Question

Why is the following gauge transformation singular?

Answer 1

Suppose I have a single particle Hamiltonian: , either for boson or fermion. I do a gauge transformation to the field operator, then the Hamiltonian will be

Answer 2

When doing this transformation, the gauge field term ℏk0σ⃗ ℏk0σ→ is also changed, due to the noncommutating relations between pauli matrix. So the new Hamiltonian is not p22mp22m, but a more complex form.