fundamental theory (although one should perhaps refrain from calling such a theory a Theory of Everything).
    There are many reasons for these two affirmative answers, which come from many directions. In fact, collecting all the evidence and all the experimental facts from the various fields of physics such as astronomy, cosmology, particle physics etc. lend quite some support to these statements. The best candidate of such a more fundamental theory is String Theory. In string theory, the elementary objects are not point-like, but are tiny strings having an extension in the order of the Planck length and which can be excited to swing. If we could develop string theory fully, i.e. to a level similar to quantum field theory, string theory could explain gravity and the Nature of space-time itself as well as the spectrum of “elementary” particles and the forces between them. Unfortunately, is seems that we Humans have not yet developed mathematics well enough to master this task.
    Besides our limited understanding of mathematics, there is one more serious obstacle. Our quest for fundamental theories has led us to consider theories on the Planck scale. Unfortunately, this scale is so tiny that an incredible large amount of energy is necessary to probe it. It appears that no current experiment, nor any experiment we Humans could start in the near future, will be able to look at this scale directly. Any experimental input which could help to strengthen the case for string theory will be more indirect, e.g. from cosmological observations.
    However, when the people of a medieval city started to build a cathedral, many of them knew that they would not live to see it finished. Theories such as string theory attempt to solve deep riddles about the structure and the origin of our Universe. We cannot expect to see it finished within our live times. We should be modest and be happy to contribute some small stones or pieces to it. And we should not fear that some time it may turn out that we contributed pieces to a theory which went into the wrong direction. This kind of mathematical physics is more speculative in the sense that we have to develop theories to a certain degree before we can see whether the theory is worth the effort.


:: Conformal Field Theory ::

Let me now come to the particular field of mathematical physics I am actually working in: conformal field theory. The degrees of freedom a string possesses may be described by a two-dimensional quantum field theory which lives on the world-sheet the string sweeps out in space when propagating in time. This quantum field theory must have a high amount of symmetry, it must be invariant under all coordinate transformations which locally preserve angles. Such coordinate transformations are called conformal transformations, and hence such quantum field theories are called conformal field theories.
    Now, there exist many such two-dimensional conformal quantum field theories. So, a natural question is, which is the one which governs the string. To answer this question, it is helpful to first classify all possible conformal field theories. It turns out that this is too hard a task to master. What seems to be feasable is, to classify all so called rational conformal field theories. These have very nice mathematical properties. In fact, they can be define in a more less mathematically rigorous way, which makes them the only non trivial quantum field theories, which enjoy this level of acceptance under mathematicians.
.     Unfortunately, the rational conformal field theories are so special, that it is very unlikely that the strings in our Universe will work with these. In the “space of all conformal field theories”, the rational ones are scattered like the galaxies in our real Universe, and they make up a subset of measure zero.
    One might ask now, how the conformal field theories living on the string world sheets might look lile. Or one might try to study conformal field theories, which are slightly more complicated than the truly rational ones, but which look more like a generic conformal field theory. In my opinion, the class of conformal field theorie I currently study is such a more suitable class. These theories are called logarithmic conformal field theories. What makes them interesting is that, on one hand, they still share many features of the nice rational theories. On the other hand, they have a much richer representation theory which makes them look like generic conformal field theories. In modern physics, Laws of Nature are understood as consequences of certain symmetries. For example, translational symmetry in space and time yields conservation of momentum and of energy, repsectively.
    Symmetries are realized by symmetry operations one can perform on the mathematical modeling of the phenoma one wishes to study. These operations can be concatenated or reversed, so they naturally form what mathematicians call a group. If the symmetry operations depend on continuous parameters, one may study only very tiny, infinitesimal, operations, i.e. operations, which deviate from the identity operation (doing nothing at all) only a tiny bit. The physics is then governed by the possible representations of these symmetry groups, or their respective symmetry algebras of infinitesimal symmetry operations. As a physicist, one soon learns that the important representations are the irreducible ones, which cannot be cut down to smaller representations. But this is a bit too simple. There is a class of representations, which is reducible, but still indecomposable. It cannot be decomposed into a direct sum of smaller representations, although it does contain smaller sub-representations. As a physics student, you usually never encounter these indecomposable representations. But Nature might be more fancyful, and so I started to study conformal field theories which have also thes indecomposable representations. These are my beloved logarithmic conformal field theories, and there exist quite a lot of very interesting individuals in this class.


:: Non-Rational Conformal Field Theory ::

The main motivation to study these logarithmic conformal field theories is that they are somewhere between the nice but very special rational conformal field theories and the ugly but generic non-rational theories. One nice feature of the rational theories is that the tensor product of two representations can always be decomposed into a finite sum of irreducible representations. In the non-rational case, one could get an integral over a whole continuum of representations instead. Interestingly, the border between these two extrem classes of conformal field theories is, at least partially, given by the logarithmic theories with their indecomposable representations.
    Why does one need these representation and this decomposing of tensor products? Now, representations somehow encode the properties of the entities in a theory, e.g. the particles it contains. Irreducible representations would then naturally correspond to elementary particles, as they cannot be decomposed into smaller units, so the particles they represent cannot be decomposed into smaller units. The tensor product decomposition then represents the one experiment, we can do in high energy physics: smashing two particles into each other and then looking for the shards. Although strings are not particles, this reasoning remains true, and so strings can have properties encoded in the representations of the conformal field theory living on their world sheets, and they can interact by coming together. Their worldsheets, which look like cylinders, can then join, and after some plumbing, a certain number of new cylinders may emerge. Ok, this is all extremely simplified and therefore not entirely precise.
    Strings have so many degrees of freedom that we cannot expect that the reasoning with the elementary particles (which in fact are not truly elementary, but made out of excitations of strings) can be carried over to the string scale in this simple fashion. Also, there are many unsolved problems in string theory. One can say that the current state of string theory is as quantum mechanics before the advent of quantum field theory. We actually do string mechanics, but we are far from being capable to write down a fully featured string field theory
    I personally believe that a deeper understanding of conformal field theory might help to pave the way for the formulation of consistent string field theories. And I believe that we still have to learn very much in conformal field theory before we will have a sufficient picture of the above mentioned space of conformal field theories. So far, we almost only know isolated points in this space, the rational theories, and even these we do not know completely. Even worse, we know almost nothing about the neighbourhood of these isolated points, and we are absolutely ignorant about the vast regions between these tiny islands.


:: Logarithmic Conformal Field Theory ::

My research seems to indicated that logarithmic conformal field theory might by a good tool to explore the space of conformal field theories beyond the small islands of rationality and its perturbations.
    I have so far contributed to the foundations of logarithmic conformal field theories. Thus, I have generalised most of the notions of rational conformal field theories to the logarithmic case, such as modular invariance of the partition function, characters and torus vacuum amplitudes, singular vectors yielding local symmetry laws, etc. We still lack a proper understanding of the geometry related to the logarithmic conformal field theories. In ordinary conformal field theory, the quantum fields can be understood as vertex operators which describe, how three closed strings join. They encode the equivalent of a three-vertex in a Feynman diagram of quantum field theory. But it is not yet clear, what geometric picture one gets for fields which transform in indecomposable representations. This is one of the questions I would like to answer.
    In the long run, I would like to contribute to the study of non-rational conformal field theories in string theory. For example, the spectra of strings ending on certain boundary manifolds in the Universe, the so called D-branes, are a very important issue. If these boundary manifolds are not just some nice group manifolds, things get complicated, as the resulting conformal field theory are no longer rational. Here, much work is still to do. But I believe that logarithmic conformal field theory will be a powerful tool to explore these questions.
    String theory is currently faced with a very hard obstacle: It seems to be a nearly insurmountable difficulty to formulate a full-fledged string field theory which would create its own ground state. The mathematics Humans so far has mastered is not sufficient to resolve this difficulty. Thus, string theorists must, for the time being, tackle a less ambitious problem, namely to formulate string theory in a given background. This means, among other things, that one has to make choices for the world sheet conformal field theories. Now, the easy choices are the rational conformal field theories. But these are so special that they certainly are not representing the generic conformal field theories from which one should presumably choose. On the other hand, logarithmic conformal field theory looks so much more like a generic conformal field theory, that it might be a superior choice to test and develop string theory further. This will be the most important future direction of my research.
    I clearly wish to stress that my approach to research in string theory is still of the fundamental kind, as I am ultimatley interested in finding a self-contained formulation of string theory in terms of string field theory. Thus, my research is more about exploring the space of conformal field theories in general than making intelligent guesses about a suitable point in it. Moreover, I am not so much concerned with guessing a suitable string background in the vast Landscape of string theories in the hope to get, by accident, phenomenological acceptable particle spectra. This latter approach to string theory is a phenomenological approach, not a fundamental one. I personally hold it with Seneca: “non est ars quae ad effectum casu venit  ::  that which achieves its effect by accident is not art.”


:: What is Mathematical Physics? ::

Nota Bene: One small remark about what is mathematical physics and what is not. Mathematical physics is defined as a methodology in theoretical physics. It is not defined by the topic one thinks about, but by the way how you think about this topic. In particular, if you think about a theory which has no experimental input so far does not automatically mean that you do mathematical physics. For example, you may think about string theory in a fundamental way, or in a phenomenological way. Both are valid methodologies, but the former is mathematical physics, and the latter is not. It is a very sad matter that many string theorists do not understand this difference. String phenomenology, certainly an important field of research, should not be called mathematical physics. This is abuse of language, and it is, in the philosophy of science, an incorrect terminology.