# Mathematics Colloquia and Seminars

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### Generalised Wick Theorem and Fredholm Integral Equations in the Planar Ising Ferromagnet.

**Mathematical Physics & Probability**

Speaker: | Douglas Abraham, University of Oxford |

Location: | 1147 MSB |

Start time: | Wed, Apr 25 2012, 4:10PM |

Using transfer matrices in the Lieb, Mattis Schultz formulation (which will be outlined briefly), it will be shown how to use discrete symmetries to derive recurrence formulae for matrix elements of observables, such as local spin, between eigenvectors. These matrix elements are needed to extract information at the level of correlation functions and generally for many developments at a microscopic level. Solutions will be obtained for these matrix elements in the limit of infinite size. The results demonstrate the need both to extend the definition of Pfaffians and to generalize Wick’s Theorem. The relevance of the Fredholm alternative for this exercise will be explained. This approach leads naturally to Fredholm Determinants and to the eponymous Integral Equations.