Mathematical Details
The operators derived above are actually a specific instance of a more generalized class of creation and annihilation operators. The more abstract form of the operators satisfy the properties below.
Let H be the one-particle Hilbert space. To get the bosonic CCR algebra, look at the algebra generated by a(f) for any f in H. The operator a(f) is called an annihilation operator and the map a(.) is antilinear. Its adjoint is a†(f) which is linear in H.
For a boson,
- ,
where we are using bra-ket notation.
For a fermion, the anticommutators are
- .
A CAR algebra.
Physically speaking, a(f) removes (i.e. annihilates) a particle in the state | f whereas a†(f) creates a particle in the state | f .
The free field vacuum state is the state with no particles. In other words,
where | 0 is the vacuum state.
If | f is normalized so that f | f = 1, then a†(f) a(f) gives the number of particles in the state | f .
Read more about this topic: Creation And Annihilation Operators
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