total angular momentum operator characterizes how a quantum system is changed when it is rotated. The relationship between angular momentum operators and rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics, as discussed further below. Just as J is the generator for rotation operators, L and S are generators for modified partial rotation operators. The operator rotates the position (in space) of all particles and fields, without rotating the internal (spin) state of any particle. Likewise, the operator rotates the internal (spin) state of all particles, without moving any particles or fields in space. The relation J = L + S comes from: i.e. if the positions are rotated, and then the internal states are rotated, then altogether the complete system has been rotated. SU(2), SO(3), and 360° rotations Although one might expect (a rotation of 360° is the identity operator), this is not assumed in quantum mechanics, and it turns out it is often not true: When the total angular momentum quantum number is a half-integer (1/2, 3/2, etc.), , and when it is an integer, . Mathematically, the structure of rotations in the universe is not SO(3), the group of three-dimensional rotations in classical mechanics. Instead, it is SU(2), which is identical to SO(3) for small rotations, but where a 360° rotation is mathematically distinguished from a rotation of 0°. (A rotation of 720° is, however, the same as a rotation of 0°.) On the other hand, in all circumstances, because a 360° rotation of a spatial configuration is the same as no rotation at all. (This is different from a 360° rotation of the internal (spin) state of the particle, which might or might not be the same as no rotation at all.) In other words, the operators carry the structure of SO(3), while and carry the structure of SU(2). From the equation , one picks an eigenstate and draws which is to say that the orbital angular momentum quantum numbers can only be integers, not half-integers. Connection to representation theory Starting with a certain quantum state , consider the set of states for all possible and , i.e. the set of states that come about from rotating the starting state in every possible way. The linear span of that set is a vector space, and therefore the manner in which the rotation operators map one state onto another is