The translation operator T ^ ^ ( x ) moves particles and fields by the amount x . Therefore, if a particle is in an eigenstate | r ⟩ ⟩ of the position operator (i.e., precisely located at the position r ), then after T ^ ^ ( x ) = ∫ ∫ d r | r + x ⟩ ⟩ ⟨ ⟨ r | acts on it, the particle is at the position r + x : T ^ ^ ( x ) | r ⟩ ⟩ = | r + x ⟩ ⟩ .