Heisenberg uncertainty principle and its analogues in higher dimension: via Wigdersons' method

The following question was proposed by Avi Wigderson and Yuval Wigderson: Is it possible to use the method in their paper(The uncertainty principle: variations on a theme) to prove Heisenberg uncertainty principle in higher dimension R^d, and get the correct dependence of the constant on d? We answer this question affirmatively, and also prove some generalizations of Heisenberg uncertainty principle in R^d via Wigdersons’ method.