Chapter 17
Examples
17.1 Spin Systems with Short-Range Interactions
As mentioned before, the quantum mechanics of infinite systems is not under mathematical control as it is in the finite-dimensional case. A non-perturbative control has
been achieved for quantum field theories in low space–time dimensions (d = 1 + 1,
d = 2 + 1), but the question is still open in d = 3 + 1 dimensions and the triviality
of the ϕ
4 theory indicates that the perturbative expansion is not reliable for existence
problems. It is clear that the existence of a non-trivial dynamics for systems with infinite degrees of freedom is not a trivial problem, but for non-relativistic systems some
result is available. As a matter of fact, for spin systems with short-range interactions
the infinite volume dynamics α t has been shown to exist.
88
To give an idea of how the problem is attacked and solved, we first consider the
simple case of a one-dimensional chain of spins (a one-dimensional “ferromagnet”)
with a formal Hamiltonian of Ising type
H = −J
i, j
σ i σ j , J > 0,
(17.1)
where the sum is over all the nearest neighbor pairs of indices i, j, which denote the
chain sites and σ denotes the component of the spin along the z direction.
The local algebra of observables is generated by the spin operators σ at the various
sites; in particular, for each volume V the algebra A(V ) is the algebra generated by
the spins sitting in the sites i ∈ V .
89 Since spins at different sites are assumed to commute, the localization condition (14.1) obviously holds and asymptotic abelianess is
satisfied by the quasi-local algebra A (the norm closure of the local algebra).
88 D.W. Robinson, Comm. Math. Phys. 7, 337 (1968).
89 For a detailed discussion of the mathematical structure of spin models, see O. Bratteli and D.W.
Robinson, loc. cit. Vol. 2, Sect. 6.2.
© 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_17
111
Examples
17.1 Spin Systems with Short-Range Interactions
As mentioned before, the quantum mechanics of infinite systems is not under mathematical control as it is in the finite-dimensional case. A non-perturbative control has
been achieved for quantum field theories in low space–time dimensions (d = 1 + 1,
d = 2 + 1), but the question is still open in d = 3 + 1 dimensions and the triviality
of the ϕ
4 theory indicates that the perturbative expansion is not reliable for existence
problems. It is clear that the existence of a non-trivial dynamics for systems with infinite degrees of freedom is not a trivial problem, but for non-relativistic systems some
result is available. As a matter of fact, for spin systems with short-range interactions
the infinite volume dynamics α t has been shown to exist.
88
To give an idea of how the problem is attacked and solved, we first consider the
simple case of a one-dimensional chain of spins (a one-dimensional “ferromagnet”)
with a formal Hamiltonian of Ising type
H = −J
i, j
σ i σ j , J > 0,
(17.1)
where the sum is over all the nearest neighbor pairs of indices i, j, which denote the
chain sites and σ denotes the component of the spin along the z direction.
The local algebra of observables is generated by the spin operators σ at the various
sites; in particular, for each volume V the algebra A(V ) is the algebra generated by
the spins sitting in the sites i ∈ V .
89 Since spins at different sites are assumed to commute, the localization condition (14.1) obviously holds and asymptotic abelianess is
satisfied by the quasi-local algebra A (the norm closure of the local algebra).
88 D.W. Robinson, Comm. Math. Phys. 7, 337 (1968).
89 For a detailed discussion of the mathematical structure of spin models, see O. Bratteli and D.W.
Robinson, loc. cit. Vol. 2, Sect. 6.2.
© 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_17
111