Chapter 13
Non-Fock Representations
As anticipated in the previous discussions, the Fock representation is very special to
the finite-dimensional case and to free fields. Actually, as a consequence of Proposition 12.1, non-Fock representations are required in order to describe many-particle
systems with non-zero density in the thermodynamical limit
N → ∞, V → ∞, N /V ≡ n = 0.
In fact, in the Fock representation, ∀ in the domain of N , if N V denotes the (operator)
number of particles in the volume V , one has
||n|| = lim
V →∞
V
−1
||N V || ≤ lim
V →∞
V
−1
||N || = 0.
Actually, for systems of non-zero density, in the thermodynamical limit, the free
Hamiltonian need not be defined even in the free case; only the energy per unit
volume is required to be finite.
59
In the following, we shall present arguments, on the basis of simple examples,
which indicate the need for non-Fock representation, also for systems with zero
density, in order to get well defined Hamiltonians.
Quite generally, in the case of interacting fields, the definition of the formal Hamiltonian, typically of the form (in a finite volume)
H =
i
ω i a
∗
i a i + g H int (a, a
∗
),
(13.1)
59 For the mathematical discussion of the free Bose gas and for the free fermion gas see H. Araki
and E. J. Woods, J. Math. Phys. 4, 637 (1963); H. Araki and W. Wyss, Helv. Phys. Acta 37, 139
(1964). For a general account, see O. Bratteli and D. W. Robinson, Operator Algebras and Quantum
Statistical Mechanics, Vol. II, Springer 1996. A simple discussion is given in Sect. 17.2.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_13
87
Non-Fock Representations
As anticipated in the previous discussions, the Fock representation is very special to
the finite-dimensional case and to free fields. Actually, as a consequence of Proposition 12.1, non-Fock representations are required in order to describe many-particle
systems with non-zero density in the thermodynamical limit
N → ∞, V → ∞, N /V ≡ n = 0.
In fact, in the Fock representation, ∀ in the domain of N , if N V denotes the (operator)
number of particles in the volume V , one has
||n|| = lim
V →∞
V
−1
||N V || ≤ lim
V →∞
V
−1
||N || = 0.
Actually, for systems of non-zero density, in the thermodynamical limit, the free
Hamiltonian need not be defined even in the free case; only the energy per unit
volume is required to be finite.
59
In the following, we shall present arguments, on the basis of simple examples,
which indicate the need for non-Fock representation, also for systems with zero
density, in order to get well defined Hamiltonians.
Quite generally, in the case of interacting fields, the definition of the formal Hamiltonian, typically of the form (in a finite volume)
H =
i
ω i a
∗
i a i + g H int (a, a
∗
),
(13.1)
59 For the mathematical discussion of the free Bose gas and for the free fermion gas see H. Araki
and E. J. Woods, J. Math. Phys. 4, 637 (1963); H. Araki and W. Wyss, Helv. Phys. Acta 37, 139
(1964). For a general account, see O. Bratteli and D. W. Robinson, Operator Algebras and Quantum
Statistical Mechanics, Vol. II, Springer 1996. A simple discussion is given in Sect. 17.2.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_13
87