12 Fock Representation
83
It should be stressed that only in the Fock representation the annihilation and
creation operators a j , a
∗
j have a simple interpretation, namely that of decreasing or
increasing the eigenvalues of N j (or of N ). Even in this case, however, their physical
meaning may not be transparent, since N j = ( p
2
j + q
2
j − 1)/2 may not be related
to a relevant observable (see, e.g. the case of the hydrogen atom or even of the free
particle). A special case is that of a system of free harmonic oscillators, where N j
is related to the Hamiltonian for the j-th degree of freedom and a j , a
∗
j , respectively,
annihilate and create elementary excitations of the system. By the same reasons, in
general the Fock state is not related to the possible ground state nor does it in general
have a simple physical meaning.
The picture emerging from the case of a system of harmonic oscillators however
suggests that the occupation number representation may be useful for describing
systems whose states can be described in terms of number of elementary excitations.
In this case, the index j may be taken to label the j-th excitation ( j may denote the
set of quantum numbers which identify such excitation) and a j , a
∗
j decrease and
increase, respectively, the number of j-th excitations. The states of the system are
then analyzed in terms of products of single excitation (or single-particle) states. As
a consequence, whereas the Hilbert (sub)space H N corresponding to a fixed number
N of particles or elementary excitations may not have a ground state, the total Hilbert
space (the direct sum of the H N , N ∈ N) has the Fock state as ground state (since
each elementary excitation has positive energy).
The message from the Proposition 12.1 is that the Fock representation is allowed
if N is a good quantum number for the description of the relevant states of the system.
This is reasonable in the case of a finite number of degrees of freedom and in the case
of non-interacting infinite degrees of freedom (with vanishing mean density). As we
shall see below, however, in the case of infinite degrees of freedom, the interaction
has generically dramatic effects, in the sense that it usually leads to a redefinition
of the degrees of the free theory, with the result that the eigenstates of the total
Hamiltonian cannot be described in terms of the eigenstates of the free Hamiltonian,
so that N is not a well defined quantum number.
In conclusion, the Fock representation for the algebra generated by the a j , a
∗
j
is convenient and physically motivated if such annihilation and creation operators
are related to the elementary excitations (or normal modes) which diagonalize the
total Hamiltonian. In general, the elementary excitations described by the a j , a
∗
j are
those which diagonalize the so-called free (or bilinear) part of the Hamiltonian and
therefore the interpretation of such annihilation and creation operators is simple if the
states of the system can be analyzed in terms of elementary excitations corresponding to the free part of the Hamiltonian; as we shall discuss below, for interacting
relativistic fields or for many-body systems with non-zero density, this is never the
case and the Fock representation is not allowed.
83
It should be stressed that only in the Fock representation the annihilation and
creation operators a j , a
∗
j have a simple interpretation, namely that of decreasing or
increasing the eigenvalues of N j (or of N ). Even in this case, however, their physical
meaning may not be transparent, since N j = ( p
2
j + q
2
j − 1)/2 may not be related
to a relevant observable (see, e.g. the case of the hydrogen atom or even of the free
particle). A special case is that of a system of free harmonic oscillators, where N j
is related to the Hamiltonian for the j-th degree of freedom and a j , a
∗
j , respectively,
annihilate and create elementary excitations of the system. By the same reasons, in
general the Fock state is not related to the possible ground state nor does it in general
have a simple physical meaning.
The picture emerging from the case of a system of harmonic oscillators however
suggests that the occupation number representation may be useful for describing
systems whose states can be described in terms of number of elementary excitations.
In this case, the index j may be taken to label the j-th excitation ( j may denote the
set of quantum numbers which identify such excitation) and a j , a
∗
j decrease and
increase, respectively, the number of j-th excitations. The states of the system are
then analyzed in terms of products of single excitation (or single-particle) states. As
a consequence, whereas the Hilbert (sub)space H N corresponding to a fixed number
N of particles or elementary excitations may not have a ground state, the total Hilbert
space (the direct sum of the H N , N ∈ N) has the Fock state as ground state (since
each elementary excitation has positive energy).
The message from the Proposition 12.1 is that the Fock representation is allowed
if N is a good quantum number for the description of the relevant states of the system.
This is reasonable in the case of a finite number of degrees of freedom and in the case
of non-interacting infinite degrees of freedom (with vanishing mean density). As we
shall see below, however, in the case of infinite degrees of freedom, the interaction
has generically dramatic effects, in the sense that it usually leads to a redefinition
of the degrees of the free theory, with the result that the eigenstates of the total
Hamiltonian cannot be described in terms of the eigenstates of the free Hamiltonian,
so that N is not a well defined quantum number.
In conclusion, the Fock representation for the algebra generated by the a j , a
∗
j
is convenient and physically motivated if such annihilation and creation operators
are related to the elementary excitations (or normal modes) which diagonalize the
total Hamiltonian. In general, the elementary excitations described by the a j , a
∗
j are
those which diagonalize the so-called free (or bilinear) part of the Hamiltonian and
therefore the interpretation of such annihilation and creation operators is simple if the
states of the system can be analyzed in terms of elementary excitations corresponding to the free part of the Hamiltonian; as we shall discuss below, for interacting
relativistic fields or for many-body systems with non-zero density, this is never the
case and the Fock representation is not allowed.