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
Famous quotes containing the words mathematical and/or details:
“It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
—Henry David Thoreau (1817–1862)
“Patience is a most necessary qualification for business; many a man would rather you heard his story than granted his request. One must seem to hear the unreasonable demands of the petulant, unmoved, and the tedious details of the dull, untired. That is the least price that a man must pay for a high station.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)