80
12 Fock Representation
One may wonder whether the above notions are physically important, since they
are not usually brought up in the standard presentations of quantum mechanics.
The reason is that, contrary to the infinite-dimensional case, for systems with a finite
number of degrees of freedom, under very general regularity conditions, there is only
one irreducible representation of the Weyl algebra, the so-called Fock representation,
all the others being unitarily equivalent to it. According to the above discussion, one
may then say that there is only one folium, or only one closed world, available for
the representations of the Weyl algebra.
From a conceptual point of view, such a result (von Neumann theorem) explains
why, for systems with a finite number of degrees of freedom, the distinction between
the algebraic structure of the canonical variables and the states is not so relevant,
since there is only one Hilbert space of states for a given quantum system. In the
language of Statistical Mechanics one could say that there is only one phase. On the
other hand, for infinite systems the occurrence of inequivalent representations, i.e.
of different “phases” or different disjoint worlds, is the generic situation.
In the following, we shall discuss a simplified version of von Neumann’s theorem,
which characterizes the Fock representation in terms of the number operator.
56
We allow for infinite degrees of freedom and we consider regular representations
of the corresponding (infinite dimensional) Weyl algebra, namely representations π
such that π(U (α)), π(V (β)) with α, β any finite component vectors, are strongly
continuous in α, β. This is the standard regularity assumption underlying the analysis
of representations of Lie groups; it appears very general since, for separable spaces,
it is equivalent to the condition that the matrix elements of π(U (α)), π(V (β)) are
measurable functions. Furthermore, by Stone’s theorem, such a regularity condition
is equivalent to the existence of the generators. Thus, we have a representation of
the (infinite dimensional) Heisenberg algebra A H and we may assume that there is a
common dense domain D for A H . The representation is said to be irreducible if any
(bounded) operator which commutes with π(A H ) on D is a multiple of the identity.
In the following, the symbol A will be used to denote both an abstract element of
A H as well its representative in the concrete representation we are considering.
For the following purposes, it is convenient to introduce the so-called annihilation
and creation operators
a j ≡ (q j + i p j )/
√
2, a
∗
j = (q j − i p j )/
√
2,
(12.2)
and the so-called number operator N j ≡ a
∗
j a j . The physical meaning of such operators will be discussed below.
The Heisenberg commutation relations give
[a j , a
∗
k ] = δ jk , [a j , a k ] = 0,
(12.3)
56 For a proof of von Neumann theorem see, e.g. [SNS 96]. For the characterization of the Fock
representation in terms of the existence of the number operator, see G.F. Dell’Antonio and S.
Doplicher, J. Math. Phys. 8, 663 (1967); J.M. Chaiken, Comm. Math. Phys. 8, 164 (1967); Ann.
Phys. 42, 23 (1968) and references therein.
12 Fock Representation
One may wonder whether the above notions are physically important, since they
are not usually brought up in the standard presentations of quantum mechanics.
The reason is that, contrary to the infinite-dimensional case, for systems with a finite
number of degrees of freedom, under very general regularity conditions, there is only
one irreducible representation of the Weyl algebra, the so-called Fock representation,
all the others being unitarily equivalent to it. According to the above discussion, one
may then say that there is only one folium, or only one closed world, available for
the representations of the Weyl algebra.
From a conceptual point of view, such a result (von Neumann theorem) explains
why, for systems with a finite number of degrees of freedom, the distinction between
the algebraic structure of the canonical variables and the states is not so relevant,
since there is only one Hilbert space of states for a given quantum system. In the
language of Statistical Mechanics one could say that there is only one phase. On the
other hand, for infinite systems the occurrence of inequivalent representations, i.e.
of different “phases” or different disjoint worlds, is the generic situation.
In the following, we shall discuss a simplified version of von Neumann’s theorem,
which characterizes the Fock representation in terms of the number operator.
56
We allow for infinite degrees of freedom and we consider regular representations
of the corresponding (infinite dimensional) Weyl algebra, namely representations π
such that π(U (α)), π(V (β)) with α, β any finite component vectors, are strongly
continuous in α, β. This is the standard regularity assumption underlying the analysis
of representations of Lie groups; it appears very general since, for separable spaces,
it is equivalent to the condition that the matrix elements of π(U (α)), π(V (β)) are
measurable functions. Furthermore, by Stone’s theorem, such a regularity condition
is equivalent to the existence of the generators. Thus, we have a representation of
the (infinite dimensional) Heisenberg algebra A H and we may assume that there is a
common dense domain D for A H . The representation is said to be irreducible if any
(bounded) operator which commutes with π(A H ) on D is a multiple of the identity.
In the following, the symbol A will be used to denote both an abstract element of
A H as well its representative in the concrete representation we are considering.
For the following purposes, it is convenient to introduce the so-called annihilation
and creation operators
a j ≡ (q j + i p j )/
√
2, a
∗
j = (q j − i p j )/
√
2,
(12.2)
and the so-called number operator N j ≡ a
∗
j a j . The physical meaning of such operators will be discussed below.
The Heisenberg commutation relations give
[a j , a
∗
k ] = δ jk , [a j , a k ] = 0,
(12.3)
56 For a proof of von Neumann theorem see, e.g. [SNS 96]. For the characterization of the Fock
representation in terms of the existence of the number operator, see G.F. Dell’Antonio and S.
Doplicher, J. Math. Phys. 8, 663 (1967); J.M. Chaiken, Comm. Math. Phys. 8, 164 (1967); Ann.
Phys. 42, 23 (1968) and references therein.