GROUP REPRESENTATIONS AND THE QUANTUM STATISTICS OF SPINS Jonathan Michael Harrison Scho ol of Mathematics Septemb er A dissertation submitted to the University of Bristol in accordance with the requirements of the degree of Doctor of Philosophy in the Faculty of Science Abstract An intuitive explanation of the connection b etween the quantum statistics of par ticles and their spins has b een sought since the early pro ofs of the spinstatistics theorem were published in the s Recently Berry and Robbins have sug gested a new approach They construct a p ositiondep endent spin basis in which exchanging the p ositions of identical particles automatically exchanges their spins In this basis the spinstatistics connection can b e derived from the singlevaluedness of the wavefunction The p ositiondep endent basis for n particles is constructed using the Schwinger representation of spin which can b e regarded as a choice of representation of the group SU n In this thesis we generalise the construction to include all representations of SU n For n vectors that can b e used to construct the p ositiondep endent basis are assembled directly using Young tableau and the sign of these vectors under the exchange of the two particles determined We nd that for a typical represen tations of SU there are several subspaces of vectors with dierent spins that can b e used to construct the p ositiondep endent basis The sign of vectors in these sub spaces under the exchange of the particles is determined not only by the spin but also by the symmetry conditions recorded in the Young tableau which lab els the representation of SU For n particles the decomp osition into subspaces that can b e used in the con struction is achieved using the characters of the relevant groups We see that typi cal representations admit parastatistics The numb er of subspaces of spin s which transform according to a given irreducible representation of S is written in terms n of Littlewo o dRichardson and ClebschGordan co ecients iii Acknowledgements There are many p eople to whom I owe a great deal of thanks for their supp ort and encouragement over the past four years Firstly and most imp ortantly to my sup ervisor Jonathan Robbins I have learnt much from him and his ideas interest and go o d humour have b een an inspiration Those sentiments also apply to the memb ers of the quantum chaos group which it has b een a pleasure to b e part of I have enjoyed talking to p eople ab out this work and in particular Michael Berrys and Aidan Schoelds comments were much appreciated Sp ecial thanks should also go to Gregory Berkolaiko for his friendship and many useful conversations some even ab out maths I would also like to thank Suhasini Scott and all my friends amongst the departments graduate students who were always the b est company These acknowledgements would not b e complete without mentioning the mem b ers of the Bristol University physics department who expanded my horizons whilst I was an undergraduate for which I am truly grateful Finally I would like to say a big thank you to my family without whom as the saying go es none of this would have b een p ossible When my parents showed me maths puzzles as a child I think they little exp ected how far it could go This research was made p ossible through the nancial supp ort of the EPSRC v Authors Declaration I declare that the work in this thesis was carried out in ac cordance with the Regulations of the University of Bristol The work is original except where indicated by sp ecial ref erence in the text and no part of the dissertation has b een submitted for any other degree Any views expressed in the dissertation are those of the author and do not necessarily represent those of the University of Bristol The thesis has not b een presented to any other university for examination either in the United Kingdom or overseas Jonathan Michael Harrison Date th Septemb er vii Contents Abstract iii Acknowledgements v Authors Declaration vii SpinStatistics and al l that The discovery of spinstatistics Paulis exclusion principle Spin and statistics The symmetrisation p ostulate Relativistic quantum eld theory Paulis pro ofs Axiomatic pro ofs Criticisms of the eld theory pro ofs Feynman and the elementary pro ofs Geometric rotation Feynmans Dirac lecture Recent candidates for an elementary pro of Parastatistics Parab osons and Parafermions Quons Exp erimental tests Conclusions ix Contents Representation Theory Preliminaries Groups and representations The symmetry group of an equilateral triangle Cosets and the quotient group Classes and characters The symmetric group The irreducible representations of S The group SU n SU Ro ots and weights of Lie algebras The highest weight classication of irreducible representations Representations of the symmetric group and Young tableau Partitions graphs and tableau Characteristic units of the group algebra Characteristic units of S n Constructing representations of SU n with characteristic units Characters of U n and SU n Characters of S n The Littlewo o dRichardson theorem Multiplying Young tableau Applying the Littlewo o dRichardson theorem to representations Decomp osing representations of SU m n into representa tions of SU m SU n Quantum indistinguishability Intro duction The p ositiondep endent spin basis Quantum mechanics in the p ositiondep endent spin basis The Schwinger representation of spin U R for two particles The exchange sign is indep endent of U r x Contents Constructing U r Smo othness of jM ri Paralleltransp ort of jM ri The Schwinger representation of SU n U R for n particles Maps from conguration space to SU n Smo othness and paralleltransp ort for n particles Alternative constructions Parastatistics The comp onents of the BR construction Summary Calculation of the exchange sign for the irreducible representations of SU The group SU The spin subspace Preparing the subspace W Basis vectors of the tensor pro duct representation Basis vectors of irreducible representations of SU Basis vectors of a representation of SU SU s s multiplets in representations of SU Spin vectors with zero weight with resp ect to the exchange algebra s s multiplets with E eigenvalue zero z Results for general multiplets Example The s s multiplets of with e z The exchange sign Dening a xed exchange rotation Selecting a vector jM i Constructing the highest weight state of a multiplet using Young tableau The eect of the symmetry conditions on the exchange sign xi Contents Evaluating the eect of the symmetry conditions for a general highest weight state Numerical calculation of the exchange sign Character decomp osition of SU n Character decomp ositions The semidirect pro duct Classes of G n H Irreducible representations of G n H The Weyl group Denition of the Weyl group The Weyl group for SU n The group n Classes of n Representations of n The action of the Weyl group on spin vectors jMi The physical subgroup H The n spin subgroup Denition of H Classes of H The volume element of H Representations of H Physical representations of H s Characters of B H Representations of SU n Characters of SU n Restricting a representation of SU n to H The decomp osition of SU The character of SU for elements of H Evaluation of A I Evaluation of A The numb er of physical representations of H in SU xii Contents The decomp osition of SU Evaluation of A I Evaluation of A Evaluation of A .