Convergence test for sigma sin ((n^2 * phi)/2)?

The series won't converge if phi is the golden ratio. I'm not sure what else it could be. If you meant "pi" instead, the series diverges because each odd term has magnitude 1. The terms don't tend to 0, hence the series doesn't converge. I believe this can only converge if phi is an even multiple of pi. Certainly it can't converge if phi is not a rational multiple of pi (such as the golden ratio, or any other algebraic number). The distance between consecutive samples is always an odd multiple of phi/2. These diffferences, modulo 2*pi, will repeatedly get close enough to pi/2 that consecutive terms will differ by approximately 1. Therefore the terms do not converge to any limit, much less 0.