In a study of the reaction e-e+ -> W-W+ with the DELPHI detector, the probabilities of the two W particles occurring in the joint polarisation states transverse-transverse (TT), longitudinal-transverse plus transverse-longitudinal (LT) and longitudinal-longitudinal (LL) have been determined using the final states WW -> l nu q qbar (l = e, mu). The two-particle joint polarisation probabilities, i.e. the spin density matrix elements rho_TT, rho_LT, rho_LL, are measured as functions of the W- production angle, theta_W-, at an average reaction energy of 198.2 GeV. Averaged over all cos(theta_W-), the following joint probabilities are obtained: rho_TT = (67 +/- 8)%, rho_LT = (30 +/- 8)%, rho_LL = (3 +/- 7)% . These results are in agreement with the Standard Model predictions of 63.0%, 28.9% and 8.1%, respectively. The related polarisation cross-sections sigma_TT, sigma_LT and sigma_LL are also presented.
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