The creation and annihilation operators obey the canonical commutation relations, in that the one operator "undoes" what the other "creates". This implies that any given solution u ( p → → , σ σ , n ) must be paired with its "anti-solution" v ( p → → , σ σ , n ) so that one undoes or cancels out the other. The pairing is to be performed so that all symmetries are preserved. As one is generally interested in Lorentz invariance, the quantum field contains an integral over all possible Lorentz coordinate frames, written above as an integral over all possible momenta (it is an integral over the fiber of the frame bundle). The pairing requires that a given u ( p → → ) is associated with a v ( p → → ) of the opposite momentum and energy. The quantum field is also a sum over all possible spin states; the dual pairing again matching opposite spins. Likewise for any other quantum numbers, these are also paired as opposites. There is a technical difficulty in carrying out this dual pairing: one must describe what it means for some given solution u to be "dual to" some other solution v , and to describe it in such a way that it remains consistently dual when integrating over the fiber of the frame bundle, when integrating (summing) over the fiber that describes the spin, and when integrating (summing) over any other fibers that occur in the theory.