Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E.P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press 1959.
E.P. Wigner, loc. cit.; V. Bargmann, J. Math. Phys. 5, 862 (1964).
V. Bargmann, Ann. Math. 59, 1 (1954).
In fact, given a fixed unit vector Ψ, one selects U(g), in a neighborhood of the identity e, by the equation (Ψ, U(g)Ψ) ≡ |(Ψ, U(g)Ψ)|; then, by the (ray) continuity condition \( U\left( g \right)\Psi \Psi \;if\;g \to g_0 = e \). This property extends to any Φ, by the continuity condition applied to \( \left| {\left( {U\left( g \right)\left( {\Psi + \lambda \Phi } \right),U\left( {g_0 } \right)\left( {\Psi + \lambda \Phi } \right)} \right)} \right| \to \left( {\Psi + \lambda \Phi ,\Psi + \lambda \Phi } \right),\;\lambda \in R \), since, for λ sufficiently small (Ψ,Ψ) +Reλ(Ψ, Φ) > 0, so that also λ2(Φ, U(g)Φ) > 0 and the convergence holds without the modulus. The extension to any g 0 follows from the unitarity of U(g). For details, see Bargmann’s paper quoted above and for a very elegant abstract proof see D.J. Simms, Lie Groups and Quantum Mechanics, Lect. Notes Math. 52, Springer 1968; Rep. Math. Phys. 2, 283 (1971).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Strocchi, F. (2008). Symmetry Breaking in Quantum Systems. In: Symmetry Breaking. Lecture Notes in Physics, vol 732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73593-9_18
Download citation
DOI: https://doi.org/10.1007/978-3-540-73593-9_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73592-2
Online ISBN: 978-3-540-73593-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)