Just as J is the generator for rotation operators, L and S are generators for modified partial rotation operators. The operator R spatial ( n ^ ^ , ϕ ϕ ) = exp ⁡ ⁡ ( − − i ϕ ϕ L n ^ ^ ℏ ℏ ) , rotates the position (in space) of all particles and fields, without rotating the internal (spin) state of any particle. Likewise, the operator R internal ( n ^ ^ , ϕ ϕ ) = exp ⁡ ⁡ ( − − i ϕ ϕ S n ^ ^ ℏ ℏ ) , rotates the internal (spin) state of all particles, without moving any particles or fields in space. The relation J = L + S comes from: R ( n ^ ^ , ϕ ϕ ) = R internal ( n ^ ^ , ϕ ϕ ) R spatial ( n ^ ^ , ϕ ϕ )