Quantum Field Theory I

1.2. MANY-BODY QUANTUM MECHANICS 21
Remark: The basis |i,j,...〉, or |n 1 ,n 2 ,...〉, arising from a particular basis
|i〉 of the 1-particle Hilbert space is perhaps the most natural, but not the only
reasonable, basis in the Fock space. Actually, any complete set of physically relevant
commuting operators defines some relevant basis (from this point of view,
|n 1 ,n 2 ,...〉 is the basis of eigenvectors of the occupation number operators).
If the eigenvalues of a complete system of commuting operators are discrete and
bounded from below, then one can label both eigenvalues and eigenvectors by
natural numbers. In such a case, the basis defined by the considered system
of operators looks like |N 1 ,N 2 ,...〉, and for a basis of this type we can define
the so-called raising and lowering operators, just as we have defined the creation
and annihilation operators: A + i (A i ) raises (lowers) the quantum number N i by
one.
Of a special interest are the cases when the Hamiltonian can be written as a sum
of two terms, one of which has eigenvectors |N 1 ,N 2 ,...〉 and the second one can
be understood as a small perturbation. If so, the system formally looks like an
almost ideal gas made of a new type of particles (created by the A + i operators
from the state in which all N i vanish). These formal particles are not to be
mistaken for the original particles, which the system is built from.
It may come as a kind of surprise that such formal particles do appear frequently
in many-body systems. They are called elementary excitations, and they come
in great variety (phonons, plasmons, magnons, etc.). Their relation to the original
particles is more or less known as the result of either detailed calculations,
or an educated guess, or some combination of the two. The description of the
system is, as a rule, much simpler in terms of the elementary excitations than
in terms of the original particles. This explains the wide use of the elementary
excitations language by both theorists and experimentalists.
Remark: The reader is perhaps familiar with the operators a + and a, satisfying
the above commutation relations, namely from the discussion of the LHO (linear
harmonic oscillator) in a basic QM course 31 . What is the relation between
the LHO a + and a operators and the ones discussed here
The famous LHO a + and a operators can be viewed as the creation and annihilation
operators of the elementary excitations for the extreme example of
many-body system presented by one LHO. A system with one particle that can
be in any of infinite number of states, is formally equivalent to the ideal gas of
arbitrary number of formal particles, all of which can be, however, in just one
state.
Do the LHO operators a + and a play any role in QFT Yes, they do, but only
an auxiliary role and only in one particular development of QFT, namely in the
canonical quantization of classical fields. But since this is still perhaps the most
common development of the theory, the role of the LHO is easy to be overestimated.
Anyway, as for the present section (with the exception of the optional
subsection 1.2.4), the reader may well forget about the LHO.
31 Recall a + = ˆx √ mω/2 − iˆp/ √ 2mω , a = ˆx √ mω/2 + iˆp/ √ 2mω. The canonical
commutation relation [ˆx, ˆp] = i implies for a + and a the commutation relation of the creation
and annihilation operators. Moreover, the eigenvalues of the operator N = a + a are natural
numbers (this follows from the commutation relation).
Note that the definition of a + and a (together with the above implications) applies to any
1-particle system, not only to the LHO. What makes the LHO special in this respect is the
Hamiltonian, which is very simple in terms of a + ,a. These operators are useful also for the
systems ”close to LHO”, where the difference can be treated as a small perturbation.