Frequently-asked questions, lectures 43 – 45

The introduction of the fermion annihilation and creation operators may be elegant, but are we losing a physical sense of what is going on? For the boson annihilation and creation operators, we at least justified them through the mathematics of handling the harmonic oscillator problem, but these fermion ones just seem to come from nowhere.

By this point, it may be best just to look at these operators as mathematical conveniences. Once you get used to them, possibly they may stop being apparently meaningless. Perhaps it is sufficient to say that these operators are just mathematical devices, but they are very useful, and they automatically take care of all of the algebra of identical particles, including all the effects of the symmetrization or antisymmetrization with respect to exchange so you don’t have to worry about explicitly about those any more. Basically, the commutation or anticommutation relations associated with these operators become the mathematics that takes care of the symmetrization or antisymmetrization with respect to exchange.

Though it is rather evocative to use the titles “creation” and “annihilation” for these operators, it is not clear that in any real sense these operators are describing for us the physical process of creation or annihilation of actual particles even if mathematically they behave as if they were actually creating and annihilating particles. That might be disappointing, but we can get over our disappointment by noting that these operators really do save us a lot of pencil lead in our algebraic manipulations here. They are a very convenient way of writing all this down!