90
13 Non-Fock Representations
the existence of the interaction picture is equivalent to the statement that the representation of the field operators at time t is unitarily equivalent to the representation
of free fields and, by the Borchers–Haag–Schroer result discussed above, the latter
require a Fock representation; in conclusion, at each time t the representation of the
interacting field operators should be equivalent to a Fock representation, contrary to
Haag’s theorem.
Thus, the solution of the dynamical problem for infinite systems is much more
difficult (especially from a mathematical point of view) than in the finite-dimensional
case, where it is essentially controlled by Kato’s theorems.
64 In the infinitedimensional case, one faces the puzzling situation that in order to give a meaning to the Hamiltonian, as an operator in a Hilbert space H, one must specify the
representation of the operators a, a
∗ (or of the fields at time zero), in terms of
which H is formally defined. On the other hand, the representation of the a, a
∗ ,
in which the dynamics is well defined, is in general non-Fock and its determination
involves a non-perturbative control on the theory. This looks as a blind alley. A possible way out of these conceptual difficulties (and also a possible way to recover
some of the results of the perturbative expansion) is provided by the constructive
strategy.
65 As already mentioned above, the strategy is to regularize the theory by
introducing UV and IR cutoffs and to determine the (cutoff-dependent) counter terms
needed to get a renormalized Hamiltonian, so that the corresponding (ground state)
correlation functions have a reasonable limit when the cutoffs are removed. This is
the content of the so-called non-perturbative renormalization, which has been successfully carried out in quantum field theory models in low space–time dimensions
(d = 1 + 1, d = 2 + 1).
66 A simple model, in which such a non-perturbative renormalization can be instructively checked to work, and which also displays the occurrence of non-Fock representations, is the so-called Yukawa model of pion–(heavy)
nucleon interaction.
67
To give at least the flavour of how non-Fock representations arise, we list a few
simple examples.
Example 13.1 Quantum field interacting with a classical source. We consider a
quantum scalar field (see, Example 12.1) interacting with a classical (time independent) real source j (x)
( + m
2
)ϕ(x) = g j (x),
(13.7)
64 For a beautiful extensive discussion, see M. Reed and B. Simon, Methods of Modern Mathematical
Physics, Vol. II (Fourier Analysis, Self-Adjointness), Academic Press 1975, Chap. X; for a sketchy
account, see e.g. [SNS 96].
65 A. S. Wightman, Introduction to some aspects of the relativistic dynamics of quantized fields,
in Cargèse Lectures in Theoretical Physics, M. Levy ed., Gordon and Breach 1967, esp. Part II,
Chap. VI; Constructive Field Theory. Introduction to the Problems, in Fundamental Interactions in
Physics and Astrophysics, G. Iverson et al. eds., Plenum 1972. Constructive Quantum Field Theory,
G. Velo and A.S. Wightman eds., Springer 1973.
66 See J. Glimm and A. Jaffe, Quantum Physics, Springer 1981 and references therein.
67 See e.g. [S 85] Part A, Sect. 2.3.
13 Non-Fock Representations
the existence of the interaction picture is equivalent to the statement that the representation of the field operators at time t is unitarily equivalent to the representation
of free fields and, by the Borchers–Haag–Schroer result discussed above, the latter
require a Fock representation; in conclusion, at each time t the representation of the
interacting field operators should be equivalent to a Fock representation, contrary to
Haag’s theorem.
Thus, the solution of the dynamical problem for infinite systems is much more
difficult (especially from a mathematical point of view) than in the finite-dimensional
case, where it is essentially controlled by Kato’s theorems.
64 In the infinitedimensional case, one faces the puzzling situation that in order to give a meaning to the Hamiltonian, as an operator in a Hilbert space H, one must specify the
representation of the operators a, a
∗ (or of the fields at time zero), in terms of
which H is formally defined. On the other hand, the representation of the a, a
∗ ,
in which the dynamics is well defined, is in general non-Fock and its determination
involves a non-perturbative control on the theory. This looks as a blind alley. A possible way out of these conceptual difficulties (and also a possible way to recover
some of the results of the perturbative expansion) is provided by the constructive
strategy.
65 As already mentioned above, the strategy is to regularize the theory by
introducing UV and IR cutoffs and to determine the (cutoff-dependent) counter terms
needed to get a renormalized Hamiltonian, so that the corresponding (ground state)
correlation functions have a reasonable limit when the cutoffs are removed. This is
the content of the so-called non-perturbative renormalization, which has been successfully carried out in quantum field theory models in low space–time dimensions
(d = 1 + 1, d = 2 + 1).
66 A simple model, in which such a non-perturbative renormalization can be instructively checked to work, and which also displays the occurrence of non-Fock representations, is the so-called Yukawa model of pion–(heavy)
nucleon interaction.
67
To give at least the flavour of how non-Fock representations arise, we list a few
simple examples.
Example 13.1 Quantum field interacting with a classical source. We consider a
quantum scalar field (see, Example 12.1) interacting with a classical (time independent) real source j (x)
( + m
2
)ϕ(x) = g j (x),
(13.7)
64 For a beautiful extensive discussion, see M. Reed and B. Simon, Methods of Modern Mathematical
Physics, Vol. II (Fourier Analysis, Self-Adjointness), Academic Press 1975, Chap. X; for a sketchy
account, see e.g. [SNS 96].
65 A. S. Wightman, Introduction to some aspects of the relativistic dynamics of quantized fields,
in Cargèse Lectures in Theoretical Physics, M. Levy ed., Gordon and Breach 1967, esp. Part II,
Chap. VI; Constructive Field Theory. Introduction to the Problems, in Fundamental Interactions in
Physics and Astrophysics, G. Iverson et al. eds., Plenum 1972. Constructive Quantum Field Theory,
G. Velo and A.S. Wightman eds., Springer 1973.
66 See J. Glimm and A. Jaffe, Quantum Physics, Springer 1981 and references therein.
67 See e.g. [S 85] Part A, Sect. 2.3.