User anonymous - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:16:46Z http://mathoverflow.net/feeds/user/3792 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16074/how-is-the-physical-meaning-of-an-irreducible-representation-justified/16082#16082 Answer by Anonymous for How is the physical meaning of an irreducible representation justified? Anonymous 2010-02-22T19:59:50Z 2010-02-22T19:59:50Z <p>I assume that you've seen most of the wikipedia articles on Wigner's classification and the like, which from what I remember do not address the interesting question you raise.</p> <p>A few years ago I thought about the closely related question: given a quantum system as a black box, how can one identify the elementary particles in this system? It is clear (in the sense described by e.g. Wigner's classification) how to do this when the system consists of a single particle, but in a multi-particle system (such as the universe) it is possible for a group of operators that commute with the Hamiltonian to act irreducibly. The answer I eventually settled upon is that identifying the elementary particles in a system is essentially as arbitrary as writing the Hamiltonian as a sum of a kinetic term and an interacting term. It would absolutely make my day if someone could correct or clarify my speculation here.</p> <p>Once this is done, I am comfortable taking the answer to your second question (about the identification of elementary particles with irreducible representations of the "symmetry group of the universe") to be: elementary particles don't exist in any real sense (as in QFT), but in a classical quantum system one can usefully define them in terms of irreducible representations in this way.</p> http://mathoverflow.net/questions/14083/modular-forms-and-the-riemann-hypothesis/14164#14164 Answer by Anonymous for Modular forms and the Riemann Hypothesis Anonymous 2010-02-04T17:06:52Z 2010-02-04T17:06:52Z <p>The RH for an L-series L(s) is equivalent to an assertion about the locations of the poles of the logarithmic derivative (log L(s))' of L(s). In this way one can relate square-root savings in estimates for log-weighted partial sums of the traces of the Hecke operators of a given form to the RH for its L-series -- for the weight 1/2 theta series you've mentioned (whose Mellin transform is zeta(2s)), this corresponds to an improved error term in the prime number theorem. As David Hansen points out above, modular forms live in linear spaces but RH is not linearly robust, so any answer to your question will have to take some additional structure of those spaces (e.g. the action of the Hecke algebra) into account.</p> <p>As for your second question, there <strong>are</strong> statements about families of modular forms that imply RH for zeta; see for instance MR0633666 (and papers that reference it).</p>