On the Euler angles for SU(N) - eScholarship

IFUM–846–FT UCB–PTH–05/22 On the Euler angles for SU (N ) Stefano Bertini 2∗ Sergio L. Cacciatori 1,3† and Bianca L. Cerchiai 4,5‡ Dipartimento di Matematica dell’Universit` di Milano, a Via Saldini 50, I-20133 Milano, Italy. Dipartimento di Fisica dell’Universit` di Milano, a Via Celoria 16, I-20133 Milano, Italy. INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy. Lawrence Berkeley National Laboratory Theory Group, Bldg 50A5104 1 Cyclotron Rd, Berkeley, CA 94720-8162, USA. Department of Physics, University of California, Berkeley, CA 94720, USA. Abstract In this paper we reconsider the problem of the Euler parametrization for the unitary groups. After constructing the generic group element in terms of generalized angles, we compute the invariant measure on SU (N ) and then we determine the full range of the parameters, using both topological and geometrical methods. In particular, we show that the given parametrization realizes the group SU (N +1) as a fibration of U (N ) over the complex projective space CP n . This justifies the interpretation of the parameters as generalized Euler angles. ∗ bertiste@tiscalinet.it † sergio.cacciatori@mi.infn.it ‡ BLCerchiai@lbl.gov