Creation and Annihilation Operators - Mathematical Details

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 .

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