LSND, SN1987A, and CPT Violation

We point out that neutrino events observed at Kamiokande and IMB from SN1987A disfavor the neutrino oscillation parameters preferred by the LSND experiment. For $\Delta m^2 > 0$ (the light side), the electron neutrinos from the neutronization burst would be lost, while the first event at Kamiokande is quite likely to be due to an electron neutrino. For $\Delta m^2 < 0$ (the dark side), the average energy of the dominantly $\bar{\nu}_e$ events is already lower than the theoretical expectations, which would get aggravated by a complete conversion from $\bar{\nu}_\mu$ to $\bar{\nu}_e$. If taken seriously, the LSND data are disfavored independent of the existence of a sterile neutrino. A possible remedy is CPT violation, which allows different mass spectra for neutrinos and anti-neutrinos and hence can accommodate atmospheric, solar and LSND data without a sterile neutrino. If this is the case, Mini-BooNE must run in $\bar{\nu}$ rather than the planned $\nu$ mode to test the LSND signal. Non-commutative geometry is briefly discussed as a possible origin of CPT violation.