Representations of Creation and Annihilation Operators
--- semantics of mathematical language of quantum field theory
Section 2.3 of Peskin and Schroeder explains canonical quantization of Klein-Gordon field,
where one can examine the parallel between harmonic oscillator and
the free field.
A main feature of quantum field theory is that it can represent the creation and
annihilation of particles. The creation and annihilation operators in canonical
quantization can be traced back to the ladder operators in the harmonic oscillator,
where the ladder operators emerge very naturally
when we factorize the Hamiltonian like (p^2 + q^2) = (q + ip)(q-ip) = (a+)(a-), and
their physical meaning emerges from the eigen-analysis of the Hamiltonian after
we promote q and p to operators (in textbooks, a- is written as "a", and a+ is written
as "a with a dagger"). The root of the quantum behavior is the quantum condition, which
leads to (a-)(a+)-(a+)(a-) = 1, so that
the Hamiltonian of a quantum field has a discrete set of eigen values, whose corresponding eigen states
are interpreted as states of different numbers of particles. The operators that transform
one eigen state to another are said to have created or annihilated particles, and thus
they are called creation and annihilation operators.
Dirac in his book deals with a+ and a- (he uses the notation eta instead of a)
in harmonic oscillator without relying on an explicit representation,
and expresses the Hamiltonian in terms of the number operator N = (a+)(a-). The symbolic approach
is simple and powerful enough to extract all the physics, such as the discrete energy
eigen values of the Hamiltonian. Dirac also interprets different energy levels as different
numbers of identical bosons in his book.
The same approach can be carried over to free field by expressing the field as a collection of
harmonic oscillators indexed by momentum. The canonical quantization of free Klein-Gordon field
gives rise to the creation and annihilation operators a+(k) and a-(k), so that a+(k) acting on the
vacuum state |0> creates a particle of momentum k, i.e., a+(k) |0> = |k>. a-(k) annihilates a
particle of momentum k, and a-(k)|0> = 0.
Despite the elegance and power of such a symbolic approach, it is still desirable to have concrete
representations of the states and the operators, just to make a+, a-, |n>, and in particular,
|0> tangible. It seems that many texts on quantum field theory do not care to tell the reader
about such concrete representations, because they all assume that the reader is already well trained in quantum
mechanics. This was certainly not the case with me. When I started reading about field theory, I only
knew about the basic notation of quantum mechanics. So when I first read about canonical quantization, |0> appeared
very unreal, and it was hard for me to believe that the calculations are not arbitrary. The rigorous
definition of Fock space did not help me at all. I really wanted to see some concrete representations,
even though they may not be entirely rigorous, or they may be tedious and not essential.
After all, we all learn our native languages by seeing and touching concrete things.
Now let us take a look at concrete representations in single harmonic oscillator in one dimension.
In Schrodinger position representation,
a+ = (x-d/dx)/sqrt(2),
a- = (x+d/dx)/sqrt(2). a+ and a- are adjoint,
and [a-, a+] = 1. The ground state |0> is represented by exp(-x^2/2)/pi^{1/4}, so that (a-) |0> = 0.
Other states can be calculated by (a+) |n> = sqrt(n+1) |n+1>. (a-) |n> = sqrt(n) |n-1>.
The state functions of |n> are related to the Hermit polynomials.
Born and Jordan (1925) treated harmonic oscillator using matrix representation.
In number representation where the number matrix
N = (a+)(a-) is diagonalized, a+ and a- are infinite matrices whose none zero
elements are (a+)(n+1, n) = sqrt(n+1), (a-)(n-1, n) = sqrt(n), n = 0, 1, ... a+ and a- are
transposes of each other. The energy eigen vector |n> is an infinite vector whose elements are 0
except that the n-th element is 1. Again, we have (a+) |n> = sqrt(n+1) |n+1>,
(a-) |n> = sqrt(n) |n-1>, and (a-) |0> = 0.
The Fock-Bargmann representation was first proposed by Fock (1927), and Bargmann in 1960s gave a more rigorous
and general treatment. It is not to be confused with the Fock space. In Fock-Bargmann representation,
z = p - iq, and the space F consists of all the functions of a
complex variable that are differentiable in the entire domain. The inner product is defined with respect to a Gaussian density.
The representation of |n> is z^{n}/sqrt(n!). The creation operator is z. The annihilation operator is d/dz.
Unlike x and d/dx in position representation, z and d/dz are adjoint to each other according to the above definition
of the inner product. The ground state |0> is 1. This representation is remarkably simple.
For quantum field, we need to extend the above representations.
By extending Schrodinger position representation, the vacuum state wave functional that represents |0> is
proportional to exp(-\int something times |x(k)|^2), See problem 8.8 of Srednicki.
Here x means field quantity, not position.
Apparently a+(k) \propto x(k) - d/dx(k), a-(k) \propto x(k)+d/dx(k).
Chapter 2 of the text of Huang gives the most concrete treatment I have seen. Since we map x in quantum mechanics to
phi in quantum field, the wave function Psi(x) becomes the wave functional Psi(phi), because phi itself
is a function. Of course, if we discretize phi into pixels, then Psi(phi) is nothing more than a high-dimensional
wave function. In the position representation, the momentum field pi(x) is changed to functional derivative
-ih d/d(phi(x)). Huang writes down explicitly the wave functional Psi(phi), the Schrodinger
equation for the wave functional, a+(k) and a-(k), in terms of functional derivative,
and vacuum wave functional Psi_0(phi) for |0>. I find it really helpful to establish such a
parallel between quantum mechanics and quantum field in terms of Schordinger equation, so that we know that
quantum field theory is really just quantum mechanics with continuously infinite many variables.
Born, Heisenberg and Jordan (1925) treated coupled harmonic oscillators such as a vibrating string
by transforming them into a system of uncoupled oscillators such as the momentum representation
mentioned above. In the original number represention, the rows and columns of the matrices a+ and a-
as well as the elements of the vector |n> are indexed by a single number n = 0, 1, 2, ... To extend
this representation, we can index the rows, columns and elements by a sequence of numbers (n_1, n_2, ..., n_k, ...),
where n_k is the number of particles of momentum k. Then we can write down the elements of a+, a-, and |n>.
Chapter 1 of Weinberg I gives an explicit formula which is essentially the same as that for the single harmonic
oscillator. It is more or less a book-keeping of all the relations.
In the Fock-Bargmann representation, the basis functions of the space F_K are in the form of
prod_{k} z_k^{n_k}/sqrt(n_{k}!), k = 1, 2,..., K, and n_k are nonnegative integers. The
inner product is defined with respect to K-dimensional Gaussian density. The Fock space
can be mapped to the limit of F_K as K --> infty, or the union of F_1, F_2, ... The vacuum state
|0> is 1, a+(k) = z(k), a-(k) = d/dz(k). In F_infty, states of different numbers of particles
are represented by multinomial products of different numbers of z_k's.
a+(k) and a-(k) add or remove one z_k from these states.
Finally, in the second quantization of Dirac (1927), the wave function of a
single particle is a(r). Let N(r) = |a(r)|^2, and phi(r) be the phase of a(r), so
a(r) = N(r)^{1/2} e^{i phi(r)/h}. We can promote N(r) to be the number operator. In
Schrodinger representation, N(r) is diagonalized as n(r), then e^{i phi(r)/h} becomes
e^{d/dn(r)}. For a wave function f(n(1), ..., n(r), ...),
e^{d/dn(r)} f(n(1), ..., n(r), ...) = f(n(1), ..., n(r)+1, ...), because
e^{a d/dx} f(x) = f(x+a), which can be seen by Taylor expanding f(x+a) at x.
So a+(r) = (n(r)+1)^{1/2}e^{d/dn(r)} = e^{d/dn(r)} n(r)^{1/2} is the creation operator,
and a-(r) = n(r)^{1/2} e^{-d/dn(r)} = e^{-d/dn(r)} (n(r)+1)^{1/2} is the annihilation operator,
with a-(r)a+(r) - a+(r)a-(r) = 1.