I’m not sure what your question means, but I suspect the problem is that there are two equally good configuration spaces.
In the symplectic hamiltonian formulation of classical mechanics, the hamiltonian must be local and differentiable on the joint position-momentum space, and that is the only constraint on the hamiltonian. If you think of the symplectic configuration space as the cotangent bundle of position space, then this amounts to saying that the hamiltonian may depend on position and the first derivative of position. But, symmetrically, it depends on momentum and the first derivative of momentum.
The lesson of the symplectic formalism is that the position and momentum configuration spaces are equally valid, but the joint symplectic configuration space is probably more valid. When you go to QM, the position and momentum configuration spaces are still there, still playing symmetric roles, but the symplectic configuration space is more problematic. (This should lead to some commentary on the “trick” of thinking of position as an operator on the Hilbert space, but I’m not sure what to say.)
I’m not sure what your question means, but I suspect the problem is that there are two equally good configuration spaces.
In the symplectic hamiltonian formulation of classical mechanics, the hamiltonian must be local and differentiable on the joint position-momentum space, and that is the only constraint on the hamiltonian. If you think of the symplectic configuration space as the cotangent bundle of position space, then this amounts to saying that the hamiltonian may depend on position and the first derivative of position. But, symmetrically, it depends on momentum and the first derivative of momentum.
The lesson of the symplectic formalism is that the position and momentum configuration spaces are equally valid, but the joint symplectic configuration space is probably more valid. When you go to QM, the position and momentum configuration spaces are still there, still playing symmetric roles, but the symplectic configuration space is more problematic. (This should lead to some commentary on the “trick” of thinking of position as an operator on the Hilbert space, but I’m not sure what to say.)