84
12 Fock Representation
The relation between the Fock representation and the free Hamiltonian can be
made more precise. To this purpose, we consider a system with infinite degrees of
freedom and the associated “free” Hamiltonian
H 0 =
i
ω i a
∗
i a i ,
where ω i denotes the energy of the free i-th excitation. We also assume that there is
an energy (or mass) gap, i.e.
ω i ≥ m > 0, ∀i.
Then, if we look for a representation of the algebra generated by the a j , a
∗
j , such
that H 0 is a self-adjoint operator (on the common dense domain), the representation
is necessarily a Fock representation. In fact, (the series which defines) H 0 dominates
(term by term the series which defines) N
N ≤ (1/m)H 0
and therefore the existence of H 0 entails the existence of N .
Example 12.1 Free scalar field. As an example, we briefly review the quantization of a free massive scalar real field. The problem is to find the (operator-valued
distributional) solution of the Klein–Gordon equation
( + m
2
)ϕ(x) = 0,
satisfying the equal time canonical commutation relations (π(x) = ˙
ϕ(x))
[ϕ(x, 0), π(y, 0)] = iδ(x − y),
[ϕ(x, 0), ϕ(y, 0)] = [π(x, 0), π(y, 0)] = 0.
(12.8)
In contrast with the classical case, the canonical relations (12.8) imply that in order
to get well defined operators one must (at least) smear the fields with test functions
of the space variables, typically f ∈ C
∞
(R
s
), s = space dimensions and of fast
decrease, (briefly f ∈ S(R
s
)). Thus, from a mathematical point of view, the fields
ϕ(x), π(x) have to be regarded as operator-valued distributions.
The algebra A W of canonical variables can be thought of as generated by the
exponentials of the real fields ϕ( f ), π(g), smeared with test functions f, g ∈ S(R
s
).
Among the many possible representations of such an infinite-dimensional Weyl algebra, a selection criterion is that one has a well defined Hamiltonian.
Now, quantization of the classical Hamiltonian
H =
1
2
d
s x[(∇ϕ)
2
(x) + π
2
(x) + m
2
ϕ
2
(x)]
12 Fock Representation
The relation between the Fock representation and the free Hamiltonian can be
made more precise. To this purpose, we consider a system with infinite degrees of
freedom and the associated “free” Hamiltonian
H 0 =
i
ω i a
∗
i a i ,
where ω i denotes the energy of the free i-th excitation. We also assume that there is
an energy (or mass) gap, i.e.
ω i ≥ m > 0, ∀i.
Then, if we look for a representation of the algebra generated by the a j , a
∗
j , such
that H 0 is a self-adjoint operator (on the common dense domain), the representation
is necessarily a Fock representation. In fact, (the series which defines) H 0 dominates
(term by term the series which defines) N
N ≤ (1/m)H 0
and therefore the existence of H 0 entails the existence of N .
Example 12.1 Free scalar field. As an example, we briefly review the quantization of a free massive scalar real field. The problem is to find the (operator-valued
distributional) solution of the Klein–Gordon equation
( + m
2
)ϕ(x) = 0,
satisfying the equal time canonical commutation relations (π(x) = ˙
ϕ(x))
[ϕ(x, 0), π(y, 0)] = iδ(x − y),
[ϕ(x, 0), ϕ(y, 0)] = [π(x, 0), π(y, 0)] = 0.
(12.8)
In contrast with the classical case, the canonical relations (12.8) imply that in order
to get well defined operators one must (at least) smear the fields with test functions
of the space variables, typically f ∈ C
∞
(R
s
), s = space dimensions and of fast
decrease, (briefly f ∈ S(R
s
)). Thus, from a mathematical point of view, the fields
ϕ(x), π(x) have to be regarded as operator-valued distributions.
The algebra A W of canonical variables can be thought of as generated by the
exponentials of the real fields ϕ( f ), π(g), smeared with test functions f, g ∈ S(R
s
).
Among the many possible representations of such an infinite-dimensional Weyl algebra, a selection criterion is that one has a well defined Hamiltonian.
Now, quantization of the classical Hamiltonian
H =
1
2
d
s x[(∇ϕ)
2
(x) + π
2
(x) + m
2
ϕ
2
(x)]