12 Fock Representation
85
requires some care, because the above formal integral involves both the definition
of the product of distributions at the same point (ultraviolet (UV) singularities) and
the integration over an infinite volume (infrared (IR) singularities). Thus, in contrast
with the standard case of finite degrees of freedom, quantization of the classical
expressions requires a regularization and/or a renormalization.
In the case of free fields, the UV renormalization of the Hamiltonian is regarded
as trivial (the problem is not even mentioned in most textbooks). It is obtained by
reordering the products of operators, say AB, so that the creation operators stay
on the left and the annihilation operators on the right as if they commute; such a
procedure is called Wick ordering and denoted by : AB :. Then, in a finite volume
with periodic boundary conditions the momentum can take only discrete values k j
and one has (ω j ≡ (k
2
j + m
2
)
1/2
),
H ren =
1
2
V
d
s x : [(∇ϕ)
2
(x) + π
2
(x) + m
2
ϕ
2
(x)] :=
j
ω j a
∗
j a j ,
where
√
2a j ≡
√
ω j ˜
ϕ(k j ) + i(
√
ω j )
−1
˜
π(k j ), and the tilde denotes the Fourier transform. By the above argument, the condition that H 0 be well defined selects the Fock
representation.
58 It is not difficult to show that one has a well defined operator also
in the infinite volume limit, when the momentum becomes a continuous variable and
H 0 =
1
2
d
s x : [(∇ϕ)
2
(x) + π
2
(x) + m
2
ϕ
2
(x)] :
=
d
s k ω(k)a
∗
(k)a(k), ω(k) = (k
2
+ m
2
)
1/2
is well defined on the dense domain obtained by applying the polynomial algebra of
the a
∗
( f ) to Ψ 0 , since H 0 Ψ 0 = 0 and the commutator is well defined [H 0 , a
∗
( f )] =
a
∗
(ω f ).
The massless case, m = 0, deserves a comment, since the number operators is no
longer dominated by the free Hamiltonian. In fact, in this case there are representations (the so-called non-Fock coherent state representations) in which H 0 is well
defined, but N is not.
58 H. J. Borchers, R. Haag and B. Schroer, Nuovo Cim. 29, 148 (1963). For a general look at free
fields from the point of view of representations of the algebra of canonical variables (canonical
quantization), see A. S. Wightman and S. Schweber, Phys. Rev. 98, 812 (1955); S. S. Schweber,
Introduction to Relativistic Quantum Field Theory, Harper and Row 1961.
85
requires some care, because the above formal integral involves both the definition
of the product of distributions at the same point (ultraviolet (UV) singularities) and
the integration over an infinite volume (infrared (IR) singularities). Thus, in contrast
with the standard case of finite degrees of freedom, quantization of the classical
expressions requires a regularization and/or a renormalization.
In the case of free fields, the UV renormalization of the Hamiltonian is regarded
as trivial (the problem is not even mentioned in most textbooks). It is obtained by
reordering the products of operators, say AB, so that the creation operators stay
on the left and the annihilation operators on the right as if they commute; such a
procedure is called Wick ordering and denoted by : AB :. Then, in a finite volume
with periodic boundary conditions the momentum can take only discrete values k j
and one has (ω j ≡ (k
2
j + m
2
)
1/2
),
H ren =
1
2
V
d
s x : [(∇ϕ)
2
(x) + π
2
(x) + m
2
ϕ
2
(x)] :=
j
ω j a
∗
j a j ,
where
√
2a j ≡
√
ω j ˜
ϕ(k j ) + i(
√
ω j )
−1
˜
π(k j ), and the tilde denotes the Fourier transform. By the above argument, the condition that H 0 be well defined selects the Fock
representation.
58 It is not difficult to show that one has a well defined operator also
in the infinite volume limit, when the momentum becomes a continuous variable and
H 0 =
1
2
d
s x : [(∇ϕ)
2
(x) + π
2
(x) + m
2
ϕ
2
(x)] :
=
d
s k ω(k)a
∗
(k)a(k), ω(k) = (k
2
+ m
2
)
1/2
is well defined on the dense domain obtained by applying the polynomial algebra of
the a
∗
( f ) to Ψ 0 , since H 0 Ψ 0 = 0 and the commutator is well defined [H 0 , a
∗
( f )] =
a
∗
(ω f ).
The massless case, m = 0, deserves a comment, since the number operators is no
longer dominated by the free Hamiltonian. In fact, in this case there are representations (the so-called non-Fock coherent state representations) in which H 0 is well
defined, but N is not.
58 H. J. Borchers, R. Haag and B. Schroer, Nuovo Cim. 29, 148 (1963). For a general look at free
fields from the point of view of representations of the algebra of canonical variables (canonical
quantization), see A. S. Wightman and S. Schweber, Phys. Rev. 98, 812 (1955); S. S. Schweber,
Introduction to Relativistic Quantum Field Theory, Harper and Row 1961.