Chapter 14
Mathematical Description of Infinitely
Extended Quantum Systems
From the discussion of the previous Chapter, it appears that the description of infinite systems looks much more difficult than in the finite-dimensional case, above
all because of the existence of (too) many possible representations of the algebra of
canonical variables. A big step in the direction of controlling the problem has been
taken by Haag et al., who emphasized the need of exploiting crucial physical properties of the algebra of observables in order to restrict their possible representations
to the physically relevant ones. The crucial ingredient is the localization property of
observable operations.
14.1 Local Structure
Any physically realizable operation is necessarily localized in space, since we cannot perform measurements or act on the system over the whole space. In order to
encode this property in the structure of the algebra of observables, it is convenient to
view it as generated by canonical variables or observables which have localization
properties.
71 Thus, for each bounded space region V , one has the C
∗ -algebra A(V )
of all observables (or canonical variables) localized in V .
A concrete realization of such a structure is obtained by considering canonical
variables, which have localization properties in the sense of (13.3). For regular test
functions f, g of compact support contained in V , (typically f, g ∈ D(V )), one
considers the set of localized canonical variables
a( f ) ≡
dx ψ(x) f (x) , a
∗
(g) =
dx ψ
∗
(x)g(x),
71 For a general discussion of this strategy, see R. Haag, Local Quantum Physics, Springer 1996.
© 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_14
95
Mathematical Description of Infinitely
Extended Quantum Systems
From the discussion of the previous Chapter, it appears that the description of infinite systems looks much more difficult than in the finite-dimensional case, above
all because of the existence of (too) many possible representations of the algebra of
canonical variables. A big step in the direction of controlling the problem has been
taken by Haag et al., who emphasized the need of exploiting crucial physical properties of the algebra of observables in order to restrict their possible representations
to the physically relevant ones. The crucial ingredient is the localization property of
observable operations.
14.1 Local Structure
Any physically realizable operation is necessarily localized in space, since we cannot perform measurements or act on the system over the whole space. In order to
encode this property in the structure of the algebra of observables, it is convenient to
view it as generated by canonical variables or observables which have localization
properties.
71 Thus, for each bounded space region V , one has the C
∗ -algebra A(V )
of all observables (or canonical variables) localized in V .
A concrete realization of such a structure is obtained by considering canonical
variables, which have localization properties in the sense of (13.3). For regular test
functions f, g of compact support contained in V , (typically f, g ∈ D(V )), one
considers the set of localized canonical variables
a( f ) ≡
dx ψ(x) f (x) , a
∗
(g) =
dx ψ
∗
(x)g(x),
71 For a general discussion of this strategy, see R. Haag, Local Quantum Physics, Springer 1996.
© 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_14
95