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To illustrate the above general ideas we discuss simple concrete models exhibiting spontaneous symmetry breaking.

Apart from simple models, like those discussed in the previous section, the existence of a symmetry breaking order parameter is a non-trivial problem which in principle requires the control on the correlation functions. In this section we briefly discuss constructive criteria for symmetry breaking.

Most of the theoretical wisdom on the phase transition of the ferromagnetic type and the related symmetry breaking is based on the two-dimensional Ising model, which also played the role of a laboratory for ideas and strategies and it is now regarded as a corner stone in the foundations of statistical mechanics. Anyone interested in critical phenomena and in the functional integral approach to quantum field theory should have a look to the model. Even if a discussion of the two-dimensional Ising model would be very appropriate for our purposes, we refer the reader to the very good accounts which can be found in literature. We restrict our discussion to the one-dimensional version of the model, which is almost trivial, but nevertheless provides an interesting simple example for testing the constructive strategies of symmetry breaking discussed above.

The physically relevant representations discussed in Chap. 5 are characterized by the existence of a lowest energy or ground state and are supposed to describe states of an infinitely extended isolated system. The situation changes if one wants to describe states of a system at non-zero temperature (), i.e. states of a system in thermal equilibrium with a reservoir. The stability of the system is now guaranteed by the reservoir and there is no need of the energy spectral condition. The role of the ground state is now taken by the and one is therefore led to discuss representations of the canonical or observable algebra defined by equilibrium states.

As an example of symmmetry breaking at non-zero temperature we discuss the free Fermi and Bose gas, starting from finite volume and then discussing the thermodynamical limit.

The general structure discussed above provides a neat and unique prescription for the quantization of relativistic fields at non-zero temperature ().

For a long time, the mechanism of spontaneous breaking of continuous symmetries has been recognized to be at the basis of many collective phenomena and in particular of phase transitions in statistical mechanics; recently, it has played a crucial role in the developments of theoretical physics, both at the level of many body physics and for the unification of elementary particle interactions.

The proof of the Goldstone theorem can be easily extended to the case of non-zero temperature = 1/, i.e. to representations defined by KMS states. In this case, the interest of the theorem is in the prediction of the Goldstone quasi particles, and the derivation of such a prediction crucially depends on the integrability of the charge density commutators. The absence of an energy gap (as in Theorem 15.1) is not very significant, since it is already implied by general properties (like timelike clustering) of the KMS states.

Relativistic systems, like elementary particles, are described by an algebra of observables which satisfies the causality condition, (4.2), and is stable under the automorphisms (, ) which describe space time translations and Lorentz transformations (with parameters , respectively). The physically relevant representations of have to satisfy the relativistic version of conditions I-III (Chap. 5).

The Goldstone theorem and its rigorous predictions on the energy spectrum at zero momentum can be extended to the case in which the Hamiltonian is not symmetric, but it has simple transformation properties in the sense that the multiple commutators of and the charge generate a finite dimensional Lie algebra, briefly

[ , ] = .