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Symmetry Breaking

Franco Strocchi

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Institución detectada Año de publicación Navegá Descargá Solicitá
No detectada 2005 SpringerLink

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Tipo de recurso:

libros

ISBN impreso

978-3-540-21318-5

ISBN electrónico

978-3-540-31536-0

Editor responsable

Springer Nature

País de edición

Reino Unido

Fecha de publicación

Información sobre derechos de publicación

© Springer-Verlag Berlin/Heidelberg 2005

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Tabla de contenidos

10 Appendix

Franco Strocchi

a)

If is the Schwartz space of test functions decreasing at infinity faster than any inverse polynomial), then the solution of the free wave equation is easily obtained by Fourier transform and one has

     (A.1)

cos || , (sin || ) / || etc. are multipliers of continuous in and

     (A.2)

The group property is easily checked.

Pp. 51-60

Introduction to Part II

Franco Strocchi

These notes arose from courses given at the International School for Advanced Studies (Trieste) and at the Scuola Normale Superiore (Pisa) in various years, with the purpose of discussing the structural features and collective effects which distinguish the quantum mechanics of systems with infinite degrees of freedom from ordinary quantum mechanics.

Pp. 63-66

1 Quantum Mechanics. Algebraic Structure and States

Franco Strocchi

We briefly review the basic structure of Quantum Mechanics (QM) with the aim of covering both the case of systems with a finite number of degrees of freedom (ordinary QM) as well as the case of systems with an infinite number of degrees of freedom (briefly ).

Pp. 67-71

2 Fock Representation

Franco Strocchi

The general lesson from the GNS theorem is that a state on the algebra of observables, namely a set of expectations, defines a realization of the system in terms of a Hilbert space of states with a reference vector which represents as a cyclic vector (so that all the other vectors of can be obtained by applying the observables to ). In this sense, a state identifies the family of states related to it by observables, equivalently accessible from it by means of physically realizable operations. Thus, one may say that describes a closed world, or phase, to which belongs. An interesting physical and mathematical question is how many closed worlds or phases are associated to a quantum system. In the mathematical language this amounts to investigating how many inequivalent (physically acceptable) representations of the observable algebra which defines the system exist.

Pp. 73-79

3 Non-Fock Representations

Franco Strocchi

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 2.1, non-Fock representations are required in order to describe many particle systems with non-zero density in the thermodynamical limit

→ ∞, → ∞, /≡ ≠ 0.

In fact, in the Fock representation, ∀ in the domain of , if denotes the (operator) number of particles in the volume , one has

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.

Pp. 81-87

4 Mathematical Description of Infinitely Extended Systems

Franco Strocchi

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.

Pp. 89-93

5 Physically Relevant Representations

Franco Strocchi

From the examples and the discussion of the previous chapter, it appears that for infinite systems the choice of the representation for the algebra of canonical variables (a basic preliminary step for even defining the dynamical problem) is a highly non-trivial problem (unless the model is exactly soluble). Among the possible representations of the relevant algebra it is therefore convenient to isolate those which are physically acceptable. For the moment we restrict our discussion to the zero temperature case. The non-zero temperature case will be briefly discussed in Chap. 12. On the basis of general physical considerations, we require the following conditions for a physically relevant representation .

Pp. 95-98

6 Cluster Property and Pure Phases

Franco Strocchi

The irreducible (physically relevant) representations selected in the previous section have a further important property, called .

Pp. 99-103

7 Examples

Franco Strocchi

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 ( = 1+1, = 2+1), but the question is still open in = 3+1 dimensions and the triviality of the 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 has been shown to exist.

Pp. 105-113

8 Symmetry Breaking in Quantum Systems

Franco Strocchi

Most of the wisdom on spontaneous symmetry breaking (SSB), especially for elementary particle theory, relies on approximations and/or a perturbative expansion. Since the mechanism of SSB is underlying most of the new developments in theoretical physics, it is worthwhile to try to understand it from a general (non-perturbative) point of view. Most of the popular explanations given in the literature are not satisfactory (if not misleading) since they do not make it clear that the crucial ingredient for the non-symmetrical behaviour of a system described by a symmetric Hamiltonian is the occurrence of infinite degrees of freedom and of inequivalent representations of the algebra of observables. We shall start by recalling a few basic facts.

Pp. 115-122