This one equation is sufficient to both identify the operator a\an as the number operator and aj, as a creation operator and permit us to proceed with the construction of the Fock states of the theory. [To see this, return to Sec. 1.5 and confirm that the above relation was all we used to construct the states and establish the properties of a*.]
We now assume that the creation operators aj, satisfy anticommutation relations (which also implies that the annihilation operators do), and use the required relation (7.22) to find the correct relations between aj, and an. Using the notation [ , to represent either a commutator or an anticommutator, the implications of (7.22) can be worked out immediately:
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