According to the standard mathematical formulation of quantum mechanics, quantum observables such as x ^ ^ and p ^ ^ should be represented as self-adjoint operators on some Hilbert space. It is relatively easy to see that two operators satisfying the above canonical commutation relations cannot both be bounded. Certainly, if x ^ ^ and p ^ ^ were trace class operators, the relation Tr ⁡ ⁡ ( A B ) = Tr ⁡ ⁡ ( B A ) gives a nonzero number on the right and zero on the left.