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:
“As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.”
—Blaise Pascal (1623–1662)
“Working women today are trying to achieve in the work world what men have achieved all along—but men have always had the help of a woman at home who took care of all the other details of living! Today the working woman is also that woman at home, and without support services in the workplace and a respect for the work women do within and outside the home, the attempt to do both is taking its toll—on women, on men, and on our children.”
—Jeanne Elium (20th century)