Chapter 18
Symmetry Breaking in Quantum Systems
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 a 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.
18.1 Wigner Symmetries
The clarification of the concept of symmetry in quantum mechanics is essentially
due to Wigner.
101 Given a quantum mechanical system, whose states are described
by rays ˆ
= {e
iλ
, λ ∈ R, , ∈ H} of a Hilbert space H, a symmetry operation g
in the sense of Wigner, briefly a Wigner symmetry, is a mapping of rays into rays,
g : ˆ
→= g ˆ
,
(18.1)
which does not change the transition probabilities, namely the modulus of the scalar
products
|(g ˆ
, g ˆ
)| = |( ˆ
, ˆ
)|.
101 E. P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra,
Academic Press 1959.
© 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_18
121
Symmetry Breaking in Quantum Systems
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 a 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.
18.1 Wigner Symmetries
The clarification of the concept of symmetry in quantum mechanics is essentially
due to Wigner.
101 Given a quantum mechanical system, whose states are described
by rays ˆ
= {e
iλ
, λ ∈ R, , ∈ H} of a Hilbert space H, a symmetry operation g
in the sense of Wigner, briefly a Wigner symmetry, is a mapping of rays into rays,
g : ˆ
→= g ˆ
,
(18.1)
which does not change the transition probabilities, namely the modulus of the scalar
products
|(g ˆ
, g ˆ
)| = |( ˆ
, ˆ
)|.
101 E. P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra,
Academic Press 1959.
© 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_18
121