What are the quantum commutation relations for the total angular momentum of light? tutorial

The total angular momentum of light has received attention for its application in a variety of phenomena such as optical communication, optical forces, and sensing. However, the quantum behavior including the commutation relations has been relatively less explored. Here, we derive the correct commutation relation for the total angular momentum of light using both relativistic and non-relativistic approaches. An important outcome of our work is the proof that the widely assumed quantum commutation relation for the total observable angular momentum of light is fundamentally incorrect. Our work will motivate experiments and lead to new insights on the quantum behavior of the angular momentum of light.