This page contains some remarks about research topics in physics which seem to me to be suitable for students. Sometimes I form this view because the topic is of the right difficulty, and sometimes because it is a current topic of research. All the topics are progressive and successful.
This topic is quite hard, but there are still some doable problems; it has been very successful in explaining the occurrence of Bosons and Fermions with the right spin (the connection between spin and statistics). It is a more general set-up than Wightman's axiomatic quantum field theory, and has at least two very comfortable ways in. I refer to the two books
Haag, R., Local Quantum Physics, Springer 1992; revised, 1996; Springer-Verlag.
Araki, H., Mathematical Theory of Quantum Fields, Oxford University Press, 1999.
A way into research is to read the articles by B. Schroer and H. J. Borchers,
Schroer, B., Particle physics and quantum field theory at the turn of the century: old principles with new concepts, J. Mathematical Phys., 41, 3801-3831, 2000.
Borchers, H. J., On revolutionizing quantum field theory with Tomita's modular theory, J. Mathematical Phys., 41, 3604-3673, 2000.
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There are lots of hard unsolved problems in this subject, even in finite-dimensional quantum theory. First, read one or better both the books
Nielson, M. A., and Cheung, Quantum Computation and Quantum Information Theory, Cambridge University Press, 2000.
Holevo, A., S., Probabilistic and Statistical Aspects of Quantum Theory, North Holland, Amsterdam, 1982. Then read some of the papers of Giacomo D'Ariano, found as the first entry when you google his name, and choose your field of interest.
Or read carefully paper 13 called "Universal simulation of Markovian quantum dynamics", by Dave Bacon, Andrew Childs et al.; you can find them on Google.
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I promised to explain the mistake in the paradox on the phase operator. It occurs because the definition of the differential operator p = -i d/dx cannot be extended to all differentiable functions on the circle in a manner which maintains the hermiticity of the operator. Thus the domain includes all differentiable periodic functions, f(x). Then the boundary condition ensures that on integration by parts the boundary terms cancel, and the operator is hermitian. If f(x) is differentiable and periodic, then x f(x) is differentiable, but not periodic. So the extension of p to include x f(x) does not lead to a hermitian operator. Dirac's bra and ket notation does not mention domains, and so the error is difficult to spot in this notation.
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M. Aizenman has assembled a list of hard, rigorous problems in mathematical physics suggested by the members of IAMP.
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The occurrence of quasiparticles carrying topological conserved quantum numbers is an inspiring idea for what elementary particles are. These objects are, justifiably, named after Tony Skyrme. He had in mind a non-linear dynamical relativistic model, in which a twist or kink in the field was stable for topological reasons. I became converted to this idea when asked to examine the Ph. D. of one of Skyrme's students, J. Williams. I concluded that a significant part of this work was correct and could be made rigorous.
It is easier to set up models in non-relativistic quantum field theory than in the relativistic case. In this case there is also some hope of producing the excitation experimentally. A rich source of results and ideas in both theory and experiment can be found here.
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The famous paper of Yang and Mills was published just as G. Shaw, of Hull, was writing up his own version of a similar model as his Ph. D. thesis. I therefore link his name to this promising model. There is some chance that Yang-Mills-Shaw theory will be understood in four space-time dimensions. This is because it is asymptotically free. See the nice article by Frank Wilczek. It has had a rigorous Euclidean treatment by Magnen, Rivasseau and Senéor, who show that (with a space cut-off) the expectation values of the Wilson loops can be defined in the limit of infinite ultraviolet cut-off. This constructive approach is very difficult, and is not recommended for the student. Their notable work took them twenty years to complete. However, there is quite a lot to do in the axiomatic side of this model: what are the observables, and what axioms do they obey? Is there a Euclidean version? Does the solution of MRS obey these axioms? How can we remove the infrared divergences?
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© Ray Streater, 11/May/2003.