226
Appendix A: Long Range Dynamics and Vacuum Seizing
quantum fields), for any local variable or observable A, α
t
V (A) attains its limit for
finite V , technically the finite volume (or infrared cutoff) dynamics α
t
V (V denoting
the size of the volume V or the infrared cutoff) converges in norm, as V → ∞.
However, for non-relativistic continuous systems, an inevitable dynamical delocalization occurs, even in the free case (see Sect. 17.2), and one must consider
algebras (of observables or more generally of dynamical variables) with a weaker
localization property, with respect to the quasi-local algebra A L (defined as the
norm closure of the local algebra A L , Eq. (14.4)), in order to guarantee the stability
under time evolution, as required by a well defined dynamical problem.
2 This issue
becomes particularly acute in the case of long range interactions (e.g. for spin system
interactions decaying slower than |x|
−3 and for Coulomb interactions in many-body
systems).
3
This problem arises also for the field algebra F in quantum gauge theories, since in
the physical (positive) gauges it is generated by fields which do not satisfy relativistic
locality, whose time evolution involves a Coulomb delocalization (see Sect. 29.3).
Hence, in general, one has to solve the problem of i) the definition of the dynamics
α
t as a (suitable) limit of α
t
V , when V → ∞, and ii) the identification of an algebra A
(of observables or more generally of dynamical variables) stable under the dynamics.
To this purpose, the guiding principle is to start with the backbone of the “local
algebra” A L ≡ ∪ V A V , A V the (weakly closed) algebra (of observables or of dynamical variables) associated with the volume V (see Chap. 14, Sect. 14.1), whose physical
relevance is undeniable.
Then, one considers as relevant states those belonging to a family Σ of factor states
on A L (i.e. defining pure phases, see Sect. 22.4), with sufficiently good behaviour
at space infinity, such that i) for any local A, α
t
V (A) converges in the correlation
functions of such states, (technically, the limit must exist in the weak topology τ
defined by Σ) and ii) defines a one-parameter group α
t of time evolution of the
algebra A (of observables or of canonical variables) stable under time evolution,
obtained as the closure of A L under (the weak topology) τ , A ≡ A
τ
L . Henceforth,
The states of Σ will be called the infrared regular states.
The role of the class of physical states for the definition of the observable algebra
and its time evolution should not look ad hoc or artificial, in view of the strict relation
between observables and states for the description of a physical system
4 ; in fact, the
observables are identified by their expectations on the physically realizable states.
The required good infrared behaviour which characterizes the infrared regular
states depends on the range of the interaction. For example, if the interaction involves
a two-body potential U (x), decaying as |x|
−d , a condition of infrared regularity for a
state ω (typically a ground or equilibrium state), allowing the removal of the infrared
2 D.A. Dubin and G.L. Sewell, J. Math. Phys. 11, 2990 (1979); G. Sewell, Lett. Math. Phys. 6, 209
(1982).
3 A general discussion and proposed solution are given in G. Morchio and F. Strocchi, J. Math. Phys.
28, 622 (1987), hereafter referred to as G. Morchio and F. Strocchi [1987].
4 F. Strocchi, The physical principles of quantum mechanics. A critical review Eur. Phys. J. Plus,
127, 12 (2012).
Appendix A: Long Range Dynamics and Vacuum Seizing
quantum fields), for any local variable or observable A, α
t
V (A) attains its limit for
finite V , technically the finite volume (or infrared cutoff) dynamics α
t
V (V denoting
the size of the volume V or the infrared cutoff) converges in norm, as V → ∞.
However, for non-relativistic continuous systems, an inevitable dynamical delocalization occurs, even in the free case (see Sect. 17.2), and one must consider
algebras (of observables or more generally of dynamical variables) with a weaker
localization property, with respect to the quasi-local algebra A L (defined as the
norm closure of the local algebra A L , Eq. (14.4)), in order to guarantee the stability
under time evolution, as required by a well defined dynamical problem.
2 This issue
becomes particularly acute in the case of long range interactions (e.g. for spin system
interactions decaying slower than |x|
−3 and for Coulomb interactions in many-body
systems).
3
This problem arises also for the field algebra F in quantum gauge theories, since in
the physical (positive) gauges it is generated by fields which do not satisfy relativistic
locality, whose time evolution involves a Coulomb delocalization (see Sect. 29.3).
Hence, in general, one has to solve the problem of i) the definition of the dynamics
α
t as a (suitable) limit of α
t
V , when V → ∞, and ii) the identification of an algebra A
(of observables or more generally of dynamical variables) stable under the dynamics.
To this purpose, the guiding principle is to start with the backbone of the “local
algebra” A L ≡ ∪ V A V , A V the (weakly closed) algebra (of observables or of dynamical variables) associated with the volume V (see Chap. 14, Sect. 14.1), whose physical
relevance is undeniable.
Then, one considers as relevant states those belonging to a family Σ of factor states
on A L (i.e. defining pure phases, see Sect. 22.4), with sufficiently good behaviour
at space infinity, such that i) for any local A, α
t
V (A) converges in the correlation
functions of such states, (technically, the limit must exist in the weak topology τ
defined by Σ) and ii) defines a one-parameter group α
t of time evolution of the
algebra A (of observables or of canonical variables) stable under time evolution,
obtained as the closure of A L under (the weak topology) τ , A ≡ A
τ
L . Henceforth,
The states of Σ will be called the infrared regular states.
The role of the class of physical states for the definition of the observable algebra
and its time evolution should not look ad hoc or artificial, in view of the strict relation
between observables and states for the description of a physical system
4 ; in fact, the
observables are identified by their expectations on the physically realizable states.
The required good infrared behaviour which characterizes the infrared regular
states depends on the range of the interaction. For example, if the interaction involves
a two-body potential U (x), decaying as |x|
−d , a condition of infrared regularity for a
state ω (typically a ground or equilibrium state), allowing the removal of the infrared
2 D.A. Dubin and G.L. Sewell, J. Math. Phys. 11, 2990 (1979); G. Sewell, Lett. Math. Phys. 6, 209
(1982).
3 A general discussion and proposed solution are given in G. Morchio and F. Strocchi, J. Math. Phys.
28, 622 (1987), hereafter referred to as G. Morchio and F. Strocchi [1987].
4 F. Strocchi, The physical principles of quantum mechanics. A critical review Eur. Phys. J. Plus,
127, 12 (2012).